Research ArticleStability Analysis for a Fractional HIV Infection Model withNonlinear Incidence

Linli Zhang1 Gang Huang2 Anping Liu2 and Ruili Fan3

1Department of Basic Course Haikou College of Economics Haikou 571127 China2School of Mathematics and Physics China University of Geosciences Wuhan 430074 China3Department of Mathematics and Statistics University of Helsinki 00014 Helsinki Finland

Correspondence should be addressed to Anping Liu wh apliusinacom

Received 30 September 2014 Accepted 16 December 2014

Academic Editor Bai-Lian Larry Li

Copyright copy 2015 Linli Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduce the fractional-order derivatives into an HIV infection model with nonlinear incidence and show that the establishedmodel in this paper possesses nonnegative solution as desired in any population dynamics We also deal with the stability of theinfection-free equilibrium the immune-absence equilibrium and the immune-presence equilibrium Numerical simulations arecarried out to illustrate the results

1 Introduction

11 The History of Fractional Calculus Fractional calculuswhich is a branch of mathematical analysis extends deriva-tives and integrals to an arbitrary order (real even orcomplex order) The study on fractional calculus started atthe end of seventeenth century and the first reference wasproposed by Leibniz and LrsquoHospital in 1695 in which half-order derivative was mentionedThe emergence of fractionalcalculus aroused interest of some famous mathematicianssuch as Lacroix Lagrange Fourier and Laplace J L Lagrangedeveloped the law of exponents for differential operatorsin 1772 which promoted indirectly the development offractional calculus P S Laplace defined a fractional deriva-tive by means of integral in 1812 and S F Lacroix and JB J Fourier in succession mentioned the arbitrary orderderivative respectively in 1819 and in 1822 However thebasic theory of fractional calculus was established with thestudies of Liouville Grunwald Letnikov and Riemann untilthe end of the nineteenth century

At the initial stage the development of fractional calculushad been restricted to the pure mathematics research Untilthe nineteen nineties the theory of fractional calculus wasapplied to nature and social sciences In the field of anoma-lous diffusion the researchers applied fractional calculus todescribe the diffusion processes which do not conform to

the brown motion and proposed the fractional anomalousdiffusionmodel According to the theoretical analysis and theexperimental data or results they found that the fractionalmodel is more reasonable to describe these processes [1]Afterwards many mathematicians and applied researchersalso have tried to demonstrate applications of fractionaldifferentials in the areas of non-Newtonian fluids [2] signalprocessing [3ndash5] viscoelasticity [6 7] fluid-dynamic trafficmodel [8] colored noise [9] bioengineering [10] solidmechanics [11] continuumand statisticalmechanics [12] andeconomics [13] and brought new research view for thosefields

12 Mathematical Modeling In biology it has been deducedthat the membranes of cells of biological organism havefractional-order electrical conductance [14] hence somemathematical models which describe cells behavior are clas-sified into groups of non-integer-order models In the fieldof rheology fractional-order derivatives embody essentialfeatures of cell rheological behavior and have enjoyed greatestsuccess [15] Some researchers also found that fractional ordi-nary differential equations are naturally related to systemswith memory which exists in most biological systems Andfractional-order equations are also closely related to fractalswhich are abundant in biological systems

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 563127 11 pageshttpdxdoiorg1011552015563127

2 Discrete Dynamics in Nature and Society

Mathematical models have been proven valuable inunderstanding the dynamics of viral infection Although alarge number of works on modeling the dynamics of viralinfection have been done [16ndash20] it has been restricted tointeger-order (delay) differential equations In recent yearsit has turned out that many phenomena in virus infectioncan be described very successfully by the models usingfractional-order differential equations [21 22] Motivated bythese references in this paper we will consider a fractional-order HIV model

Among a large number of virus infection modelsdescribed with integer-order or delay differential equationsa typical model is given as the following simple three-dimensional system

1199091015840

(119905) = 120582 minus 119889119909(119905) minus 120573119909(119905)V(119905)

1199101015840

(119905) = 120573119909(119905)V(119905) minus 119901119910(119905)

V1015840(119905) = 119896119910(119905) minus 119906V(119905)

(1)

Here 119909(119905) represents the concentration of uninfected cells attime 119905 119910(119905) represents the concentration of infected cells thatproduce virus at time 119905 V(119905) represents the concentration ofviruses at time 119905 119889 (119889 gt 0) and 120573 (120573 gt 0) are the death rate ofuninfected cells and the rate constant characterizing infectionof the cells respectively119901 (119901 gt 0) is the death rate of infectedcells due to either the virus or the immune system Free virusis produced from infected cells at the rate 119896119910(119905) and removedat rate 119906V(119905)

The infection rate in model (1) is usually assumed tobe bilinear with respect to virus V(t) and uninfected targetcells 119909(119905) However there are some evidences which showthat a bilinear infection might not be an effective assumptionwhen the number of target cells is large enough [23ndash25]AndHuang et al [17] incorporated Holling type II functionalresponse infection rate 119892(119909(119905) V(119905)) into basic virus dynamics(1) which was expressed as

119892(119909(119905) V(119905)) =

120573119909(119905)V(119905)1 + 119902119909(119905)

(2)

where 119902 (119902 ge 0) is a constant Hence system (1) can bemodified into the following system

1199091015840

(119905) = 120582 minus 119889119909(119905) minus

120573119909(119905)V(119905)1 + 119902119909(119905)

1199101015840

(119905) =

120573119909(119905)V(119905)1 + 119902119909(119905)

minus 119901119910(119905)

V1015840(119905) = 119896119910(119905) minus 119906V(119905)

(3)

The immune response following viral infection is uni-versal and necessary to eliminate or control the diseaseAntibodies cytokines natural killer cells B cells and T cellsare all essential components of a normal immune responseto viral infection Since cytotoxic T lymphocytes (CTLs) playa critical role in antiviral defense by attacking the virus-infected cells Huang et al [17] considered the immuneresponse to model (3) with 119911

1015840(119905) = 119888119910(119905)119911(119905) minus 119887119911(119905)

System (3) with 1199111015840(119905) = 119888119910(119905)119911(119905) minus 119887119911(119905) can be further

simplified if we take into consideration the fact that anaverage life span of viral particles is usually significantlyshorter than one of the infected cells It can be assumedtherefore that compared with a ldquoslowrdquo variation of theinfected cells level the virus load V(119905) relatively quicklyreaches a quasiequilibrium level The equality V(119905) = 0 holdsin the quasiequilibrium state and hence V(119905) = 119896119910(119905)119906This assumption is referred to as ldquoseparation of time scalesrdquoand is in common use in the virus dynamics [18] We haveto stress that this assumption does not imply that the virusconcentration V(119905) remains constant on the contrary it isassumed to be proportional to the varying concentrationof infected cells 119910(119905) Accordingly system (3) with 119911

1015840(119905) =

119888119910(119905)119911(119905)minus119887119911(119905) can now be reformulated as a system of threedifferential equations

1199091015840

(119905) = 120582 minus 119889119909 (119905) minus

120573119909 (119905) 119910 (119905)

1 + 119902119909 (119905)

1199101015840

(119905) =

120573119909 (119905) 119910 (119905)

1 + 119902119909 (119905)

minus 119886119910 (119905) minus 119901119910 (119905) 119911 (119905)

1199111015840

(119905) = 119888119910(119905)119911(119905) minus 119887119911(119905)

(4)

Further we introduce the fractional order into model (4)and a new system can be described by the following set ofFODEs of order 120572 gt 0

119909120572

(119905) = 120582 minus 119889119909 minus

120573119909119910

1 + 119902119909

119910120572

(119905) =

120573119909119910

1 + 119902119909

minus 119886119910 minus 119901119910119911

119911120572

(119905) = 119888119910119911 minus 119887119911

(5)

119909 (0) = 1199090 119910 (0) = 119910

0 119911 (0) = 119911

0 (6)

where the immune response is assumed to get stronger ata rate 119888119910119911 which is proportional to the number of infectedcells and their current concentration the immune responsedecays exponentially at a rate 119887119911 which is proportional totheir current concentration the parameter 119902 expresses theefficacy of nonlytic component

The paper is organized as follows In the next sectionwe give two definitions and two lemmas about fractionalcalculus In Section 3 we show the nonnegativeness of thesolutions of system (5) with initial condition (6) In Section 4we give a detailed stability analysis for three equilibria Finallynumerical simulations are presented to illustrate the obtainedresults in Section 5

2 Fractional Calculus

Definition 1 (see [26]) The Riemann-Liouville (R-L) frac-tional integral operator of order 120572 gt 0 of a function119891 119877

+rarr

119877 is defined as

119868120572119891 (119909) =

1

Γ (120572)

int

119909

0

(119909 minus 119905)120572minus1

119891 (119905) 119889119905 (7)

Discrete Dynamics in Nature and Society 3

Here Γ(sdot) is the Euler gamma function which is defined as

Γ (119899) = int

infin

0

119905119899minus1

119890minus119905119889119905 (8)

This function is generalization of a factorial in the followingform

Γ (119899) = (119899 minus 1) (9)

Definition 2 (see [26]) The Caputo (C) fractional derivativeof order 120572 gt 0 119899 minus 1 lt 120572 lt 119899 119899 isin 119873 is defined as

119863120572119891 (119905) = 119868

119899minus120572119863119899119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0

119891(119899)

(119904)

(119905 minus 119904)120572+1minus119899

119889119904

(10)

where the function119891(119905) has absolutely continuous derivativesup to order (119899 minus 1) In particular when 0 lt 120572 lt 1 one has

119863120572119891 (119905) =

1

Γ (1 minus 120572)

int

119905

0

1198911015840(119904)

(119905 minus 119904)120572119889119904 (11)

In this paper we use Caputo fractional derivative defini-tion The main advantage of Caputorsquos definition is that theinitial conditions for fractional differential equations withCaputo derivatives take the same form as that for integer-order differential equations

Lemma 3 (see [26]) Consider the following commensuratefractional-order system

119863120572119909 = 119891 (119909)

119909 (0) = 1199090

(12)

with 0 lt 120572 le 1 and 119909 isin 119877119899 The equilibrium points of system

(12) are calculated by solving the following equation 119891(119909) = 0These points are locally asymptotically stable if all eigenvalues119903119894of Jacobian matrix 119869 = 120597119891120597119909 evaluated at the equilibrium

points satisfy

1003816100381610038161003816arg(119903119894)1003816100381610038161003816gt

120579120587

2

(13)

Definition 4 (see [27]) The discriminant 119863(119891) of a polyno-mial

119891(119909) = 119909119899+ 1198881119909119899minus1

+ 1198882119909119899minus2

+ sdot sdot sdot + 119888119899

(14)

is defined by 119863(119891) = (minus1)119899(119899minus1)2

119877(119891 1198911015840) where 119891

1015840 is thederivative of119891 If119892(119909) = 119909

119897+1198891119909119897minus1

+1198892119909119897minus2

+sdot sdot sdot+119889119899119877(119891 119892) is

the determinant of the corresponding Sylvester (119899+119897)otimes(119899+119897)

matrix The Sylvester matrix is formed by filling the matrixbeginning with the upper left corner with the coefficients of119891(119909) and then shifting down one row and one column to theright and filling in the coefficients starting there until they hitthe right sideThe process is then repeated for the coefficientsof 119892(119909)

Lemma 5 (see [27]) For the polynomial equation

119875 (120582) = 120582119899+ ℎ1120582119899minus1

+ ℎ2120582119899minus2

+ sdot sdot sdot + ℎ119899= 0 (15)

the conditions which make all the roots of (15) satisfy (13) aredisplayed as follows

(i) for 119899 = 1 the condition is ℎ1gt 0

(ii) for 119899 = 2 the conditions are either Routh-Hurwitzconditions or

ℎ1lt 0 4ℎ

2gt (ℎ1)2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

tanminus1(radic4ℎ2minus (ℎ1)2

ℎ1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt

120572120587

2

(16)

(iii) for 119899 = 3

(a) if the discriminant of 119875(120582) 119863(119875) is positive thenRouth-Hurwitz conditions are the necessary andsufficient conditions that is ℎ

1gt 0 ℎ

3gt 0 and

ℎ1ℎ2gt ℎ3if 119863(119875) gt 0

(b) if119863(119875) lt 0 ℎ1ge 0 ℎ

2ge 0 and ℎ

3gt 0 then (13)

for (15) holds when 120572 lt 23(c) if119863(119875) lt 0 ℎ

1lt 0 and ℎ

2lt 0 then (13) for (15)

holds when 120572 gt 23(d) if 119863(119875) lt 0 ℎ

1gt 0 ℎ

2gt 0 and ℎ

1ℎ2= ℎ3 then

(13) for (15) holds for all 120572 isin [0 1)

3 Nonnegative Solutions

Let 1198773+

= 119882 isin 1198773

119882 ge 0 and 119882(119905) = (119909(119905) 119910(119905) 119911(119905))119879

For the proof of the theorem about nonnegative solutions wewould need the following lemma

Lemma 6 (see [28] generalized mean value theorem) Let119891(119909) isin 119862[119886 119887] and 119863

120572119891(119909) isin 119862[119886 119887] for 0 lt 120572 le 1 Then

one has

119891(119909) = 119891(119886) +

1

Γ(120572)

119863120572119891(120585)(119909 minus 119886)

120572 (17)

with 0 le 120585 le 119909 forall119909 isin (119886 119887] where Γ(119909) = int

+infin

0119905119909minus1

119890minus119905119889119905

Remark 7 Suppose that 119891(119909) isin 119862[119886 119887] and 119863120572119891(119909) isin

119862[119886 119887] for 0 lt 120572 le 1 It is clear from Lemma 6 that if119863120572119891(119909) ge 0 forall119909 isin (119886 119887) then 119891(119909) is nondecreasing for

each 119909 isin [119886 119887] If 119863120572119891(119909) le 0 forall119909 isin (119886 119887) then 119891(119909) is

nonincreasing for each 119909 isin [119886 119887]

We now prove the main theorem

Theorem 8 There is a unique solution for the initial valueproblem (5) with (6) and the solution remains in 119877

3

+

Proof From Theorem 31 and Remark 32 of [29] we obtainthe solution on (0infin) solving the initial value problem (5)with (6) which is not only existent but also unique Next wewill show the nonnegative orthant 1198773

+is a positively invariant

region What is needed for this is to show that on eachhyperplane bounding the nonnegative orthant the vectorfield points into 119877

3

+ From (5) we find

119909120572(119905)

1003816100381610038161003816119909=0

= 120582 ge 0 119910120572(119905)

1003816100381610038161003816119910=0

= 0 119911120572(119905)

1003816100381610038161003816119911=0

= 0

(18)

By Remark 7 the solution will remain in 1198773

+

4 Discrete Dynamics in Nature and Society

4 Equilibrium States and Their Stability

In this section we investigate the stability of the fractional-ordermodel of HIV infection of CD4 + T cells that is system(5) with (6) Consider the initial value problem (5) with (6)with 120572 satisfying 0 lt 120572 le 1

In order to obtain the equilibria of system (5) we set119909120572(119905) = 0 119910120572(119905) = 0 and 119911

120572(119905) = 0 and we have

120582 minus 119889119909 minus

120573119909119910

1 + 119902119909

= 0

120573119909119910

1 + 119902119909

minus 119886119910 minus 119901119910119911 = 0

119888119910119911 minus 119887119911 = 0

(19)

System (5) has three types of relevant nonnegative equilib-rium states System (5) always has an infection-free equilib-rium 119864

0= (1199090 0 0) where

1199090=

120582

119889

(20)

which means that the infected cells are cleared The basicreproductive number of the viruses for system (5) is given by

1198770=

120582120573

119886 (119889 + 119902120582)

(21)

This number describes the average number of newly gener-ated infected cells from one infected cell at the beginningof the infection process When 119877

0gt 1 in addition to the

infection-free equilibrium 1198640 there is an immune-absence

equilibrium 1198641= (1199091 1199101 1199111) where

1199091=

119886

120573 minus 119886119902

1199101=

119889 + 119902120582

120573 minus 119886119902

(1198770minus 1) 119911

1= 0

(22)Note that 119877

0gt 1 implies 120573 gt 119886119902 When the HIV infection

is in the immune-absence equilibrium 1198641 the infected cells

and virus exist but the immune response is not activated yetFurther we denote

1198771= 1198770minus

119887(120573 minus 119886119902)

119888(119889 + 119902120582)

(23)

Note that 1198771

= ((120573 minus 119886119902)119888(119889 + 119902120582))(1198881199101minus 119887) + 1 which

implies that 1198771

gt 1 is equivalent to 1198881199101119887 gt 1 The

latter 1198881199101119887 is seen as the immune reproductive number

which expresses the average number of activated CTLsgenerated from one CTL during its life time 1119887 throughthe stimulation of the infected cells 119910

1 It is reasonable that

immune response is activated in the case where 1198771

gt 1When119877

1gt 1 in addition to the infection-free equilibrium119864

0

and the immune-absence equilibrium 1198641 there is an interior

immune-presence equilibrium 119864lowast

= (119909lowast 119910lowast 119911lowast) where

119909lowast

=

minus (119887120573 + 119888119889 minus 119888119902120582) + radic(119887120573 + 119888119889 minus 119888119902120582)2

+ 41198882119889119902120582

2119888119889119902

119910lowast

=

119887

119888

119911lowast

=

119888

119887119901

(120582 minus 119889119909lowast) minus

119886

119901

(24)

119864lowast expresses the state where CTLs immune response is

presentNext we establish the local asymptotic stability of model

(5) by the characteristic equation

Theorem 9 Consider system (5)

(1) The infection-free equilibrium 1198640is locally asymptoti-

cally stable if 1198770lt 1

(2) If 1198770gt 1 the equilibrium 119864

0is unstable and if 119877

0= 1

it is a critical case

Proof The characteristic equation for the infection-free equi-librium 119864

0is given as follows

det(

minus119889 minus 119903 minus

1205731199090

1 + 1199021199090

0

0

1205731199090

1 + 1199021199090

minus 119886 minus 119903 0

0 0 minus119887 minus 119903

) = 0 (25)

It is reduced to

(119903 + 119889)(119903 + 119887)(119903 + 119886 minus

1205731199090

1 + 1199021199090

) = 0 (26)

It is clear that (26) has the characteristic roots 1199031= minus119889 lt 0

which means | arg 1199031| = 120587 gt 120572(1205872) 119903

2= minus119887 lt 0 which

means | arg 1199032| = 120587 gt 120572(1205872) and 119903

3= (120573119909

0(1 + 119902119909

0)) minus 119886 =

119886(1198770minus 1) Since the imaginary part of characteristic root 119903

3is

zero 1198770lt 1 which means | arg 119903

3| = 120587 gt 120572(1205872) is necessary

and sufficient to ensure the local asymptotic stability of 1198640 If

1198770gt 1 | arg 119903

3| = 0 lt 120572(1205872) hence 119864

0is unstable If 119877

0= 1

1199033= 0 which is a critical case

Theorem 10 Consider system (5)

(1) The immune-free equilibrium 1198641is locally asymptoti-

cally stable if 1198771lt 1

(2) If 1198771gt 1 the immune-free equilibrium 119864

1is unstable

if 1198771= 1 it is a critical state

Proof The characteristic equation for the immune-free equi-librium 119864

1is given as follows

det(

minus119889 minus

1205731199101

(1 + 1199021199091)2minus 119903 minus

1205731199091

1 + 1199021199091

0

1205731199101

(1 + 1199021199091)2

1205731199091

1 + 1199021199091

minus 119886 minus 119903 minus1199011199101

0 0 1198881199101minus 119887 minus 119903

)

= 0

(27)

Using 1205731199091(1 + 119902119909

1) = 119886 it is reduced to

(119903 + 119887 minus 1198881199101)[1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2] = 0

(28)

Discrete Dynamics in Nature and Society 5

The root of (28) 1199031

= 1198881199101minus 119887 is negative and | arg 119903

1| = 120587 gt

120572(1205872)when1198771lt 1 positive and | arg 119903

1| = 0 lt 120572(1205872)when

1198771gt 1 and zero when 119877

1= 1 which is a critical case

Now we consider the equation

1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2

= 0 (29)

Denote

119861 =

1205731199101

(1 + 1199021199091)2

119862 =

1198861205731199101

(1 + 1199021199091)2

(30)

Since 119861 gt 0 and 119862 gt 0 (29) has two negative real roots andwe denote them by 119903

2and 1199033 It is easy to see | arg 119903

2| = 120587 gt

120572(1205872) and | arg 1199033| = 120587 gt 120572(1205872) Hence when 119877

1lt 1 the

immune-free equilibrium 1198641is locally asymptotically stable

when 1198771gt 1 119864

1is unstable and when 119877

1= 1 it is a critical

case

To discuss the local stability of the immune-presentequilibrium state 119864

lowast we consider the linearized system of (5)at 119864lowast The Jacobian matrix at 119864lowast is given by

119869 (119864lowast) = (

minus119889 minus

120573119910lowast

(1 + 119902119909lowast)2

minus

120573119909lowast

1 + 119902119909lowast

0

120573119910lowast

(1 + 119902119909lowast)2

0 minus119901119910lowast

0 119888119911lowast

0

) (31)

The characteristic equation of the linearized system is

119875 (119903) = 1199033+ 11988611199032+ 1198862119903 + 1198863= 0 (32)

where

1198861= 119889 +

120573119910lowast

(1 + 119902119909lowast)2

gt 0

1198862= 119888119901119910

lowast119911lowast+

1205732119909lowast119910lowast

(1 + 119902119909lowast)3

gt 0

1198863= 119888119901119910

lowast119911lowast(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

11988611198862minus 1198863=

1205732119909lowast119910lowast

(1 + 119902119909lowast)3(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

(33)

Based on Definition 4 we obtain the discriminant of (32)

119863(119875) = minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1198861

1198862

1198863

0

0 1 1198861

1198862

1198863

3 21198861

1198862

0 0

0 3 21198861

1198862

0

0 0 3 21198861

1198862

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 18119886111988621198863+ (11988611198862)2

minus 41198863

11198863minus 41198863

2minus 27119886

2

3

(34)

Using the result (iii) of Lemmas 5 and 3 we have the followingtheorem

Theorem 11 Consider system (5) Under the condition of 1198771gt

1(1) if the discriminant of 119875(119903) 119863(119875) is positive namely

119863(119875) gt 0 then the immune-present equilibrium 119864lowast

is locally asymptotically stable for 0 lt 120572 le 1(2) if 119863(119875) lt 0 then the immune-present equilibrium 119864

lowast

is locally asymptotically stable for 0 lt 120572 lt 23

5 Numerical Method

Atanackovic and Stankovic introduced a numerical methodto solve the single linear FDE in 2004 [1] A few years laterthey developed again a method to solve the nonlinear FDE[30] It was shown that the fractional derivative of a function119891(119905) with order 120572 satisfying 0 lt 120572 lt 1 may be expressed as119863120572119891(119905)

=

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

infin

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

infin

sum

119898=2

Γ (119898 minus 1 + 120572)

Γ (120572 minus 1) (119898 minus 1)

times(

119891 (119905)

119905120572

+

119881119898

(119891) (119905)

119905119898minus1+120572

)]

(35)where

119881119898(119891)(119905) = minus(119898 minus 1)int

119905

0

120591119898minus2

119891(120591)119889120591 119898 = 2 3

(36)with the following properties

119889

119889119905

119881119898

(119891) = minus (119898 minus 1) 119905119898minus2

119891 (119905) 119898 = 2 3 (37)

We approximate119863120572119891(119905) by using119872 terms in sums appearing

in (35) as follows119863120572119891(119905)

≃

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

119872

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

119872

sum

119898=2

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)(119898 minus 1)

times(

119891(119905)

119905120572

+

119881119898(119891)(119905)

119905119898minus1+120572

)]

(38)

6 Discrete Dynamics in Nature and Society

We can rewrite (38) as follows

119863120572119891(119905) ≃ Ω(120579 119905119872)119891

(1)

(119905) + Φ(120579 119905119872)119891(119905)

+

119872

sum

119898=2

119860 (120579 119905 119898)

119881119898

(119891) (119905)

119905119898minus1+120572

(39)

where

Ω (120572 119905119872) =

1 + sum119872

119898=1(Γ (119898 minus 1 + 120572) Γ (120572 minus 1)119898)

Γ (2 minus 120572) 119905120572minus1

119877 (119886 119905) =

1 minus 120572

119905120572Γ(2 minus 120572)

119860 (120572 119905 119898) = minus

Γ (119898 minus 1 + 120572)

Γ (2 minus 120572) Γ (120572 minus 1) (119898 minus 1)

Φ (120572 119905119872) = 119877 (119886 119905) +

119872

sum

119898=2

119860 (120572 119905 119898)

119905120572

(40)

We set

Θ1(119905) = 119909 (119905) Θ

119898(119905) = 119881

119898(119909) (119905)

Θ119872+1

(119905) = 119910 (119905) Θ119872+119898

(119905) = 119881119898

(119910) (119905)

Θ2119872+1

(119905) = 119911 (119905) Θ2119872+119898

(119905) = 119881119898

(119911) (119905)

(41)

for 119898 = 2 3 We can rewrite system (19) as the followingform

Ω(120572 119905119872)Θ1015840

1(119905) + Φ(120572 119905119872)Θ

1(119905) +

119872

sum

119898=2

119860(120572 119905 119898)

Θ119898(119905)

119905119898minus1+120572

= 120582 minus 119889Θ1(119905) minus

120573Θ1(119905) Θ119872+1

(119905)

1 + 119902Θ1(119905)

Ω(120572 119905119872)Θ1015840

119872+1(119905) + Φ(120572 119905119872)Θ

119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

=

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus 119886Θ119872+1

(119905) minus 119901Θ119872+1

(119905) Θ2119872+1

(119905)

Ω(120572 119905119872)Θ1015840

2119872+1(119905) + Φ(120572 119905119872)Θ

2119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

= 119888Θ119872+1

(119905) Θ2119872+1

(119905) minus 119887Θ2119872+1

(119905)

(42)

where

Θ119898(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ1(120591)119889120591

Θ119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ119872+1

(120591)119889120591

Θ2119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ2119872+1

(120591)119889120591

119898 = 2 3 119872

(43)

Now we can rewrite (39) and (42) as the following form

Θ1015840

1(119905) =

1

Ω(120572 119905119872)

(120582 minus (119889 + Φ(120572 119905119872))Θ1(119905)

minus

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119898

(119905)

119905119898minus1+120572

)

Θ1015840

119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ1(119905) 119898 = 2 3 119872

Θ1015840

119872+1(119905) =

1

Ω(120572 119905119872)

(

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus (119886 + Φ(120572 119905119872))Θ119872+1

(119905)

minus 119901Θ119872+1

(119905)Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ119872+1

(119905) 119898 = 2 3 119872

Θ1015840

2119872+1(119905) =

1

Ω(120572 119905119872)

(119888Θ119872+1

(119905)Θ2119872+1

(119905)

minus (119887 + Φ(120572 119905119872))Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

2119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ2119872+1

(119905) 119898 = 2 3 119872

(44)with the following initial conditions

Θ1(120575) = 119909

0 Θ119898

(120575) = 0 119898 = 2 3 119872

Θ119872+1

(120575) = 1199100 Θ119872+119898

(120575) = 0 119898 = 2 3 119872

Θ2119872+1

(120575) = 1199110 Θ2119872+119898

(120575) = 0 119898 = 2 3 119872

(45)Now we consider the numerical solution of system of ordi-nary differential equations (44) with the initial conditions(45) by using the well-known Runge-Kutta method of fourthorder

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

Mathematical models have been proven valuable inunderstanding the dynamics of viral infection Although alarge number of works on modeling the dynamics of viralinfection have been done [16ndash20] it has been restricted tointeger-order (delay) differential equations In recent yearsit has turned out that many phenomena in virus infectioncan be described very successfully by the models usingfractional-order differential equations [21 22] Motivated bythese references in this paper we will consider a fractional-order HIV model

Among a large number of virus infection modelsdescribed with integer-order or delay differential equationsa typical model is given as the following simple three-dimensional system

1199091015840

(119905) = 120582 minus 119889119909(119905) minus 120573119909(119905)V(119905)

1199101015840

(119905) = 120573119909(119905)V(119905) minus 119901119910(119905)

V1015840(119905) = 119896119910(119905) minus 119906V(119905)

(1)

Here 119909(119905) represents the concentration of uninfected cells attime 119905 119910(119905) represents the concentration of infected cells thatproduce virus at time 119905 V(119905) represents the concentration ofviruses at time 119905 119889 (119889 gt 0) and 120573 (120573 gt 0) are the death rate ofuninfected cells and the rate constant characterizing infectionof the cells respectively119901 (119901 gt 0) is the death rate of infectedcells due to either the virus or the immune system Free virusis produced from infected cells at the rate 119896119910(119905) and removedat rate 119906V(119905)

The infection rate in model (1) is usually assumed tobe bilinear with respect to virus V(t) and uninfected targetcells 119909(119905) However there are some evidences which showthat a bilinear infection might not be an effective assumptionwhen the number of target cells is large enough [23ndash25]AndHuang et al [17] incorporated Holling type II functionalresponse infection rate 119892(119909(119905) V(119905)) into basic virus dynamics(1) which was expressed as

119892(119909(119905) V(119905)) =

120573119909(119905)V(119905)1 + 119902119909(119905)

(2)

where 119902 (119902 ge 0) is a constant Hence system (1) can bemodified into the following system

1199091015840

(119905) = 120582 minus 119889119909(119905) minus

120573119909(119905)V(119905)1 + 119902119909(119905)

1199101015840

(119905) =

120573119909(119905)V(119905)1 + 119902119909(119905)

minus 119901119910(119905)

V1015840(119905) = 119896119910(119905) minus 119906V(119905)

(3)

The immune response following viral infection is uni-versal and necessary to eliminate or control the diseaseAntibodies cytokines natural killer cells B cells and T cellsare all essential components of a normal immune responseto viral infection Since cytotoxic T lymphocytes (CTLs) playa critical role in antiviral defense by attacking the virus-infected cells Huang et al [17] considered the immuneresponse to model (3) with 119911

1015840(119905) = 119888119910(119905)119911(119905) minus 119887119911(119905)

System (3) with 1199111015840(119905) = 119888119910(119905)119911(119905) minus 119887119911(119905) can be further

simplified if we take into consideration the fact that anaverage life span of viral particles is usually significantlyshorter than one of the infected cells It can be assumedtherefore that compared with a ldquoslowrdquo variation of theinfected cells level the virus load V(119905) relatively quicklyreaches a quasiequilibrium level The equality V(119905) = 0 holdsin the quasiequilibrium state and hence V(119905) = 119896119910(119905)119906This assumption is referred to as ldquoseparation of time scalesrdquoand is in common use in the virus dynamics [18] We haveto stress that this assumption does not imply that the virusconcentration V(119905) remains constant on the contrary it isassumed to be proportional to the varying concentrationof infected cells 119910(119905) Accordingly system (3) with 119911

1015840(119905) =

119888119910(119905)119911(119905)minus119887119911(119905) can now be reformulated as a system of threedifferential equations

1199091015840

(119905) = 120582 minus 119889119909 (119905) minus

120573119909 (119905) 119910 (119905)

1 + 119902119909 (119905)

1199101015840

(119905) =

120573119909 (119905) 119910 (119905)

1 + 119902119909 (119905)

minus 119886119910 (119905) minus 119901119910 (119905) 119911 (119905)

1199111015840

(119905) = 119888119910(119905)119911(119905) minus 119887119911(119905)

(4)

Further we introduce the fractional order into model (4)and a new system can be described by the following set ofFODEs of order 120572 gt 0

119909120572

(119905) = 120582 minus 119889119909 minus

120573119909119910

1 + 119902119909

119910120572

(119905) =

120573119909119910

1 + 119902119909

minus 119886119910 minus 119901119910119911

119911120572

(119905) = 119888119910119911 minus 119887119911

(5)

119909 (0) = 1199090 119910 (0) = 119910

0 119911 (0) = 119911

0 (6)

where the immune response is assumed to get stronger ata rate 119888119910119911 which is proportional to the number of infectedcells and their current concentration the immune responsedecays exponentially at a rate 119887119911 which is proportional totheir current concentration the parameter 119902 expresses theefficacy of nonlytic component

The paper is organized as follows In the next sectionwe give two definitions and two lemmas about fractionalcalculus In Section 3 we show the nonnegativeness of thesolutions of system (5) with initial condition (6) In Section 4we give a detailed stability analysis for three equilibria Finallynumerical simulations are presented to illustrate the obtainedresults in Section 5

2 Fractional Calculus

Definition 1 (see [26]) The Riemann-Liouville (R-L) frac-tional integral operator of order 120572 gt 0 of a function119891 119877

+rarr

119877 is defined as

119868120572119891 (119909) =

1

Γ (120572)

int

119909

0

(119909 minus 119905)120572minus1

119891 (119905) 119889119905 (7)

Discrete Dynamics in Nature and Society 3

Here Γ(sdot) is the Euler gamma function which is defined as

Γ (119899) = int

infin

0

119905119899minus1

119890minus119905119889119905 (8)

This function is generalization of a factorial in the followingform

Γ (119899) = (119899 minus 1) (9)

Definition 2 (see [26]) The Caputo (C) fractional derivativeof order 120572 gt 0 119899 minus 1 lt 120572 lt 119899 119899 isin 119873 is defined as

119863120572119891 (119905) = 119868

119899minus120572119863119899119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0

119891(119899)

(119904)

(119905 minus 119904)120572+1minus119899

119889119904

(10)

where the function119891(119905) has absolutely continuous derivativesup to order (119899 minus 1) In particular when 0 lt 120572 lt 1 one has

119863120572119891 (119905) =

1

Γ (1 minus 120572)

int

119905

0

1198911015840(119904)

(119905 minus 119904)120572119889119904 (11)

In this paper we use Caputo fractional derivative defini-tion The main advantage of Caputorsquos definition is that theinitial conditions for fractional differential equations withCaputo derivatives take the same form as that for integer-order differential equations

Lemma 3 (see [26]) Consider the following commensuratefractional-order system

119863120572119909 = 119891 (119909)

119909 (0) = 1199090

(12)

with 0 lt 120572 le 1 and 119909 isin 119877119899 The equilibrium points of system

(12) are calculated by solving the following equation 119891(119909) = 0These points are locally asymptotically stable if all eigenvalues119903119894of Jacobian matrix 119869 = 120597119891120597119909 evaluated at the equilibrium

points satisfy

1003816100381610038161003816arg(119903119894)1003816100381610038161003816gt

120579120587

2

(13)

Definition 4 (see [27]) The discriminant 119863(119891) of a polyno-mial

119891(119909) = 119909119899+ 1198881119909119899minus1

+ 1198882119909119899minus2

+ sdot sdot sdot + 119888119899

(14)

is defined by 119863(119891) = (minus1)119899(119899minus1)2

119877(119891 1198911015840) where 119891

1015840 is thederivative of119891 If119892(119909) = 119909

119897+1198891119909119897minus1

+1198892119909119897minus2

+sdot sdot sdot+119889119899119877(119891 119892) is

the determinant of the corresponding Sylvester (119899+119897)otimes(119899+119897)

matrix The Sylvester matrix is formed by filling the matrixbeginning with the upper left corner with the coefficients of119891(119909) and then shifting down one row and one column to theright and filling in the coefficients starting there until they hitthe right sideThe process is then repeated for the coefficientsof 119892(119909)

Lemma 5 (see [27]) For the polynomial equation

119875 (120582) = 120582119899+ ℎ1120582119899minus1

+ ℎ2120582119899minus2

+ sdot sdot sdot + ℎ119899= 0 (15)

the conditions which make all the roots of (15) satisfy (13) aredisplayed as follows

(i) for 119899 = 1 the condition is ℎ1gt 0

(ii) for 119899 = 2 the conditions are either Routh-Hurwitzconditions or

ℎ1lt 0 4ℎ

2gt (ℎ1)2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

tanminus1(radic4ℎ2minus (ℎ1)2

ℎ1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt

120572120587

2

(16)

(iii) for 119899 = 3

(a) if the discriminant of 119875(120582) 119863(119875) is positive thenRouth-Hurwitz conditions are the necessary andsufficient conditions that is ℎ

1gt 0 ℎ

3gt 0 and

ℎ1ℎ2gt ℎ3if 119863(119875) gt 0

(b) if119863(119875) lt 0 ℎ1ge 0 ℎ

2ge 0 and ℎ

3gt 0 then (13)

for (15) holds when 120572 lt 23(c) if119863(119875) lt 0 ℎ

1lt 0 and ℎ

2lt 0 then (13) for (15)

holds when 120572 gt 23(d) if 119863(119875) lt 0 ℎ

1gt 0 ℎ

2gt 0 and ℎ

1ℎ2= ℎ3 then

(13) for (15) holds for all 120572 isin [0 1)

3 Nonnegative Solutions

Let 1198773+

= 119882 isin 1198773

119882 ge 0 and 119882(119905) = (119909(119905) 119910(119905) 119911(119905))119879

For the proof of the theorem about nonnegative solutions wewould need the following lemma

Lemma 6 (see [28] generalized mean value theorem) Let119891(119909) isin 119862[119886 119887] and 119863

120572119891(119909) isin 119862[119886 119887] for 0 lt 120572 le 1 Then

one has

119891(119909) = 119891(119886) +

1

Γ(120572)

119863120572119891(120585)(119909 minus 119886)

120572 (17)

with 0 le 120585 le 119909 forall119909 isin (119886 119887] where Γ(119909) = int

+infin

0119905119909minus1

119890minus119905119889119905

Remark 7 Suppose that 119891(119909) isin 119862[119886 119887] and 119863120572119891(119909) isin

119862[119886 119887] for 0 lt 120572 le 1 It is clear from Lemma 6 that if119863120572119891(119909) ge 0 forall119909 isin (119886 119887) then 119891(119909) is nondecreasing for

each 119909 isin [119886 119887] If 119863120572119891(119909) le 0 forall119909 isin (119886 119887) then 119891(119909) is

nonincreasing for each 119909 isin [119886 119887]

We now prove the main theorem

Theorem 8 There is a unique solution for the initial valueproblem (5) with (6) and the solution remains in 119877

3

+

Proof From Theorem 31 and Remark 32 of [29] we obtainthe solution on (0infin) solving the initial value problem (5)with (6) which is not only existent but also unique Next wewill show the nonnegative orthant 1198773

+is a positively invariant

region What is needed for this is to show that on eachhyperplane bounding the nonnegative orthant the vectorfield points into 119877

3

+ From (5) we find

119909120572(119905)

1003816100381610038161003816119909=0

= 120582 ge 0 119910120572(119905)

1003816100381610038161003816119910=0

= 0 119911120572(119905)

1003816100381610038161003816119911=0

= 0

(18)

By Remark 7 the solution will remain in 1198773

+

4 Discrete Dynamics in Nature and Society

4 Equilibrium States and Their Stability

In this section we investigate the stability of the fractional-ordermodel of HIV infection of CD4 + T cells that is system(5) with (6) Consider the initial value problem (5) with (6)with 120572 satisfying 0 lt 120572 le 1

In order to obtain the equilibria of system (5) we set119909120572(119905) = 0 119910120572(119905) = 0 and 119911

120572(119905) = 0 and we have

120582 minus 119889119909 minus

120573119909119910

1 + 119902119909

= 0

120573119909119910

1 + 119902119909

minus 119886119910 minus 119901119910119911 = 0

119888119910119911 minus 119887119911 = 0

(19)

System (5) has three types of relevant nonnegative equilib-rium states System (5) always has an infection-free equilib-rium 119864

0= (1199090 0 0) where

1199090=

120582

119889

(20)

which means that the infected cells are cleared The basicreproductive number of the viruses for system (5) is given by

1198770=

120582120573

119886 (119889 + 119902120582)

(21)

This number describes the average number of newly gener-ated infected cells from one infected cell at the beginningof the infection process When 119877

0gt 1 in addition to the

infection-free equilibrium 1198640 there is an immune-absence

equilibrium 1198641= (1199091 1199101 1199111) where

1199091=

119886

120573 minus 119886119902

1199101=

119889 + 119902120582

120573 minus 119886119902

(1198770minus 1) 119911

1= 0

(22)Note that 119877

0gt 1 implies 120573 gt 119886119902 When the HIV infection

is in the immune-absence equilibrium 1198641 the infected cells

and virus exist but the immune response is not activated yetFurther we denote

1198771= 1198770minus

119887(120573 minus 119886119902)

119888(119889 + 119902120582)

(23)

Note that 1198771

= ((120573 minus 119886119902)119888(119889 + 119902120582))(1198881199101minus 119887) + 1 which

implies that 1198771

gt 1 is equivalent to 1198881199101119887 gt 1 The

latter 1198881199101119887 is seen as the immune reproductive number

which expresses the average number of activated CTLsgenerated from one CTL during its life time 1119887 throughthe stimulation of the infected cells 119910

1 It is reasonable that

immune response is activated in the case where 1198771

gt 1When119877

1gt 1 in addition to the infection-free equilibrium119864

0

and the immune-absence equilibrium 1198641 there is an interior

immune-presence equilibrium 119864lowast

= (119909lowast 119910lowast 119911lowast) where

119909lowast

=

minus (119887120573 + 119888119889 minus 119888119902120582) + radic(119887120573 + 119888119889 minus 119888119902120582)2

+ 41198882119889119902120582

2119888119889119902

119910lowast

=

119887

119888

119911lowast

=

119888

119887119901

(120582 minus 119889119909lowast) minus

119886

119901

(24)

119864lowast expresses the state where CTLs immune response is

presentNext we establish the local asymptotic stability of model

(5) by the characteristic equation

Theorem 9 Consider system (5)

(1) The infection-free equilibrium 1198640is locally asymptoti-

cally stable if 1198770lt 1

(2) If 1198770gt 1 the equilibrium 119864

0is unstable and if 119877

0= 1

it is a critical case

Proof The characteristic equation for the infection-free equi-librium 119864

0is given as follows

det(

minus119889 minus 119903 minus

1205731199090

1 + 1199021199090

0

0

1205731199090

1 + 1199021199090

minus 119886 minus 119903 0

0 0 minus119887 minus 119903

) = 0 (25)

It is reduced to

(119903 + 119889)(119903 + 119887)(119903 + 119886 minus

1205731199090

1 + 1199021199090

) = 0 (26)

It is clear that (26) has the characteristic roots 1199031= minus119889 lt 0

which means | arg 1199031| = 120587 gt 120572(1205872) 119903

2= minus119887 lt 0 which

means | arg 1199032| = 120587 gt 120572(1205872) and 119903

3= (120573119909

0(1 + 119902119909

0)) minus 119886 =

119886(1198770minus 1) Since the imaginary part of characteristic root 119903

3is

zero 1198770lt 1 which means | arg 119903

3| = 120587 gt 120572(1205872) is necessary

and sufficient to ensure the local asymptotic stability of 1198640 If

1198770gt 1 | arg 119903

3| = 0 lt 120572(1205872) hence 119864

0is unstable If 119877

0= 1

1199033= 0 which is a critical case

Theorem 10 Consider system (5)

(1) The immune-free equilibrium 1198641is locally asymptoti-

cally stable if 1198771lt 1

(2) If 1198771gt 1 the immune-free equilibrium 119864

1is unstable

if 1198771= 1 it is a critical state

Proof The characteristic equation for the immune-free equi-librium 119864

1is given as follows

det(

minus119889 minus

1205731199101

(1 + 1199021199091)2minus 119903 minus

1205731199091

1 + 1199021199091

0

1205731199101

(1 + 1199021199091)2

1205731199091

1 + 1199021199091

minus 119886 minus 119903 minus1199011199101

0 0 1198881199101minus 119887 minus 119903

)

= 0

(27)

Using 1205731199091(1 + 119902119909

1) = 119886 it is reduced to

(119903 + 119887 minus 1198881199101)[1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2] = 0

(28)

Discrete Dynamics in Nature and Society 5

The root of (28) 1199031

= 1198881199101minus 119887 is negative and | arg 119903

1| = 120587 gt

120572(1205872)when1198771lt 1 positive and | arg 119903

1| = 0 lt 120572(1205872)when

1198771gt 1 and zero when 119877

1= 1 which is a critical case

Now we consider the equation

1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2

= 0 (29)

Denote

119861 =

1205731199101

(1 + 1199021199091)2

119862 =

1198861205731199101

(1 + 1199021199091)2

(30)

Since 119861 gt 0 and 119862 gt 0 (29) has two negative real roots andwe denote them by 119903

2and 1199033 It is easy to see | arg 119903

2| = 120587 gt

120572(1205872) and | arg 1199033| = 120587 gt 120572(1205872) Hence when 119877

1lt 1 the

immune-free equilibrium 1198641is locally asymptotically stable

when 1198771gt 1 119864

1is unstable and when 119877

1= 1 it is a critical

case

To discuss the local stability of the immune-presentequilibrium state 119864

lowast we consider the linearized system of (5)at 119864lowast The Jacobian matrix at 119864lowast is given by

119869 (119864lowast) = (

minus119889 minus

120573119910lowast

(1 + 119902119909lowast)2

minus

120573119909lowast

1 + 119902119909lowast

0

120573119910lowast

(1 + 119902119909lowast)2

0 minus119901119910lowast

0 119888119911lowast

0

) (31)

The characteristic equation of the linearized system is

119875 (119903) = 1199033+ 11988611199032+ 1198862119903 + 1198863= 0 (32)

where

1198861= 119889 +

120573119910lowast

(1 + 119902119909lowast)2

gt 0

1198862= 119888119901119910

lowast119911lowast+

1205732119909lowast119910lowast

(1 + 119902119909lowast)3

gt 0

1198863= 119888119901119910

lowast119911lowast(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

11988611198862minus 1198863=

1205732119909lowast119910lowast

(1 + 119902119909lowast)3(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

(33)

Based on Definition 4 we obtain the discriminant of (32)

119863(119875) = minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1198861

1198862

1198863

0

0 1 1198861

1198862

1198863

3 21198861

1198862

0 0

0 3 21198861

1198862

0

0 0 3 21198861

1198862

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 18119886111988621198863+ (11988611198862)2

minus 41198863

11198863minus 41198863

2minus 27119886

2

3

(34)

Using the result (iii) of Lemmas 5 and 3 we have the followingtheorem

Theorem 11 Consider system (5) Under the condition of 1198771gt

1(1) if the discriminant of 119875(119903) 119863(119875) is positive namely

119863(119875) gt 0 then the immune-present equilibrium 119864lowast

is locally asymptotically stable for 0 lt 120572 le 1(2) if 119863(119875) lt 0 then the immune-present equilibrium 119864

lowast

is locally asymptotically stable for 0 lt 120572 lt 23

5 Numerical Method

Atanackovic and Stankovic introduced a numerical methodto solve the single linear FDE in 2004 [1] A few years laterthey developed again a method to solve the nonlinear FDE[30] It was shown that the fractional derivative of a function119891(119905) with order 120572 satisfying 0 lt 120572 lt 1 may be expressed as119863120572119891(119905)

=

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

infin

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

infin

sum

119898=2

Γ (119898 minus 1 + 120572)

Γ (120572 minus 1) (119898 minus 1)

times(

119891 (119905)

119905120572

+

119881119898

(119891) (119905)

119905119898minus1+120572

)]

(35)where

119881119898(119891)(119905) = minus(119898 minus 1)int

119905

0

120591119898minus2

119891(120591)119889120591 119898 = 2 3

(36)with the following properties

119889

119889119905

119881119898

(119891) = minus (119898 minus 1) 119905119898minus2

119891 (119905) 119898 = 2 3 (37)

We approximate119863120572119891(119905) by using119872 terms in sums appearing

in (35) as follows119863120572119891(119905)

≃

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

119872

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

119872

sum

119898=2

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)(119898 minus 1)

times(

119891(119905)

119905120572

+

119881119898(119891)(119905)

119905119898minus1+120572

)]

(38)

6 Discrete Dynamics in Nature and Society

We can rewrite (38) as follows

119863120572119891(119905) ≃ Ω(120579 119905119872)119891

(1)

(119905) + Φ(120579 119905119872)119891(119905)

+

119872

sum

119898=2

119860 (120579 119905 119898)

119881119898

(119891) (119905)

119905119898minus1+120572

(39)

where

Ω (120572 119905119872) =

1 + sum119872

119898=1(Γ (119898 minus 1 + 120572) Γ (120572 minus 1)119898)

Γ (2 minus 120572) 119905120572minus1

119877 (119886 119905) =

1 minus 120572

119905120572Γ(2 minus 120572)

119860 (120572 119905 119898) = minus

Γ (119898 minus 1 + 120572)

Γ (2 minus 120572) Γ (120572 minus 1) (119898 minus 1)

Φ (120572 119905119872) = 119877 (119886 119905) +

119872

sum

119898=2

119860 (120572 119905 119898)

119905120572

(40)

We set

Θ1(119905) = 119909 (119905) Θ

119898(119905) = 119881

119898(119909) (119905)

Θ119872+1

(119905) = 119910 (119905) Θ119872+119898

(119905) = 119881119898

(119910) (119905)

Θ2119872+1

(119905) = 119911 (119905) Θ2119872+119898

(119905) = 119881119898

(119911) (119905)

(41)

for 119898 = 2 3 We can rewrite system (19) as the followingform

Ω(120572 119905119872)Θ1015840

1(119905) + Φ(120572 119905119872)Θ

1(119905) +

119872

sum

119898=2

119860(120572 119905 119898)

Θ119898(119905)

119905119898minus1+120572

= 120582 minus 119889Θ1(119905) minus

120573Θ1(119905) Θ119872+1

(119905)

1 + 119902Θ1(119905)

Ω(120572 119905119872)Θ1015840

119872+1(119905) + Φ(120572 119905119872)Θ

119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

=

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus 119886Θ119872+1

(119905) minus 119901Θ119872+1

(119905) Θ2119872+1

(119905)

Ω(120572 119905119872)Θ1015840

2119872+1(119905) + Φ(120572 119905119872)Θ

2119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

= 119888Θ119872+1

(119905) Θ2119872+1

(119905) minus 119887Θ2119872+1

(119905)

(42)

where

Θ119898(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ1(120591)119889120591

Θ119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ119872+1

(120591)119889120591

Θ2119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ2119872+1

(120591)119889120591

119898 = 2 3 119872

(43)

Now we can rewrite (39) and (42) as the following form

Θ1015840

1(119905) =

1

Ω(120572 119905119872)

(120582 minus (119889 + Φ(120572 119905119872))Θ1(119905)

minus

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119898

(119905)

119905119898minus1+120572

)

Θ1015840

119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ1(119905) 119898 = 2 3 119872

Θ1015840

119872+1(119905) =

1

Ω(120572 119905119872)

(

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus (119886 + Φ(120572 119905119872))Θ119872+1

(119905)

minus 119901Θ119872+1

(119905)Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ119872+1

(119905) 119898 = 2 3 119872

Θ1015840

2119872+1(119905) =

1

Ω(120572 119905119872)

(119888Θ119872+1

(119905)Θ2119872+1

(119905)

minus (119887 + Φ(120572 119905119872))Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

2119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ2119872+1

(119905) 119898 = 2 3 119872

(44)with the following initial conditions

Θ1(120575) = 119909

0 Θ119898

(120575) = 0 119898 = 2 3 119872

Θ119872+1

(120575) = 1199100 Θ119872+119898

(120575) = 0 119898 = 2 3 119872

Θ2119872+1

(120575) = 1199110 Θ2119872+119898

(120575) = 0 119898 = 2 3 119872

(45)Now we consider the numerical solution of system of ordi-nary differential equations (44) with the initial conditions(45) by using the well-known Runge-Kutta method of fourthorder

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

Here Γ(sdot) is the Euler gamma function which is defined as

Γ (119899) = int

infin

0

119905119899minus1

119890minus119905119889119905 (8)

This function is generalization of a factorial in the followingform

Γ (119899) = (119899 minus 1) (9)

Definition 2 (see [26]) The Caputo (C) fractional derivativeof order 120572 gt 0 119899 minus 1 lt 120572 lt 119899 119899 isin 119873 is defined as

119863120572119891 (119905) = 119868

119899minus120572119863119899119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0

119891(119899)

(119904)

(119905 minus 119904)120572+1minus119899

119889119904

(10)

where the function119891(119905) has absolutely continuous derivativesup to order (119899 minus 1) In particular when 0 lt 120572 lt 1 one has

119863120572119891 (119905) =

1

Γ (1 minus 120572)

int

119905

0

1198911015840(119904)

(119905 minus 119904)120572119889119904 (11)

In this paper we use Caputo fractional derivative defini-tion The main advantage of Caputorsquos definition is that theinitial conditions for fractional differential equations withCaputo derivatives take the same form as that for integer-order differential equations

Lemma 3 (see [26]) Consider the following commensuratefractional-order system

119863120572119909 = 119891 (119909)

119909 (0) = 1199090

(12)

with 0 lt 120572 le 1 and 119909 isin 119877119899 The equilibrium points of system

(12) are calculated by solving the following equation 119891(119909) = 0These points are locally asymptotically stable if all eigenvalues119903119894of Jacobian matrix 119869 = 120597119891120597119909 evaluated at the equilibrium

points satisfy

1003816100381610038161003816arg(119903119894)1003816100381610038161003816gt

120579120587

2

(13)

Definition 4 (see [27]) The discriminant 119863(119891) of a polyno-mial

119891(119909) = 119909119899+ 1198881119909119899minus1

+ 1198882119909119899minus2

+ sdot sdot sdot + 119888119899

(14)

is defined by 119863(119891) = (minus1)119899(119899minus1)2

119877(119891 1198911015840) where 119891

1015840 is thederivative of119891 If119892(119909) = 119909

119897+1198891119909119897minus1

+1198892119909119897minus2

+sdot sdot sdot+119889119899119877(119891 119892) is

the determinant of the corresponding Sylvester (119899+119897)otimes(119899+119897)

matrix The Sylvester matrix is formed by filling the matrixbeginning with the upper left corner with the coefficients of119891(119909) and then shifting down one row and one column to theright and filling in the coefficients starting there until they hitthe right sideThe process is then repeated for the coefficientsof 119892(119909)

Lemma 5 (see [27]) For the polynomial equation

119875 (120582) = 120582119899+ ℎ1120582119899minus1

+ ℎ2120582119899minus2

+ sdot sdot sdot + ℎ119899= 0 (15)

the conditions which make all the roots of (15) satisfy (13) aredisplayed as follows

(i) for 119899 = 1 the condition is ℎ1gt 0

(ii) for 119899 = 2 the conditions are either Routh-Hurwitzconditions or

ℎ1lt 0 4ℎ

2gt (ℎ1)2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

tanminus1(radic4ℎ2minus (ℎ1)2

ℎ1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt

120572120587

2

(16)

(iii) for 119899 = 3

(a) if the discriminant of 119875(120582) 119863(119875) is positive thenRouth-Hurwitz conditions are the necessary andsufficient conditions that is ℎ

1gt 0 ℎ

3gt 0 and

ℎ1ℎ2gt ℎ3if 119863(119875) gt 0

(b) if119863(119875) lt 0 ℎ1ge 0 ℎ

2ge 0 and ℎ

3gt 0 then (13)

for (15) holds when 120572 lt 23(c) if119863(119875) lt 0 ℎ

1lt 0 and ℎ

2lt 0 then (13) for (15)

holds when 120572 gt 23(d) if 119863(119875) lt 0 ℎ

1gt 0 ℎ

2gt 0 and ℎ

1ℎ2= ℎ3 then

(13) for (15) holds for all 120572 isin [0 1)

3 Nonnegative Solutions

Let 1198773+

= 119882 isin 1198773

119882 ge 0 and 119882(119905) = (119909(119905) 119910(119905) 119911(119905))119879

For the proof of the theorem about nonnegative solutions wewould need the following lemma

Lemma 6 (see [28] generalized mean value theorem) Let119891(119909) isin 119862[119886 119887] and 119863

120572119891(119909) isin 119862[119886 119887] for 0 lt 120572 le 1 Then

one has

119891(119909) = 119891(119886) +

1

Γ(120572)

119863120572119891(120585)(119909 minus 119886)

120572 (17)

with 0 le 120585 le 119909 forall119909 isin (119886 119887] where Γ(119909) = int

+infin

0119905119909minus1

119890minus119905119889119905

Remark 7 Suppose that 119891(119909) isin 119862[119886 119887] and 119863120572119891(119909) isin

119862[119886 119887] for 0 lt 120572 le 1 It is clear from Lemma 6 that if119863120572119891(119909) ge 0 forall119909 isin (119886 119887) then 119891(119909) is nondecreasing for

each 119909 isin [119886 119887] If 119863120572119891(119909) le 0 forall119909 isin (119886 119887) then 119891(119909) is

nonincreasing for each 119909 isin [119886 119887]

We now prove the main theorem

Theorem 8 There is a unique solution for the initial valueproblem (5) with (6) and the solution remains in 119877

3

+

Proof From Theorem 31 and Remark 32 of [29] we obtainthe solution on (0infin) solving the initial value problem (5)with (6) which is not only existent but also unique Next wewill show the nonnegative orthant 1198773

+is a positively invariant

region What is needed for this is to show that on eachhyperplane bounding the nonnegative orthant the vectorfield points into 119877

3

+ From (5) we find

119909120572(119905)

1003816100381610038161003816119909=0

= 120582 ge 0 119910120572(119905)

1003816100381610038161003816119910=0

= 0 119911120572(119905)

1003816100381610038161003816119911=0

= 0

(18)

By Remark 7 the solution will remain in 1198773

+

4 Discrete Dynamics in Nature and Society

4 Equilibrium States and Their Stability

In this section we investigate the stability of the fractional-ordermodel of HIV infection of CD4 + T cells that is system(5) with (6) Consider the initial value problem (5) with (6)with 120572 satisfying 0 lt 120572 le 1

In order to obtain the equilibria of system (5) we set119909120572(119905) = 0 119910120572(119905) = 0 and 119911

120572(119905) = 0 and we have

120582 minus 119889119909 minus

120573119909119910

1 + 119902119909

= 0

120573119909119910

1 + 119902119909

minus 119886119910 minus 119901119910119911 = 0

119888119910119911 minus 119887119911 = 0

(19)

System (5) has three types of relevant nonnegative equilib-rium states System (5) always has an infection-free equilib-rium 119864

0= (1199090 0 0) where

1199090=

120582

119889

(20)

which means that the infected cells are cleared The basicreproductive number of the viruses for system (5) is given by

1198770=

120582120573

119886 (119889 + 119902120582)

(21)

This number describes the average number of newly gener-ated infected cells from one infected cell at the beginningof the infection process When 119877

0gt 1 in addition to the

infection-free equilibrium 1198640 there is an immune-absence

equilibrium 1198641= (1199091 1199101 1199111) where

1199091=

119886

120573 minus 119886119902

1199101=

119889 + 119902120582

120573 minus 119886119902

(1198770minus 1) 119911

1= 0

(22)Note that 119877

0gt 1 implies 120573 gt 119886119902 When the HIV infection

is in the immune-absence equilibrium 1198641 the infected cells

and virus exist but the immune response is not activated yetFurther we denote

1198771= 1198770minus

119887(120573 minus 119886119902)

119888(119889 + 119902120582)

(23)

Note that 1198771

= ((120573 minus 119886119902)119888(119889 + 119902120582))(1198881199101minus 119887) + 1 which

implies that 1198771

gt 1 is equivalent to 1198881199101119887 gt 1 The

latter 1198881199101119887 is seen as the immune reproductive number

which expresses the average number of activated CTLsgenerated from one CTL during its life time 1119887 throughthe stimulation of the infected cells 119910

1 It is reasonable that

immune response is activated in the case where 1198771

gt 1When119877

1gt 1 in addition to the infection-free equilibrium119864

0

and the immune-absence equilibrium 1198641 there is an interior

immune-presence equilibrium 119864lowast

= (119909lowast 119910lowast 119911lowast) where

119909lowast

=

minus (119887120573 + 119888119889 minus 119888119902120582) + radic(119887120573 + 119888119889 minus 119888119902120582)2

+ 41198882119889119902120582

2119888119889119902

119910lowast

=

119887

119888

119911lowast

=

119888

119887119901

(120582 minus 119889119909lowast) minus

119886

119901

(24)

119864lowast expresses the state where CTLs immune response is

presentNext we establish the local asymptotic stability of model

(5) by the characteristic equation

Theorem 9 Consider system (5)

(1) The infection-free equilibrium 1198640is locally asymptoti-

cally stable if 1198770lt 1

(2) If 1198770gt 1 the equilibrium 119864

0is unstable and if 119877

0= 1

it is a critical case

Proof The characteristic equation for the infection-free equi-librium 119864

0is given as follows

det(

minus119889 minus 119903 minus

1205731199090

1 + 1199021199090

0

0

1205731199090

1 + 1199021199090

minus 119886 minus 119903 0

0 0 minus119887 minus 119903

) = 0 (25)

It is reduced to

(119903 + 119889)(119903 + 119887)(119903 + 119886 minus

1205731199090

1 + 1199021199090

) = 0 (26)

It is clear that (26) has the characteristic roots 1199031= minus119889 lt 0

which means | arg 1199031| = 120587 gt 120572(1205872) 119903

2= minus119887 lt 0 which

means | arg 1199032| = 120587 gt 120572(1205872) and 119903

3= (120573119909

0(1 + 119902119909

0)) minus 119886 =

119886(1198770minus 1) Since the imaginary part of characteristic root 119903

3is

zero 1198770lt 1 which means | arg 119903

3| = 120587 gt 120572(1205872) is necessary

and sufficient to ensure the local asymptotic stability of 1198640 If

1198770gt 1 | arg 119903

3| = 0 lt 120572(1205872) hence 119864

0is unstable If 119877

0= 1

1199033= 0 which is a critical case

Theorem 10 Consider system (5)

(1) The immune-free equilibrium 1198641is locally asymptoti-

cally stable if 1198771lt 1

(2) If 1198771gt 1 the immune-free equilibrium 119864

1is unstable

if 1198771= 1 it is a critical state

Proof The characteristic equation for the immune-free equi-librium 119864

1is given as follows

det(

minus119889 minus

1205731199101

(1 + 1199021199091)2minus 119903 minus

1205731199091

1 + 1199021199091

0

1205731199101

(1 + 1199021199091)2

1205731199091

1 + 1199021199091

minus 119886 minus 119903 minus1199011199101

0 0 1198881199101minus 119887 minus 119903

)

= 0

(27)

Using 1205731199091(1 + 119902119909

1) = 119886 it is reduced to

(119903 + 119887 minus 1198881199101)[1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2] = 0

(28)

Discrete Dynamics in Nature and Society 5

The root of (28) 1199031

= 1198881199101minus 119887 is negative and | arg 119903

1| = 120587 gt

120572(1205872)when1198771lt 1 positive and | arg 119903

1| = 0 lt 120572(1205872)when

1198771gt 1 and zero when 119877

1= 1 which is a critical case

Now we consider the equation

1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2

= 0 (29)

Denote

119861 =

1205731199101

(1 + 1199021199091)2

119862 =

1198861205731199101

(1 + 1199021199091)2

(30)

Since 119861 gt 0 and 119862 gt 0 (29) has two negative real roots andwe denote them by 119903

2and 1199033 It is easy to see | arg 119903

2| = 120587 gt

120572(1205872) and | arg 1199033| = 120587 gt 120572(1205872) Hence when 119877

1lt 1 the

immune-free equilibrium 1198641is locally asymptotically stable

when 1198771gt 1 119864

1is unstable and when 119877

1= 1 it is a critical

case

To discuss the local stability of the immune-presentequilibrium state 119864

lowast we consider the linearized system of (5)at 119864lowast The Jacobian matrix at 119864lowast is given by

119869 (119864lowast) = (

minus119889 minus

120573119910lowast

(1 + 119902119909lowast)2

minus

120573119909lowast

1 + 119902119909lowast

0

120573119910lowast

(1 + 119902119909lowast)2

0 minus119901119910lowast

0 119888119911lowast

0

) (31)

The characteristic equation of the linearized system is

119875 (119903) = 1199033+ 11988611199032+ 1198862119903 + 1198863= 0 (32)

where

1198861= 119889 +

120573119910lowast

(1 + 119902119909lowast)2

gt 0

1198862= 119888119901119910

lowast119911lowast+

1205732119909lowast119910lowast

(1 + 119902119909lowast)3

gt 0

1198863= 119888119901119910

lowast119911lowast(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

11988611198862minus 1198863=

1205732119909lowast119910lowast

(1 + 119902119909lowast)3(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

(33)

Based on Definition 4 we obtain the discriminant of (32)

119863(119875) = minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1198861

1198862

1198863

0

0 1 1198861

1198862

1198863

3 21198861

1198862

0 0

0 3 21198861

1198862

0

0 0 3 21198861

1198862

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 18119886111988621198863+ (11988611198862)2

minus 41198863

11198863minus 41198863

2minus 27119886

2

3

(34)

Using the result (iii) of Lemmas 5 and 3 we have the followingtheorem

Theorem 11 Consider system (5) Under the condition of 1198771gt

1(1) if the discriminant of 119875(119903) 119863(119875) is positive namely

119863(119875) gt 0 then the immune-present equilibrium 119864lowast

is locally asymptotically stable for 0 lt 120572 le 1(2) if 119863(119875) lt 0 then the immune-present equilibrium 119864

lowast

is locally asymptotically stable for 0 lt 120572 lt 23

5 Numerical Method

Atanackovic and Stankovic introduced a numerical methodto solve the single linear FDE in 2004 [1] A few years laterthey developed again a method to solve the nonlinear FDE[30] It was shown that the fractional derivative of a function119891(119905) with order 120572 satisfying 0 lt 120572 lt 1 may be expressed as119863120572119891(119905)

=

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

infin

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

infin

sum

119898=2

Γ (119898 minus 1 + 120572)

Γ (120572 minus 1) (119898 minus 1)

times(

119891 (119905)

119905120572

+

119881119898

(119891) (119905)

119905119898minus1+120572

)]

(35)where

119881119898(119891)(119905) = minus(119898 minus 1)int

119905

0

120591119898minus2

119891(120591)119889120591 119898 = 2 3

(36)with the following properties

119889

119889119905

119881119898

(119891) = minus (119898 minus 1) 119905119898minus2

119891 (119905) 119898 = 2 3 (37)

We approximate119863120572119891(119905) by using119872 terms in sums appearing

in (35) as follows119863120572119891(119905)

≃

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

119872

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

119872

sum

119898=2

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)(119898 minus 1)

times(

119891(119905)

119905120572

+

119881119898(119891)(119905)

119905119898minus1+120572

)]

(38)

6 Discrete Dynamics in Nature and Society

We can rewrite (38) as follows

119863120572119891(119905) ≃ Ω(120579 119905119872)119891

(1)

(119905) + Φ(120579 119905119872)119891(119905)

+

119872

sum

119898=2

119860 (120579 119905 119898)

119881119898

(119891) (119905)

119905119898minus1+120572

(39)

where

Ω (120572 119905119872) =

1 + sum119872

119898=1(Γ (119898 minus 1 + 120572) Γ (120572 minus 1)119898)

Γ (2 minus 120572) 119905120572minus1

119877 (119886 119905) =

1 minus 120572

119905120572Γ(2 minus 120572)

119860 (120572 119905 119898) = minus

Γ (119898 minus 1 + 120572)

Γ (2 minus 120572) Γ (120572 minus 1) (119898 minus 1)

Φ (120572 119905119872) = 119877 (119886 119905) +

119872

sum

119898=2

119860 (120572 119905 119898)

119905120572

(40)

We set

Θ1(119905) = 119909 (119905) Θ

119898(119905) = 119881

119898(119909) (119905)

Θ119872+1

(119905) = 119910 (119905) Θ119872+119898

(119905) = 119881119898

(119910) (119905)

Θ2119872+1

(119905) = 119911 (119905) Θ2119872+119898

(119905) = 119881119898

(119911) (119905)

(41)

for 119898 = 2 3 We can rewrite system (19) as the followingform

Ω(120572 119905119872)Θ1015840

1(119905) + Φ(120572 119905119872)Θ

1(119905) +

119872

sum

119898=2

119860(120572 119905 119898)

Θ119898(119905)

119905119898minus1+120572

= 120582 minus 119889Θ1(119905) minus

120573Θ1(119905) Θ119872+1

(119905)

1 + 119902Θ1(119905)

Ω(120572 119905119872)Θ1015840

119872+1(119905) + Φ(120572 119905119872)Θ

119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

=

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus 119886Θ119872+1

(119905) minus 119901Θ119872+1

(119905) Θ2119872+1

(119905)

Ω(120572 119905119872)Θ1015840

2119872+1(119905) + Φ(120572 119905119872)Θ

2119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

= 119888Θ119872+1

(119905) Θ2119872+1

(119905) minus 119887Θ2119872+1

(119905)

(42)

where

Θ119898(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ1(120591)119889120591

Θ119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ119872+1

(120591)119889120591

Θ2119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ2119872+1

(120591)119889120591

119898 = 2 3 119872

(43)

Now we can rewrite (39) and (42) as the following form

Θ1015840

1(119905) =

1

Ω(120572 119905119872)

(120582 minus (119889 + Φ(120572 119905119872))Θ1(119905)

minus

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119898

(119905)

119905119898minus1+120572

)

Θ1015840

119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ1(119905) 119898 = 2 3 119872

Θ1015840

119872+1(119905) =

1

Ω(120572 119905119872)

(

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus (119886 + Φ(120572 119905119872))Θ119872+1

(119905)

minus 119901Θ119872+1

(119905)Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ119872+1

(119905) 119898 = 2 3 119872

Θ1015840

2119872+1(119905) =

1

Ω(120572 119905119872)

(119888Θ119872+1

(119905)Θ2119872+1

(119905)

minus (119887 + Φ(120572 119905119872))Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

2119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ2119872+1

(119905) 119898 = 2 3 119872

(44)with the following initial conditions

Θ1(120575) = 119909

0 Θ119898

(120575) = 0 119898 = 2 3 119872

Θ119872+1

(120575) = 1199100 Θ119872+119898

(120575) = 0 119898 = 2 3 119872

Θ2119872+1

(120575) = 1199110 Θ2119872+119898

(120575) = 0 119898 = 2 3 119872

(45)Now we consider the numerical solution of system of ordi-nary differential equations (44) with the initial conditions(45) by using the well-known Runge-Kutta method of fourthorder

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

4 Equilibrium States and Their Stability

In this section we investigate the stability of the fractional-ordermodel of HIV infection of CD4 + T cells that is system(5) with (6) Consider the initial value problem (5) with (6)with 120572 satisfying 0 lt 120572 le 1

In order to obtain the equilibria of system (5) we set119909120572(119905) = 0 119910120572(119905) = 0 and 119911

120572(119905) = 0 and we have

120582 minus 119889119909 minus

120573119909119910

1 + 119902119909

= 0

120573119909119910

1 + 119902119909

minus 119886119910 minus 119901119910119911 = 0

119888119910119911 minus 119887119911 = 0

(19)

System (5) has three types of relevant nonnegative equilib-rium states System (5) always has an infection-free equilib-rium 119864

0= (1199090 0 0) where

1199090=

120582

119889

(20)

which means that the infected cells are cleared The basicreproductive number of the viruses for system (5) is given by

1198770=

120582120573

119886 (119889 + 119902120582)

(21)

This number describes the average number of newly gener-ated infected cells from one infected cell at the beginningof the infection process When 119877

0gt 1 in addition to the

infection-free equilibrium 1198640 there is an immune-absence

equilibrium 1198641= (1199091 1199101 1199111) where

1199091=

119886

120573 minus 119886119902

1199101=

119889 + 119902120582

120573 minus 119886119902

(1198770minus 1) 119911

1= 0

(22)Note that 119877

0gt 1 implies 120573 gt 119886119902 When the HIV infection

is in the immune-absence equilibrium 1198641 the infected cells

and virus exist but the immune response is not activated yetFurther we denote

1198771= 1198770minus

119887(120573 minus 119886119902)

119888(119889 + 119902120582)

(23)

Note that 1198771

= ((120573 minus 119886119902)119888(119889 + 119902120582))(1198881199101minus 119887) + 1 which

implies that 1198771

gt 1 is equivalent to 1198881199101119887 gt 1 The

latter 1198881199101119887 is seen as the immune reproductive number

which expresses the average number of activated CTLsgenerated from one CTL during its life time 1119887 throughthe stimulation of the infected cells 119910

1 It is reasonable that

immune response is activated in the case where 1198771

gt 1When119877

1gt 1 in addition to the infection-free equilibrium119864

0

and the immune-absence equilibrium 1198641 there is an interior

immune-presence equilibrium 119864lowast

= (119909lowast 119910lowast 119911lowast) where

119909lowast

=

minus (119887120573 + 119888119889 minus 119888119902120582) + radic(119887120573 + 119888119889 minus 119888119902120582)2

+ 41198882119889119902120582

2119888119889119902

119910lowast

=

119887

119888

119911lowast

=

119888

119887119901

(120582 minus 119889119909lowast) minus

119886

119901

(24)

119864lowast expresses the state where CTLs immune response is

presentNext we establish the local asymptotic stability of model

(5) by the characteristic equation

Theorem 9 Consider system (5)

(1) The infection-free equilibrium 1198640is locally asymptoti-

cally stable if 1198770lt 1

(2) If 1198770gt 1 the equilibrium 119864

0is unstable and if 119877

0= 1

it is a critical case

Proof The characteristic equation for the infection-free equi-librium 119864

0is given as follows

det(

minus119889 minus 119903 minus

1205731199090

1 + 1199021199090

0

0

1205731199090

1 + 1199021199090

minus 119886 minus 119903 0

0 0 minus119887 minus 119903

) = 0 (25)

It is reduced to

(119903 + 119889)(119903 + 119887)(119903 + 119886 minus

1205731199090

1 + 1199021199090

) = 0 (26)

It is clear that (26) has the characteristic roots 1199031= minus119889 lt 0

which means | arg 1199031| = 120587 gt 120572(1205872) 119903

2= minus119887 lt 0 which

means | arg 1199032| = 120587 gt 120572(1205872) and 119903

3= (120573119909

0(1 + 119902119909

0)) minus 119886 =

119886(1198770minus 1) Since the imaginary part of characteristic root 119903

3is

zero 1198770lt 1 which means | arg 119903

3| = 120587 gt 120572(1205872) is necessary

and sufficient to ensure the local asymptotic stability of 1198640 If

1198770gt 1 | arg 119903

3| = 0 lt 120572(1205872) hence 119864

0is unstable If 119877

0= 1

1199033= 0 which is a critical case

Theorem 10 Consider system (5)

(1) The immune-free equilibrium 1198641is locally asymptoti-

cally stable if 1198771lt 1

(2) If 1198771gt 1 the immune-free equilibrium 119864

1is unstable

if 1198771= 1 it is a critical state

Proof The characteristic equation for the immune-free equi-librium 119864

1is given as follows

det(

minus119889 minus

1205731199101

(1 + 1199021199091)2minus 119903 minus

1205731199091

1 + 1199021199091

0

1205731199101

(1 + 1199021199091)2

1205731199091

1 + 1199021199091

minus 119886 minus 119903 minus1199011199101

0 0 1198881199101minus 119887 minus 119903

)

= 0

(27)

Using 1205731199091(1 + 119902119909

1) = 119886 it is reduced to

(119903 + 119887 minus 1198881199101)[1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2] = 0

(28)

Discrete Dynamics in Nature and Society 5

The root of (28) 1199031

= 1198881199101minus 119887 is negative and | arg 119903

1| = 120587 gt

120572(1205872)when1198771lt 1 positive and | arg 119903

1| = 0 lt 120572(1205872)when

1198771gt 1 and zero when 119877

1= 1 which is a critical case

Now we consider the equation

1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2

= 0 (29)

Denote

119861 =

1205731199101

(1 + 1199021199091)2

119862 =

1198861205731199101

(1 + 1199021199091)2

(30)

Since 119861 gt 0 and 119862 gt 0 (29) has two negative real roots andwe denote them by 119903

2and 1199033 It is easy to see | arg 119903

2| = 120587 gt

120572(1205872) and | arg 1199033| = 120587 gt 120572(1205872) Hence when 119877

1lt 1 the

immune-free equilibrium 1198641is locally asymptotically stable

when 1198771gt 1 119864

1is unstable and when 119877

1= 1 it is a critical

case

To discuss the local stability of the immune-presentequilibrium state 119864

lowast we consider the linearized system of (5)at 119864lowast The Jacobian matrix at 119864lowast is given by

119869 (119864lowast) = (

minus119889 minus

120573119910lowast

(1 + 119902119909lowast)2

minus

120573119909lowast

1 + 119902119909lowast

0

120573119910lowast

(1 + 119902119909lowast)2

0 minus119901119910lowast

0 119888119911lowast

0

) (31)

The characteristic equation of the linearized system is

119875 (119903) = 1199033+ 11988611199032+ 1198862119903 + 1198863= 0 (32)

where

1198861= 119889 +

120573119910lowast

(1 + 119902119909lowast)2

gt 0

1198862= 119888119901119910

lowast119911lowast+

1205732119909lowast119910lowast

(1 + 119902119909lowast)3

gt 0

1198863= 119888119901119910

lowast119911lowast(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

11988611198862minus 1198863=

1205732119909lowast119910lowast

(1 + 119902119909lowast)3(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

(33)

Based on Definition 4 we obtain the discriminant of (32)

119863(119875) = minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1198861

1198862

1198863

0

0 1 1198861

1198862

1198863

3 21198861

1198862

0 0

0 3 21198861

1198862

0

0 0 3 21198861

1198862

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 18119886111988621198863+ (11988611198862)2

minus 41198863

11198863minus 41198863

2minus 27119886

2

3

(34)

Using the result (iii) of Lemmas 5 and 3 we have the followingtheorem

Theorem 11 Consider system (5) Under the condition of 1198771gt

1(1) if the discriminant of 119875(119903) 119863(119875) is positive namely

119863(119875) gt 0 then the immune-present equilibrium 119864lowast

is locally asymptotically stable for 0 lt 120572 le 1(2) if 119863(119875) lt 0 then the immune-present equilibrium 119864

lowast

is locally asymptotically stable for 0 lt 120572 lt 23

5 Numerical Method

Atanackovic and Stankovic introduced a numerical methodto solve the single linear FDE in 2004 [1] A few years laterthey developed again a method to solve the nonlinear FDE[30] It was shown that the fractional derivative of a function119891(119905) with order 120572 satisfying 0 lt 120572 lt 1 may be expressed as119863120572119891(119905)

=

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

infin

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

infin

sum

119898=2

Γ (119898 minus 1 + 120572)

Γ (120572 minus 1) (119898 minus 1)

times(

119891 (119905)

119905120572

+

119881119898

(119891) (119905)

119905119898minus1+120572

)]

(35)where

119881119898(119891)(119905) = minus(119898 minus 1)int

119905

0

120591119898minus2

119891(120591)119889120591 119898 = 2 3

(36)with the following properties

119889

119889119905

119881119898

(119891) = minus (119898 minus 1) 119905119898minus2

119891 (119905) 119898 = 2 3 (37)

We approximate119863120572119891(119905) by using119872 terms in sums appearing

in (35) as follows119863120572119891(119905)

≃

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

119872

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

119872

sum

119898=2

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)(119898 minus 1)

times(

119891(119905)

119905120572

+

119881119898(119891)(119905)

119905119898minus1+120572

)]

(38)

6 Discrete Dynamics in Nature and Society

We can rewrite (38) as follows

119863120572119891(119905) ≃ Ω(120579 119905119872)119891

(1)

(119905) + Φ(120579 119905119872)119891(119905)

+

119872

sum

119898=2

119860 (120579 119905 119898)

119881119898

(119891) (119905)

119905119898minus1+120572

(39)

where

Ω (120572 119905119872) =

1 + sum119872

119898=1(Γ (119898 minus 1 + 120572) Γ (120572 minus 1)119898)

Γ (2 minus 120572) 119905120572minus1

119877 (119886 119905) =

1 minus 120572

119905120572Γ(2 minus 120572)

119860 (120572 119905 119898) = minus

Γ (119898 minus 1 + 120572)

Γ (2 minus 120572) Γ (120572 minus 1) (119898 minus 1)

Φ (120572 119905119872) = 119877 (119886 119905) +

119872

sum

119898=2

119860 (120572 119905 119898)

119905120572

(40)

We set

Θ1(119905) = 119909 (119905) Θ

119898(119905) = 119881

119898(119909) (119905)

Θ119872+1

(119905) = 119910 (119905) Θ119872+119898

(119905) = 119881119898

(119910) (119905)

Θ2119872+1

(119905) = 119911 (119905) Θ2119872+119898

(119905) = 119881119898

(119911) (119905)

(41)

for 119898 = 2 3 We can rewrite system (19) as the followingform

Ω(120572 119905119872)Θ1015840

1(119905) + Φ(120572 119905119872)Θ

1(119905) +

119872

sum

119898=2

119860(120572 119905 119898)

Θ119898(119905)

119905119898minus1+120572

= 120582 minus 119889Θ1(119905) minus

120573Θ1(119905) Θ119872+1

(119905)

1 + 119902Θ1(119905)

Ω(120572 119905119872)Θ1015840

119872+1(119905) + Φ(120572 119905119872)Θ

119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

=

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus 119886Θ119872+1

(119905) minus 119901Θ119872+1

(119905) Θ2119872+1

(119905)

Ω(120572 119905119872)Θ1015840

2119872+1(119905) + Φ(120572 119905119872)Θ

2119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

= 119888Θ119872+1

(119905) Θ2119872+1

(119905) minus 119887Θ2119872+1

(119905)

(42)

where

Θ119898(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ1(120591)119889120591

Θ119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ119872+1

(120591)119889120591

Θ2119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ2119872+1

(120591)119889120591

119898 = 2 3 119872

(43)

Now we can rewrite (39) and (42) as the following form

Θ1015840

1(119905) =

1

Ω(120572 119905119872)

(120582 minus (119889 + Φ(120572 119905119872))Θ1(119905)

minus

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119898

(119905)

119905119898minus1+120572

)

Θ1015840

119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ1(119905) 119898 = 2 3 119872

Θ1015840

119872+1(119905) =

1

Ω(120572 119905119872)

(

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus (119886 + Φ(120572 119905119872))Θ119872+1

(119905)

minus 119901Θ119872+1

(119905)Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ119872+1

(119905) 119898 = 2 3 119872

Θ1015840

2119872+1(119905) =

1

Ω(120572 119905119872)

(119888Θ119872+1

(119905)Θ2119872+1

(119905)

minus (119887 + Φ(120572 119905119872))Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

2119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ2119872+1

(119905) 119898 = 2 3 119872

(44)with the following initial conditions

Θ1(120575) = 119909

0 Θ119898

(120575) = 0 119898 = 2 3 119872

Θ119872+1

(120575) = 1199100 Θ119872+119898

(120575) = 0 119898 = 2 3 119872

Θ2119872+1

(120575) = 1199110 Θ2119872+119898

(120575) = 0 119898 = 2 3 119872

(45)Now we consider the numerical solution of system of ordi-nary differential equations (44) with the initial conditions(45) by using the well-known Runge-Kutta method of fourthorder

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

The root of (28) 1199031

= 1198881199101minus 119887 is negative and | arg 119903

1| = 120587 gt

120572(1205872)when1198771lt 1 positive and | arg 119903

1| = 0 lt 120572(1205872)when

1198771gt 1 and zero when 119877

1= 1 which is a critical case

Now we consider the equation

1199032+ (119889 +

1205731199101

(1 + 1199021199091)2)119903 +

1198861205731199101

(1 + 1199021199091)2

= 0 (29)

Denote

119861 =

1205731199101

(1 + 1199021199091)2

119862 =

1198861205731199101

(1 + 1199021199091)2

(30)

Since 119861 gt 0 and 119862 gt 0 (29) has two negative real roots andwe denote them by 119903

2and 1199033 It is easy to see | arg 119903

2| = 120587 gt

120572(1205872) and | arg 1199033| = 120587 gt 120572(1205872) Hence when 119877

1lt 1 the

immune-free equilibrium 1198641is locally asymptotically stable

when 1198771gt 1 119864

1is unstable and when 119877

1= 1 it is a critical

case

To discuss the local stability of the immune-presentequilibrium state 119864

lowast we consider the linearized system of (5)at 119864lowast The Jacobian matrix at 119864lowast is given by

119869 (119864lowast) = (

minus119889 minus

120573119910lowast

(1 + 119902119909lowast)2

minus

120573119909lowast

1 + 119902119909lowast

0

120573119910lowast

(1 + 119902119909lowast)2

0 minus119901119910lowast

0 119888119911lowast

0

) (31)

The characteristic equation of the linearized system is

119875 (119903) = 1199033+ 11988611199032+ 1198862119903 + 1198863= 0 (32)

where

1198861= 119889 +

120573119910lowast

(1 + 119902119909lowast)2

gt 0

1198862= 119888119901119910

lowast119911lowast+

1205732119909lowast119910lowast

(1 + 119902119909lowast)3

gt 0

1198863= 119888119901119910

lowast119911lowast(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

11988611198862minus 1198863=

1205732119909lowast119910lowast

(1 + 119902119909lowast)3(119889 +

120573119910lowast

(1 + 119902119909lowast)2) gt 0

(33)

Based on Definition 4 we obtain the discriminant of (32)

119863(119875) = minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1198861

1198862

1198863

0

0 1 1198861

1198862

1198863

3 21198861

1198862

0 0

0 3 21198861

1198862

0

0 0 3 21198861

1198862

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 18119886111988621198863+ (11988611198862)2

minus 41198863

11198863minus 41198863

2minus 27119886

2

3

(34)

Using the result (iii) of Lemmas 5 and 3 we have the followingtheorem

Theorem 11 Consider system (5) Under the condition of 1198771gt

1(1) if the discriminant of 119875(119903) 119863(119875) is positive namely

119863(119875) gt 0 then the immune-present equilibrium 119864lowast

is locally asymptotically stable for 0 lt 120572 le 1(2) if 119863(119875) lt 0 then the immune-present equilibrium 119864

lowast

is locally asymptotically stable for 0 lt 120572 lt 23

5 Numerical Method

Atanackovic and Stankovic introduced a numerical methodto solve the single linear FDE in 2004 [1] A few years laterthey developed again a method to solve the nonlinear FDE[30] It was shown that the fractional derivative of a function119891(119905) with order 120572 satisfying 0 lt 120572 lt 1 may be expressed as119863120572119891(119905)

=

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

infin

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

infin

sum

119898=2

Γ (119898 minus 1 + 120572)

Γ (120572 minus 1) (119898 minus 1)

times(

119891 (119905)

119905120572

+

119881119898

(119891) (119905)

119905119898minus1+120572

)]

(35)where

119881119898(119891)(119905) = minus(119898 minus 1)int

119905

0

120591119898minus2

119891(120591)119889120591 119898 = 2 3

(36)with the following properties

119889

119889119905

119881119898

(119891) = minus (119898 minus 1) 119905119898minus2

119891 (119905) 119898 = 2 3 (37)

We approximate119863120572119891(119905) by using119872 terms in sums appearing

in (35) as follows119863120572119891(119905)

≃

1

Γ(2 minus 120572)

times

119891(1)

(119905)

119905120572minus1

[1 +

119872

sum

119898=1

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)119898

]

minus [

120572 minus 1

119905120572

119891(119905) +

119872

sum

119898=2

Γ(119898 minus 1 + 120572)

Γ(120572 minus 1)(119898 minus 1)

times(

119891(119905)

119905120572

+

119881119898(119891)(119905)

119905119898minus1+120572

)]

(38)

6 Discrete Dynamics in Nature and Society

We can rewrite (38) as follows

119863120572119891(119905) ≃ Ω(120579 119905119872)119891

(1)

(119905) + Φ(120579 119905119872)119891(119905)

+

119872

sum

119898=2

119860 (120579 119905 119898)

119881119898

(119891) (119905)

119905119898minus1+120572

(39)

where

Ω (120572 119905119872) =

1 + sum119872

119898=1(Γ (119898 minus 1 + 120572) Γ (120572 minus 1)119898)

Γ (2 minus 120572) 119905120572minus1

119877 (119886 119905) =

1 minus 120572

119905120572Γ(2 minus 120572)

119860 (120572 119905 119898) = minus

Γ (119898 minus 1 + 120572)

Γ (2 minus 120572) Γ (120572 minus 1) (119898 minus 1)

Φ (120572 119905119872) = 119877 (119886 119905) +

119872

sum

119898=2

119860 (120572 119905 119898)

119905120572

(40)

We set

Θ1(119905) = 119909 (119905) Θ

119898(119905) = 119881

119898(119909) (119905)

Θ119872+1

(119905) = 119910 (119905) Θ119872+119898

(119905) = 119881119898

(119910) (119905)

Θ2119872+1

(119905) = 119911 (119905) Θ2119872+119898

(119905) = 119881119898

(119911) (119905)

(41)

for 119898 = 2 3 We can rewrite system (19) as the followingform

Ω(120572 119905119872)Θ1015840

1(119905) + Φ(120572 119905119872)Θ

1(119905) +

119872

sum

119898=2

119860(120572 119905 119898)

Θ119898(119905)

119905119898minus1+120572

= 120582 minus 119889Θ1(119905) minus

120573Θ1(119905) Θ119872+1

(119905)

1 + 119902Θ1(119905)

Ω(120572 119905119872)Θ1015840

119872+1(119905) + Φ(120572 119905119872)Θ

119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

=

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus 119886Θ119872+1

(119905) minus 119901Θ119872+1

(119905) Θ2119872+1

(119905)

Ω(120572 119905119872)Θ1015840

2119872+1(119905) + Φ(120572 119905119872)Θ

2119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

= 119888Θ119872+1

(119905) Θ2119872+1

(119905) minus 119887Θ2119872+1

(119905)

(42)

where

Θ119898(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ1(120591)119889120591

Θ119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ119872+1

(120591)119889120591

Θ2119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ2119872+1

(120591)119889120591

119898 = 2 3 119872

(43)

Now we can rewrite (39) and (42) as the following form

Θ1015840

1(119905) =

1

Ω(120572 119905119872)

(120582 minus (119889 + Φ(120572 119905119872))Θ1(119905)

minus

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119898

(119905)

119905119898minus1+120572

)

Θ1015840

119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ1(119905) 119898 = 2 3 119872

Θ1015840

119872+1(119905) =

1

Ω(120572 119905119872)

(

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus (119886 + Φ(120572 119905119872))Θ119872+1

(119905)

minus 119901Θ119872+1

(119905)Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ119872+1

(119905) 119898 = 2 3 119872

Θ1015840

2119872+1(119905) =

1

Ω(120572 119905119872)

(119888Θ119872+1

(119905)Θ2119872+1

(119905)

minus (119887 + Φ(120572 119905119872))Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

2119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ2119872+1

(119905) 119898 = 2 3 119872

(44)with the following initial conditions

Θ1(120575) = 119909

0 Θ119898

(120575) = 0 119898 = 2 3 119872

Θ119872+1

(120575) = 1199100 Θ119872+119898

(120575) = 0 119898 = 2 3 119872

Θ2119872+1

(120575) = 1199110 Θ2119872+119898

(120575) = 0 119898 = 2 3 119872

(45)Now we consider the numerical solution of system of ordi-nary differential equations (44) with the initial conditions(45) by using the well-known Runge-Kutta method of fourthorder

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

We can rewrite (38) as follows

119863120572119891(119905) ≃ Ω(120579 119905119872)119891

(1)

(119905) + Φ(120579 119905119872)119891(119905)

+

119872

sum

119898=2

119860 (120579 119905 119898)

119881119898

(119891) (119905)

119905119898minus1+120572

(39)

where

Ω (120572 119905119872) =

1 + sum119872

119898=1(Γ (119898 minus 1 + 120572) Γ (120572 minus 1)119898)

Γ (2 minus 120572) 119905120572minus1

119877 (119886 119905) =

1 minus 120572

119905120572Γ(2 minus 120572)

119860 (120572 119905 119898) = minus

Γ (119898 minus 1 + 120572)

Γ (2 minus 120572) Γ (120572 minus 1) (119898 minus 1)

Φ (120572 119905119872) = 119877 (119886 119905) +

119872

sum

119898=2

119860 (120572 119905 119898)

119905120572

(40)

We set

Θ1(119905) = 119909 (119905) Θ

119898(119905) = 119881

119898(119909) (119905)

Θ119872+1

(119905) = 119910 (119905) Θ119872+119898

(119905) = 119881119898

(119910) (119905)

Θ2119872+1

(119905) = 119911 (119905) Θ2119872+119898

(119905) = 119881119898

(119911) (119905)

(41)

for 119898 = 2 3 We can rewrite system (19) as the followingform

Ω(120572 119905119872)Θ1015840

1(119905) + Φ(120572 119905119872)Θ

1(119905) +

119872

sum

119898=2

119860(120572 119905 119898)

Θ119898(119905)

119905119898minus1+120572

= 120582 minus 119889Θ1(119905) minus

120573Θ1(119905) Θ119872+1

(119905)

1 + 119902Θ1(119905)

Ω(120572 119905119872)Θ1015840

119872+1(119905) + Φ(120572 119905119872)Θ

119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

=

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus 119886Θ119872+1

(119905) minus 119901Θ119872+1

(119905) Θ2119872+1

(119905)

Ω(120572 119905119872)Θ1015840

2119872+1(119905) + Φ(120572 119905119872)Θ

2119872+1(119905)

+

119872

sum

119898=2

119860(120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

= 119888Θ119872+1

(119905) Θ2119872+1

(119905) minus 119887Θ2119872+1

(119905)

(42)

where

Θ119898(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ1(120591)119889120591

Θ119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ119872+1

(120591)119889120591

Θ2119872+119898

(119905) = minus(119898 minus 1) int

119905

0

120591119898minus2

Θ2119872+1

(120591)119889120591

119898 = 2 3 119872

(43)

Now we can rewrite (39) and (42) as the following form

Θ1015840

1(119905) =

1

Ω(120572 119905119872)

(120582 minus (119889 + Φ(120572 119905119872))Θ1(119905)

minus

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119898

(119905)

119905119898minus1+120572

)

Θ1015840

119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ1(119905) 119898 = 2 3 119872

Θ1015840

119872+1(119905) =

1

Ω(120572 119905119872)

(

120573Θ1(119905)Θ119872+1

(119905)

1 + 119902Θ1(119905)

minus (119886 + Φ(120572 119905119872))Θ119872+1

(119905)

minus 119901Θ119872+1

(119905)Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ119872+1

(119905) 119898 = 2 3 119872

Θ1015840

2119872+1(119905) =

1

Ω(120572 119905119872)

(119888Θ119872+1

(119905)Θ2119872+1

(119905)

minus (119887 + Φ(120572 119905119872))Θ2119872+1

(119905)

minus

119872

sum

119898=2

119860 (120572 119905 119898)

Θ2119872+119898

(119905)

119905119898minus1+120572

)

Θ1015840

2119872+119898(119905) = minus (119898 minus 1) 119905

119898minus2Θ2119872+1

(119905) 119898 = 2 3 119872

(44)with the following initial conditions

Θ1(120575) = 119909

0 Θ119898

(120575) = 0 119898 = 2 3 119872

Θ119872+1

(120575) = 1199100 Θ119872+119898

(120575) = 0 119898 = 2 3 119872

Θ2119872+1

(120575) = 1199110 Θ2119872+119898

(120575) = 0 119898 = 2 3 119872

(45)Now we consider the numerical solution of system of ordi-nary differential equations (44) with the initial conditions(45) by using the well-known Runge-Kutta method of fourthorder

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

0 10 20 30 40 50 60 70 80 90 1000

20406080

100

Num

ber o

f hea

lthy

cells

Time t

120572 = 085

120572 = 095

120572 = 1

(a) Time evolution of the state variables

Num

ber o

f inf

ecte

d ce

lls

050

100150200

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(b) Time evolution of the state variablesN

umbe

r of C

TL

0

002

004

006

0 10 20 30 40 50 60 70 80 90 100Time t

120572 = 085

120572 = 095

120572 = 1

(c) Time evolution of the state variables

Figure 1 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 085 120572 = 095 and 120572 = 1 in condition of 119863(119875) gt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and119911(119905) in (c) for 120572 = 085 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 095 and the blue lines for 120572 = 1

0 20 40 60 80 100 120 140 160 180 20040

60

80

100

Time t

120572 = 056 120572 = 065

Num

ber o

f hea

lthy

cells

(a) Time evolution of the state variables

0 50 100 150 200 2500

5

10

15

120572 = 056 120572 = 065

Time t

Num

ber o

f inf

ecte

d ce

lls

(b) Time evolution of the state variables

0 50 100 150 200 250 3000

05

1

15

120572 = 056 120572 = 065

Time t

Num

ber o

f CTL

(c) Time evolution of the state variables

Figure 2 The diagrams show the time evolution of the trajectory of system (5) for healthy cells 119909(119905) infected cells 119910(119905) and the CTL 119911(119905)respectively with respect to 120572 = 056 and 120572 = 065 in condition of 119863(119875) lt 0 The red lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in(c) for 120572 = 056 the green lines correspond to 119909(119905) in (a) 119910(119905) in (b) and 119911(119905) in (c) for 120572 = 065

6 Numerical Simulation and Discussion

Firstly by using GEM (generalized Euler method) [21] wesimulate system (5) with the parameter values as shown inTable 1

By direct calculation we have 1198770

= 314916 1198771

=

302250 and 119863(119875) = 000076142 gt 0 and the simulations

display that the immune-present equilibrium119864lowast is asymptot-

ically stable for 120572 = 085 120572 = 095 and 120572 = 1 (see Figure 1)FromFigure 1 we can clearly see that comparedwith the caseof order 120572 = 085 the trajectory of model with order 120572 = 095

is closer to the trajectory of the model with the integer-order1 That is the farther from 120572 to 1 the bigger of the trajectorydifference of them

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

500

0

51015200

002

004

006

008

0

20

4005

1015

0

002

004

006

008

050 0510

0

002

004

006

008

01

050 02468

0

002

004

006

008

01

z(t

)

z(t

)

y(t)y(t)

y(t)y(t)

x(t)x(t)

120572 = 06

z(t

)

z(t

)

x(t)x(t)

120572 = 08 120572 = 09

120572 = 07

Figure 3 The three-dimensional diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09

in condition of 119863(119875) lt 0

Table 1

Parameter Value Parameter Value120582 233 119888 00031120573 05 119889 009119886 002 119901 10119887 015 119902 079

We choose the parameter values as shown in Table 2Based on the parameter values in Table 2 we have 119877

0=

35714 1198771

= 34769 and 119863(119875) = minus000069 lt 0 With thesame simulatedmethod it is shown that the immune-presentequilibrium 119864

lowast is asymptotically stable for 120572 = 056 and 120572 =

065 (see Figure 2)

Next we use the method that is shown in the previoussection to simulate system (5) by transforming system (5)to one order ordinary differential equation Here we set theparameters values as shown in Table 3 with 119872 = 10 Δ119905 =

0005 We have 1198770

= 90445 1198771

= 87438 and 119863(119875) =

minus100374 lt 0 The approximate solutions 119909(119905) 119910(119905) and 119911(119905)

for 120572 = 06 120572 = 07 120572 = 08 and 120572 = 09 are displayed inFigure 3 It shows that the immune-present equilibrium 119864

lowast isasymptotically stable for 120572 = 06 lt 23 and there exists thelimit circle for 120572 gt 23

We also simulate the situation of system (5) by themethodin the previous section when 120572 = 1with the parameter valuesof Table 3 in Figure 4 Both Figures 1 and 4 show that theimmune-present equilibrium 119864

lowast is asymptotically stable for

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

0 500 1000 150025

30

35

40

45

50

55

60x

(t) h

ealth

y ce

lls

Time t

(a)

0 500 1000 15000

5

10

15

20

y(t

) inf

ecte

d ce

lls

Time t

(b)

0 500 1000 15000

002

004

006

008

01

012

014

z(t

) ant

igen

-spe

cific

CTL

Time t

(c)

2040

60 051015

0

002

004

006

008

01

012

014

y(t) infected cells

x(t) healthy cells

z(t

) ant

igen

-spe

cific

CTL

(d)

Figure 4 The diagrams show the approximate solutions of 119909(119905) 119910(119905) and 119911(119905) for 120572 = 1 in condition of 1198771gt 1 and 119863(119875) lt 0 The panel (a)

corresponds to healthy cells (b) to infected cells and (c) to antigen-specific CTL The panel (d) shows the three-dimensional trajectory ofsystem (5) with time evolution

Table 2

Parameter Value Parameter Value120582 10 119888 0051120573 001 119889 018119886 01 119901 06119887 015 119902 001

120572 = 1 and that is nothing to do with 119863(119875) gt 0 or 119863(119875) lt 0Hence the results of fractional-order system when 120572 = 1 areconsistent with the result of integer-order HIV model (4)

Table 3

Parameter Value Parameter Value120582 2946 119888 009120573 0286 119889 09862119886 00384 119901 085119887 2568 119902 079

7 Conclusion

Fractional differential equations have garnered a lot ofattention and appreciation due to their ability to providean exact description of different nonlinear phenomena

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

The advantage of fractional-order systems is that they allowgreater degrees of freedom in themodelNowadaysmore andmore investigators begin to study the qualitative propertiesand numerical solutions of fractional-order virus infectionmodels In this paper we introduced a fractional-order HIVinfection model with nonlinear incidence and dealt with themathematical behaviors of the model

We showed that system (5) possesses nonnegative solu-tions and studied the stability behavior of the infection-free equilibrium the immune-absence equilibrium and theimmune-presence equilibrium We found that the stabilityof the infection-free equilibrium and the immune-absenceequilibrium of system (5) is the same as that of system (4)When the basic reproduction number of viruses (119877

0) is less

than one the infection-free equilibrium is stable howeverwhen 119877

0is more than one the infection-free equilibrium is

unstable and when the immune reproduction number (1198771)

is less than one the immune-absence equilibrium is stablehowever when 119877

1is more than one the immune-absence

equilibrium is unstable However the results for the immune-presence equilibrium of system (5) are different to those ofsystem (4) In system (4) the immune-presence equilibriumis stable when119877

1is more than one while in system (5) when

1198771is more than one the immune-presence equilibrium is not

always stable In the condition of 1198771gt 1 when119863(119875) gt 0 the

immune-presence equilibrium is stable for 0 lt 120572 le 1 whilewhen 119863(119875) lt 0 the immune-presence equilibrium is stableonly for 0 lt 120572 lt 23 But using the simulation we foundwhen 119863(119875) lt 0 the immune-presence equilibrium is stablefor 120572 = 1 From the simulation we also found the fartherfrom 120572 to 1 the bigger of their trajectory difference Theseresults show that the integer-order model can be viewed asa special case from the more general fractional-order modelAlthough a large part of results is illustrated by both theoryanalysis andnumerical simulation the result for the immune-presence equilibrium when 119863(119875) lt 0 and 120572 = 1 can be justverified by the simulation in this paper

In this paper we introduce the fractional calculus intothe HIV infection model with nonlinear incidence and fromthe theory analysis and numerical simulations it is illustratedthat the integer-orderHIV infectionmodel can be viewed as aspecial case of fractional-ordermodelWehope that thisworkcan create interest and further do research effort in this fieldsince the fractionalmodelingmight providemore insight intounderstanding the dynamical behaviors of such systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the Hainan Natural Sci-ence Foundation (112006) andNatural Science Foundation ofHainan Provincial Department of Education (Hjkj2013-47)Gang Huang was supported by the Fundamental ResearchFunds for the Central Universities (no CUG130415) andthe National Natural Science Foundation of China (no11201435)

References

[1] T M Atanackovic and B Stankovic ldquoAn expansion formula forfractional derivatives and its applicationrdquo Fractional Calculus ampApplied Analysis vol 7 no 3 pp 365ndash378 2004

[2] G Bognar ldquoSimilarity solution of boundary layer flows for non-Newtonian fluidsrdquo International Journal of Nonlinear SciencesandNumerical Simulation vol 10 no 11-12 pp 1555ndash1566 2009

[3] M Benmalek and A Charef ldquoDigital fractional order operatorsfor R-wave detection in electrocardiogram signalrdquo IET SignalProcessing vol 3 no 5 pp 381ndash391 2009

[4] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 2012

[5] Y Ferdi A Taleb-Ahmed and M R Lakehal ldquoEfficient gen-eration of 1119891120573 noise using signal modeling techniquesrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 55no 6 pp 1704ndash1710 2008

[6] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[7] G L Jia and Y X Ming ldquoStudy on the viscoelasticity ofcancellous bone based on higher-order fractional modelsrdquo inProceedings of the 2nd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo08) pp 1733ndash17362006

[8] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 pp 86ndash90 1999

[9] B Mandelbrot ldquoSome noises with 1119891 spectrum a bridgebetween direct current and white noiserdquo IEEE Transactions onInformation Theory vol 13 no 2 pp 289ndash298 1967

[10] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 pp 1ndash377 2004

[11] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[12] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer Berlin Germany 1997

[13] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[14] K S Cole ldquoElectric conductance of biological systemsrdquo inProceedings of the Cold Spring Harbor Symposia on QuantitativeBiology pp 107ndash116 Cold Spring Harbor NY USA 1993

[15] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[16] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[17] G Huang H Yokoi Y Takeuchi T Kajiwara and T SasakildquoImpact of intracellular delay immune activation delay andnonlinear incidence on viral dynamicsrdquo Japan Journal of Indus-trial and Applied Mathematics vol 28 no 3 pp 383ndash411 2011

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

[18] G Huang Y Takeuchi and A Korobeinikov ldquoHIV evolutionand progression of the infection to AIDSrdquo Journal ofTheoreticalBiology vol 307 pp 149ndash159 2012

[19] A S Perelson D E Kirschner and R de Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[20] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999

[21] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 article 1 2012

[22] Y Ding and H Ye ldquoA fractional-order differential equationmodel of HIV infection of CD4+ T-cellsrdquo Mathematical andComputer Modelling vol 50 no 3-4 pp 386ndash392 2009

[23] A Korobeinikov ldquoGlobal properties of infectious disease mod-els with nonlinear incidencerdquo Bulletin of Mathematical Biologyvol 69 no 6 pp 1871ndash1886 2007

[24] A Korobeinikov ldquoGlobal asymptotic properties of virusdynamics models with dose dependent parasite reproductionand virulence and nonlinear incidence raterdquo MathematicalMedicine and Biology vol 26 pp 225ndash239 2009

[25] A Korobeinikov ldquoStability of ecosystem global properties ofa general predator-prey modelrdquo Mathematical Medicine andBiology vol 26 no 4 pp 309ndash321 2009

[26] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer New York NY USA 2011

[27] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and Its Applications vol 379 no 2 pp 607ndash614 2007

[28] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[29] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[30] T M Atanackovic and B Stankovic ldquoOn a numerical schemefor solving differential equations of fractional orderrdquoMechanicsResearch Communications vol 35 no 7 pp 429ndash438 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of