Research Article An Impulse Dynamic Model for Computer Wormsdownloads.hindawi.com/journals/aaa/2013/286209.pdf · replicate itself and spread from one computer to others including

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 286209 8 pageshttpdxdoiorg1011552013286209

Research ArticleAn Impulse Dynamic Model for Computer Worms

Chunming Zhang Yun Zhao and Yingjiang Wu

School of Information Engineering Guangdong Medical College Dongguan 523808 China

Correspondence should be addressed to Chunming Zhang chunfei2002163com

Received 3 May 2013 Accepted 2 June 2013

Academic Editor Luca Guerrini

Copyright copy 2013 Chunming Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A worm spread model concerning impulsive control strategy is proposed and analyzed We prove that there exists a globallyattractive virus-free periodic solution when the vaccination rate is larger than 120579

1 Moreover we show that the system is uniformly

persistent if the vaccination rate is less than 1205791 Some numerical simulations are also given to illustrate our main results

1 Introduction

Computer virus is a kind of computer program that canreplicate itself and spread from one computer to othersincluding viruses worms and trojan horses Worms usesystem vulnerability to search and attack computers Ashardware and software technologies develop and computernetworks become an essential tool for daily life worms startto be a major threat In June 2010 the Belarusian securityfirm Virus Block Ada discovered deadly Stuxnet worm TheStuxnet worm is the first known example of a cyber-weaponthat is designed not just to steal and manipulate data but toattack a processing system and cause physical damage TheStuxnet worm is the first cyber-attack of its kind and hasinfected thousands of computer systems worldwide

Consequently the trial on better understanding thewormpropagation dynamics is an important matter for improvingthe safety and reliability in computer systems and networksSimilar to the biological viruses there are two ways to studythis problem microscopic and macroscopic Following amacroscopic approach since [1 2] took the first step towardsmodeling the spread behavior of worms much effort hasbeen done in the area of developing a mathematical modelfor the worms propagation [3ndash13] These models provide areasonable qualitative understanding of the conditions underwhich viruses spread much faster than others and why

In [7] the authors investigated a differential SEIR modelby making the following assumptions (Figure 1)

A population size119873(119905) that is the total nodes at any time119905 in the computer network is partitioned into subclasses of

nodes which are susceptible exposed (infected but not yetinfectious) infectious and recovered with sizes denoted by119878(119905) 119864(119905) 119868(119905) and 119877(119905) respectively

One has

119873(119905) = 119878 (119905) + 119864 (119905) + 119868 (119905) + 119877 (119905)

1198781015840(119905) = 119887 minus 120582119868119878 minus 119901119887119864 minus 119902119887119868 minus 119889119878 + 120577119877

1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864

1198681015840(119905) = 120576119864 minus 120574119868 minus 119889119868 minus 120578119868

1198771015840(119905) = 120574119868 minus 120577119877 minus 119889119877

(1)

where 119887 119889 and 120582 are positive constants and 120576 120578 120574 120577 arenonnegative constants The constant 119887 is the recruitment rateof susceptible nodes to the computer network 119889 is the percapita natural mortality rate (ie the crashing of nodes dueto the reason other than the attack of worms) 120576 is the rateconstant for nodes leaving the exposed class 119864 for infectiveclass 119868 120574 is the rate constant for nodes leaving the infectiveclass 119868 for recovered class 119877 120578 is the disease related death rate(ie crashing of nodes due to the attack of worms) in the class119868 and 120577 is the rate constant for nodes becoming susceptibleagain after recovering

In the SEIRS model the flow is from class 119878 to class 119864class 119864 to class 119868 class 119868 to class 119877 and again class 119877 toclass 119878 For the vertical transformation we assume that afraction119901 and a fraction 119902 of the new nodes from the exposedand the infectious classes respectively are introduced into

2 Abstract and Applied Analysis

S I R120582IS 120574I

120577R

dS dI dR

b 120576E

pbE

E

dE

qbI+

Figure 1 Original model

the exposed class 119864 Consequently the birth flux into theexposed class is given by 119901119887119864 + 119902119887119868 and the birth flux intothe susceptible class is given by 119887 minus 119901119887119864 minus 119902119887119868

As we know antivirus software is a kind of computerprogramwhich can detect and eliminate knownwormThereare two common methods to detect worms using a list ofworm signature definition and using a heuristic algorithm tofindworm based on common behaviors It has been observedthat it does not always work in detecting a novel worm byusing the heuristic algorithm On the other hand obviouslyit is impossible for antivirus software to find a new wormsignature definition on the dated list So to keep the antivirussoftware in high efficiency it is important to ensure thatit is updated Based on the previous facts we propose animpulsive system to model the process of periodic installingor updating antivirus software on susceptible computers atfixed time for controlling the spread of worm

Based on the previous facts we propose the followingassumptions

(H1) the antivirus software is installed or updated at time119905 = 119896120591 (119896 isin 119873) where 120591 is the period of the impulsiveeffect

(H2) 119878 computers are successfully vaccinated from 119878 classto 119877 class with rate 120579 (0 lt 120579 lt 1)

According to the previous assumptions (H1)-(H2) andfor the reason of simplicity we propose the following model(Figure 2)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864


1198771015840(119905) = 120574119868 minus 120577119877 minus 119889119877

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119868 (119905+) = 119868 (119905)

119877 (119905+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896120591 119896 isin 119885+

(2)

The total population size 119873(119905) can be determined by 119873(119905) =119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) to form the differential equation

1198731015840(119905) = 119887 minus 119889119873 (119905) minus 120578119868 (119905) (3)

S I R120582IS 120574I

120577R

dS dI dR

b 120576E

pbE

E

dE

120579S

qbI+

Figure 2 Impulse model

which is derived by adding the equations in system (1) Thusthe total population size 119873 may vary in time From (2) wehave

119887 minus (119889 + 120578)119873 (119905) le 1198731015840(119905) le 119887 minus 119889119873 (119905) (4)

It follows that119887

(119889 + 120578)le lim119909rarrinfin

inf 119873(119905) le lim119909rarrinfin

sup119873(119905) le119887

119889

(5)

The system (2) can be reduced to the equivalent system

1198781015840(119905) = 119887 minus 120582119868119878 minus 119901119887119864 minus 119902119887119868 minus 119889119878

+ 120577 (119873 minus 119878 minus 119864 minus 119868)

1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868

minus 120576119864 minus 119889119864


1198731015840(119905) = 119887 minus 119889119873 (119905) minus 120578119868 (119905)

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119868 (119905+) = 119868 (119905)

119873 (119905+) = 119873 (119905)

119905 = 119896120591 119896 isin 119885+

(6)

The initial conditions for (6) are

119878 (0+) gt 0 119864 (0

+) gt 0 119868 (0

+) gt 0 119873 (0

+) gt 0

(7)

From physical considerations we discuss system (6) in theclosed set

Ω = (119878 119864 119868119873) isin 1198774

+| 0 le 119878 + 119864 + 119868 le

119887

119889 0 le 119873 le

119887

119889

(8)

The organization of this paper is as follows In Section 2we establish sufficient condition for the local and globalattractivity of virus-free periodic solution The sufficientcondition for the permanence of the model is obtained inSection 3 Some numerical simulations are performed inSection 4 In the final section a brief conclusion is given andsome future research directions are also pointed out

Abstract and Applied Analysis 3

2 Global Attractivity of Virus-FreePeriodic Solution

To prove our main results we state three lemmas which willbe essential to our proofs

Lemma 1 (see [14]) Consider the following impulsive differen-tial equations

(119905) = 119886 minus 119887119906 (119905) 119905 = 119896120591

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896120591

(9)

where 119886 gt 0 119887 gt 0 and 0 lt 120579 lt 1 Then system (9) has aunique positive periodic solution

119906119890(119905) =

119886

119887+ (119906lowastminus119886

119887) 119890minus119887(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

(10)

which is globally asymptotically stable there 119906lowast = 119886(1minus120579)(1minus

119890minus119887120591

)119887(1 minus (1 minus 120579)119890minus119887120591

)

If 119868(119905) equiv 0 we have the following limit systems


minus 119889119878 + 120577 (119873 minus 119878 minus 119864)

1198641015840(119905) = 119901119887119864 minus 120576119864 minus 119889119864

1198731015840(119905) = 119887 minus 119889119873

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119873 (119905+) = 119873 (119905)

119905 = 119896120591 119896 isin 119885+

(11)

When119901119887minus119889minus120576 lt 0 there exists 1199051when 119905 gt 119905

1 lim119905rarrinfin

119864(119905) =

0 From the third and sixth equations of system (11) we havelim119905rarrinfin

119873(119905) = 119887119889 We have the following limit systems

119889119878

119889119905= (1 minus 119901) 119887 +

V119887

120583minus (120583 + V) 119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879

(12)

According to Lemma 1 we know that periodic solution ofsystem (12) is of the form

119878119890(119905) =

(120583 (1 minus 119901) + V) 119887

120583 (120583 + V)

+ (119878lowastminus(120583 (1 minus 119901) + V) 119887

120583 (120583 + V)) 119890minus(120583+V)(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

(13)

and it is globally asymptotically stable where 119878lowast = ((1minus119901)119887+

(V119887120583))(1 minus 120579)(119890(120583+V)120591 minus 1)(120583 + V)(119890(120583+V)120591 minus 1 + 120579)

Theorem 2 Let (119878(119905) 119864(119905) 119868(119905)119873(119905)) be any solution ofsystem (6) with initial values 119878(0+) gt 0 119864(0+) gt 0 119868(0+) gt 0

and 119873(0+) gt 0 then (119878119890(119905) 0 0 119887119889) is locally asymptotically

stable provided that 119901119887 minus 119889 minus 120576 lt 0 and 1198770lt 1 where

1198770=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591

times 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)]

(14)

Proof The local stability of virus-free periodic solution maybe determined by considering the behaviors of a smallamplitude perturbation of the solution Define 119908(119905) = 119878(119905) minus

119878119890(119905) 119909(119905) = 119864(119905) 119910(119905) = 119868(119905) and 119911(119905) = 119873(119905)minus119887119889 and then

the linearized system of system (6) reads as

1199081015840(119905) = minus (119889 + 120577)119908 (119905) minus (119901119887 + 120577) 119909 (119905)

minus (120582119878119890(119905) + 120577 + 119902119887) 119910 (119905) + 120577119911 (119905)

1199091015840(119905) = (119901119887 minus 119889 minus 120576) 119909 (119905)

+ (119902119887 + 120582119878119890(119905)) 119910 (119905)

1199101015840(119905) = 120576119909 (119905) minus (119903 + 119889 + 120578) 119910 (119905)

1199111015840(119905) = minus120578119910 (119905) minus 119889119911 (119905)

119905 = 119896120591 119896isin119885+

119908 (119905+) = 119908 (119905)

119909 (119905+) = 119909 (119905)

119910 (119905+) = 119910 (119905)

119911 (119905+) = 119911 (119905)

119905 = 119896120591 119896 isin 119885+

(15)

Let Φ(119905) be the fundamental solution matrix of system (15)and then Φ(119905)must satisfy

119889Φ (119905)

119889119905

= (

minus119889 minus 120577 minus119901119887 minus 120577 minus120582119878119890(119905) minus 120577 minus 119902119887 120577

0 119901119887 minus 119889 minus 120576 119902119887 + 120582119878119890(119905) 0

0 120576 minus119903 minus 119889 minus 120578 0

0 0 minus120578 minus119889

)Φ(119905)

= 119860Φ (119905)

(16)

and Φ(0) = 119868 the identity matrix We can easily see thattwo eigenvalues of the matrix 119860 are minus119889 minus 120577 and minus119889 and theother two eigenvalues are determined by the 2 times 2 matrix 119861 =

(119901119887minus119889minus120576 119902119887+120582119878

119890(119905)

120576 minus119903minus119889minus120578) Denote the eigenvalues of 119861 as 120582

1 1205822 and

then as119901119887minus119889minus120576 lt 0 we have1205821+1205822= 119901119887minus119889minus120576minus119903minus119889minus120578 lt 0

12058211205822= (119901119887 minus 119889 minus 120576) (minus119903 minus 119889 minus 120578) minus 120576 (119902119887 + 120582119878

119890(119905))

(17)


Therefore by the Floquet theorem [15] (119878119890(119905) 0 0 119887119889) is

locally asymptotically stable provided that

119866 = int

120591

0

[(119901119887 minus 119889 minus 120576) (minus120574 minus 119889 minus 120578) minus 120576 (119902119887 + 120582119878119890(119905))] 119889119905

= (119901119887 minus 119889 minus 120576) (minus120574 minus 119889 minus 120578) 120591 minus 120576119902119887120591

minus 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)] gt 0

(18)

When1198770= (1((119889+120576minus119901119887)(120574+119889+120578)120591minus120576119902119887120591))120576120582(119887119889)[120591+120579(1minus

119890(120577+119889)120591

)(120577 + 119889)(119890(120577+119889)120591

minus 1 + 120579)] le 1 the previous inequalityis satisfied for 119901119887 minus 119889 minus 120576 lt 0

The proof is complete

Theorem 3 If 1198771le 1 and 119901119887minus119889minus120576 lt 0 then (119878

119890(119905) 0 0 119887119889)

is globally asymptotically stable for system (11) where

1198771=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591120576120582119887120591

119889

times [1 minus120579

(119890(120577+119889)120591 minus 1 + 120579)]

(19)

Proof Because 119890119909 minus 1 gt 119909 for 119909 gt 0 we have

1198770lt 1198771 (20)

ByTheorem 2 we know that (119878119890(119905) 0 0 119887119889) is locally asymp-

totically stable In the following we will prove the globalattraction of (119878

119890(119905) 0 0 119887119889)

Let

119871 = 120576119864 minus (119901119887 minus 120576 minus 119889) 119868 (21)

Then

1198711015840= 1205761198641015840minus (119901119887 minus 120576 minus 119889) 119868

1015840

1198711015840= [120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868

(22)

Therefore (119878119890(119905) 0 0 119887119889) is globally asymptotically stable

provided that

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868 119889119905 lt 0

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119889119905

lt [120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 120591

+ 120576120582119887

119889(1 minus

120579

119890(119889+120577)120591 minus 1 + 120579) 120591 lt 0

(23)

When 1198771= (1((119889 + 120576 minus119901119887)(120574 + 119889+ 120578)120591 minus 120576119902119887120591))120576120582(119887120591119889)[1 minus

120579(119890(120577+119889)120591

minus 1 + 120579)] le 1 the previous inequality is satisfiedThe proof is complete

Corollary 4 The virus-free periodic solution (Se(t) 0 0 bd)of system (6) is globally attractive if 120579 gt 120579

1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591

minus 1)119889[(119889 + 120576 minus 119901119887)(119903 + 119889 + 120578) minus 120576119902119887]

Theorem 2 determines the global attractivity of (6) inΩ for the case 119877

0lt 1 Its realistic implication is that the

infected computers vanish so the worms are removed fromthe network Corollary 4 implies that the computer virus willdisappear if the vaccination rate is less than 120579

1

3 Permanence

In this section we say that the worm is local if the infectedpopulation persists above a certain positive level for suffi-ciently large timeThe local of worm can be well captured andstudied through the notion of permanence

Definition 5 System (6) is said to be uniformly persistent ifthere is an 120593 gt 0 (independent of the initial data) such thatevery solution (119878(119905) 119868(119905) 119877(119905)119873(119905)) with initial conditions(8) of system (6) satisfies

lim119905rarrinfin

inf 119878 (119905) ge 120593 lim119905rarrinfin

inf 119868 (119905) ge 120593

lim119905rarrinfin


inf 119873(119905) ge 120593

(24)

Theorem 6 Suppose that 1198771gt 1 and 119901119887 minus 119889 minus 120576 lt 0 Then

there is a positive constant 119898119868such that each positive solution

(119878(119905) 119864(119905) 119868(119905) 119877(119905)) of system (6) satisfies 119868(119905) ge 119898119868 for t

large enough

Proof Now we will prove that there exist 119898119868gt 0 and a

sufficiently large 119905119901such that 119868(119905) ge 119898

119868holds for all 119905 gt 119905

119901

Suppose that 119868(119905) lt 119868lowast for all 119905 gt 119905

0 From the forth equation

of (6) we have

119887 minus 119889119873 (119905) gt 1198731015840(119905) gt 119887 minus 119889119873 (119905) minus 120578119868

lowast

119887

119889gt 119873 (119905) gt

119887 minus 120578119868lowast

119889

(25)

From the second equation of (6) we have

1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864

lt 120582119868lowast119878 + 119902119887119868

lowast+ (119901119887 minus 120576 minus 119889) 119864

lt 120582119868lowast 119887

119889+ 119902119887119868lowast+ (119901119887 minus 120576 minus 119889) 119864

(26)

As 119901119887 minus 119889 minus 120576 lt 0

119864 (119905) lt(119902119887 + 120582) 119887119868

lowast

119889 (120576 + 119889 minus 119901119887) (27)


From the first equation of (6) we have

1198781015840(119905) = 119887 minus 120582119868119878 minus 119901119887119864 minus 119902119887119868 minus 119889119878 + 120577 (119873 minus 119878 minus 119864 minus 119868)

gt 119887 + 120577119887 minus 120578119868

lowast

119889minus 119902119887119868lowastminus 120577119868lowast

minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast

119889 (120576 + 119889 minus 119901119887)minus (120582119868lowast+ 119889 + 120577) 119878 (119905)

(28)

Consider the following comparison system

V1015840 (119905) = 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast

119889 (120576 + 119889 minus 119901119887)minus (120582119868lowast+ 119889 + 120577) V (119905)

119905 = 119896120591

V (119905+) = (1 minus 120579) V (119905) 119905 = 119896120591

V (0+) = 119878 (0+)

(29)

let119860 = 119887+120577((119887minus120578119868lowast)119889)minus119902119887119868

lowastminus120577119868lowastminus(119901119887+120577)((119902119887+120582)119887119868

lowast119889(120576+

119889 minus 119901119887)) 119861 = 120582119868lowast + 119889 + 120577By Lemma 1 we know that there exists 119905

1gt 1199050such that

V (119905) =119860

119861+ (Vlowast minus

119860

119861) 119890minus119861(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

Vlowast =119860 (1 minus 120579) (1 minus 119890

minus119861120591)

119861 (1 minus (1 minus 120579) 119890minus119861120591)

119878 (119905) gt V (119905) = 119878lowast gt 0 for 119905 gt 1199051

(30)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864

gt 120582119868lowast119878lowast+ 119902119887119868lowastminus (120576 + 119889 minus 119901119887) 119864

119864 (119905) gt120582119868lowast119878lowast+ 119902119887119868lowast

120576 + 119889 minus 119901119887

(31)

From the third equation of (6) we have


gt 120576120582119868lowast119878lowast+ 119902119887119868lowast

120576 + 119889 minus 119901119887minus (120574 + 119889 + 120578) 119868

(32)

Note that 1198771gt 1 we have

119868 (119905) gt120576

120574 + 119889 + 120578

120582119868lowast119878lowast+ 119902119887119868lowast

120576 + 119889 minus 119901119887gt 119868lowast (33)

This contradicts 119868(119905) le 119868lowast Hence we can claim that for any

1199050gt 0 it is impossible that

119868 (119905) lt 119868lowast

forall119905 ge t0 (34)

By the claim we are left to consider two cases First 119868(119905) ge 119868lowast

for 119905 large enough Second 119868(119905) oscillates about 119868lowast for 119905 largeenough Obviously there is nothing to prove for the first caseFor the second case we can choose 119905

2gt 1199051 and 120585 gt 0 satisfy

119868 (1199052) = 119868 (119905

2+ 120585) = 119868

lowast for 119905

2lt 119905 lt 119905

2+ 120585 (35)

119868(119905) is uniformly continuous since the positive solutions of (6)are ultimately bounded and 119868(119905) is not affected by impulses

Therefore it is certain that there exists a 120578 (0 lt 120578 lt 120591 and120578 is independent of the choice of 119905

2) such that

119868 (119905) ge119868lowast

2 for 119905

2le 119905 le 119905

2+ 120578 (36)

In this case we consider the following three possible cases interm of the sizes of 120578 120585 and 120591Case 1 If 120585 le 120578 lt 120591 then it is obvious that 119868(119905) ge Ilowast2 for119905 isin [1199052 1199052+ 120585]

Case 2 If 120578 lt 120585 le 120591 then from the second equation of system(6) we obtain 119868(119905) gt minus(119889 + 120574 + 120578)119868(119905)

Since 119868(1199052) = 119868lowast it is obvious that 119868(119905) gt 119868

lowast119890minus(119889+120574+120578)120591

= 119868lowastlowast

for 119905 isin [1199052 1199052+ 120585]

Case 3 If 120578 lt 120591 le 120585 it is easy to obtain that 119868(119905) gt 119868lowastlowast for

119905 isin [1199052 1199052+ 120591] Then proceeding exactly as the proof for the

previous claim we have that 119868(119905) gt 119868lowastlowast for 119905 isin [119905

2+ 120591 1199052+ 120585]

Owing to the randomicity of 1199052 we can obtain that there

exists119898119868= min119868lowast2 119868lowastlowast such 119868(119905) gt 119898

119868holds for all 119905 gt 119905

119901

The proof of Theorem 6 is completed

Theorem 7 Suppose 1198771gt 1 Then system (6) is permanent

Proof Let (119878(119905) 119864(119905) 119868(119905)119873(119905)) be any solution of system (6)First from the first equation of system (6) we have

1198781015840(119905) gt 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119878 (37)


1(119905) = 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119911

1(119905) 119905 = 119896120591

1199111(119905+) = (1 minus 120579) 119911

1(119905) 119905 = 119896120591

(38)

By Lemma 1 we know that for any sufficiently small 120576 gt 0there exists a 119905

1(1199051is sufficiently large) such that

119878 (119905) ge 1199111(119905) gt 119911

lowast

1(119905) minus 120576

ge(1 minus 119901119887 minus 119902119887) 119887 (119890

(119889+120582119887)120591minus 1)

(119889 + 120582119887) (119890(119889+120582119887)120591 minus 1 + 120579)minus 120576 =119898

119878gt 0

119896120591 lt 119905 le (119896 + 1) 120591

(39)

From (31) we have

119864 (119905) gt120582119898119868119898119878+ 119902119887119898

119868

120576 + 119889 minus 119901119887minus 120576 =119898

119864 (40)


0 10 20 30 40 502

3

4

5

6

7

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

t

(a)

0 10 20 30 40 500

05

1

15

The p

opul

atio

n of

expo

sed

com

pute

rs

t

(b)

0 10 20 30 40 500

1

01

02

03

04

05

06

07

08

09

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 502

3

4

5

6

7

8

9

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)

Figure 3 Global attractivity of virus-free periodic solution of system


(119905) gt 119887 minus 120583119873 (119905) minus 120572119873 (119905) (41)

It is easy to see that

119873(119905) gt119887

120583 + 120572minus 120576 =119898

119873 (42)

We letΩ0= (119878 119868119873) isin 119877

3

+| 119898119878le 119878119898

119868le 119868119898

119873le 119873 le 119887120583

119878 + 119868 le 119887120583 By Theorem 6 and the previous discussions weknow that the setΩ

0is a global attractor inΩ and of course

every solution of system (6) with initial conditions (8) willeventually enter and remain in region Ω

0 Therefore system

(6) is permanentThe proof of Theorem 7 is completed

Corollary 8 It follows from Theorem 6 that the system (6) isuniformly persistent provided that 120579 lt 120579

1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591


4 Numerical Simulations

In this section we have performed some numerical simula-tions to show the geometric impression of our results

To demonstrate the global attractivity of virus-free peri-odic solution of system (6) we take following set parametervalues 119887 = 21 120582 = 02 119901 = 01 119902 = 015 119889 = 03 120577=06 120576 = 04 120574 = 06 120591 = 05 120578 = 03 and 120579 = 04 In thiscase we have 119877

0= 06886 lt 1 In Figures 3(a) 3(b) 3(c) and

3(d) we have displayed respectively the susceptible exposedinfected and recovered population of system (6) with initialconditions 119878(0) = 6 119864(0) = 1 119868(0) = 1 and 119877(0) = 8

To demonstrate the permanence of system (6) we takethe following set parameter values 119887 = 27 120582 = 02 119901 = 01119902 = 015 119889 = 03 120577 = 06 120576 = 04 120574 = 06 120591 = 05 120578 = 03

and 120579 = 01 In this case we have 1198771= 16496 gt 1 In Figures

4(a) 4(b) 4(c) and 4(d) we have displayed respectively thesusceptible exposed infected and recovered populations ofsystem (6) with initial conditions 119878(0) = 6119864(0) = 1 119868(0) = 1

and 119877(0) = 8


0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

(a)

0 10 20 30 40 50t

1

12

14

16

18

2

22

24

26

The p

opul

atio

n of

expo

sed

com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

4

5

6

7

8

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)

Figure 4 Permanence of system

5 Conclusion

We have analyzed the SEIRS model with pulse vaccinationand varying total population sizeWe have shown that119877

1gt 1

or 120579 lt 1205791implies that the worm will be local whereas 119877

1lt 1

or 120579 gt 1205791implies that the worm will fade out We have also

established sufficient condition for the permanence of themodel Our results indicate that a large pulse vaccination ratewill lead to eradication of the worm

References

[1] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of IEEE Symposiumon Security and Privacy pp 343ndash359 1991

[2] J O Kephart S R White and D M Chess ldquoComputers andepidemiologyrdquo IEEE Spectrum vol 30 no 5 pp 20ndash26 1993

[3] L Billings W M Spears and I B Schwartz ldquoA unifiedprediction of computer virus spread in connected networksrdquoPhysics Letters A vol 297 no 3-4 pp 261ndash266 2002

[4] X Han and Q Tan ldquoDynamical behavior of computer virus onInternetrdquoAppliedMathematics and Computation vol 217 no 6pp 2520ndash2526 2010

[5] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007

[6] B K Mishra and D Saini ldquoMathematical models on computervirusesrdquo Applied Mathematics and Computation vol 187 no 2pp 929ndash936 2007

[7] B K Mishra and S K Pandey ldquoDynamic model of worms withvertical transmission in computer networkrdquoAppliedMathemat-ics and Computation vol 217 no 21 pp 8438ndash8446 2011

[8] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009

[9] J R C Piqueira A A Vasconcelos C E C J Gabriel and V OAraujo ldquoDynamic models for computer virusesrdquo Computers ampSecurity vol 27 no 7-8 pp 355ndash359 2008


[10] J Ren X Yang L-X Yang Y Xu and F Yang ldquoA delayedcomputer virus propagation model and its dynamicsrdquo ChaosSolitons amp Fractals vol 45 no 1 pp 74ndash79 2012

[11] J Ren X Yang Q Zhu L X Yang and C Zhang ldquoA novelcomputer virus model and its dynamicsrdquo Nonlinear Analysisvol 13 no 1 pp 376ndash384 2012

[12] J C Wierman and D J Marchette ldquoModeling computer virusprevalence with a susceptible-infected-susceptible model withreintroductionrdquo Computational Statistics amp Data Analysis vol45 no 1 pp 3ndash23 2004

[13] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008

[14] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 1 pp 731ndash745 2007

[15] R Shi and L Chen ldquoStage-structured impulsive 119878119868 model forpest managementrdquo Discrete Dynamics in Nature and SocietyArticle ID 97608 11 pages 2007

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


S I R120582IS 120574I

120577R

dS dI dR

b 120576E

pbE

E

dE

qbI+

Figure 1 Original model

the exposed class 119864 Consequently the birth flux into theexposed class is given by 119901119887119864 + 119902119887119868 and the birth flux intothe susceptible class is given by 119887 minus 119901119887119864 minus 119902119887119868

As we know antivirus software is a kind of computerprogramwhich can detect and eliminate knownwormThereare two common methods to detect worms using a list ofworm signature definition and using a heuristic algorithm tofindworm based on common behaviors It has been observedthat it does not always work in detecting a novel worm byusing the heuristic algorithm On the other hand obviouslyit is impossible for antivirus software to find a new wormsignature definition on the dated list So to keep the antivirussoftware in high efficiency it is important to ensure thatit is updated Based on the previous facts we propose animpulsive system to model the process of periodic installingor updating antivirus software on susceptible computers atfixed time for controlling the spread of worm

Based on the previous facts we propose the followingassumptions

(H1) the antivirus software is installed or updated at time119905 = 119896120591 (119896 isin 119873) where 120591 is the period of the impulsiveeffect

(H2) 119878 computers are successfully vaccinated from 119878 classto 119877 class with rate 120579 (0 lt 120579 lt 1)

According to the previous assumptions (H1)-(H2) andfor the reason of simplicity we propose the following model(Figure 2)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864


1198771015840(119905) = 120574119868 minus 120577119877 minus 119889119877

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119868 (119905+) = 119868 (119905)

119877 (119905+) = 119877 (119905) + 120579119878 (119905)

119905 = 119896120591 119896 isin 119885+

(2)

The total population size 119873(119905) can be determined by 119873(119905) =119878(119905) + 119864(119905) + 119868(119905) + 119877(119905) to form the differential equation

1198731015840(119905) = 119887 minus 119889119873 (119905) minus 120578119868 (119905) (3)

S I R120582IS 120574I

120577R

dS dI dR

b 120576E

pbE

E

dE

120579S

qbI+

Figure 2 Impulse model

which is derived by adding the equations in system (1) Thusthe total population size 119873 may vary in time From (2) wehave

119887 minus (119889 + 120578)119873 (119905) le 1198731015840(119905) le 119887 minus 119889119873 (119905) (4)

It follows that119887

(119889 + 120578)le lim119909rarrinfin

inf 119873(119905) le lim119909rarrinfin

sup119873(119905) le119887

119889

(5)

The system (2) can be reduced to the equivalent system

1198781015840(119905) = 119887 minus 120582119868119878 minus 119901119887119864 minus 119902119887119868 minus 119889119878

+ 120577 (119873 minus 119878 minus 119864 minus 119868)

1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868

minus 120576119864 minus 119889119864


1198731015840(119905) = 119887 minus 119889119873 (119905) minus 120578119868 (119905)

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119868 (119905+) = 119868 (119905)

119873 (119905+) = 119873 (119905)

119905 = 119896120591 119896 isin 119885+

(6)

The initial conditions for (6) are

119878 (0+) gt 0 119864 (0

+) gt 0 119868 (0

+) gt 0 119873 (0

+) gt 0

(7)

From physical considerations we discuss system (6) in theclosed set

Ω = (119878 119864 119868119873) isin 1198774

+| 0 le 119878 + 119864 + 119868 le

119887

119889 0 le 119873 le

119887

119889

(8)

The organization of this paper is as follows In Section 2we establish sufficient condition for the local and globalattractivity of virus-free periodic solution The sufficientcondition for the permanence of the model is obtained inSection 3 Some numerical simulations are performed inSection 4 In the final section a brief conclusion is given andsome future research directions are also pointed out





(119905) = 119886 minus 119887119906 (119905) 119905 = 119896120591

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896120591

(9)


119906119890(119905) =

119886

119887+ (119906lowastminus119886

119887) 119890minus119887(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

(10)


119890minus119887120591


)




1198641015840(119905) = 119901119887119864 minus 120576119864 minus 119889119864

1198731015840(119905) = 119887 minus 119889119873

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119873 (119905+) = 119873 (119905)

119905 = 119896120591 119896 isin 119885+

(11)



119864(119905) =



119889119878

119889119905= (1 minus 119901) 119887 +

V119887

120583minus (120583 + V) 119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879

(12)


119878119890(119905) =

(120583 (1 minus 119901) + V) 119887

120583 (120583 + V)


120583 (120583 + V)) 119890minus(120583+V)(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

(13)






1198770=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591

times 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)]

(14)




1199081015840(119905) = minus (119889 + 120577)119908 (119905) minus (119901119887 + 120577) 119909 (119905)

minus (120582119878119890(119905) + 120577 + 119902119887) 119910 (119905) + 120577119911 (119905)

1199091015840(119905) = (119901119887 minus 119889 minus 120576) 119909 (119905)

+ (119902119887 + 120582119878119890(119905)) 119910 (119905)

1199101015840(119905) = 120576119909 (119905) minus (119903 + 119889 + 120578) 119910 (119905)

1199111015840(119905) = minus120578119910 (119905) minus 119889119911 (119905)

119905 = 119896120591 119896isin119885+

119908 (119905+) = 119908 (119905)

119909 (119905+) = 119909 (119905)

119910 (119905+) = 119910 (119905)

119911 (119905+) = 119911 (119905)

119905 = 119896120591 119896 isin 119885+

(15)


119889Φ (119905)

119889119905

= (


0 119901119887 minus 119889 minus 120576 119902119887 + 120582119878119890(119905) 0


0 0 minus120578 minus119889

)Φ(119905)

= 119860Φ (119905)

(16)


(119901119887minus119889minus120576 119902119887+120582119878

119890(119905)


1 1205822 and



119890(119905))

(17)




119866 = int

120591

0



minus 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)] gt 0

(18)


119890(120577+119889)120591

)(120577 + 119889)(119890(120577+119889)120591




119890(119905) 0 0 119887119889)


1198771=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591120576120582119887120591

119889


(119890(120577+119889)120591 minus 1 + 120579)]

(19)


1198770lt 1198771 (20)



119890(119905) 0 0 119887119889)

Let


Then


1015840

1198711015840= [120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868

(22)


provided that

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868 119889119905 lt 0

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119889119905

lt [120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 120591

+ 120576120582119887

119889(1 minus

120579

119890(119889+120577)120591 minus 1 + 120579) 120591 lt 0

(23)


120579(119890(120577+119889)120591



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591





1

3 Permanence



lim119905rarrinfin


inf 119868 (119905) ge 120593

lim119905rarrinfin


inf 119873(119905) ge 120593

(24)




large enough



119868holds for all 119905 gt 119905

119901



of (6) we have


lowast

119887

119889gt 119873 (119905) gt

119887 minus 120578119868lowast

119889

(25)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864

lt 120582119868lowast119878 + 119902119887119868


lt 120582119868lowast 119887


(26)

As 119901119887 minus 119889 minus 120576 lt 0

119864 (119905) lt(119902119887 + 120582) 119887119868

lowast

119889 (120576 + 119889 minus 119901119887) (27)




gt 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


(28)


V1015840 (119905) = 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


119905 = 119896120591

V (119905+) = (1 minus 120579) V (119905) 119905 = 119896120591

V (0+) = 119878 (0+)

(29)



lowast119889(120576+



V (119905) =119860


119860

119861) 119890minus119861(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591


minus119861120591)



(30)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864



120576 + 119889 minus 119901119887

(31)




120576 + 119889 minus 119901119887minus (120574 + 119889 + 120578) 119868

(32)


119868 (119905) gt120576

120574 + 119889 + 120578


120576 + 119889 minus 119901119887gt 119868lowast (33)



119868 (119905) lt 119868lowast

forall119905 ge t0 (34)




119868 (1199052) = 119868 (119905

2+ 120585) = 119868

lowast for 119905

2lt 119905 lt 119905

2+ 120585 (35)



2) such that

119868 (119905) ge119868lowast

2 for 119905

2le 119905 le 119905

2+ 120578 (36)




lowast119890minus(119889+120574+120578)120591


for 119905 isin [1199052 1199052+ 120585]




2+ 120591 1199052+ 120585]



119868holds for all 119905 gt 119905

119901




1198781015840(119905) gt 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119878 (37)


1(119905) = 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119911

1(119905) 119905 = 119896120591

1199111(119905+) = (1 minus 120579) 119911

1(119905) 119905 = 119896120591

(38)



119878 (119905) ge 1199111(119905) gt 119911

lowast

1(119905) minus 120576

ge(1 minus 119901119887 minus 119902119887) 119887 (119890

(119889+120582119887)120591minus 1)

(119889 + 120582119887) (119890(119889+120582119887)120591 minus 1 + 120579)minus 120576 =119898

119878gt 0

119896120591 lt 119905 le (119896 + 1) 120591

(39)

From (31) we have

119864 (119905) gt120582119898119868119898119878+ 119902119887119898

119868

120576 + 119889 minus 119901119887minus 120576 =119898

119864 (40)


0 10 20 30 40 502

3

4

5

6

7

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

t

(a)

0 10 20 30 40 500

05

1

15

The p

opul

atio

n of

expo

sed

com

pute

rs

t

(b)

0 10 20 30 40 500

1

01

02

03

04

05

06

07

08

09

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 502

3

4

5

6

7

8

9

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)



(119905) gt 119887 minus 120583119873 (119905) minus 120572119873 (119905) (41)


119873(119905) gt119887

120583 + 120572minus 120576 =119898

119873 (42)

We letΩ0= (119878 119868119873) isin 119877

3

+| 119898119878le 119878119898

119868le 119868119898

119873le 119873 le 119887120583




0 Therefore system



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591










and 119877(0) = 8


0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

(a)

0 10 20 30 40 50t

1

12

14

16

18

2

22

24

26

The p

opul

atio

n of

expo

sed

com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

4

5

6

7

8

The p

opul

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reco

vere

d co

mpu

ters

t

(d)


5 Conclusion


1gt 1


1lt 1



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(119905) = 119886 minus 119887119906 (119905) 119905 = 119896120591

119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896120591

(9)


119906119890(119905) =

119886

119887+ (119906lowastminus119886

119887) 119890minus119887(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

(10)


119890minus119887120591


)




1198641015840(119905) = 119901119887119864 minus 120576119864 minus 119889119864

1198731015840(119905) = 119887 minus 119889119873

119905 = 119896120591 119896 isin 119885+

119878 (119905+) = (1 minus 120579) 119878 (119905)

119864 (119905+) = 119864 (119905)

119873 (119905+) = 119873 (119905)

119905 = 119896120591 119896 isin 119885+

(11)



119864(119905) =



119889119878

119889119905= (1 minus 119901) 119887 +

V119887

120583minus (120583 + V) 119878 119905 = 119896119879 119896 isin 119885

+

119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879

(12)


119878119890(119905) =

(120583 (1 minus 119901) + V) 119887

120583 (120583 + V)


120583 (120583 + V)) 119890minus(120583+V)(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591

(13)






1198770=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591

times 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)]

(14)




1199081015840(119905) = minus (119889 + 120577)119908 (119905) minus (119901119887 + 120577) 119909 (119905)

minus (120582119878119890(119905) + 120577 + 119902119887) 119910 (119905) + 120577119911 (119905)

1199091015840(119905) = (119901119887 minus 119889 minus 120576) 119909 (119905)

+ (119902119887 + 120582119878119890(119905)) 119910 (119905)

1199101015840(119905) = 120576119909 (119905) minus (119903 + 119889 + 120578) 119910 (119905)

1199111015840(119905) = minus120578119910 (119905) minus 119889119911 (119905)

119905 = 119896120591 119896isin119885+

119908 (119905+) = 119908 (119905)

119909 (119905+) = 119909 (119905)

119910 (119905+) = 119910 (119905)

119911 (119905+) = 119911 (119905)

119905 = 119896120591 119896 isin 119885+

(15)


119889Φ (119905)

119889119905

= (


0 119901119887 minus 119889 minus 120576 119902119887 + 120582119878119890(119905) 0


0 0 minus120578 minus119889

)Φ(119905)

= 119860Φ (119905)

(16)


(119901119887minus119889minus120576 119902119887+120582119878

119890(119905)


1 1205822 and



119890(119905))

(17)




119866 = int

120591

0



minus 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)] gt 0

(18)


119890(120577+119889)120591

)(120577 + 119889)(119890(120577+119889)120591




119890(119905) 0 0 119887119889)


1198771=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591120576120582119887120591

119889


(119890(120577+119889)120591 minus 1 + 120579)]

(19)


1198770lt 1198771 (20)



119890(119905) 0 0 119887119889)

Let


Then


1015840

1198711015840= [120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868

(22)


provided that

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868 119889119905 lt 0

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119889119905

lt [120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 120591

+ 120576120582119887

119889(1 minus

120579

119890(119889+120577)120591 minus 1 + 120579) 120591 lt 0

(23)


120579(119890(120577+119889)120591



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591





1

3 Permanence



lim119905rarrinfin


inf 119868 (119905) ge 120593

lim119905rarrinfin


inf 119873(119905) ge 120593

(24)




large enough



119868holds for all 119905 gt 119905

119901



of (6) we have


lowast

119887

119889gt 119873 (119905) gt

119887 minus 120578119868lowast

119889

(25)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864

lt 120582119868lowast119878 + 119902119887119868


lt 120582119868lowast 119887


(26)

As 119901119887 minus 119889 minus 120576 lt 0

119864 (119905) lt(119902119887 + 120582) 119887119868

lowast

119889 (120576 + 119889 minus 119901119887) (27)




gt 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


(28)


V1015840 (119905) = 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


119905 = 119896120591

V (119905+) = (1 minus 120579) V (119905) 119905 = 119896120591

V (0+) = 119878 (0+)

(29)



lowast119889(120576+



V (119905) =119860


119860

119861) 119890minus119861(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591


minus119861120591)



(30)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864



120576 + 119889 minus 119901119887

(31)




120576 + 119889 minus 119901119887minus (120574 + 119889 + 120578) 119868

(32)


119868 (119905) gt120576

120574 + 119889 + 120578


120576 + 119889 minus 119901119887gt 119868lowast (33)



119868 (119905) lt 119868lowast

forall119905 ge t0 (34)




119868 (1199052) = 119868 (119905

2+ 120585) = 119868

lowast for 119905

2lt 119905 lt 119905

2+ 120585 (35)



2) such that

119868 (119905) ge119868lowast

2 for 119905

2le 119905 le 119905

2+ 120578 (36)




lowast119890minus(119889+120574+120578)120591


for 119905 isin [1199052 1199052+ 120585]




2+ 120591 1199052+ 120585]



119868holds for all 119905 gt 119905

119901




1198781015840(119905) gt 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119878 (37)


1(119905) = 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119911

1(119905) 119905 = 119896120591

1199111(119905+) = (1 minus 120579) 119911

1(119905) 119905 = 119896120591

(38)



119878 (119905) ge 1199111(119905) gt 119911

lowast

1(119905) minus 120576

ge(1 minus 119901119887 minus 119902119887) 119887 (119890

(119889+120582119887)120591minus 1)

(119889 + 120582119887) (119890(119889+120582119887)120591 minus 1 + 120579)minus 120576 =119898

119878gt 0

119896120591 lt 119905 le (119896 + 1) 120591

(39)

From (31) we have

119864 (119905) gt120582119898119868119898119878+ 119902119887119898

119868

120576 + 119889 minus 119901119887minus 120576 =119898

119864 (40)


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8Th

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05

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com

pute

rs

t

(b)

0 10 20 30 40 500

1

01

02

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cted

com

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t

(c)

0 10 20 30 40 502

3

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6

7

8

9

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vere

d co

mpu

ters

t

(d)



(119905) gt 119887 minus 120583119873 (119905) minus 120572119873 (119905) (41)


119873(119905) gt119887

120583 + 120572minus 120576 =119898

119873 (42)

We letΩ0= (119878 119868119873) isin 119877

3

+| 119898119878le 119878119898

119868le 119868119898

119873le 119873 le 119887120583




0 Therefore system



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591










and 119877(0) = 8


0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

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of su

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tible

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(a)

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1

12

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16

18

2

22

24

26

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expo

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com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

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infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

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5

6

7

8

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)


5 Conclusion


1gt 1


1lt 1



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119866 = int

120591

0



minus 120576120582119887

119889[120591 +

120579 (1 minus 119890(120577+119889)120591

)

(120577 + 119889) (119890(120577+119889)120591 minus 1 + 120579)] gt 0

(18)


119890(120577+119889)120591

)(120577 + 119889)(119890(120577+119889)120591




119890(119905) 0 0 119887119889)


1198771=

1

(119889 + 120576 minus 119901119887) (120574 + 119889 + 120578) 120591 minus 120576119902119887120591120576120582119887120591

119889


(119890(120577+119889)120591 minus 1 + 120579)]

(19)


1198770lt 1198771 (20)



119890(119905) 0 0 119887119889)

Let


Then


1015840

1198711015840= [120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868

(22)


provided that

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119868 119889119905 lt 0

1198711015840= int

120591

0

[120576120582119878 + 120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 119889119905

lt [120576119902119887 + (119901119887 + 120576 minus 119889) (119903 + 119889 + 120578)] 120591

+ 120576120582119887

119889(1 minus

120579

119890(119889+120577)120591 minus 1 + 120579) 120591 lt 0

(23)


120579(119890(120577+119889)120591



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591





1

3 Permanence



lim119905rarrinfin


inf 119868 (119905) ge 120593

lim119905rarrinfin


inf 119873(119905) ge 120593

(24)




large enough



119868holds for all 119905 gt 119905

119901



of (6) we have


lowast

119887

119889gt 119873 (119905) gt

119887 minus 120578119868lowast

119889

(25)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864

lt 120582119868lowast119878 + 119902119887119868


lt 120582119868lowast 119887


(26)

As 119901119887 minus 119889 minus 120576 lt 0

119864 (119905) lt(119902119887 + 120582) 119887119868

lowast

119889 (120576 + 119889 minus 119901119887) (27)




gt 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


(28)


V1015840 (119905) = 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


119905 = 119896120591

V (119905+) = (1 minus 120579) V (119905) 119905 = 119896120591

V (0+) = 119878 (0+)

(29)



lowast119889(120576+



V (119905) =119860


119860

119861) 119890minus119861(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591


minus119861120591)



(30)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864



120576 + 119889 minus 119901119887

(31)




120576 + 119889 minus 119901119887minus (120574 + 119889 + 120578) 119868

(32)


119868 (119905) gt120576

120574 + 119889 + 120578


120576 + 119889 minus 119901119887gt 119868lowast (33)



119868 (119905) lt 119868lowast

forall119905 ge t0 (34)




119868 (1199052) = 119868 (119905

2+ 120585) = 119868

lowast for 119905

2lt 119905 lt 119905

2+ 120585 (35)



2) such that

119868 (119905) ge119868lowast

2 for 119905

2le 119905 le 119905

2+ 120578 (36)




lowast119890minus(119889+120574+120578)120591


for 119905 isin [1199052 1199052+ 120585]




2+ 120591 1199052+ 120585]



119868holds for all 119905 gt 119905

119901




1198781015840(119905) gt 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119878 (37)


1(119905) = 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119911

1(119905) 119905 = 119896120591

1199111(119905+) = (1 minus 120579) 119911

1(119905) 119905 = 119896120591

(38)



119878 (119905) ge 1199111(119905) gt 119911

lowast

1(119905) minus 120576

ge(1 minus 119901119887 minus 119902119887) 119887 (119890

(119889+120582119887)120591minus 1)

(119889 + 120582119887) (119890(119889+120582119887)120591 minus 1 + 120579)minus 120576 =119898

119878gt 0

119896120591 lt 119905 le (119896 + 1) 120591

(39)

From (31) we have

119864 (119905) gt120582119898119868119898119878+ 119902119887119898

119868

120576 + 119889 minus 119901119887minus 120576 =119898

119864 (40)


0 10 20 30 40 502

3

4

5

6

7

8Th

e pop

ulat

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of su

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tible

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pute

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t

(a)

0 10 20 30 40 500

05

1

15

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opul

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expo

sed

com

pute

rs

t

(b)

0 10 20 30 40 500

1

01

02

03

04

05

06

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08

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cted

com

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t

(c)

0 10 20 30 40 502

3

4

5

6

7

8

9

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vere

d co

mpu

ters

t

(d)



(119905) gt 119887 minus 120583119873 (119905) minus 120572119873 (119905) (41)


119873(119905) gt119887

120583 + 120572minus 120576 =119898

119873 (42)

We letΩ0= (119878 119868119873) isin 119877

3

+| 119898119878le 119878119898

119868le 119868119898

119873le 119873 le 119887120583




0 Therefore system



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591










and 119877(0) = 8


0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

(a)

0 10 20 30 40 50t

1

12

14

16

18

2

22

24

26

The p

opul

atio

n of

expo

sed

com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

4

5

6

7

8

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)


5 Conclusion


1gt 1


1lt 1



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












gt 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


(28)


V1015840 (119905) = 119887 + 120577119887 minus 120578119868

lowast


minus (119901119887 + 120577)(119902119887 + 120582) 119887119868

lowast


119905 = 119896120591

V (119905+) = (1 minus 120579) V (119905) 119905 = 119896120591

V (0+) = 119878 (0+)

(29)



lowast119889(120576+



V (119905) =119860


119860

119861) 119890minus119861(119905minus119896120591)

119896120591 lt 119905 le (119896 + 1) 120591


minus119861120591)



(30)


1198641015840(119905) = 120582119868119878 + 119901119887119864 + 119902119887119868 minus 120576119864 minus 119889119864



120576 + 119889 minus 119901119887

(31)




120576 + 119889 minus 119901119887minus (120574 + 119889 + 120578) 119868

(32)


119868 (119905) gt120576

120574 + 119889 + 120578


120576 + 119889 minus 119901119887gt 119868lowast (33)



119868 (119905) lt 119868lowast

forall119905 ge t0 (34)




119868 (1199052) = 119868 (119905

2+ 120585) = 119868

lowast for 119905

2lt 119905 lt 119905

2+ 120585 (35)



2) such that

119868 (119905) ge119868lowast

2 for 119905

2le 119905 le 119905

2+ 120578 (36)




lowast119890minus(119889+120574+120578)120591


for 119905 isin [1199052 1199052+ 120585]




2+ 120591 1199052+ 120585]



119868holds for all 119905 gt 119905

119901




1198781015840(119905) gt 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119878 (37)


1(119905) = 119887 minus 119901119887

2minus 1199021198872minus (119889 + 120582119887) 119911

1(119905) 119905 = 119896120591

1199111(119905+) = (1 minus 120579) 119911

1(119905) 119905 = 119896120591

(38)



119878 (119905) ge 1199111(119905) gt 119911

lowast

1(119905) minus 120576

ge(1 minus 119901119887 minus 119902119887) 119887 (119890

(119889+120582119887)120591minus 1)

(119889 + 120582119887) (119890(119889+120582119887)120591 minus 1 + 120579)minus 120576 =119898

119878gt 0

119896120591 lt 119905 le (119896 + 1) 120591

(39)

From (31) we have

119864 (119905) gt120582119898119868119898119878+ 119902119887119898

119868

120576 + 119889 minus 119901119887minus 120576 =119898

119864 (40)


0 10 20 30 40 502

3

4

5

6

7

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

t

(a)

0 10 20 30 40 500

05

1

15

The p

opul

atio

n of

expo

sed

com

pute

rs

t

(b)

0 10 20 30 40 500

1

01

02

03

04

05

06

07

08

09

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opul

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n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 502

3

4

5

6

7

8

9

The p

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n of

reco

vere

d co

mpu

ters

t

(d)



(119905) gt 119887 minus 120583119873 (119905) minus 120572119873 (119905) (41)


119873(119905) gt119887

120583 + 120572minus 120576 =119898

119873 (42)

We letΩ0= (119878 119868119873) isin 119877

3

+| 119898119878le 119878119898

119868le 119868119898

119873le 119873 le 119887120583




0 Therefore system



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591










and 119877(0) = 8


0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

(a)

0 10 20 30 40 50t

1

12

14

16

18

2

22

24

26

The p

opul

atio

n of

expo

sed

com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

4

5

6

7

8

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)


5 Conclusion


1gt 1


1lt 1



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 502

3

4

5

6

7

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

t

(a)

0 10 20 30 40 500

05

1

15

The p

opul

atio

n of

expo

sed

com

pute

rs

t

(b)

0 10 20 30 40 500

1

01

02

03

04

05

06

07

08

09

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 502

3

4

5

6

7

8

9

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)



(119905) gt 119887 minus 120583119873 (119905) minus 120572119873 (119905) (41)


119873(119905) gt119887

120583 + 120572minus 120576 =119898

119873 (42)

We letΩ0= (119878 119868119873) isin 119877

3

+| 119898119878le 119878119898

119868le 119868119898

119873le 119873 le 119887120583




0 Therefore system



1 where 120579

1= 1 minus

119890(120577+119889)120591

+ 120576119887120582(119890(120577+119889)120591










and 119877(0) = 8


0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

(a)

0 10 20 30 40 50t

1

12

14

16

18

2

22

24

26

The p

opul

atio

n of

expo

sed

com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

4

5

6

7

8

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)


5 Conclusion


1gt 1


1lt 1



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50t

35

4

45

5

55

6

65

7

75

8Th

e pop

ulat

ion

of su

scep

tible

com

pute

rs

(a)

0 10 20 30 40 50t

1

12

14

16

18

2

22

24

26

The p

opul

atio

n of

expo

sed

com

pute

rs

(b)

0 10 20 30 40 50

065

07

075

08

085

09

095

1

The p

opul

atio

n of

infe

cted

com

pute

rs

t

(c)

0 10 20 30 40 501

2

3

4

5

6

7

8

The p

opul

atio

n of

reco

vere

d co

mpu

ters

t

(d)


5 Conclusion


1gt 1


1lt 1



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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