Hindawi Publishing CorporationISRN Applied MathematicsVolume 2013 Article ID 710643 11 pageshttpdxdoiorg1011552013710643

Research ArticleGlobal Stability of an SEIS Epidemic Model withGeneral Saturation Incidence

Hui Zhang1 Li Yingqi1 and Wenxiong Xu2

1 Mathematics and OR Section Xirsquoan Research Institute of High-Tech Hongqing Town Shaanxi Xirsquoan 710025 China2 School of Mathematics and Statistics Xirsquoan Jiaotong University Shaanxi Xirsquoan 710049 China

Correspondence should be addressed to Hui Zhang zh53054958163com

Received 18 February 2013 Accepted 11 March 2013

Academic Editors J R Fernandez C Lu E Skubalska-Rafajlowicz and F Tadeo

Copyright copy 2013 Hui 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 present an SEIS epidemic model with infective force in both latent period and infected period which has different generalsaturation incidence rates It is shown that the global dynamics are completely determined by the basic reproductive number 119877

0

If 1198770le 1 the disease-free equilibrium is globally asymptotically stable in 119879 by LaSallersquos Invariance Principle and the disease dies

out Moreover using the method of autonomous convergence theorem we obtain that the unique epidemic equilibrium is globallyasymptotically stable in 1198790 and the disease spreads to be endemic

1 Introduction

Epidemiology is the study of hot spots of the spread of infec-tious disease with the objective to trace factors that con-tribute to their occurrence Mathematical epidemiologymodels describing the population dynamics of infectious dis-eases have been playing an important role in better under-standing of epidemiological patterns and disease control fora long time Epidemiological models are now widely used asmore epidemiologists realize the role that modeling can playin basic understanding and policy development In recentyears many epidemiological models of ordinary differentialequations have been studies by a number of authors [1ndash4]

The most general form of an epidemiological model isan SEIRS model consisting of four population subclasses 119878-susceptible 119864mdashexposed 119868mdashinfected and 119877mdashrecovered Allother models are limiting cases of the SEIRS model undervarious parameter restrictions If there is no immunity andhence no R class the SEIS model is obtained which can beregarded when the average period of immunity tends to zero

Many epidemic models with the infectious force in thelatent period have been performed Guihua and Zhen [5 6]studied global stability of an SEI model with general inci-dence or standard incidence Mukhopadhyay and Bhattach-aryya [7] discussed global stability of an SEIS model withstandard incidence Global dynamics of an SEI model withacute and chronic stages were given by Yuan and Yang [8]

Incidence rate plays a very important role in the researchof epidemiological models Comparing with bilinear andstandard incidence rate saturating incidence rate may bemore suitable for our real word which should generallybe written as 120573119862(119873)119878119868119873 where 119873 is the total populationsize Michaelis and Menten combined the two previousapproaches by assuming that if the number of available part-ners119873 is low the number of actual per capita partners 119862(119873)

is proportional to119873 whereas if the number of available part-ners is large there is a saturation effect whichmakes the num-ber of actual partners constant Specifically it has the form(Michaelis-Menten contact rate)

119862 (119873) =119886119873

1 + 119887119873 (1)

Obviously incidence with above form suggests that the num-ber of new cases per unit time is saturated with the total pop-ulation Using a mechanistic argument Heesterbeek andMetz [9] derived the expression for the saturating contact rateof individual contacts in a population that mixes randomlythat is

119862 (119873) =119887119873

1 + 119887119873 + radic1 + 2119887119873

(2)

Furthermore 119862(119873) is nondecreasing and 119862(119873)119873 is nonin-creasing

2 ISRN Applied Mathematics

The above discussion reveals the importance of incidencefunctions in epidemic models Different nonlinear forms ofincidence can exhibit very dynamics and hence are able tounearth some otherwise unknown features of disease dynam-ics Though the aspect of nonlinearity in incidence hasfound a significant importance in the existing literature thefact that population subclasses with different infection stat-uses should have different incidence rates has received littleattention among mathematical epidemiologists Thus in anSEIS epidemic model since there is a difference in relativemeasure of infectiousness between the exposed and theinfected populations the incidence rate between the suscep-tible fraction 119878 and the infected fraction 119868 should be differentfrom that between 119878 with the exposed fraction 119864

The present analysis aims to explore the impact of this dis-tinct incidence for exposed and infected populations underthe influence of spatial heterogeneity As a model systemWe have divided the population in researched area into threeclasses 119878mdashsusceptible 119864mdashexposed with the infectious forceand 119868mdashinfected

In the next section we establish the model discussed inthis paper and determine the basic reproductive number InSection 3 we analyze the global stability of the disease-freeequilibrium In Section 4 we resolve the unique existenceand global stability of the epidemic equilibrium In Section5 we present some numerical simulation of examples whichvalidate these theoretical results The paper ends with a briefdiscussion in Section 6

2 The Model and the BasicReproductive Number

The model we consider has the following population sub-classes (i) 119878mdashthe susceptible (ii) 119864mdashthe exposed and (iii)119868mdashthe infected The total population size denoted by 119873 is119873(119905) = 119878(119905) + 119864(119905) + 119868(119904) The transfer mechanism from theclass 119878 to the class 119864 is guided by the function

119891 (119905) =12057311198621(119873)

119873119878119864 +

12057321198622(119873)

119873119878119868 (3)

where 1205731and 120573

2are average numbers of adequate contacts of

an exposed individual and an infectious individual per unittime respectively and119862

119894(119873) (119894 = 1 2) are relevant saturation

contact rate which satisfy the following assumptions for119873 gt

0

(i) 119862119894(119873) gt 0

(ii) 1198621015840119894(119873) ge 0

(iii) [119862119894(119873)119873]

1015840le 0

The assumptions (i) and (ii) are biologically motivated Asthe total population119873 increases the probability of a contactwith a susceptible individual decreases and thus the force ofthe exposed or the infected is expected to be a decreasingfunction of 119873 And the assumption (iii) implies that thecontact rate 119862

119894(119873) is saturated

The population transfer among compartments is sche-matically depicted in the transfer diagram in Figure 1

120583119868120583119864 120572119868

120575119868

Λ

12057321198622(119873)

119873119878119868

12057311198621(119873)

119873119878119864

120583119878

119878 119868119864120576119864

Figure 1 The transfer diagram for model (4)

The transfer diagram leads to the following SEIS epidemicmodel of ordinary differential equations

1198781015840= Λ minus 120583119878 minus

12057311198621 (119873)

119873119878119864 minus

12057321198622 (119873)

119873119878119868 + 120575119868

1198641015840=12057311198621(119873)

119873119878119864 +

12057321198622(119873)

119873119878119868 minus (120583 + 120576) 119864

1198681015840= 120576119864 minus (120583 + 120572 + 120575) 119868

(4)

where Λ is the recruitment rate of the population 120583 is thenatural death rate and 120572 is the death rate for the infected 119864individuals move to the class 119868 at the rate 120576 and 119868 individualsrecover at the rate 120575 which are assumed to join the susceptibleclass The above parameters are positive

Summing up the three equations in system (4) then thetime derivative of119873(119905) along a solution of system (4) is

1198731015840= Λ minus 120583 (119878 + 119864 + 119868) minus 120572119868 (5)

Therefore1198731015840 le Λminus120583119873 equivalently1198731015840+120583119873 le Λ Applyinga theorem on differential inequalities [10] we get 0 le 119873 le

Λ120583 for 119905 rarr +infin Thus the three-dimensional simplex

119879 = (119878 119864 119868) isin R3+| 0 le 119878 + 119864 + 119868 le

Λ

120583 (6)

is positively invariant with respect to system (4) whereR3+denotes the nonnegative cone of R3 including its lower

dimensional facesBy using 119878 = 119873minus119864minus119868 and (5) we get the following system

1198641015840=12057311198621 (119873) 119864 + 120573

21198622 (119873) 119868

119873(119873 minus 119864 minus 119868) minus (120583 + 120576) 119864

1198681015840= 120576119864 minus (120583 + 120572 + 120575) 119868

1198731015840= Λ minus 120583119873 minus 120572119868

(7)

The dynamical behavior of system (4) in 119879 is equivalent tothat of system (7)Thus in the rest of the paper we will studythe system (7) in the feasible region

119866 = (119864 119868119873) isin R3+| 0 le 119864 + 119868 le 119873 le

Λ

120583 (8)

which can be shown to be a positive invariant set for system(7)

Now we derive the basic reproductive number of system(4) by the method of next-generation matrix formulated in[11]

ISRN Applied Mathematics 3

Let 119909 = (119864 119868 119878)119879 then system (4) can be written as

1199091015840= F (119909) minusV (119909) (9)

where

F (119909) = (

[12057311198621(119873) 119864 + 120573

21198622(119873) 119868]

119878

119873

0

0

)

V (119909) = (

(120583 + 120576) 119864

minus120576119864 + (120583 + 120572 + 120575) 119868

minusΛ + 120583119878 + [12057311198621 (119873) 119864 + 120573

21198622 (119873) 119868]

119878

119873minus 120575119868

)

(10)

Then1199090= (0 0 Λ120583)

119879 is the unique disease-free equilibriumof system (9) and the Jacobian matrices ofF(119909) andV(119909) atequilibrium 119909

0are respectively

119863F (1199090) = (

119865 1198742times1

1198741times2

0

) 119863V (1199090) = (

119881 1198742times1

1198691

120583

)

(11)

where

119865 = (12057311198621(Λ

120583) 12057321198622(Λ

120583)

0 0

)

119881 = (120583 + 120576 0

minus120576 120583 + 120572 + 120575)

1198691= (12057311198621(Λ

120583) 12057321198622(Λ

120583) minus 120575)

(12)

Obviously all eigenvalues of minus119863V(1199090) have negative real

partsWe call

119865119881minus1=

1

(120583 + 120576) (120583 + 120572 + 120575)(12057311198621(Λ

120583) (120583 + 120572 + 120575) + 120576120573

21198622(Λ

120583) 12057321198622(Λ

120583) (120583 + 120576)

0 0

) (13)

the next generation matrix for system (9) According to [11Theorem 2] the basic reproductive number of system (4)which is the number of secondary infectious cases producedby an exposed individual and an infectious individual duringtheir effective infectious period when introduced in a popu-lation of susceptible is

1198770= 120588 (119865119881

minus1) =

12057311198621(Λ120583) (120583 + 120572 + 120575) + 120576120573

21198622(Λ120583)

(120583 + 120576) (120583 + 120572 + 120575)

(14)

where 120588(119860) denotes the spectral radius of matrix 119860

3 Stability Analysis ofthe Disease-Free Equilibrium

In this section we discuss the global stability of the disease-free equilibrium It is obvious that system (7) always has theunique disease-free equilibrium 119875

0= (0 0 Λ120583) in 119866 About

1198750 we have the following main results

Theorem 1 The disease-free equilibrium 1198750is globally asymp-

totically stable in 119866 if 1198770le 1 and it is unstable if 119877

0gt 1

Proof The Jacobian matrix of system (7) at 1198750= (0 0 Λ120583)

goes as follows

119869 (1198750) = (

12057311198621(Λ

120583) minus (120583 + 120576) 120573

21198622(Λ

120583) 0

120576 minus120596 0

0 minus120572 minus120583

) (15)

which has a eigenvalue 1205821= minus120583 lt 0 obviously The other

two eigenvalues 1205822and 120582

3are determined by the following

equation

1205822minus [12057311198621(Λ

120583) minus (120583 + 120596 + 120576)] 120582

minus 120596[12057311198621(Λ

120583) minus (120583 + 120576)] minus 120576120573

21198622(Λ

120583) = 0

(16)

If 1198770gt 1 we can have easily

12058221205823= minus 120596[120573

11198621(Λ

120583) minus (120583 + 120576)] minus 120576120573

21198622(Λ

120583)

= 120596 (120583 + 120576) (1 minus 1198770) lt 0

(17)

Therefore 1205822and 120582

3are two opposite-sign real roots Thus

1198750is unstable

4 ISRN Applied Mathematics

Since 1198770lt 1 implies 120573

11198621(Λ120583) lt (120583+120576)minus(120576120596)120573

21198622(Λ

120583) then we get

1205822+ 1205823= 12057311198621(Λ

120583) minus (120583 + 120596 + 120576)

lt minus 120596 minus120576

12059612057321198622(Λ

120583) lt 0

12058221205823= 120596 (120583 + 120576) (1 minus 119877

0) gt 0

(18)

Therefore 1205822and 120582

3have negative real parts Hence 119875

0is

locally asymptotically stableWhen 119877

0= 1 it implies that 120582

21205823= 0 120573

11198621(Λ120583) minus (120583 +

120576) = minus(120576120596)12057321198622(Λ120583) We may as well assume that 120582

2= 0

then 1205823= minus120596minus (120576120596)120573

21198622(Λ120583) The characteristic matrix of

119869(1198750) has three invariable factors 1 1 and 120582(120582 + 120583)(120582 + 120596 +

(120576120596)12057321198622(Λ120583)) Because the elementary factor with respect

to 1205822= 0 is 120582 which is single 119875

0is stable

Constructing a suitable Lyapunov function

119881 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (19)

then the time derivative of 119881 along a solution of system (7)gives

= 1198770

12057311198621 (119873) 119864 + 12057321198622 (119873) 119868

119873(119873 minus 119864 minus 119868)

minus 1198770(120583 + 120576) 119864 +

1205761205732

1205961198622(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

= 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

le 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(119873) 119864 minus 120573

21198622(119873) 119868

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873[(1198770minus 1)119873 minus 119877

0119864 minus 119877

0119868]

(20)

Hence le 0 holds if1198770le 1 Furthermore = 0 if and only

if 119864 = 119868 = 0 Let 119865 = (119864 119868119873) isin 119866 | = 0 = (0 0119873)then the largest compact invariant set in119865 for system (7) is theset (0 0119873) Thus the solution of system (7) satisfies 119864 rarr

0 119868 rarr 0 as 119905 rarr +infin by LaSallersquos Invariance Principle [12]Therefore the limit system of system (7) is

1198641015840= 0

1198681015840= 0

1198731015840= Λ minus 120583119873

(21)

It is obviously known that the equilibrium (0 0 Λ120583) ofsystem (21) is globally asymptotically stable thus the disease-free equilibrium119875

0of system (7) is globally attractive in119866 On

the basis of local stability 1198750is globally asymptotically stable

in 119866 if 1198770le 1 This completes the proof

About system (4) we also obtain

Theorem 2 The unique disease-free equilibrium 1198750= (Λ120583

0 0) of system (4) is globally asymptotically stable in119879 if1198770le 1

and it is unstable if 1198770gt 1

4 Existence and Stability ofthe Endemic Equilibrium

In this section we first discuss the existence and uniquenessof the endemic equilibrium 119875

lowast of system (7) when 1198770gt 1

Whereafter we focus on investigating the local stability of119875lowastWe have to prove that the Jacobian matrix 119869(119875

lowast) is stable

namely all its eigenvalues have negative real parts This isroutinely done by verifying the Routh-Hurwitz conditionsFinally we study the global stability of the endemic equilib-rium 119875

lowast of system (4) with the method of autonomous con-vergence theorem of Li and Muldowney in [13]

The coordinates of the endemic equilibrium (positiveequilibrium) of system (7) are the positive solutions of equa-tions

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868) minus (120583 + 120576) 119864 = 0

120576119864 minus (120583 + 120572 + 120575) 119868 = 0

Λ minus 120583119873 minus 120572119868 = 0

(22)

in 119866119900Let 120596 = 120583+120572+120575 by the direct calculation we can get the

following equation of119873 easily as

120593 (119873) =120572120576 + 120583120596 + 120583120576

120572120576[12057311198621(119873) +

120576

12059612057321198622(119873)]

minusΛ (120596 + 120576)

120572120576[12057311198621(119873)

119873+120576

120596

12057321198622(119873)

119873]

minus (120583 + 120576) = 0

(23)

Because 119862119894(119873) (119894 = 1 2) satisfy conditions (i) (ii) and

(iii) thus 120593(119873) is an increasing continuous function and120593(Λ120583) = (120583 + 120576)(119877

0minus 1) When 119873 is sufficiently small

120593(119873) lt 0 If 1198770gt 1 then 120593(Λ120583) gt 0 According to the

zero-point theorem 120593(119873) has the unique positive solution119873lowast in the open interval (0 Λ120583) Then 119868lowast = (Λ minus 120583119873

lowast)120572

119864lowast= (120596120576)119868

lowast Otherwise if 1198770le 1 119873lowast does not exist in (0

Λ120583) Therefore we have the following theorem

Theorem 3 When 1198770gt 1 system (7) has the unique end-

emic equilibrium 119875lowast= (119864lowast 119868lowast 119873lowast) besides the disease-free

equilibrium 1198750in 119866

Theorem 4 When 1198770gt 1 the unique endemic equilibrium

119875lowast is locally asymptotically stable in 119866119900

ISRN Applied Mathematics 5

Proof The Jacobian matrix of system (7) at 119875lowast = (119864lowast 119868lowast 119873lowast)

is

119869 (119875lowast) = (

11988611

11988612

11988613

120576 minus120596 0

0 minus120572 minus120583

) (24)

where

11988611= minus

12057612057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

120596119873lowastminus119882lowastlt 0

11988612=12057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

119873lowastminus119882lowast

11988613= [120573

1(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= 1205731119864lowast[1198621015840

1(119873lowast) minus (119864

lowast+ 119868lowast) (

1198621(119873lowast)

119873lowast)

1015840

]

+ 1205732119868lowast[1198621015840

2(119873lowast) minus (119864

lowast+ 119868lowast) (

1198622(119873lowast)

119873lowast)

1015840

] gt 0

(25)

thereinto119882lowast = (12057311198621(119873lowast)119864lowast+ 12057321198622(119873lowast)119868lowast)119873lowast

Therefore the characteristic equation of 119869(119875lowast) is

1205823+ 11988611205822+ 1198862120582 + 1198863= 0 (26)

where

1198861= 120583 + 120596 minus 119886

11gt 0

1198862= (120583 minus 119886

11) 120596 minus 120583119886

11minus 12057611988612

= (120596 minus 11988611) 120583 + (120596 + 120576)119882

lowastgt 0

1198863= minus 120596120583119886

11minus 12058312057611988612+ 120572120576119886

13

= 120583 (120596 + 120576)119882lowast+ 120572120576119886

13gt 0

(27)

By calculation we have

1198671= 1198861gt 0

1198672= 11988611198862minus 1198863

= (120583 + 120596 minus 11988611) [(120596 minus 119886

11) 120583 + (120596 + 120576)119882

lowast]

minus 120583 (120596 + 120576)119882lowast

minus 120572120576[1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120596 minus 11988611) (120596 + 120576) minus 120583 (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast)

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120572 + 120575 minus 11988611) (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) gt 0

1198673= 11988631198672gt 0

(28)

By Routh-Hurwitz stability theorem [10] all the three eigen-values of 119869(119875lowast) have negative real parts Thus the endemicequilibrium 119875

lowast is locally asymptotically stable in 119866119900 when

1198770gt 1

Denote the boundary and the interior of 119879 by 120597119879 and 119879119900we also obtain for system (4)

Theorem 5 When 1198770gt 1 system (4) has a unique endemic

equilibrium 119875lowast

= (119878lowast 119864lowast 119868lowast) and it is locally asymptotically

stable in 119879119900 thereinto 119878lowast = 119873lowastminus 119864lowastminus 119868lowast

Now we briefly outline the autonomous convergencetheorem in [13] for proving global stability of the endemicequilibrium 119875

lowastLet 119863 sub R119899 be an open set and let 119909 rarr 119891(119909) isin R119899 be a

1198621 function defined in 119863 We consider the autonomous sys-

tem in R119899 = 119891 (119909) (29)

Let 119909 be an equilibrium of (29) that is 119891(119909) = 0 We recallthat 119909 is said to be globally stable in119863 if it is locally stable andall trajectories in119863 converge to 119909

Assume that the following hypothesis hold

(H1) 119863 is simply connected(H2) there exists a compact absorbing set Γ sub 119863(H3) 119909 is the only equilibrium of (29) and is locally stable

in119863The basic job is to find conditions under which the

global stability of 119909 with respect to 119863 is implied by itslocal stability The difficulty associated with this problem islargely due to the lack of practical tools A new approachto the global stability problem has emerged from a series ofpapers on higher-dimensional generalizations of the criteriaof Bendixson and Dulac for planar systems and on so-calledautonomous convergence theorems First we now introducea definition which will appear in the following context

6 ISRN Applied Mathematics

Definition 6 (see [13]) Suppose system (29) has a periodicsolution 119909 = 119901(119905)with least period120596 gt 0 and orbit 120574 = 119901(119905)

0 le 119905 le 120596 This orbit is orbitally stable if for each 120576 gt 0there exists a 120575 gt 0 such that any solution 119909(119905) for which thedistance of119909(0) from 120574 is less than 120575 remains at a distance lessthan 120576 from 120574 for all 119905 ge 0 It is asymptotically orbitally stable ifthe distance of 119909(119905) from 120574 also tends to zero as 119905 rarr infin Thisorbit 120574 is asymptotically orbitally stable with asymptotic phaseif it is asymptotically orbitally stable and there is a 119887 gt 0 suchthat any solution 119909(119905) for which the distance of 119909(0) from 120574

is less than 119887 satisfies |119909(119905) minus 119901(119905 minus 120591)| rarr 0 as 119905 rarr infin forsome 120591 which may depend on 119909(0)

Theorem 7 (see [14]) A sufficient condition for a period orbit120574 = 119901(119905) 0 le 119905 le 120596 of (29) is asymptotically orbitally stablewith asymptotic phase such that the linear system

1199111015840(119905) = (

120597119891[2]

120597119909(119901 (119905))) 119911 (119905) (30)

is asymptotically stable

Remark 8 Equation (30) is called the second compoundequation of (29) and 120597119891[2]120597119909 is the second compoundmatrixof the Jacobian matrix 120597119891120597119909 of 119891

It is also demonstrated thatTheorem 7 generalizes a classof Poincare for the orbital stability of periodic solutions toplanar autonomous systems

Theorem 9 (see [13]) Under assumptions (H1) (H2) and(H3) 119909 is globally asymptotically stable in 119863 provided that

(H4) the system (29) satisfies a Poincare-Bendixson crite-rion

(H5) a periodic orbit of the system (29) is asymptoticallyorbitally stable

As a matter of fact the condition (H2) is true if and only ifthe system (4) is uniformly persistent in 119879119900

Definition 10 (see [15 16]) System (4) is said to be uniformlypersistent if there exists a constant 120578 isin (0 1) such that anysolution (119878(119905) 119864(119905) 119868(119905)) with initial point (119878(0) 119864(0) 119868(0)) isin119879119900 satisfies

min lim119905rarrinfin

inf 119878 (119905) lim119905rarrinfin

inf 119864 (119905) lim119905rarrinfin

inf 119868 (119905) ge 120578 (31)

Lemma 11 When 1198770gt 1 system (4) is uniformly persistent in

119879119900

Proof Any solution of system (4)which begins from (119878 0 0)

0 le 119878 le Λ120583 always in fact converges at the point 1198750=

(Λ120583 0 0) along the 119878-axis Except the 119878-axis the solution ofsystem (4) which begin from 120597119879 will converge in the region119879119900 Thus 119875

0is the unique 120596-limit point in 120597119879 of system (4)

Let

119880 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (32)

then the time derivative of 119880 along a solution of system (4)gives

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

1198731198781198770

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

ge [12057311198621(Λ

120583)119864 + 120573

21198622(Λ

120583) 119868] (

120583

Λ1198770119878 minus 1)

(33)

When 1198770gt 1 if the trajectories (119878 119864 119868) in 119879119900 sufficiently

converge to1198750 it implies that gt 0That is to say there exists

a neighborhood U(1198750) of 1198750 such that when the trajectories

of system (4) begin from119879119900capU(119875

0) it will come out ofU(119875

0)

Therefore 1198750is not a 120596-limit point of any trajectory in 119879

119900Thus 119872 = (119878 0 0) | 0 le 119878 le Λ120583 is the largest invariantset in 120597119879 of system (4) When 119877

0gt 1 119872 is isolated Also

the invariant set 119882119904(119872) sube 120597119879 where 119882119904(119872) = 119909 isin 119863

119891119899(119909) rarr 119872 as 119899 rarr +infin [15] is the stable set of119872 Accord-

ing to [15 Theorem 41] system (4) is uniformly persist-ent in119879119900 when119877

0gt 1Thus there exists a compact absorbing

subset in 119879119900 for system (4)

Lemma 12 When 1198770gt 1 system (4) satisfies the Poincare-

Bendixson criterion in 119879119900

Proof Because the system (4) is not quasimonotone wecannot verify that the system (4) is competitive by examiningits Jacobian matrix Thus we can replace the system (4) by

1199091015840

1= Λ minus 120583119909

1minus12057311198621(119873)

119873119878 (119873 minus 119909

1minus 1199093)

minus12057321198622 (119873)

119873119878 (119873 minus 119909

1minus 1199092) + 120575119909

3

1199091015840

2=12057311198621(119873)

1198731198781199092+12057321198622(119873)

1198731198781199093minus (120583 + 120576) 119909

2

1199091015840

3= 1205761199092minus (120583 + 120572 + 120575) 119909

3

(34)

Then system (34) has a solution 119906(119905) = (119878(119905) 119864(119905) 119868(119905))Let x = (119909

1 1199092 1199093)119879isin R3 we have

x1015840 = (119861 minus 120583119868) x + 119862 (119878 119864 119868) (35)

where 119868 denotes the 3 times 3 unit matrix 119862(119878 119864 119868) is a functionthat need not concern us and

119861 = (

12057311198621 (119873) + 12057321198622 (119873)

11987311987812057321198622 (119873)

11987311987812057311198621 (119873)

119873119878 + 120575

012057311198621 (119873)

119873119878 minus 120576

12057321198622 (119873)

119873119878

0 120576 minus (120572 + 120575)

)

(36)

The off-diagonal entries in this matrix are nonnegative thusthe system (34) as a whole is quasimonotone [17] Then wecan verify that the system (34) is competitive [18] with respectto the partial ordering defined by the orthant119870 = (119878 119864 119868) isin

R3+ Since 119879119900 is convex system (4) satisfies the Poincare-

Bendixson criterion [10 19] in 119879119900 when 1198770gt 1

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 2008

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Applied Mathematics

The above discussion reveals the importance of incidencefunctions in epidemic models Different nonlinear forms ofincidence can exhibit very dynamics and hence are able tounearth some otherwise unknown features of disease dynam-ics Though the aspect of nonlinearity in incidence hasfound a significant importance in the existing literature thefact that population subclasses with different infection stat-uses should have different incidence rates has received littleattention among mathematical epidemiologists Thus in anSEIS epidemic model since there is a difference in relativemeasure of infectiousness between the exposed and theinfected populations the incidence rate between the suscep-tible fraction 119878 and the infected fraction 119868 should be differentfrom that between 119878 with the exposed fraction 119864

The present analysis aims to explore the impact of this dis-tinct incidence for exposed and infected populations underthe influence of spatial heterogeneity As a model systemWe have divided the population in researched area into threeclasses 119878mdashsusceptible 119864mdashexposed with the infectious forceand 119868mdashinfected

In the next section we establish the model discussed inthis paper and determine the basic reproductive number InSection 3 we analyze the global stability of the disease-freeequilibrium In Section 4 we resolve the unique existenceand global stability of the epidemic equilibrium In Section5 we present some numerical simulation of examples whichvalidate these theoretical results The paper ends with a briefdiscussion in Section 6

2 The Model and the BasicReproductive Number

The model we consider has the following population sub-classes (i) 119878mdashthe susceptible (ii) 119864mdashthe exposed and (iii)119868mdashthe infected The total population size denoted by 119873 is119873(119905) = 119878(119905) + 119864(119905) + 119868(119904) The transfer mechanism from theclass 119878 to the class 119864 is guided by the function

119891 (119905) =12057311198621(119873)

119873119878119864 +

12057321198622(119873)

119873119878119868 (3)

where 1205731and 120573

2are average numbers of adequate contacts of

an exposed individual and an infectious individual per unittime respectively and119862

119894(119873) (119894 = 1 2) are relevant saturation

contact rate which satisfy the following assumptions for119873 gt

0

(i) 119862119894(119873) gt 0

(ii) 1198621015840119894(119873) ge 0

(iii) [119862119894(119873)119873]

1015840le 0

The assumptions (i) and (ii) are biologically motivated Asthe total population119873 increases the probability of a contactwith a susceptible individual decreases and thus the force ofthe exposed or the infected is expected to be a decreasingfunction of 119873 And the assumption (iii) implies that thecontact rate 119862

119894(119873) is saturated

The population transfer among compartments is sche-matically depicted in the transfer diagram in Figure 1

120583119868120583119864 120572119868

120575119868

Λ

12057321198622(119873)

119873119878119868

12057311198621(119873)

119873119878119864

120583119878

119878 119868119864120576119864

Figure 1 The transfer diagram for model (4)

The transfer diagram leads to the following SEIS epidemicmodel of ordinary differential equations

1198781015840= Λ minus 120583119878 minus

12057311198621 (119873)

119873119878119864 minus

12057321198622 (119873)

119873119878119868 + 120575119868

1198641015840=12057311198621(119873)

119873119878119864 +

12057321198622(119873)

119873119878119868 minus (120583 + 120576) 119864

1198681015840= 120576119864 minus (120583 + 120572 + 120575) 119868

(4)

where Λ is the recruitment rate of the population 120583 is thenatural death rate and 120572 is the death rate for the infected 119864individuals move to the class 119868 at the rate 120576 and 119868 individualsrecover at the rate 120575 which are assumed to join the susceptibleclass The above parameters are positive

Summing up the three equations in system (4) then thetime derivative of119873(119905) along a solution of system (4) is

1198731015840= Λ minus 120583 (119878 + 119864 + 119868) minus 120572119868 (5)

Therefore1198731015840 le Λminus120583119873 equivalently1198731015840+120583119873 le Λ Applyinga theorem on differential inequalities [10] we get 0 le 119873 le

Λ120583 for 119905 rarr +infin Thus the three-dimensional simplex

119879 = (119878 119864 119868) isin R3+| 0 le 119878 + 119864 + 119868 le

Λ

120583 (6)

is positively invariant with respect to system (4) whereR3+denotes the nonnegative cone of R3 including its lower

dimensional facesBy using 119878 = 119873minus119864minus119868 and (5) we get the following system

1198641015840=12057311198621 (119873) 119864 + 120573

21198622 (119873) 119868

119873(119873 minus 119864 minus 119868) minus (120583 + 120576) 119864

1198681015840= 120576119864 minus (120583 + 120572 + 120575) 119868

1198731015840= Λ minus 120583119873 minus 120572119868

(7)

The dynamical behavior of system (4) in 119879 is equivalent tothat of system (7)Thus in the rest of the paper we will studythe system (7) in the feasible region

119866 = (119864 119868119873) isin R3+| 0 le 119864 + 119868 le 119873 le

Λ

120583 (8)

which can be shown to be a positive invariant set for system(7)

Now we derive the basic reproductive number of system(4) by the method of next-generation matrix formulated in[11]

ISRN Applied Mathematics 3

Let 119909 = (119864 119868 119878)119879 then system (4) can be written as

1199091015840= F (119909) minusV (119909) (9)

where

F (119909) = (

[12057311198621(119873) 119864 + 120573

21198622(119873) 119868]

119878

119873

0

0

)

V (119909) = (

(120583 + 120576) 119864

minus120576119864 + (120583 + 120572 + 120575) 119868

minusΛ + 120583119878 + [12057311198621 (119873) 119864 + 120573

21198622 (119873) 119868]

119878

119873minus 120575119868

)

(10)

Then1199090= (0 0 Λ120583)

119879 is the unique disease-free equilibriumof system (9) and the Jacobian matrices ofF(119909) andV(119909) atequilibrium 119909

0are respectively

119863F (1199090) = (

119865 1198742times1

1198741times2

0

) 119863V (1199090) = (

119881 1198742times1

1198691

120583

)

(11)

where

119865 = (12057311198621(Λ

120583) 12057321198622(Λ

120583)

0 0

)

119881 = (120583 + 120576 0

minus120576 120583 + 120572 + 120575)

1198691= (12057311198621(Λ

120583) 12057321198622(Λ

120583) minus 120575)

(12)

Obviously all eigenvalues of minus119863V(1199090) have negative real

partsWe call

119865119881minus1=

1

(120583 + 120576) (120583 + 120572 + 120575)(12057311198621(Λ

120583) (120583 + 120572 + 120575) + 120576120573

21198622(Λ

120583) 12057321198622(Λ

120583) (120583 + 120576)

0 0

) (13)

the next generation matrix for system (9) According to [11Theorem 2] the basic reproductive number of system (4)which is the number of secondary infectious cases producedby an exposed individual and an infectious individual duringtheir effective infectious period when introduced in a popu-lation of susceptible is

1198770= 120588 (119865119881

minus1) =

12057311198621(Λ120583) (120583 + 120572 + 120575) + 120576120573

21198622(Λ120583)

(120583 + 120576) (120583 + 120572 + 120575)

(14)

where 120588(119860) denotes the spectral radius of matrix 119860

3 Stability Analysis ofthe Disease-Free Equilibrium

In this section we discuss the global stability of the disease-free equilibrium It is obvious that system (7) always has theunique disease-free equilibrium 119875

0= (0 0 Λ120583) in 119866 About

1198750 we have the following main results

Theorem 1 The disease-free equilibrium 1198750is globally asymp-

totically stable in 119866 if 1198770le 1 and it is unstable if 119877

0gt 1

Proof The Jacobian matrix of system (7) at 1198750= (0 0 Λ120583)

goes as follows

119869 (1198750) = (

12057311198621(Λ

120583) minus (120583 + 120576) 120573

21198622(Λ

120583) 0

120576 minus120596 0

0 minus120572 minus120583

) (15)

which has a eigenvalue 1205821= minus120583 lt 0 obviously The other

two eigenvalues 1205822and 120582

3are determined by the following

equation

1205822minus [12057311198621(Λ

120583) minus (120583 + 120596 + 120576)] 120582

minus 120596[12057311198621(Λ

120583) minus (120583 + 120576)] minus 120576120573

21198622(Λ

120583) = 0

(16)

If 1198770gt 1 we can have easily

12058221205823= minus 120596[120573

11198621(Λ

120583) minus (120583 + 120576)] minus 120576120573

21198622(Λ

120583)

= 120596 (120583 + 120576) (1 minus 1198770) lt 0

(17)

Therefore 1205822and 120582

3are two opposite-sign real roots Thus

1198750is unstable

4 ISRN Applied Mathematics

Since 1198770lt 1 implies 120573

11198621(Λ120583) lt (120583+120576)minus(120576120596)120573

21198622(Λ

120583) then we get

1205822+ 1205823= 12057311198621(Λ

120583) minus (120583 + 120596 + 120576)

lt minus 120596 minus120576

12059612057321198622(Λ

120583) lt 0

12058221205823= 120596 (120583 + 120576) (1 minus 119877

0) gt 0

(18)

Therefore 1205822and 120582

3have negative real parts Hence 119875

0is

locally asymptotically stableWhen 119877

0= 1 it implies that 120582

21205823= 0 120573

11198621(Λ120583) minus (120583 +

120576) = minus(120576120596)12057321198622(Λ120583) We may as well assume that 120582

2= 0

then 1205823= minus120596minus (120576120596)120573

21198622(Λ120583) The characteristic matrix of

119869(1198750) has three invariable factors 1 1 and 120582(120582 + 120583)(120582 + 120596 +

(120576120596)12057321198622(Λ120583)) Because the elementary factor with respect

to 1205822= 0 is 120582 which is single 119875

0is stable

Constructing a suitable Lyapunov function

119881 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (19)

then the time derivative of 119881 along a solution of system (7)gives

= 1198770

12057311198621 (119873) 119864 + 12057321198622 (119873) 119868

119873(119873 minus 119864 minus 119868)

minus 1198770(120583 + 120576) 119864 +

1205761205732

1205961198622(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

= 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

le 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(119873) 119864 minus 120573

21198622(119873) 119868

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873[(1198770minus 1)119873 minus 119877

0119864 minus 119877

0119868]

(20)

Hence le 0 holds if1198770le 1 Furthermore = 0 if and only

if 119864 = 119868 = 0 Let 119865 = (119864 119868119873) isin 119866 | = 0 = (0 0119873)then the largest compact invariant set in119865 for system (7) is theset (0 0119873) Thus the solution of system (7) satisfies 119864 rarr

0 119868 rarr 0 as 119905 rarr +infin by LaSallersquos Invariance Principle [12]Therefore the limit system of system (7) is

1198641015840= 0

1198681015840= 0

1198731015840= Λ minus 120583119873

(21)

It is obviously known that the equilibrium (0 0 Λ120583) ofsystem (21) is globally asymptotically stable thus the disease-free equilibrium119875

0of system (7) is globally attractive in119866 On

the basis of local stability 1198750is globally asymptotically stable

in 119866 if 1198770le 1 This completes the proof

About system (4) we also obtain

Theorem 2 The unique disease-free equilibrium 1198750= (Λ120583

0 0) of system (4) is globally asymptotically stable in119879 if1198770le 1

and it is unstable if 1198770gt 1

4 Existence and Stability ofthe Endemic Equilibrium

In this section we first discuss the existence and uniquenessof the endemic equilibrium 119875

lowast of system (7) when 1198770gt 1

Whereafter we focus on investigating the local stability of119875lowastWe have to prove that the Jacobian matrix 119869(119875

lowast) is stable

namely all its eigenvalues have negative real parts This isroutinely done by verifying the Routh-Hurwitz conditionsFinally we study the global stability of the endemic equilib-rium 119875

lowast of system (4) with the method of autonomous con-vergence theorem of Li and Muldowney in [13]

The coordinates of the endemic equilibrium (positiveequilibrium) of system (7) are the positive solutions of equa-tions

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868) minus (120583 + 120576) 119864 = 0

120576119864 minus (120583 + 120572 + 120575) 119868 = 0

Λ minus 120583119873 minus 120572119868 = 0

(22)

in 119866119900Let 120596 = 120583+120572+120575 by the direct calculation we can get the

following equation of119873 easily as

120593 (119873) =120572120576 + 120583120596 + 120583120576

120572120576[12057311198621(119873) +

120576

12059612057321198622(119873)]

minusΛ (120596 + 120576)

120572120576[12057311198621(119873)

119873+120576

120596

12057321198622(119873)

119873]

minus (120583 + 120576) = 0

(23)

Because 119862119894(119873) (119894 = 1 2) satisfy conditions (i) (ii) and

(iii) thus 120593(119873) is an increasing continuous function and120593(Λ120583) = (120583 + 120576)(119877

0minus 1) When 119873 is sufficiently small

120593(119873) lt 0 If 1198770gt 1 then 120593(Λ120583) gt 0 According to the

zero-point theorem 120593(119873) has the unique positive solution119873lowast in the open interval (0 Λ120583) Then 119868lowast = (Λ minus 120583119873

lowast)120572

119864lowast= (120596120576)119868

lowast Otherwise if 1198770le 1 119873lowast does not exist in (0

Λ120583) Therefore we have the following theorem

Theorem 3 When 1198770gt 1 system (7) has the unique end-

emic equilibrium 119875lowast= (119864lowast 119868lowast 119873lowast) besides the disease-free

equilibrium 1198750in 119866

Theorem 4 When 1198770gt 1 the unique endemic equilibrium

119875lowast is locally asymptotically stable in 119866119900

ISRN Applied Mathematics 5

Proof The Jacobian matrix of system (7) at 119875lowast = (119864lowast 119868lowast 119873lowast)

is

119869 (119875lowast) = (

11988611

11988612

11988613

120576 minus120596 0

0 minus120572 minus120583

) (24)

where

11988611= minus

12057612057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

120596119873lowastminus119882lowastlt 0

11988612=12057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

119873lowastminus119882lowast

11988613= [120573

1(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= 1205731119864lowast[1198621015840

1(119873lowast) minus (119864

lowast+ 119868lowast) (

1198621(119873lowast)

119873lowast)

1015840

]

+ 1205732119868lowast[1198621015840

2(119873lowast) minus (119864

lowast+ 119868lowast) (

1198622(119873lowast)

119873lowast)

1015840

] gt 0

(25)

thereinto119882lowast = (12057311198621(119873lowast)119864lowast+ 12057321198622(119873lowast)119868lowast)119873lowast

Therefore the characteristic equation of 119869(119875lowast) is

1205823+ 11988611205822+ 1198862120582 + 1198863= 0 (26)

where

1198861= 120583 + 120596 minus 119886

11gt 0

1198862= (120583 minus 119886

11) 120596 minus 120583119886

11minus 12057611988612

= (120596 minus 11988611) 120583 + (120596 + 120576)119882

lowastgt 0

1198863= minus 120596120583119886

11minus 12058312057611988612+ 120572120576119886

13

= 120583 (120596 + 120576)119882lowast+ 120572120576119886

13gt 0

(27)

By calculation we have

1198671= 1198861gt 0

1198672= 11988611198862minus 1198863

= (120583 + 120596 minus 11988611) [(120596 minus 119886

11) 120583 + (120596 + 120576)119882

lowast]

minus 120583 (120596 + 120576)119882lowast

minus 120572120576[1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120596 minus 11988611) (120596 + 120576) minus 120583 (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast)

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120572 + 120575 minus 11988611) (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) gt 0

1198673= 11988631198672gt 0

(28)

By Routh-Hurwitz stability theorem [10] all the three eigen-values of 119869(119875lowast) have negative real parts Thus the endemicequilibrium 119875

lowast is locally asymptotically stable in 119866119900 when

1198770gt 1

Denote the boundary and the interior of 119879 by 120597119879 and 119879119900we also obtain for system (4)

Theorem 5 When 1198770gt 1 system (4) has a unique endemic

equilibrium 119875lowast

= (119878lowast 119864lowast 119868lowast) and it is locally asymptotically

stable in 119879119900 thereinto 119878lowast = 119873lowastminus 119864lowastminus 119868lowast

Now we briefly outline the autonomous convergencetheorem in [13] for proving global stability of the endemicequilibrium 119875

lowastLet 119863 sub R119899 be an open set and let 119909 rarr 119891(119909) isin R119899 be a

1198621 function defined in 119863 We consider the autonomous sys-

tem in R119899 = 119891 (119909) (29)

Let 119909 be an equilibrium of (29) that is 119891(119909) = 0 We recallthat 119909 is said to be globally stable in119863 if it is locally stable andall trajectories in119863 converge to 119909

Assume that the following hypothesis hold

(H1) 119863 is simply connected(H2) there exists a compact absorbing set Γ sub 119863(H3) 119909 is the only equilibrium of (29) and is locally stable

in119863The basic job is to find conditions under which the

global stability of 119909 with respect to 119863 is implied by itslocal stability The difficulty associated with this problem islargely due to the lack of practical tools A new approachto the global stability problem has emerged from a series ofpapers on higher-dimensional generalizations of the criteriaof Bendixson and Dulac for planar systems and on so-calledautonomous convergence theorems First we now introducea definition which will appear in the following context

6 ISRN Applied Mathematics

Definition 6 (see [13]) Suppose system (29) has a periodicsolution 119909 = 119901(119905)with least period120596 gt 0 and orbit 120574 = 119901(119905)

0 le 119905 le 120596 This orbit is orbitally stable if for each 120576 gt 0there exists a 120575 gt 0 such that any solution 119909(119905) for which thedistance of119909(0) from 120574 is less than 120575 remains at a distance lessthan 120576 from 120574 for all 119905 ge 0 It is asymptotically orbitally stable ifthe distance of 119909(119905) from 120574 also tends to zero as 119905 rarr infin Thisorbit 120574 is asymptotically orbitally stable with asymptotic phaseif it is asymptotically orbitally stable and there is a 119887 gt 0 suchthat any solution 119909(119905) for which the distance of 119909(0) from 120574

is less than 119887 satisfies |119909(119905) minus 119901(119905 minus 120591)| rarr 0 as 119905 rarr infin forsome 120591 which may depend on 119909(0)

Theorem 7 (see [14]) A sufficient condition for a period orbit120574 = 119901(119905) 0 le 119905 le 120596 of (29) is asymptotically orbitally stablewith asymptotic phase such that the linear system

1199111015840(119905) = (

120597119891[2]

120597119909(119901 (119905))) 119911 (119905) (30)

is asymptotically stable

Remark 8 Equation (30) is called the second compoundequation of (29) and 120597119891[2]120597119909 is the second compoundmatrixof the Jacobian matrix 120597119891120597119909 of 119891

It is also demonstrated thatTheorem 7 generalizes a classof Poincare for the orbital stability of periodic solutions toplanar autonomous systems

Theorem 9 (see [13]) Under assumptions (H1) (H2) and(H3) 119909 is globally asymptotically stable in 119863 provided that

(H4) the system (29) satisfies a Poincare-Bendixson crite-rion

(H5) a periodic orbit of the system (29) is asymptoticallyorbitally stable

As a matter of fact the condition (H2) is true if and only ifthe system (4) is uniformly persistent in 119879119900

Definition 10 (see [15 16]) System (4) is said to be uniformlypersistent if there exists a constant 120578 isin (0 1) such that anysolution (119878(119905) 119864(119905) 119868(119905)) with initial point (119878(0) 119864(0) 119868(0)) isin119879119900 satisfies

min lim119905rarrinfin

inf 119878 (119905) lim119905rarrinfin

inf 119864 (119905) lim119905rarrinfin

inf 119868 (119905) ge 120578 (31)

Lemma 11 When 1198770gt 1 system (4) is uniformly persistent in

119879119900

Proof Any solution of system (4)which begins from (119878 0 0)

0 le 119878 le Λ120583 always in fact converges at the point 1198750=

(Λ120583 0 0) along the 119878-axis Except the 119878-axis the solution ofsystem (4) which begin from 120597119879 will converge in the region119879119900 Thus 119875

0is the unique 120596-limit point in 120597119879 of system (4)

Let

119880 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (32)

then the time derivative of 119880 along a solution of system (4)gives

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

1198731198781198770

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

ge [12057311198621(Λ

120583)119864 + 120573

21198622(Λ

120583) 119868] (

120583

Λ1198770119878 minus 1)

(33)

When 1198770gt 1 if the trajectories (119878 119864 119868) in 119879119900 sufficiently

converge to1198750 it implies that gt 0That is to say there exists

a neighborhood U(1198750) of 1198750 such that when the trajectories

of system (4) begin from119879119900capU(119875

0) it will come out ofU(119875

0)

Therefore 1198750is not a 120596-limit point of any trajectory in 119879

119900Thus 119872 = (119878 0 0) | 0 le 119878 le Λ120583 is the largest invariantset in 120597119879 of system (4) When 119877

0gt 1 119872 is isolated Also

the invariant set 119882119904(119872) sube 120597119879 where 119882119904(119872) = 119909 isin 119863

119891119899(119909) rarr 119872 as 119899 rarr +infin [15] is the stable set of119872 Accord-

ing to [15 Theorem 41] system (4) is uniformly persist-ent in119879119900 when119877

0gt 1Thus there exists a compact absorbing

subset in 119879119900 for system (4)

Lemma 12 When 1198770gt 1 system (4) satisfies the Poincare-

Bendixson criterion in 119879119900

Proof Because the system (4) is not quasimonotone wecannot verify that the system (4) is competitive by examiningits Jacobian matrix Thus we can replace the system (4) by

1199091015840

1= Λ minus 120583119909

1minus12057311198621(119873)

119873119878 (119873 minus 119909

1minus 1199093)

minus12057321198622 (119873)

119873119878 (119873 minus 119909

1minus 1199092) + 120575119909

3

1199091015840

2=12057311198621(119873)

1198731198781199092+12057321198622(119873)

1198731198781199093minus (120583 + 120576) 119909

2

1199091015840

3= 1205761199092minus (120583 + 120572 + 120575) 119909

3

(34)

Then system (34) has a solution 119906(119905) = (119878(119905) 119864(119905) 119868(119905))Let x = (119909

1 1199092 1199093)119879isin R3 we have

x1015840 = (119861 minus 120583119868) x + 119862 (119878 119864 119868) (35)

where 119868 denotes the 3 times 3 unit matrix 119862(119878 119864 119868) is a functionthat need not concern us and

119861 = (

12057311198621 (119873) + 12057321198622 (119873)

11987311987812057321198622 (119873)

11987311987812057311198621 (119873)

119873119878 + 120575

012057311198621 (119873)

119873119878 minus 120576

12057321198622 (119873)

119873119878

0 120576 minus (120572 + 120575)

)

(36)

The off-diagonal entries in this matrix are nonnegative thusthe system (34) as a whole is quasimonotone [17] Then wecan verify that the system (34) is competitive [18] with respectto the partial ordering defined by the orthant119870 = (119878 119864 119868) isin

R3+ Since 119879119900 is convex system (4) satisfies the Poincare-

Bendixson criterion [10 19] in 119879119900 when 1198770gt 1

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 2008

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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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

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 3

Let 119909 = (119864 119868 119878)119879 then system (4) can be written as

1199091015840= F (119909) minusV (119909) (9)

where

F (119909) = (

[12057311198621(119873) 119864 + 120573

21198622(119873) 119868]

119878

119873

0

0

)

V (119909) = (

(120583 + 120576) 119864

minus120576119864 + (120583 + 120572 + 120575) 119868

minusΛ + 120583119878 + [12057311198621 (119873) 119864 + 120573

21198622 (119873) 119868]

119878

119873minus 120575119868

)

(10)

Then1199090= (0 0 Λ120583)

119879 is the unique disease-free equilibriumof system (9) and the Jacobian matrices ofF(119909) andV(119909) atequilibrium 119909

0are respectively

119863F (1199090) = (

119865 1198742times1

1198741times2

0

) 119863V (1199090) = (

119881 1198742times1

1198691

120583

)

(11)

where

119865 = (12057311198621(Λ

120583) 12057321198622(Λ

120583)

0 0

)

119881 = (120583 + 120576 0

minus120576 120583 + 120572 + 120575)

1198691= (12057311198621(Λ

120583) 12057321198622(Λ

120583) minus 120575)

(12)

Obviously all eigenvalues of minus119863V(1199090) have negative real

partsWe call

119865119881minus1=

1

(120583 + 120576) (120583 + 120572 + 120575)(12057311198621(Λ

120583) (120583 + 120572 + 120575) + 120576120573

21198622(Λ

120583) 12057321198622(Λ

120583) (120583 + 120576)

0 0

) (13)

the next generation matrix for system (9) According to [11Theorem 2] the basic reproductive number of system (4)which is the number of secondary infectious cases producedby an exposed individual and an infectious individual duringtheir effective infectious period when introduced in a popu-lation of susceptible is

1198770= 120588 (119865119881

minus1) =

12057311198621(Λ120583) (120583 + 120572 + 120575) + 120576120573

21198622(Λ120583)

(120583 + 120576) (120583 + 120572 + 120575)

(14)

where 120588(119860) denotes the spectral radius of matrix 119860

3 Stability Analysis ofthe Disease-Free Equilibrium

In this section we discuss the global stability of the disease-free equilibrium It is obvious that system (7) always has theunique disease-free equilibrium 119875

0= (0 0 Λ120583) in 119866 About

1198750 we have the following main results

Theorem 1 The disease-free equilibrium 1198750is globally asymp-

totically stable in 119866 if 1198770le 1 and it is unstable if 119877

0gt 1

Proof The Jacobian matrix of system (7) at 1198750= (0 0 Λ120583)

goes as follows

119869 (1198750) = (

12057311198621(Λ

120583) minus (120583 + 120576) 120573

21198622(Λ

120583) 0

120576 minus120596 0

0 minus120572 minus120583

) (15)

which has a eigenvalue 1205821= minus120583 lt 0 obviously The other

two eigenvalues 1205822and 120582

3are determined by the following

equation

1205822minus [12057311198621(Λ

120583) minus (120583 + 120596 + 120576)] 120582

minus 120596[12057311198621(Λ

120583) minus (120583 + 120576)] minus 120576120573

21198622(Λ

120583) = 0

(16)

If 1198770gt 1 we can have easily

12058221205823= minus 120596[120573

11198621(Λ

120583) minus (120583 + 120576)] minus 120576120573

21198622(Λ

120583)

= 120596 (120583 + 120576) (1 minus 1198770) lt 0

(17)

Therefore 1205822and 120582

3are two opposite-sign real roots Thus

1198750is unstable

4 ISRN Applied Mathematics

Since 1198770lt 1 implies 120573

11198621(Λ120583) lt (120583+120576)minus(120576120596)120573

21198622(Λ

120583) then we get

1205822+ 1205823= 12057311198621(Λ

120583) minus (120583 + 120596 + 120576)

lt minus 120596 minus120576

12059612057321198622(Λ

120583) lt 0

12058221205823= 120596 (120583 + 120576) (1 minus 119877

0) gt 0

(18)

Therefore 1205822and 120582

3have negative real parts Hence 119875

0is

locally asymptotically stableWhen 119877

0= 1 it implies that 120582

21205823= 0 120573

11198621(Λ120583) minus (120583 +

120576) = minus(120576120596)12057321198622(Λ120583) We may as well assume that 120582

2= 0

then 1205823= minus120596minus (120576120596)120573

21198622(Λ120583) The characteristic matrix of

119869(1198750) has three invariable factors 1 1 and 120582(120582 + 120583)(120582 + 120596 +

(120576120596)12057321198622(Λ120583)) Because the elementary factor with respect

to 1205822= 0 is 120582 which is single 119875

0is stable

Constructing a suitable Lyapunov function

119881 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (19)

then the time derivative of 119881 along a solution of system (7)gives

= 1198770

12057311198621 (119873) 119864 + 12057321198622 (119873) 119868

119873(119873 minus 119864 minus 119868)

minus 1198770(120583 + 120576) 119864 +

1205761205732

1205961198622(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

= 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

le 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(119873) 119864 minus 120573

21198622(119873) 119868

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873[(1198770minus 1)119873 minus 119877

0119864 minus 119877

0119868]

(20)

Hence le 0 holds if1198770le 1 Furthermore = 0 if and only

if 119864 = 119868 = 0 Let 119865 = (119864 119868119873) isin 119866 | = 0 = (0 0119873)then the largest compact invariant set in119865 for system (7) is theset (0 0119873) Thus the solution of system (7) satisfies 119864 rarr

0 119868 rarr 0 as 119905 rarr +infin by LaSallersquos Invariance Principle [12]Therefore the limit system of system (7) is

1198641015840= 0

1198681015840= 0

1198731015840= Λ minus 120583119873

(21)

It is obviously known that the equilibrium (0 0 Λ120583) ofsystem (21) is globally asymptotically stable thus the disease-free equilibrium119875

0of system (7) is globally attractive in119866 On

the basis of local stability 1198750is globally asymptotically stable

in 119866 if 1198770le 1 This completes the proof

About system (4) we also obtain

Theorem 2 The unique disease-free equilibrium 1198750= (Λ120583

0 0) of system (4) is globally asymptotically stable in119879 if1198770le 1

and it is unstable if 1198770gt 1

4 Existence and Stability ofthe Endemic Equilibrium

In this section we first discuss the existence and uniquenessof the endemic equilibrium 119875

lowast of system (7) when 1198770gt 1

Whereafter we focus on investigating the local stability of119875lowastWe have to prove that the Jacobian matrix 119869(119875

lowast) is stable

namely all its eigenvalues have negative real parts This isroutinely done by verifying the Routh-Hurwitz conditionsFinally we study the global stability of the endemic equilib-rium 119875

lowast of system (4) with the method of autonomous con-vergence theorem of Li and Muldowney in [13]

The coordinates of the endemic equilibrium (positiveequilibrium) of system (7) are the positive solutions of equa-tions

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868) minus (120583 + 120576) 119864 = 0

120576119864 minus (120583 + 120572 + 120575) 119868 = 0

Λ minus 120583119873 minus 120572119868 = 0

(22)

in 119866119900Let 120596 = 120583+120572+120575 by the direct calculation we can get the

following equation of119873 easily as

120593 (119873) =120572120576 + 120583120596 + 120583120576

120572120576[12057311198621(119873) +

120576

12059612057321198622(119873)]

minusΛ (120596 + 120576)

120572120576[12057311198621(119873)

119873+120576

120596

12057321198622(119873)

119873]

minus (120583 + 120576) = 0

(23)

Because 119862119894(119873) (119894 = 1 2) satisfy conditions (i) (ii) and

(iii) thus 120593(119873) is an increasing continuous function and120593(Λ120583) = (120583 + 120576)(119877

0minus 1) When 119873 is sufficiently small

120593(119873) lt 0 If 1198770gt 1 then 120593(Λ120583) gt 0 According to the

zero-point theorem 120593(119873) has the unique positive solution119873lowast in the open interval (0 Λ120583) Then 119868lowast = (Λ minus 120583119873

lowast)120572

119864lowast= (120596120576)119868

lowast Otherwise if 1198770le 1 119873lowast does not exist in (0

Λ120583) Therefore we have the following theorem

Theorem 3 When 1198770gt 1 system (7) has the unique end-

emic equilibrium 119875lowast= (119864lowast 119868lowast 119873lowast) besides the disease-free

equilibrium 1198750in 119866

Theorem 4 When 1198770gt 1 the unique endemic equilibrium

119875lowast is locally asymptotically stable in 119866119900

ISRN Applied Mathematics 5

Proof The Jacobian matrix of system (7) at 119875lowast = (119864lowast 119868lowast 119873lowast)

is

119869 (119875lowast) = (

11988611

11988612

11988613

120576 minus120596 0

0 minus120572 minus120583

) (24)

where

11988611= minus

12057612057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

120596119873lowastminus119882lowastlt 0

11988612=12057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

119873lowastminus119882lowast

11988613= [120573

1(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= 1205731119864lowast[1198621015840

1(119873lowast) minus (119864

lowast+ 119868lowast) (

1198621(119873lowast)

119873lowast)

1015840

]

+ 1205732119868lowast[1198621015840

2(119873lowast) minus (119864

lowast+ 119868lowast) (

1198622(119873lowast)

119873lowast)

1015840

] gt 0

(25)

thereinto119882lowast = (12057311198621(119873lowast)119864lowast+ 12057321198622(119873lowast)119868lowast)119873lowast

Therefore the characteristic equation of 119869(119875lowast) is

1205823+ 11988611205822+ 1198862120582 + 1198863= 0 (26)

where

1198861= 120583 + 120596 minus 119886

11gt 0

1198862= (120583 minus 119886

11) 120596 minus 120583119886

11minus 12057611988612

= (120596 minus 11988611) 120583 + (120596 + 120576)119882

lowastgt 0

1198863= minus 120596120583119886

11minus 12058312057611988612+ 120572120576119886

13

= 120583 (120596 + 120576)119882lowast+ 120572120576119886

13gt 0

(27)

By calculation we have

1198671= 1198861gt 0

1198672= 11988611198862minus 1198863

= (120583 + 120596 minus 11988611) [(120596 minus 119886

11) 120583 + (120596 + 120576)119882

lowast]

minus 120583 (120596 + 120576)119882lowast

minus 120572120576[1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120596 minus 11988611) (120596 + 120576) minus 120583 (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast)

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120572 + 120575 minus 11988611) (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) gt 0

1198673= 11988631198672gt 0

(28)

By Routh-Hurwitz stability theorem [10] all the three eigen-values of 119869(119875lowast) have negative real parts Thus the endemicequilibrium 119875

lowast is locally asymptotically stable in 119866119900 when

1198770gt 1

Denote the boundary and the interior of 119879 by 120597119879 and 119879119900we also obtain for system (4)

Theorem 5 When 1198770gt 1 system (4) has a unique endemic

equilibrium 119875lowast

= (119878lowast 119864lowast 119868lowast) and it is locally asymptotically

stable in 119879119900 thereinto 119878lowast = 119873lowastminus 119864lowastminus 119868lowast

Now we briefly outline the autonomous convergencetheorem in [13] for proving global stability of the endemicequilibrium 119875

lowastLet 119863 sub R119899 be an open set and let 119909 rarr 119891(119909) isin R119899 be a

1198621 function defined in 119863 We consider the autonomous sys-

tem in R119899 = 119891 (119909) (29)

Let 119909 be an equilibrium of (29) that is 119891(119909) = 0 We recallthat 119909 is said to be globally stable in119863 if it is locally stable andall trajectories in119863 converge to 119909

Assume that the following hypothesis hold

(H1) 119863 is simply connected(H2) there exists a compact absorbing set Γ sub 119863(H3) 119909 is the only equilibrium of (29) and is locally stable

in119863The basic job is to find conditions under which the

global stability of 119909 with respect to 119863 is implied by itslocal stability The difficulty associated with this problem islargely due to the lack of practical tools A new approachto the global stability problem has emerged from a series ofpapers on higher-dimensional generalizations of the criteriaof Bendixson and Dulac for planar systems and on so-calledautonomous convergence theorems First we now introducea definition which will appear in the following context

6 ISRN Applied Mathematics

Definition 6 (see [13]) Suppose system (29) has a periodicsolution 119909 = 119901(119905)with least period120596 gt 0 and orbit 120574 = 119901(119905)

0 le 119905 le 120596 This orbit is orbitally stable if for each 120576 gt 0there exists a 120575 gt 0 such that any solution 119909(119905) for which thedistance of119909(0) from 120574 is less than 120575 remains at a distance lessthan 120576 from 120574 for all 119905 ge 0 It is asymptotically orbitally stable ifthe distance of 119909(119905) from 120574 also tends to zero as 119905 rarr infin Thisorbit 120574 is asymptotically orbitally stable with asymptotic phaseif it is asymptotically orbitally stable and there is a 119887 gt 0 suchthat any solution 119909(119905) for which the distance of 119909(0) from 120574

is less than 119887 satisfies |119909(119905) minus 119901(119905 minus 120591)| rarr 0 as 119905 rarr infin forsome 120591 which may depend on 119909(0)

Theorem 7 (see [14]) A sufficient condition for a period orbit120574 = 119901(119905) 0 le 119905 le 120596 of (29) is asymptotically orbitally stablewith asymptotic phase such that the linear system

1199111015840(119905) = (

120597119891[2]

120597119909(119901 (119905))) 119911 (119905) (30)

is asymptotically stable

Remark 8 Equation (30) is called the second compoundequation of (29) and 120597119891[2]120597119909 is the second compoundmatrixof the Jacobian matrix 120597119891120597119909 of 119891

It is also demonstrated thatTheorem 7 generalizes a classof Poincare for the orbital stability of periodic solutions toplanar autonomous systems

Theorem 9 (see [13]) Under assumptions (H1) (H2) and(H3) 119909 is globally asymptotically stable in 119863 provided that

(H4) the system (29) satisfies a Poincare-Bendixson crite-rion

(H5) a periodic orbit of the system (29) is asymptoticallyorbitally stable

As a matter of fact the condition (H2) is true if and only ifthe system (4) is uniformly persistent in 119879119900

Definition 10 (see [15 16]) System (4) is said to be uniformlypersistent if there exists a constant 120578 isin (0 1) such that anysolution (119878(119905) 119864(119905) 119868(119905)) with initial point (119878(0) 119864(0) 119868(0)) isin119879119900 satisfies

min lim119905rarrinfin

inf 119878 (119905) lim119905rarrinfin

inf 119864 (119905) lim119905rarrinfin

inf 119868 (119905) ge 120578 (31)

Lemma 11 When 1198770gt 1 system (4) is uniformly persistent in

119879119900

Proof Any solution of system (4)which begins from (119878 0 0)

0 le 119878 le Λ120583 always in fact converges at the point 1198750=

(Λ120583 0 0) along the 119878-axis Except the 119878-axis the solution ofsystem (4) which begin from 120597119879 will converge in the region119879119900 Thus 119875

0is the unique 120596-limit point in 120597119879 of system (4)

Let

119880 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (32)

then the time derivative of 119880 along a solution of system (4)gives

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

1198731198781198770

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

ge [12057311198621(Λ

120583)119864 + 120573

21198622(Λ

120583) 119868] (

120583

Λ1198770119878 minus 1)

(33)

When 1198770gt 1 if the trajectories (119878 119864 119868) in 119879119900 sufficiently

converge to1198750 it implies that gt 0That is to say there exists

a neighborhood U(1198750) of 1198750 such that when the trajectories

of system (4) begin from119879119900capU(119875

0) it will come out ofU(119875

0)

Therefore 1198750is not a 120596-limit point of any trajectory in 119879

119900Thus 119872 = (119878 0 0) | 0 le 119878 le Λ120583 is the largest invariantset in 120597119879 of system (4) When 119877

0gt 1 119872 is isolated Also

the invariant set 119882119904(119872) sube 120597119879 where 119882119904(119872) = 119909 isin 119863

119891119899(119909) rarr 119872 as 119899 rarr +infin [15] is the stable set of119872 Accord-

ing to [15 Theorem 41] system (4) is uniformly persist-ent in119879119900 when119877

0gt 1Thus there exists a compact absorbing

subset in 119879119900 for system (4)

Lemma 12 When 1198770gt 1 system (4) satisfies the Poincare-

Bendixson criterion in 119879119900

Proof Because the system (4) is not quasimonotone wecannot verify that the system (4) is competitive by examiningits Jacobian matrix Thus we can replace the system (4) by

1199091015840

1= Λ minus 120583119909

1minus12057311198621(119873)

119873119878 (119873 minus 119909

1minus 1199093)

minus12057321198622 (119873)

119873119878 (119873 minus 119909

1minus 1199092) + 120575119909

3

1199091015840

2=12057311198621(119873)

1198731198781199092+12057321198622(119873)

1198731198781199093minus (120583 + 120576) 119909

2

1199091015840

3= 1205761199092minus (120583 + 120572 + 120575) 119909

3

(34)

Then system (34) has a solution 119906(119905) = (119878(119905) 119864(119905) 119868(119905))Let x = (119909

1 1199092 1199093)119879isin R3 we have

x1015840 = (119861 minus 120583119868) x + 119862 (119878 119864 119868) (35)

where 119868 denotes the 3 times 3 unit matrix 119862(119878 119864 119868) is a functionthat need not concern us and

119861 = (

12057311198621 (119873) + 12057321198622 (119873)

11987311987812057321198622 (119873)

11987311987812057311198621 (119873)

119873119878 + 120575

012057311198621 (119873)

119873119878 minus 120576

12057321198622 (119873)

119873119878

0 120576 minus (120572 + 120575)

)

(36)

The off-diagonal entries in this matrix are nonnegative thusthe system (34) as a whole is quasimonotone [17] Then wecan verify that the system (34) is competitive [18] with respectto the partial ordering defined by the orthant119870 = (119878 119864 119868) isin

R3+ Since 119879119900 is convex system (4) satisfies the Poincare-

Bendixson criterion [10 19] in 119879119900 when 1198770gt 1

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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

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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

4 ISRN Applied Mathematics

Since 1198770lt 1 implies 120573

11198621(Λ120583) lt (120583+120576)minus(120576120596)120573

21198622(Λ

120583) then we get

1205822+ 1205823= 12057311198621(Λ

120583) minus (120583 + 120596 + 120576)

lt minus 120596 minus120576

12059612057321198622(Λ

120583) lt 0

12058221205823= 120596 (120583 + 120576) (1 minus 119877

0) gt 0

(18)

Therefore 1205822and 120582

3have negative real parts Hence 119875

0is

locally asymptotically stableWhen 119877

0= 1 it implies that 120582

21205823= 0 120573

11198621(Λ120583) minus (120583 +

120576) = minus(120576120596)12057321198622(Λ120583) We may as well assume that 120582

2= 0

then 1205823= minus120596minus (120576120596)120573

21198622(Λ120583) The characteristic matrix of

119869(1198750) has three invariable factors 1 1 and 120582(120582 + 120583)(120582 + 120596 +

(120576120596)12057321198622(Λ120583)) Because the elementary factor with respect

to 1205822= 0 is 120582 which is single 119875

0is stable

Constructing a suitable Lyapunov function

119881 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (19)

then the time derivative of 119881 along a solution of system (7)gives

= 1198770

12057311198621 (119873) 119864 + 12057321198622 (119873) 119868

119873(119873 minus 119864 minus 119868)

minus 1198770(120583 + 120576) 119864 +

1205761205732

1205961198622(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

= 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

le 1198770

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868)

minus 12057311198621(119873) 119864 minus 120573

21198622(119873) 119868

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873[(1198770minus 1)119873 minus 119877

0119864 minus 119877

0119868]

(20)

Hence le 0 holds if1198770le 1 Furthermore = 0 if and only

if 119864 = 119868 = 0 Let 119865 = (119864 119868119873) isin 119866 | = 0 = (0 0119873)then the largest compact invariant set in119865 for system (7) is theset (0 0119873) Thus the solution of system (7) satisfies 119864 rarr

0 119868 rarr 0 as 119905 rarr +infin by LaSallersquos Invariance Principle [12]Therefore the limit system of system (7) is

1198641015840= 0

1198681015840= 0

1198731015840= Λ minus 120583119873

(21)

It is obviously known that the equilibrium (0 0 Λ120583) ofsystem (21) is globally asymptotically stable thus the disease-free equilibrium119875

0of system (7) is globally attractive in119866 On

the basis of local stability 1198750is globally asymptotically stable

in 119866 if 1198770le 1 This completes the proof

About system (4) we also obtain

Theorem 2 The unique disease-free equilibrium 1198750= (Λ120583

0 0) of system (4) is globally asymptotically stable in119879 if1198770le 1

and it is unstable if 1198770gt 1

4 Existence and Stability ofthe Endemic Equilibrium

In this section we first discuss the existence and uniquenessof the endemic equilibrium 119875

lowast of system (7) when 1198770gt 1

Whereafter we focus on investigating the local stability of119875lowastWe have to prove that the Jacobian matrix 119869(119875

lowast) is stable

namely all its eigenvalues have negative real parts This isroutinely done by verifying the Routh-Hurwitz conditionsFinally we study the global stability of the endemic equilib-rium 119875

lowast of system (4) with the method of autonomous con-vergence theorem of Li and Muldowney in [13]

The coordinates of the endemic equilibrium (positiveequilibrium) of system (7) are the positive solutions of equa-tions

12057311198621(119873) 119864 + 120573

21198622(119873) 119868

119873(119873 minus 119864 minus 119868) minus (120583 + 120576) 119864 = 0

120576119864 minus (120583 + 120572 + 120575) 119868 = 0

Λ minus 120583119873 minus 120572119868 = 0

(22)

in 119866119900Let 120596 = 120583+120572+120575 by the direct calculation we can get the

following equation of119873 easily as

120593 (119873) =120572120576 + 120583120596 + 120583120576

120572120576[12057311198621(119873) +

120576

12059612057321198622(119873)]

minusΛ (120596 + 120576)

120572120576[12057311198621(119873)

119873+120576

120596

12057321198622(119873)

119873]

minus (120583 + 120576) = 0

(23)

Because 119862119894(119873) (119894 = 1 2) satisfy conditions (i) (ii) and

(iii) thus 120593(119873) is an increasing continuous function and120593(Λ120583) = (120583 + 120576)(119877

0minus 1) When 119873 is sufficiently small

120593(119873) lt 0 If 1198770gt 1 then 120593(Λ120583) gt 0 According to the

zero-point theorem 120593(119873) has the unique positive solution119873lowast in the open interval (0 Λ120583) Then 119868lowast = (Λ minus 120583119873

lowast)120572

119864lowast= (120596120576)119868

lowast Otherwise if 1198770le 1 119873lowast does not exist in (0

Λ120583) Therefore we have the following theorem

Theorem 3 When 1198770gt 1 system (7) has the unique end-

emic equilibrium 119875lowast= (119864lowast 119868lowast 119873lowast) besides the disease-free

equilibrium 1198750in 119866

Theorem 4 When 1198770gt 1 the unique endemic equilibrium

119875lowast is locally asymptotically stable in 119866119900

ISRN Applied Mathematics 5

Proof The Jacobian matrix of system (7) at 119875lowast = (119864lowast 119868lowast 119873lowast)

is

119869 (119875lowast) = (

11988611

11988612

11988613

120576 minus120596 0

0 minus120572 minus120583

) (24)

where

11988611= minus

12057612057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

120596119873lowastminus119882lowastlt 0

11988612=12057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

119873lowastminus119882lowast

11988613= [120573

1(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= 1205731119864lowast[1198621015840

1(119873lowast) minus (119864

lowast+ 119868lowast) (

1198621(119873lowast)

119873lowast)

1015840

]

+ 1205732119868lowast[1198621015840

2(119873lowast) minus (119864

lowast+ 119868lowast) (

1198622(119873lowast)

119873lowast)

1015840

] gt 0

(25)

thereinto119882lowast = (12057311198621(119873lowast)119864lowast+ 12057321198622(119873lowast)119868lowast)119873lowast

Therefore the characteristic equation of 119869(119875lowast) is

1205823+ 11988611205822+ 1198862120582 + 1198863= 0 (26)

where

1198861= 120583 + 120596 minus 119886

11gt 0

1198862= (120583 minus 119886

11) 120596 minus 120583119886

11minus 12057611988612

= (120596 minus 11988611) 120583 + (120596 + 120576)119882

lowastgt 0

1198863= minus 120596120583119886

11minus 12058312057611988612+ 120572120576119886

13

= 120583 (120596 + 120576)119882lowast+ 120572120576119886

13gt 0

(27)

By calculation we have

1198671= 1198861gt 0

1198672= 11988611198862minus 1198863

= (120583 + 120596 minus 11988611) [(120596 minus 119886

11) 120583 + (120596 + 120576)119882

lowast]

minus 120583 (120596 + 120576)119882lowast

minus 120572120576[1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120596 minus 11988611) (120596 + 120576) minus 120583 (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast)

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120572 + 120575 minus 11988611) (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) gt 0

1198673= 11988631198672gt 0

(28)

By Routh-Hurwitz stability theorem [10] all the three eigen-values of 119869(119875lowast) have negative real parts Thus the endemicequilibrium 119875

lowast is locally asymptotically stable in 119866119900 when

1198770gt 1

Denote the boundary and the interior of 119879 by 120597119879 and 119879119900we also obtain for system (4)

Theorem 5 When 1198770gt 1 system (4) has a unique endemic

equilibrium 119875lowast

= (119878lowast 119864lowast 119868lowast) and it is locally asymptotically

stable in 119879119900 thereinto 119878lowast = 119873lowastminus 119864lowastminus 119868lowast

Now we briefly outline the autonomous convergencetheorem in [13] for proving global stability of the endemicequilibrium 119875

lowastLet 119863 sub R119899 be an open set and let 119909 rarr 119891(119909) isin R119899 be a

1198621 function defined in 119863 We consider the autonomous sys-

tem in R119899 = 119891 (119909) (29)

Let 119909 be an equilibrium of (29) that is 119891(119909) = 0 We recallthat 119909 is said to be globally stable in119863 if it is locally stable andall trajectories in119863 converge to 119909

Assume that the following hypothesis hold

(H1) 119863 is simply connected(H2) there exists a compact absorbing set Γ sub 119863(H3) 119909 is the only equilibrium of (29) and is locally stable

in119863The basic job is to find conditions under which the

global stability of 119909 with respect to 119863 is implied by itslocal stability The difficulty associated with this problem islargely due to the lack of practical tools A new approachto the global stability problem has emerged from a series ofpapers on higher-dimensional generalizations of the criteriaof Bendixson and Dulac for planar systems and on so-calledautonomous convergence theorems First we now introducea definition which will appear in the following context

6 ISRN Applied Mathematics

Definition 6 (see [13]) Suppose system (29) has a periodicsolution 119909 = 119901(119905)with least period120596 gt 0 and orbit 120574 = 119901(119905)

0 le 119905 le 120596 This orbit is orbitally stable if for each 120576 gt 0there exists a 120575 gt 0 such that any solution 119909(119905) for which thedistance of119909(0) from 120574 is less than 120575 remains at a distance lessthan 120576 from 120574 for all 119905 ge 0 It is asymptotically orbitally stable ifthe distance of 119909(119905) from 120574 also tends to zero as 119905 rarr infin Thisorbit 120574 is asymptotically orbitally stable with asymptotic phaseif it is asymptotically orbitally stable and there is a 119887 gt 0 suchthat any solution 119909(119905) for which the distance of 119909(0) from 120574

is less than 119887 satisfies |119909(119905) minus 119901(119905 minus 120591)| rarr 0 as 119905 rarr infin forsome 120591 which may depend on 119909(0)

Theorem 7 (see [14]) A sufficient condition for a period orbit120574 = 119901(119905) 0 le 119905 le 120596 of (29) is asymptotically orbitally stablewith asymptotic phase such that the linear system

1199111015840(119905) = (

120597119891[2]

120597119909(119901 (119905))) 119911 (119905) (30)

is asymptotically stable

Remark 8 Equation (30) is called the second compoundequation of (29) and 120597119891[2]120597119909 is the second compoundmatrixof the Jacobian matrix 120597119891120597119909 of 119891

It is also demonstrated thatTheorem 7 generalizes a classof Poincare for the orbital stability of periodic solutions toplanar autonomous systems

Theorem 9 (see [13]) Under assumptions (H1) (H2) and(H3) 119909 is globally asymptotically stable in 119863 provided that

(H4) the system (29) satisfies a Poincare-Bendixson crite-rion

(H5) a periodic orbit of the system (29) is asymptoticallyorbitally stable

As a matter of fact the condition (H2) is true if and only ifthe system (4) is uniformly persistent in 119879119900

Definition 10 (see [15 16]) System (4) is said to be uniformlypersistent if there exists a constant 120578 isin (0 1) such that anysolution (119878(119905) 119864(119905) 119868(119905)) with initial point (119878(0) 119864(0) 119868(0)) isin119879119900 satisfies

min lim119905rarrinfin

inf 119878 (119905) lim119905rarrinfin

inf 119864 (119905) lim119905rarrinfin

inf 119868 (119905) ge 120578 (31)

Lemma 11 When 1198770gt 1 system (4) is uniformly persistent in

119879119900

Proof Any solution of system (4)which begins from (119878 0 0)

0 le 119878 le Λ120583 always in fact converges at the point 1198750=

(Λ120583 0 0) along the 119878-axis Except the 119878-axis the solution ofsystem (4) which begin from 120597119879 will converge in the region119879119900 Thus 119875

0is the unique 120596-limit point in 120597119879 of system (4)

Let

119880 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (32)

then the time derivative of 119880 along a solution of system (4)gives

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

1198731198781198770

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

ge [12057311198621(Λ

120583)119864 + 120573

21198622(Λ

120583) 119868] (

120583

Λ1198770119878 minus 1)

(33)

When 1198770gt 1 if the trajectories (119878 119864 119868) in 119879119900 sufficiently

converge to1198750 it implies that gt 0That is to say there exists

a neighborhood U(1198750) of 1198750 such that when the trajectories

of system (4) begin from119879119900capU(119875

0) it will come out ofU(119875

0)

Therefore 1198750is not a 120596-limit point of any trajectory in 119879

119900Thus 119872 = (119878 0 0) | 0 le 119878 le Λ120583 is the largest invariantset in 120597119879 of system (4) When 119877

0gt 1 119872 is isolated Also

the invariant set 119882119904(119872) sube 120597119879 where 119882119904(119872) = 119909 isin 119863

119891119899(119909) rarr 119872 as 119899 rarr +infin [15] is the stable set of119872 Accord-

ing to [15 Theorem 41] system (4) is uniformly persist-ent in119879119900 when119877

0gt 1Thus there exists a compact absorbing

subset in 119879119900 for system (4)

Lemma 12 When 1198770gt 1 system (4) satisfies the Poincare-

Bendixson criterion in 119879119900

Proof Because the system (4) is not quasimonotone wecannot verify that the system (4) is competitive by examiningits Jacobian matrix Thus we can replace the system (4) by

1199091015840

1= Λ minus 120583119909

1minus12057311198621(119873)

119873119878 (119873 minus 119909

1minus 1199093)

minus12057321198622 (119873)

119873119878 (119873 minus 119909

1minus 1199092) + 120575119909

3

1199091015840

2=12057311198621(119873)

1198731198781199092+12057321198622(119873)

1198731198781199093minus (120583 + 120576) 119909

2

1199091015840

3= 1205761199092minus (120583 + 120572 + 120575) 119909

3

(34)

Then system (34) has a solution 119906(119905) = (119878(119905) 119864(119905) 119868(119905))Let x = (119909

1 1199092 1199093)119879isin R3 we have

x1015840 = (119861 minus 120583119868) x + 119862 (119878 119864 119868) (35)

where 119868 denotes the 3 times 3 unit matrix 119862(119878 119864 119868) is a functionthat need not concern us and

119861 = (

12057311198621 (119873) + 12057321198622 (119873)

11987311987812057321198622 (119873)

11987311987812057311198621 (119873)

119873119878 + 120575

012057311198621 (119873)

119873119878 minus 120576

12057321198622 (119873)

119873119878

0 120576 minus (120572 + 120575)

)

(36)

The off-diagonal entries in this matrix are nonnegative thusthe system (34) as a whole is quasimonotone [17] Then wecan verify that the system (34) is competitive [18] with respectto the partial ordering defined by the orthant119870 = (119878 119864 119868) isin

R3+ Since 119879119900 is convex system (4) satisfies the Poincare-

Bendixson criterion [10 19] in 119879119900 when 1198770gt 1

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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

ISRN Applied Mathematics 5

Proof The Jacobian matrix of system (7) at 119875lowast = (119864lowast 119868lowast 119873lowast)

is

119869 (119875lowast) = (

11988611

11988612

11988613

120576 minus120596 0

0 minus120572 minus120583

) (24)

where

11988611= minus

12057612057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

120596119873lowastminus119882lowastlt 0

11988612=12057321198622(119873lowast) (119873lowastminus 119864lowastminus 119868lowast)

119873lowastminus119882lowast

11988613= [120573

1(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= 1205731119864lowast[1198621015840

1(119873lowast) minus (119864

lowast+ 119868lowast) (

1198621(119873lowast)

119873lowast)

1015840

]

+ 1205732119868lowast[1198621015840

2(119873lowast) minus (119864

lowast+ 119868lowast) (

1198622(119873lowast)

119873lowast)

1015840

] gt 0

(25)

thereinto119882lowast = (12057311198621(119873lowast)119864lowast+ 12057321198622(119873lowast)119868lowast)119873lowast

Therefore the characteristic equation of 119869(119875lowast) is

1205823+ 11988611205822+ 1198862120582 + 1198863= 0 (26)

where

1198861= 120583 + 120596 minus 119886

11gt 0

1198862= (120583 minus 119886

11) 120596 minus 120583119886

11minus 12057611988612

= (120596 minus 11988611) 120583 + (120596 + 120576)119882

lowastgt 0

1198863= minus 120596120583119886

11minus 12058312057611988612+ 120572120576119886

13

= 120583 (120596 + 120576)119882lowast+ 120572120576119886

13gt 0

(27)

By calculation we have

1198671= 1198861gt 0

1198672= 11988611198862minus 1198863

= (120583 + 120596 minus 11988611) [(120596 minus 119886

11) 120583 + (120596 + 120576)119882

lowast]

minus 120583 (120596 + 120576)119882lowast

minus 120572120576[1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) + 119882

lowast

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120596 minus 11988611) (120596 + 120576) minus 120583 (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast)

= (120596 minus 11988611) 1205832+ (120596 minus 119886

11)2120583

+ [(120583 + 120572 + 120575 minus 11988611) (120596 + 120576) minus 120572120576]119882

lowast

minus 120572120576 [1205731(1198621(119873lowast)

119873lowast)

1015840

119864lowast+ 1205732(1198622(119873lowast)

119873lowast)

1015840

119868lowast]

times (119873lowastminus 119864lowastminus 119868lowast) gt 0

1198673= 11988631198672gt 0

(28)

By Routh-Hurwitz stability theorem [10] all the three eigen-values of 119869(119875lowast) have negative real parts Thus the endemicequilibrium 119875

lowast is locally asymptotically stable in 119866119900 when

1198770gt 1

Denote the boundary and the interior of 119879 by 120597119879 and 119879119900we also obtain for system (4)

Theorem 5 When 1198770gt 1 system (4) has a unique endemic

equilibrium 119875lowast

= (119878lowast 119864lowast 119868lowast) and it is locally asymptotically

stable in 119879119900 thereinto 119878lowast = 119873lowastminus 119864lowastminus 119868lowast

Now we briefly outline the autonomous convergencetheorem in [13] for proving global stability of the endemicequilibrium 119875

lowastLet 119863 sub R119899 be an open set and let 119909 rarr 119891(119909) isin R119899 be a

1198621 function defined in 119863 We consider the autonomous sys-

tem in R119899 = 119891 (119909) (29)

Let 119909 be an equilibrium of (29) that is 119891(119909) = 0 We recallthat 119909 is said to be globally stable in119863 if it is locally stable andall trajectories in119863 converge to 119909

Assume that the following hypothesis hold

(H1) 119863 is simply connected(H2) there exists a compact absorbing set Γ sub 119863(H3) 119909 is the only equilibrium of (29) and is locally stable

in119863The basic job is to find conditions under which the

global stability of 119909 with respect to 119863 is implied by itslocal stability The difficulty associated with this problem islargely due to the lack of practical tools A new approachto the global stability problem has emerged from a series ofpapers on higher-dimensional generalizations of the criteriaof Bendixson and Dulac for planar systems and on so-calledautonomous convergence theorems First we now introducea definition which will appear in the following context

6 ISRN Applied Mathematics

Definition 6 (see [13]) Suppose system (29) has a periodicsolution 119909 = 119901(119905)with least period120596 gt 0 and orbit 120574 = 119901(119905)

0 le 119905 le 120596 This orbit is orbitally stable if for each 120576 gt 0there exists a 120575 gt 0 such that any solution 119909(119905) for which thedistance of119909(0) from 120574 is less than 120575 remains at a distance lessthan 120576 from 120574 for all 119905 ge 0 It is asymptotically orbitally stable ifthe distance of 119909(119905) from 120574 also tends to zero as 119905 rarr infin Thisorbit 120574 is asymptotically orbitally stable with asymptotic phaseif it is asymptotically orbitally stable and there is a 119887 gt 0 suchthat any solution 119909(119905) for which the distance of 119909(0) from 120574

is less than 119887 satisfies |119909(119905) minus 119901(119905 minus 120591)| rarr 0 as 119905 rarr infin forsome 120591 which may depend on 119909(0)

Theorem 7 (see [14]) A sufficient condition for a period orbit120574 = 119901(119905) 0 le 119905 le 120596 of (29) is asymptotically orbitally stablewith asymptotic phase such that the linear system

1199111015840(119905) = (

120597119891[2]

120597119909(119901 (119905))) 119911 (119905) (30)

is asymptotically stable

Remark 8 Equation (30) is called the second compoundequation of (29) and 120597119891[2]120597119909 is the second compoundmatrixof the Jacobian matrix 120597119891120597119909 of 119891

It is also demonstrated thatTheorem 7 generalizes a classof Poincare for the orbital stability of periodic solutions toplanar autonomous systems

Theorem 9 (see [13]) Under assumptions (H1) (H2) and(H3) 119909 is globally asymptotically stable in 119863 provided that

(H4) the system (29) satisfies a Poincare-Bendixson crite-rion

(H5) a periodic orbit of the system (29) is asymptoticallyorbitally stable

As a matter of fact the condition (H2) is true if and only ifthe system (4) is uniformly persistent in 119879119900

Definition 10 (see [15 16]) System (4) is said to be uniformlypersistent if there exists a constant 120578 isin (0 1) such that anysolution (119878(119905) 119864(119905) 119868(119905)) with initial point (119878(0) 119864(0) 119868(0)) isin119879119900 satisfies

min lim119905rarrinfin

inf 119878 (119905) lim119905rarrinfin

inf 119864 (119905) lim119905rarrinfin

inf 119868 (119905) ge 120578 (31)

Lemma 11 When 1198770gt 1 system (4) is uniformly persistent in

119879119900

Proof Any solution of system (4)which begins from (119878 0 0)

0 le 119878 le Λ120583 always in fact converges at the point 1198750=

(Λ120583 0 0) along the 119878-axis Except the 119878-axis the solution ofsystem (4) which begin from 120597119879 will converge in the region119879119900 Thus 119875

0is the unique 120596-limit point in 120597119879 of system (4)

Let

119880 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (32)

then the time derivative of 119880 along a solution of system (4)gives

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

1198731198781198770

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

ge [12057311198621(Λ

120583)119864 + 120573

21198622(Λ

120583) 119868] (

120583

Λ1198770119878 minus 1)

(33)

When 1198770gt 1 if the trajectories (119878 119864 119868) in 119879119900 sufficiently

converge to1198750 it implies that gt 0That is to say there exists

a neighborhood U(1198750) of 1198750 such that when the trajectories

of system (4) begin from119879119900capU(119875

0) it will come out ofU(119875

0)

Therefore 1198750is not a 120596-limit point of any trajectory in 119879

119900Thus 119872 = (119878 0 0) | 0 le 119878 le Λ120583 is the largest invariantset in 120597119879 of system (4) When 119877

0gt 1 119872 is isolated Also

the invariant set 119882119904(119872) sube 120597119879 where 119882119904(119872) = 119909 isin 119863

119891119899(119909) rarr 119872 as 119899 rarr +infin [15] is the stable set of119872 Accord-

ing to [15 Theorem 41] system (4) is uniformly persist-ent in119879119900 when119877

0gt 1Thus there exists a compact absorbing

subset in 119879119900 for system (4)

Lemma 12 When 1198770gt 1 system (4) satisfies the Poincare-

Bendixson criterion in 119879119900

Proof Because the system (4) is not quasimonotone wecannot verify that the system (4) is competitive by examiningits Jacobian matrix Thus we can replace the system (4) by

1199091015840

1= Λ minus 120583119909

1minus12057311198621(119873)

119873119878 (119873 minus 119909

1minus 1199093)

minus12057321198622 (119873)

119873119878 (119873 minus 119909

1minus 1199092) + 120575119909

3

1199091015840

2=12057311198621(119873)

1198731198781199092+12057321198622(119873)

1198731198781199093minus (120583 + 120576) 119909

2

1199091015840

3= 1205761199092minus (120583 + 120572 + 120575) 119909

3

(34)

Then system (34) has a solution 119906(119905) = (119878(119905) 119864(119905) 119868(119905))Let x = (119909

1 1199092 1199093)119879isin R3 we have

x1015840 = (119861 minus 120583119868) x + 119862 (119878 119864 119868) (35)

where 119868 denotes the 3 times 3 unit matrix 119862(119878 119864 119868) is a functionthat need not concern us and

119861 = (

12057311198621 (119873) + 12057321198622 (119873)

11987311987812057321198622 (119873)

11987311987812057311198621 (119873)

119873119878 + 120575

012057311198621 (119873)

119873119878 minus 120576

12057321198622 (119873)

119873119878

0 120576 minus (120572 + 120575)

)

(36)

The off-diagonal entries in this matrix are nonnegative thusthe system (34) as a whole is quasimonotone [17] Then wecan verify that the system (34) is competitive [18] with respectto the partial ordering defined by the orthant119870 = (119878 119864 119868) isin

R3+ Since 119879119900 is convex system (4) satisfies the Poincare-

Bendixson criterion [10 19] in 119879119900 when 1198770gt 1

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 2008

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Applied Mathematics

Definition 6 (see [13]) Suppose system (29) has a periodicsolution 119909 = 119901(119905)with least period120596 gt 0 and orbit 120574 = 119901(119905)

0 le 119905 le 120596 This orbit is orbitally stable if for each 120576 gt 0there exists a 120575 gt 0 such that any solution 119909(119905) for which thedistance of119909(0) from 120574 is less than 120575 remains at a distance lessthan 120576 from 120574 for all 119905 ge 0 It is asymptotically orbitally stable ifthe distance of 119909(119905) from 120574 also tends to zero as 119905 rarr infin Thisorbit 120574 is asymptotically orbitally stable with asymptotic phaseif it is asymptotically orbitally stable and there is a 119887 gt 0 suchthat any solution 119909(119905) for which the distance of 119909(0) from 120574

is less than 119887 satisfies |119909(119905) minus 119901(119905 minus 120591)| rarr 0 as 119905 rarr infin forsome 120591 which may depend on 119909(0)

Theorem 7 (see [14]) A sufficient condition for a period orbit120574 = 119901(119905) 0 le 119905 le 120596 of (29) is asymptotically orbitally stablewith asymptotic phase such that the linear system

1199111015840(119905) = (

120597119891[2]

120597119909(119901 (119905))) 119911 (119905) (30)

is asymptotically stable

Remark 8 Equation (30) is called the second compoundequation of (29) and 120597119891[2]120597119909 is the second compoundmatrixof the Jacobian matrix 120597119891120597119909 of 119891

It is also demonstrated thatTheorem 7 generalizes a classof Poincare for the orbital stability of periodic solutions toplanar autonomous systems

Theorem 9 (see [13]) Under assumptions (H1) (H2) and(H3) 119909 is globally asymptotically stable in 119863 provided that

(H4) the system (29) satisfies a Poincare-Bendixson crite-rion

(H5) a periodic orbit of the system (29) is asymptoticallyorbitally stable

As a matter of fact the condition (H2) is true if and only ifthe system (4) is uniformly persistent in 119879119900

Definition 10 (see [15 16]) System (4) is said to be uniformlypersistent if there exists a constant 120578 isin (0 1) such that anysolution (119878(119905) 119864(119905) 119868(119905)) with initial point (119878(0) 119864(0) 119868(0)) isin119879119900 satisfies

min lim119905rarrinfin

inf 119878 (119905) lim119905rarrinfin

inf 119864 (119905) lim119905rarrinfin

inf 119868 (119905) ge 120578 (31)

Lemma 11 When 1198770gt 1 system (4) is uniformly persistent in

119879119900

Proof Any solution of system (4)which begins from (119878 0 0)

0 le 119878 le Λ120583 always in fact converges at the point 1198750=

(Λ120583 0 0) along the 119878-axis Except the 119878-axis the solution ofsystem (4) which begin from 120597119879 will converge in the region119879119900 Thus 119875

0is the unique 120596-limit point in 120597119879 of system (4)

Let

119880 = 1198770119864 +

1205732

1205961198622(Λ

120583) 119868 (32)

then the time derivative of 119880 along a solution of system (4)gives

=12057311198621(119873) 119864 + 120573

21198622(119873) 119868

1198731198781198770

minus 12057311198621(Λ

120583)119864 minus 120573

21198622(Λ

120583) 119868

ge [12057311198621(Λ

120583)119864 + 120573

21198622(Λ

120583) 119868] (

120583

Λ1198770119878 minus 1)

(33)

When 1198770gt 1 if the trajectories (119878 119864 119868) in 119879119900 sufficiently

converge to1198750 it implies that gt 0That is to say there exists

a neighborhood U(1198750) of 1198750 such that when the trajectories

of system (4) begin from119879119900capU(119875

0) it will come out ofU(119875

0)

Therefore 1198750is not a 120596-limit point of any trajectory in 119879

119900Thus 119872 = (119878 0 0) | 0 le 119878 le Λ120583 is the largest invariantset in 120597119879 of system (4) When 119877

0gt 1 119872 is isolated Also

the invariant set 119882119904(119872) sube 120597119879 where 119882119904(119872) = 119909 isin 119863

119891119899(119909) rarr 119872 as 119899 rarr +infin [15] is the stable set of119872 Accord-

ing to [15 Theorem 41] system (4) is uniformly persist-ent in119879119900 when119877

0gt 1Thus there exists a compact absorbing

subset in 119879119900 for system (4)

Lemma 12 When 1198770gt 1 system (4) satisfies the Poincare-

Bendixson criterion in 119879119900

Proof Because the system (4) is not quasimonotone wecannot verify that the system (4) is competitive by examiningits Jacobian matrix Thus we can replace the system (4) by

1199091015840

1= Λ minus 120583119909

1minus12057311198621(119873)

119873119878 (119873 minus 119909

1minus 1199093)

minus12057321198622 (119873)

119873119878 (119873 minus 119909

1minus 1199092) + 120575119909

3

1199091015840

2=12057311198621(119873)

1198731198781199092+12057321198622(119873)

1198731198781199093minus (120583 + 120576) 119909

2

1199091015840

3= 1205761199092minus (120583 + 120572 + 120575) 119909

3

(34)

Then system (34) has a solution 119906(119905) = (119878(119905) 119864(119905) 119868(119905))Let x = (119909

1 1199092 1199093)119879isin R3 we have

x1015840 = (119861 minus 120583119868) x + 119862 (119878 119864 119868) (35)

where 119868 denotes the 3 times 3 unit matrix 119862(119878 119864 119868) is a functionthat need not concern us and

119861 = (

12057311198621 (119873) + 12057321198622 (119873)

11987311987812057321198622 (119873)

11987311987812057311198621 (119873)

119873119878 + 120575

012057311198621 (119873)

119873119878 minus 120576

12057321198622 (119873)

119873119878

0 120576 minus (120572 + 120575)

)

(36)

The off-diagonal entries in this matrix are nonnegative thusthe system (34) as a whole is quasimonotone [17] Then wecan verify that the system (34) is competitive [18] with respectto the partial ordering defined by the orthant119870 = (119878 119864 119868) isin

R3+ Since 119879119900 is convex system (4) satisfies the Poincare-

Bendixson criterion [10 19] in 119879119900 when 1198770gt 1

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 2008

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Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Applied Mathematics 7

Lemma 13 When 1198770gt 1 the trajectory of any nonconstant

periodic solution 119901(119905) = (119878(119905) 119864(119905) 119868(119905)) to system (4) ifit exists is asymptotically orbitally stable with asymptoticallyphase

Proof Suppose that the period solution 119901(119905) is periodic ofleast period 120591 gt 0 such that (119878(0) 119864(0) 119868(0)) isin 119879

119900 Theperiod orbit is 119901 = 119901(119905) 0 le 119905 le 120591 The Jacobian matrix ofsystem (4) at (119878 119864 119868) is given by

J (119901 (119905)) =(

minus120583 minusM minus N minus12057311198621(119873)

119873119878 minus N 120575 minus

12057321198622(119873)

119873119878 minus N

M + N 12057311198621 (119873)

119873119878 + N minus (120583 + 120576)

12057321198622 (119873)

119873119878 + N

0 120576 minus (120583 + 120572 + 120575)

) (37)

where

M =12057311198621(119873)

119873119864 +

12057321198622(119873)

119873119868 ge 0

N = (12057311198621(119873)

119873)

1015840

119878119864 + (12057321198622(119873)

119873)

1015840

119878119868 le 0

(38)

Then the second compound matrix of J(119901(119905)) is

J[2] (119901 (119905)) =((

(

minus(2120583 + 120576) +12057311198621 (119873)

119873119878 minusM 120573

21198622 (119873)

119873119878 + N minus120575 +

12057321198622 (119873)

119873119878 + N

120576 minus (120583 + 120596) minusM minus N minus12057311198621(119873)

119873119878 minus N

0 M + N minus (120583 + 120576 + 120596) +12057311198621(119873)

119873119878 + N

))

)

(39)

whose definition can be found in the appendixFurthermore the second compound system of (4) is the

following periodic linear system

1198831015840= minus [Mminus

12057311198621(119873)

119873119878+2120583 + 120576]119883 + [

12057321198622(119873)

119873119878 + N]119884

+ [12057321198622(119873)

119873119878 + N minus 120575]119885

1198841015840=120576119883 minus (M + N + 120583 + 120596)119884 minus [

12057311198621(119873)

119873119878 + N]119885

1198851015840= (M + N) 119884 + [

12057311198621 (119873)

119873119878 + N minus 120583 minus 120576 minus 120596]119885

(40)

Let (119909 119910 119911) be a vector in R3 We choose a vector normin R3 as

1003817100381710038171003817(119909 (119905) 119910 (119905) 119911 (119905))1003817100381710038171003817 = sup |119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816 + |119911 (119905)| (41)

Let

119871 (119905) = sup|119883 (119905)| 119864 (119905)

119868 (119905)(|119884 (119905)| + |119885 (119905)|) (42)

When 1198770gt 1 system (4) is uniformly persistent in 119879119900 Then

there exists constant 119896 gt 0 such that

119871 (119905) ge 119896 sup |119883 (119905)| |119884 (119905)| + |119885 (119905)| (43)

for all (119883(119905) 119884(119905) 119885(119905)) isin R3By direct calculations we can obtain the following differ-

ential inequalities

119863+ |119883 (119905)| le minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+ [12057321198622(119873)

119873119878 + N] |119884 (119905)|

+ [12057321198622(119873)

119873119878 + N minus 120575] |119885 (119905)|

le minus [M minus12057311198621 (119873)

119873119878 + 2120583 + 120576] |119883 (119905)|

+119868

119864[12057321198622(119873)

119873119878 + N] 119864

119868(|119884| + |119885|)

(44)

8 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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 ISRN Applied Mathematics

119863+ |119884 (119905)| le 120576 |119883 (119905)| minus (M + N + 120583 + 120596) |119884 (119905)|

minus [12057311198621 (119873)

119873119878 + N] |119885 (119905)|

(45)

119863+ |119885 (119905)| le (M + N) |119884 (119905)|

+ [12057311198621 (119873)

119873119878 + N minus 120583 minus 120596] |119885 (119905)|

(46)

Using (45) and (46) we have

119863+

119864

119868(|119884 (119905)| + |119885 (119905)|)

= (1198641015840

119864minus1198681015840

119868)119864

119868(|119884 (119905)| + |119885 (119905)|)

+119864

119868(119863+ |119884 (119905)| + 119863+ |119885 (119905)|)

le120576119864

119868|119883 (119905)| + (

1198641015840

119864minus1198681015840

119868minus 120583 minus 120596)

119864

119868

times (|119884 (119905)| + |119885 (119905)|)

(47)

Therefore we obtain from (44) and (47)

119863+119871 (119905) le sup 119892

1(119905) 1198922(119905) 119871 (119905) (48)

where

1198921(119905) = minus [M minus

12057311198621 (119873)

119873119878 + 2120583 + 120576]

+119868

119864[12057321198622(119873)

119873119878 + N]

(49)

1198922 (119905) =

120576119864

119868+1198641015840

119864minus1198681015840

119868minus 120583 minus 120596 (50)

The system (4) implies

12057321198622(119873) 119878119868

119873119864=1198641015840

119864minus12057311198621(119873) 119878

119873+ 120583 + 120576 (51)

120576119864

119868=1198681015840

119868+ 120596 (52)

Substituting (51) into (49) and (52) into (50) we have

1198921 (119905) =

1198641015840

119864minus 120583 minusM +

119868

119864N le

1198641015840

119864minus 120583

1198922(119905) =

1198641015840

119864minus 120583

(53)

Thus

sup 1198921(119905) 1198922(119905) le

1198641015840

119864minus 120583

119863+119871 (119905) le (

1198641015840

119864minus 120583)119871 (119905)

int

120591

0

sup 1198921 (119905) 1198922 (119905) 119889119905 le int

120591

0

(1198641015840

119864minus 120583)119889119905

= ln119864 (119905)|1205910minus 120583120591 = minus120583120591 lt 0

(54)

which implies that 119871(119905) rarr 0 as 119905 rarr infin and in turn that(119883(119905) 119884(119905) 119885(119905)) rarr 0 as 119905 rarr infin Aa a result the secondcompound system (40) is asymptotically stable Thus theperiodic solution (119878(119905) 119864(119905) 119868(119905)) is asymptotically orbitallystable with asymptotically phase

By Lemmas 11ndash13 we know that system (4) is satisfiedwith every condition of Theorem 9 thus we can obtain thefollowing

Theorem 14 If 1198770gt 1 the unique endemic equilibrium 119875

lowast

=

(119878lowast 119864lowast 119868lowast) of system (4) is globally asymptotically stable in 119879119900

Theorem 15 If 1198770gt 1 the unique endemic equilibrium 119875

lowast=

(119864lowast 119868lowast 119873lowast) of system (7) is globally asymptotically stable in

119866119900

5 Example and Numerical Simulation

In this paper we considered an SEIS model with saturationincidence Now we give the number simulations for system(4) (see Figures 2 and 3)

Choose 1198621(119873) = (101198733)(1 + 101198733 + radic1 + 201198733)

and 1198622(119873) = (1198733)(1 + 1198733 + radic1 + 21198733) Assume that

Λ = 06 120583 = 005 120576 = 015 120575 = 015 1205732= 036 and 120572 = 01

We choose randomly six initial values (1 22 57) (51 2213) (33 18 27) (44 21 06) (08 54 22) and (54 1316) in 119879119900 = (119878 119864 119868) isin R3

+| 0 lt 119878 + 119864 + 119868 lt 12

If 1205731= 01 119877

0= 085 We give the trajectory plot and its

tridimensional figure by Matlab softwareIf 1205731= 036 119877

0= 14 We give the trajectory plot and its

tridimensional figure by Matlab software

6 Discussion

In this paper we present a complete mathematical analysisfor the global stability problem at the equilibria of an SEISepidemic model with saturation incidence The basic repro-ductive number 119877

0is obtained as a sharp threshold parame-

ter which represents the average number of secondary infec-tions from a single exposed host and infectious host If 119877

0le

1 the disease-free equilibrium 1198750is globally asymptotically

stable in the feasible region119879 by Lyapunov function and thusthe disease always dies out If119877

0gt 1 the unique disease equi-

librium 119875lowast is globally asymptotically stable in 119879119900 so that the

disease if initially present will persist at the unique endemic

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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

ISRN Applied Mathematics 9

0 50 100 150 200 2500

2

4

6

8

10

12

119878(119905)

119864(119905)

119868(119905)

119905

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

119905

119878(119905)

119864(119905)

119868(119905)

(b)

Figure 2 Movement paths of 119878 119864 and 119868 as functions of time 119905 For (a) we have 1198770= 085 and 119875

0is globally stableThe disease is extinct For

(b) we have 1198770= 14 and 119875lowast is globally stable The disease spreads to be endemic

05

1015

02

460

1

2

3

4

5

6

119878119864

119868

(a)

0 2 4 68

10

02

4

605

1

15

2

25

3

35

119878119864

119868

(b)

Figure 3 The graph of the trajectory in (119878 119864 119868)-space (a) and (b) correspond with Figures 2(a) and 2(b) respectively

equilibrium levelThe global stability of119875lowast inmodel is provedusing a geometrical approach in [13] We expect that theseapproaches can be applied to solve global stability problemsin many other epidemic models

Appendix

Compound Matrices

Let 119860 be a linear operator on R119899 and also denote its matrixrepresentation with respect to the standard basis of R119899Let and2R119899 denote the exterior product of R119899 119860 induces

canonically a linear operator 119860[2] on and2R119899 for 119906

1 1199062isin R119899

define

119860[2](1199061and 1199062) = 119860 (119906

1) and 1199062+ 1199061and 119860 (119906

2) (A1)

and extend the definition over and2R119899 by linearity The matrixrepresentation of 119860[2] with respect to the canonical basis inand2R119899 is called the second additive compound matrix of 119860

This is an (1198992)times(119899

2)matrix and satisfies the property (119860+119861)[2] =

119860[2]+119861[2]The entries in119860[2] are linear relations of those in119860

Let 119860 = (119886119894119895) For any integer 119894 = 1 2 (

119899

2) let (119894) = (119894

1 1198942)

be the 119894th member in the lexicographic ordering of integer

10 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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 ISRN Applied Mathematics

pairs such that 1 le 1198941lt 1198942le 119898 Then the entry in the 119895th

column of 119885 = 119860[2] is

119911119894119895=

11988611989411198941

+ 11988611989421198942

if (119894) = (119895)

(minus1)119903+119904119886119894119904119895119903

if exactly one entry 119894119904of

(119894) does not occur in (119895)

and 119895119903does not occur in (119894)

0 if (119894) differs from (119895) in twoor more entries

(A2)

For any integer 1 le 119896 le 119899 the 119896th additive compound matrix119860119896 of 119860 is defined canonically For detailed discussions of

compound matrices and their properties we refer the readerto [20] A comprehensive survey on compound matrices andtheir relations to differential equations is given in [20] For119899 = 2 3 and 4 the second additive compound matrix 119860[2] ofan 119899 times 119899matrix 119860 = (119886

119894119895) is respectively

119899 = 2 11988611+ 11988622

119899 = 3 (

11988611+ 11988622

11988623

minus11988613

11988632

11988611+ 11988633

11988612

minus11988631

11988621

11988622+ 11988633

)

119899 = 4 (

(

11988611+ 11988622

11988623

minus11988624

minus11988613

minus11988614

0

11988632

11988611+ 11988633

11988634

11988612

0 minus11988614

11988642

11988643

11988611+ 11988644

0 11988612

11988613

minus11988631

11988621

0 11988622+ 11988633

11988634

minus11988624

minus11988641

0 11988621

11988643

11988622+ 11988644

11988623

0 minus11988641

11988631

minus11988642

11988632

11988633+ 11988644

)

)

(A3)

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (10531030)There is no financial conflictof interests between the authors and the commercial identityAlso it partially contains the results obtained in [21] anddevelops the ideas formulated in [21]

References

[1] S Busenberg and P van den Driessche ldquoAnalysis of a diseasetransmission model in a population with varying sizerdquo Journalof Mathematical Biology vol 28 no 3 pp 257ndash270 1990

[2] M Y Li and J S Muldowney ldquoGlobal stability for the SEIRmodel in epidemiologyrdquoMathematical Biosciences vol 125 no2 pp 155ndash164 1995

[3] H Guo and M Y Li ldquoGlobal dynamics of a staged progressionmodel for infectious diseasesrdquo Mathematical Biosciences andEngineering vol 3 no 3 pp 513ndash525 2006

[4] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinctincidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[5] L Guihua and J Zhen ldquoGlobal stability of an SEI epidemicmodel with general contact raterdquo Chaos Solitons and Fractalsvol 23 no 3 pp 997ndash1004 2005

[6] LGuihua and J Zhen ldquoGlobal stability of an SEI epidemicmod-elrdquoChaos Solitons and Fractals vol 21 no 4 pp 925ndash931 2004

[7] B Mukhopadhyay and R Bhattacharyya ldquoAnalysis of a spa-tially extended nonlinear SEIS epidemic model with distinct

incidence for exposed and infectivesrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 585ndash598 2008

[8] J Yuan and Z Yang ldquoGlobal dynamics of an SEI model withacute and chronic stagesrdquo Journal of Computational and AppliedMathematics vol 213 no 2 pp 465ndash476 2008

[9] J A PHeesterbeek and J A JMetz ldquoThe saturating contact ratein marriage- and epidemic modelsrdquo Journal of MathematicalBiology vol 31 no 5 pp 529ndash539 1993

[10] J K Hale Ordinary Differential Equations Wiley-InterscienceNew York NY USA 1969

[11] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[12] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer New York NY USA 2nd edition2003

[13] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[14] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[15] J Hofbauer and J W-H So ldquoUniform persistence and repellorsfor mapsrdquo Proceedings of the American Mathematical Societyvol 107 no 4 pp 1137ndash1142 1989

[16] G J Butler H I Freedman and P Waltman ldquoUniformly per-sistent systemsrdquo Proceedings of the American MathematicalSociety vol 96 no 3 pp 425ndash430 1986

[17] G Herzog and R Redheffer ldquoNonautonomous SEIRS andThron models for epidemiology and cell biologyrdquo NonlinearAnalysis Real World Applications vol 5 no 1 pp 33ndash44 2004

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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

ISRN Applied Mathematics 11

[18] H L Smith ldquoSystems of ordinary differential equations whichgenerate an order preserving flow A survey of resultsrdquo SIAMReview vol 30 no 1 pp 87ndash113 1988

[19] M W Hirsch ldquoSystems of differential equations that are com-petitive or cooperative IV Structural stability in three-dimen-sional systemsrdquo SIAM Journal onMathematical Analysis vol 21no 5 pp 1225ndash1234 1990

[20] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal of Mathematics vol 20no 4 pp 857ndash872 1990

[21] H Zhang andWX Xu ldquoAsymptotic stability analysis of an SEISepidemic model with infectious force in both latent period andinfected periodrdquo Journal of Shaanxi Normal University vol 36no 6 pp 5ndash9 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