Research ArticleStability and Bifurcation of a Computer Virus PropagationModel with Delay and Incomplete Antivirus Ability

Jianguo Ren1 and Yonghong Xu2

1 College of Computer, Jiangsu Normal University, Xuzhou 221116, China2 College of Live Science, Jiangsu Normal University, Xuzhou 221116, China

Correspondence should be addressed to Jianguo Ren; rjgrjgrjgrjg@126.com

Received 12 March 2014; Revised 11 August 2014; Accepted 11 August 2014; Published 30 September 2014

Academic Editor: Jose R. C. Piqueira

Copyright © 2014 J. Ren and Y. Xu. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new computer virus propagation model with delay and incomplete antivirus ability is formulated and its global dynamics isanalyzed.The existence and stability of the equilibria are investigated by resorting to the threshold value 𝑅

0. By analysis, it is found

that the model may undergo a Hopf bifurcation induced by the delay. Correspondingly, the critical value of the Hopf bifurcationis obtained. Using Lyapunov functional approach, it is proved that, under suitable conditions, the unique virus-free equilibriumis globally asymptotically stable if 𝑅

0< 1, whereas the virus equilibrium is globally asymptotically stable if 𝑅

0> 1. Numerical

examples are presented to illustrate possible behavioral scenarios of the mode.

1. Introduction

With the rapid developments of information and communi-cation technologies, computer has brought great convenienceto our life. While enjoying the convenience from Internet,people have to confront the threat of virus intrusions. As thedamaging programs, computer viruses parasitize themselveson a host mainly through the Internet and have also becomean enormous threat to computers and network resources.So, understanding and predicting the dynamics of computervirus propagation are, therefore, an important pursuit. Con-sequently, a number of computer virus propagation models,ranging from conventional SIR compartment model [1–3]to its extensions [4–16], were proposed by borrowing fromclassical epidemic models to investigate the behaviors ofcomputer virus propagation over network.

There is something strikingly different between computerviruses and biological viruses: computer viruses in latentstatus possess infectivity [17–19]. Consequently, recentlyproposed models can distinguish latent computers frominfected ones by introducing the 𝐿 and 𝐵 compartments [17–19], named as the S(susceptible)-L(latent)-B(breaking-out)-S(susceptible) model, which represents the dynamics of virusby systems of ordinary differential equations.

One common feature shared by a computer virus islatency [20], which means that, when viruses enter in a host,they do not always immediately break out, but they hidethemselves and only become active after a certain period.It is therefore easy to show that there is an inevitable delayfrom virus invasion to its outbreak. On the other hand, in realnetworks, the limited cost results in the incomplete antivirusability. Indeed, when attempting to model computer viruspropagation, some of characteristics of viruses and networksshould be taken into consideration.

In this paper, a new computer virus propagation model,which incorporates simultaneously the above-mentionedaspects, is established. The aim is to extend and analyzethe SLBS computer virus propagation model without delayand incomplete antivirus ability first proposed by Yang etal. [17–19]. This study is motivated by the fact that thedelay plays a key role and is inevitably a complex impacton the investigation of computer virus spreading behaviors[21]. The incorporation of the delay and incomplete antivirusability of networks makes the model more realistic butits mathematical qualitative analysis may be difficult. Inour model, the existence and stability of the equilibria areinvestigated by resorting to the threshold value 𝑅

0, a certain

condition. By analysis, it is found that themodelmay undergo

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 475934, 9 pageshttp://dx.doi.org/10.1155/2014/475934

2 Mathematical Problems in Engineering

aHopf bifurcation induced by the delay. Correspondingly, thecritical value 𝜏

0of the Hopf bifurcation is obtained. When

delay 𝜏 < 𝜏0, the virus spreading is stable and easy to

protect; whereas 𝜏 > 𝜏0, the virus spreading is unstable and

out of control. Applying Lyapunov functional approach, itis proven that the unique virus-free equilibrium is globallyasymptotically stable under certain condition if 𝑅

0< 1,

whereas the virus equilibrium is globally asymptoticallystable if 𝑅

0> 1. Numerical examples are presented to

demonstrate the analytical results and to illustrate possiblebehavioral scenarios of the mode. Our results may providesome understanding of the spreading behaviors of computerviruses.

The organization of this paper is as follows. In the nextsection, we present the mathematical model to be discussed.In Section 3, we study the existence and local and globalstability of the virus-free and virus equilibria, respectively,and investigate the Hopf bifurcation. In Section 4, numericalexamples are presented to demonstrate the analytical results.Finally, some conclusions are given in Section 5.

2. Mathematical Model

Consider the typical SLBSmode [17–19], which is formulatedas the following system of ordinary differential equations:

d𝑆d𝑡

= 𝜇 − 𝛽𝑆 (𝑡) 𝐿 (𝑡) − 𝛽𝑆 (𝑡) 𝐵 (𝑡) + 𝛾𝐵 (𝑡) − 𝜇𝑆 (𝑡) ,

d𝐿d𝑡

= 𝛽𝑆 (𝑡) 𝐿 (𝑡) + 𝛽𝑆 (𝑡) 𝐵 (𝑡) − 𝛼𝐿 (𝑡) − 𝜇𝐿 (𝑡) ,

d𝐵d𝑡

= 𝛼𝐿 (𝑡) − 𝛾𝐵 (𝑡) − 𝜇 (𝑡) 𝐵.

(1)

Here, it is assumed that a computer (or node) is categorizedas internal or external depending on whether or not itis currently connected to the network. The total numberof computers connected to the network is divided intothree compartments: internal uninfected compartment, (i.e.,virus-free computers), internal infected compartment wherecomputers are currently latent (latent computers, for short),and internal infected compartment where computers are cur-rently breaking out (breaking-out computers, for short). Let𝑆(𝑡), 𝐿(𝑡), and 𝐵(𝑡) denote their corresponding percentagesat time 𝑡, respectively. This model involves four positiveparameters: 𝜇 denotes the rate at which external virus-freecomputers are connected to the network and at which aninternal node is disconnected from the network; 𝛽 denotesthe rate at which, when having connection to one latent orbreaking-out computer, one virus-free computer can becomeinfected; 𝛾 denotes the rate at which a breaking-out computergets a scan by running the antivirus software; 𝛼 denotes therate at which the latent computer is triggered.

By carefully considering the natures of computer virus,the following assumptions are made.

(i) At time 𝑡, the transition from the latent to thebreaking-out is given by 𝐿(𝑡−𝜏), whichmeans a latentcomputer moves into the breaking-out compartmentafter a period of time 𝜏.

(ii) Since the antivirus ability is incomplete, at time 𝑡, thebreaking-out computers may either be temporarilysuppressed in their latency with probability 𝑒 or becured into the virus-free ones with probability (1− 𝑒),where 𝑒 > 0 is a constant. If 𝑒 = 0, then the antivirusability is fully effective, whereas 𝑒 = 1 means thatantivirus ability is utterly ineffective.

Based on the assumptions above, one can obtain the followingcomputer virus propagation model:

d𝑆d𝑡

= 𝜇 − 𝛽𝑆 (𝑡) 𝐿 (𝑡) − 𝛽𝑆 (𝑡) 𝐵 (𝑡)

+ (1 − 𝑒) 𝛾𝐵 (𝑡) − 𝜇𝑆 (𝑡) ,

d𝐿d𝑡

= 𝛽𝑆 (𝑡) 𝐿 (𝑡) + 𝛽𝑆 (𝑡) 𝐵 (𝑡)

− 𝛼𝐿 (𝑡 − 𝜏) + 𝑒𝛾𝐵 (𝑡) − 𝜇𝐿 (𝑡) ,

d𝐵d𝑡

= 𝛼𝐿 (𝑡 − 𝜏) − 𝛾𝐵 (𝑡) − 𝜇 (𝑡) 𝐵.

(2)

Let 𝑆(𝑡) + 𝐿(𝑡) + 𝐵(𝑡) = 1. Thus, model (2) can be written asthe following:

d𝐿d𝑡

= 𝛽 [1 − 𝐿 (𝑡) − 𝐵 (𝑡)] [𝐿 (𝑡) + 𝐵 (𝑡)]

− 𝛼𝐿 (𝑡 − 𝜏) + 𝑒𝛾𝐵 (𝑡) − 𝜇𝐿 (𝑡) ,

d𝐵d𝑡

= 𝛼𝐿 (𝑡 − 𝜏) − 𝛾𝐵 (𝑡) − 𝜇𝐵 (𝑡) .

(3)

All the parameters are positive constants. The initial condi-tions are

(𝜙1(𝜃) , 𝜙

2(𝜃)) ∈ 𝐶

+= ((−𝜏, 0] , 𝑅

2

+) , 𝜙

𝑖(𝜃) > 0, 𝑖 = 1, 2,

(4)

where 𝑅2

+= {(𝐿, 𝐵) ∈ 𝑅

2, 𝐿 ≥ 0, 𝐵 ≥ 0}.

3. Model Analysis

In this section, we intended to study the dynamical behaviorsof model (3). First, a threshold value 𝑅

0is defined as the

number of virus-free computers that are infected by a singlecomputer virus during its lift span. The threshold value 𝑅

0

plays a key role in the epidemic dynamics. By resorting to it,the existence and stability of the equilibria can be determined.Generally speaking, if 𝑅

0< 1, the virus-free equilibrium is

globally asymptotically stable, and when 𝑅0

> 1, the virusequilibrium exists and is globally asymptotically stable. Adirect computation gives

𝑅0=

𝛽 (𝛼 + 𝜇 + 𝛾)

(𝜇 + 𝛼) (𝜇 + 𝛾) − 𝛼𝑒𝛾. (5)

3.1. Stability of Virus-Free Equilibrium. It is clear that model(3) always admits unique virus-free equilibrium 𝐸

0(0, 0). We

Mathematical Problems in Engineering 3

first consider its stability. For model (3), the correspondingcharacteristic equation at 𝐸

0is

𝜆2+ 𝑚1𝜆 + 𝑚

0+ (𝛼𝜆 + 𝑛

0) 𝑒−𝜆𝜏

= 0, (6)

which is similar to the relative forms in [22–24], where

𝑚1= 2𝜇 + 𝛾 − 𝛽,

𝑚0= (𝜇 − 𝛽) (𝜇 + 𝛾) ,

𝑛0= [(𝜇 + 𝛾) − (𝛽 + 𝑒𝛾)] 𝛼.

(7)

Our aim is to investigate the stability behavior in the case 𝜏 =

0. Obviously, (6) is a transcendental equation, and 𝑖𝜂 (𝜂 > 0)

is its root if and only if 𝜂 satisfies

−𝜂2+ 𝑚1𝑖𝜂 + 𝑚

0= − (𝛼𝑖𝜂 + 𝑛

0) (cos 𝜂𝜏 − 𝑖 sin 𝜂𝜏) . (8)

Separating the real and imaginary parts, we have

−𝜂2+ 𝑚0= −𝑛0cos 𝜂𝜏 − 𝛼𝜂 sin 𝜂𝜏

𝑚1𝜂 = −𝛼𝜂 cos 𝜂𝜏 + 𝑛

0sin 𝜂𝜏.

(9)

Eliminating 𝜏 by squaring and adding (9), we obtain apolynomial in 𝜂 as

𝜂4+ (𝑚2

1− 𝛼2− 2𝑚0) 𝜂2+ 𝑚2

0− 𝑛2

0= 0, (10)

where

𝑚2

1− 2𝑚0− 𝛼2= [(𝜇 − 𝛽) + (𝜇 + 𝛾)]

2

− 2 (𝜇 − 𝛽) (𝜇 + 𝛾) − 𝛼2

= (𝜇 + 𝛾)2+ (𝜇 − 𝛽)

2− 𝛼2,

𝑚0+ 𝑛0= (𝜇 − 𝛽) (𝜇 + 𝛾) + (𝜇 + 𝛾) 𝛼 − (𝛽 + 𝑒𝛾) 𝛼

= (𝜇 + 𝛾) (𝜇 + 𝛼) − 𝛼𝑒𝛾 − 𝛽 (𝜇 + 𝛾 + 𝛼)

= [(𝜇 + 𝛾) (𝜇 + 𝛼) − 𝛼𝑒𝛾] (1 − 𝑅0) > 0,

𝑚0− 𝑛0= (𝜇 − 𝛽) (𝜇 + 𝛾) − (𝜇 + 𝛾) 𝛼 + (𝛽 + 𝑒𝛾) 𝛼

= (𝜇 − 𝛽 − 𝛼) (𝜇 + 𝛾) + 𝛼𝛽 + 𝛼𝑒𝛾,

𝑚2

0− 𝑛2

0= (𝑚0+ 𝑛0) (𝑚0− 𝑛0)

= {[(𝜇 + 𝛾) (𝜇 + 𝛼) − 𝛼𝑒𝛾] (1 − 𝑅0)}

× {(𝜇 − 𝛽 − 𝛼) (𝜇 + 𝛾) + 𝛼𝛽 + 𝛼𝑒𝛾} .

(11)

Clearly, if 𝜇 > 𝛼+𝛽, then𝑚2

1−2𝑚0−𝛼2> 0 and𝑚

2

0−𝑛2

0> 0. It

follows from the Hurwitz criterion that the roots of (10) havenegative real parts. Hence, we have the following.

Theorem 1. When 𝑅0

< 1, the virus-free equilibrium 𝐸0is

locally asymptotically stable for all 𝜏 > 0 provided that 𝜇 >

𝛼 + 𝛽.Now, it is the turn to examine the global stability of virus-

free equilibrium. The following theorem is obtained.

Theorem 2. When 𝑅0

< 1, the virus-free equilibrium 𝐸0is

globally asymptotically stable for all 𝜏 > 0.

Proof. By use of the Lyapunov Direct Method, consider thefollowing function:

𝑉 (𝐿, 𝐵) = 𝐿 + 𝐵. (12)

It is clear that 𝑉 is a positive definite. Then the derivative of𝑉 is

�� (𝐿, 𝐵) =dd𝑡

(1 − 𝑆)

= −𝜇 + 𝛽𝑆 (𝐿 + 𝐵) − (1 − 𝑒) 𝛾𝐵 + 𝜇𝑆

= 𝜇 (𝑆 − 1) + 𝛽𝑆 (1 − 𝑆) − (1 − 𝑒) 𝛾𝐵

= (𝑆 − 1) (𝜇 − 𝛽𝑆) − (1 − 𝑒) 𝛾𝐵.

(13)

If 𝜇 > 𝛼 + 𝛽 hold, then (𝑆 − 1)(𝜇 − 𝛽𝑆) < 0. Furthermore,��(𝐿, 𝐵) < 0 by the Lyapunov-LaSalle type theorem shows thatlim𝑡→∞

𝐿(𝑡) = 0 and lim𝑡→∞

𝐵(𝑡) = 0. Hence, when 𝑅0< 1,

the virus-free equilibrium 𝐸∗is globally asymptotically sta-

ble.

3.2. Stability of Virus Equilibrium. Next, we examine thestability of virus equilibrium. After direct computations, theunique virus equilibrium 𝐸

∗(𝐿∗, 𝐵∗) of model (3) reads

𝐿∗=

(𝛾 + 𝜇) 𝐵∗

𝛼=

(𝛾 + 𝜇) (𝑅0− 1)

𝑅0(𝛾 + 𝜇 + 𝛼)

,

𝐵∗=

𝛽𝛼 (𝛾 + 𝜇 + 𝛼) − 𝛼 (𝛾 + 𝜇) (𝜇 + 𝛼) + 𝛼2𝑒𝛾

𝛽(𝛾 + 𝜇 + 𝛼)2

=𝛼 (𝑅0− 1)

𝑅0(𝛾 + 𝜇 + 𝛼)

.

(14)

If 𝑅0

< 1, the 𝐸∗(𝐿∗, 𝐵∗) does not exist. It suffices to show

the local asymptotical stability of 𝐸∗for model (3). Indeed,

the Jacobian matrix of the linearized system of this systemevaluated at 𝐸

∗is

(𝛽 − 2𝛽 (𝐿

∗+ 𝐵∗) − 𝜇 − 𝛼𝑒

−𝜆𝜏− 𝜆 𝛽 − 2𝛽 (𝐿

∗+ 𝐵∗) + 𝑒𝛾

𝛼𝑒−𝜆𝜏

− (𝜇 + 𝛾) − 𝜆) .

(15)

Its characteristic equation is

𝜆2+ 𝑝1𝜆 + 𝑝0+ (𝛼𝜆 + 𝑞

0) 𝑒−𝜆𝜏

= 0, (16)

where

𝑝1= 2𝜇 + 𝛾 − 𝛽 + 2𝛽 (𝐿

∗+ 𝐵∗) ,

𝑝0= (𝜇 − 𝛽) (𝜇 + 𝛾) + 2𝛽 (𝜇 + 𝛾) (𝐿

∗+ 𝐵∗) ,

𝑞0= (𝜇 + 𝛾) 𝛼 − [𝛽 + 𝑒𝛾 − 2𝛽 (𝐿

∗+ 𝐵∗)] 𝛼.

(17)

In the case 𝜏 = 0, for our purpose, if 𝑖𝜂 (𝜂 > 0) is a solutionof (16), separating real and imaginary parts, we derive that

−𝜂2+ 𝑝1𝑖𝜂 + 𝑝

0= − (𝛼𝑖𝜂 + 𝑞

0) (cos 𝜂𝜏 − 𝑖 sin 𝜂𝜏) . (18)

4 Mathematical Problems in Engineering

Separating the real and imaginary parts yields

−𝜂2+ 𝑝0= −𝑞0cos 𝜂𝜏 − 𝛼𝜂 sin 𝜂𝜏,

𝑝1𝜂 = −𝛼𝜂 cos 𝜂𝜏 + 𝑞

0sin 𝜂𝜏.

(19)

Eliminating 𝜏 by squaring and adding (19), we obtain apolynomial in 𝜂 as

𝜂4+ (𝑝2

1− 𝛼2− 2𝑝0) 𝜂2+ 𝑝2

0− 𝑞2

0= 0. (20)

For convenience, let ℎ1

≜ 𝑝2

1− 𝛼2− 2𝑝0and ℎ

2≜ 𝑝2

0− 𝑞2

0.

If ℎ1

> 0 and ℎ2

> 0, then both of the two roots of (20)have negative real parts. By theHurwitz criterion,𝐸

∗is locally

asymptotically stable for all 𝜏 > 0. However, if ℎ1

< 0 andℎ2> 0, then (20) has the positive root 𝜂

1= (1/2)(𝛼

2+ 2𝑝0−

𝑝2

1+ √(𝑝

2

1− 𝛼2 − 2𝑝

0)2− 4(𝑝2

0− 𝑞2

0)). It follows that (16) has

a positive root. Say, the characteristic equation (16) has a pairof imaginary roots ±𝑖𝜂

0, and corresponding delay 𝜏

0is given

by (19):

𝜏0=

1

𝜂arccos[

𝜂2(1 − 𝛼𝑝

1) − 𝑝0

𝜂2𝛼2 + 𝑞0

] +2𝑘𝜋

𝜂,

𝑘 = 0, 1, 2, 3, . . . .

(21)

Furthermore, we can also verify the transversality conditiondR((𝜆(𝜏)))/d𝜏‖

𝜏=𝜏0> 0. Then, we establish the following

theorem.

Theorem 3. With 𝑅0> 1, model (3) has a unique virus equi-

librium 𝐸∗. Furthermore,

(1) 𝐸∗is locally asymptotically stable when 𝜏 < 𝜏

0and is

unstable when 𝜏 > 𝜏0, where

𝜏0=

1

𝜂arccos[

𝜂2(1 − 𝛼𝑝

1) − 𝑝0

𝜂2𝛼2 + 𝑞0

] +2𝑘𝜋

𝜂,

𝑘 = 0, 1, 2, 3, . . . ;

(22)

(2) when 𝜏 = 𝜏0, the model (3) undergoes a Hopf bifurca-

tion.Now, we are ready to examine the global stability of virus equi-librium 𝐸

∗. Then we have the following.

Theorem4. When𝑅0> 1, the virus equilibrium𝐸

∗is globally

asymptotically stable.

Proof. We consider the following function:

𝑉1(𝐿, 𝐵) =

1

2[(𝐿 − 𝐿

∗) + (𝐵 − 𝐵

∗)]2+

1

2𝜔(𝐿 − 𝐿

∗)2, (23)

where 𝜔 is a positive constant to be determined. The deriva-tive of 𝑉

1is

��1(𝐿, 𝐵) = (𝑆 − 𝑆

∗)d𝑆d𝑡

+ 𝜔 (𝐿 − 𝐿∗)d𝐿d𝑡

= (𝑆 − 𝑆∗) [−𝛽 (𝑆 − 𝑆

∗) (1 − 𝑆 − 𝑆

∗)

− (1 − 𝑒) 𝛾 (𝐿 − 𝐿∗)

− [(1 − 𝑒) 𝛾 + 𝜇] (𝑆 − 𝑆∗)]

+ 𝜔 (𝐿 − 𝐿∗) [𝛽 (𝑆 − 𝑆

∗) (1 − 𝑆 − 𝑆

∗)

− 𝛼 (𝐿𝜏 − 𝐿∗) − 𝑒𝛾 (𝑆 − 𝑆

∗)

− (𝑒𝛾 + 𝜇) (𝐿 − 𝐿∗)]

= [𝛽 (𝑆 + 𝑆∗− 1) − (1 − 𝑒) 𝛾 − 𝜇]

× (𝑆 − 𝑆∗)2− 𝜔 (𝑒𝛾 + 𝜇) (𝐿 − 𝐿

∗)2

− 𝜔𝛼 (𝐿 − 𝐿∗) (𝐿𝜏 − 𝐿

∗)

+ [𝜔𝛽 (1 − 𝑆 − 𝑆∗) − 𝜔𝑒𝛾 − (1 − 𝑒) 𝛾]

× (𝑆 − 𝑆∗) (𝐿 − 𝐿

∗)

= 𝛽 (𝑆 − 1) (𝑆 − 𝑆∗)2+ [𝛽𝑆∗− (1 − 𝑒) 𝛾 − 𝜇]

× (𝑆 − 𝑆∗)2− 𝜔 (𝑒𝛾 + 𝜇) (𝐿 − 𝐿

∗)2

− 𝜔𝛼 (𝐿 − 𝐿∗) (𝐿𝜏 − 𝐿

∗)

+ [𝜔𝛽 (1 − 𝑆 − 𝑆∗) − 𝜔𝑒𝛾 − (1 − 𝑒) 𝛾]

× (𝑆 − 𝑆∗) (𝐿 − 𝐿

∗)

= 𝛽 (𝑆 − 1) (𝑆 − 𝑆∗)2−

(𝜇 + 𝛾) (1 − 𝑒) 𝛾

𝛼 + 𝜇 + 𝛾

× (𝑆 − 𝑆∗)2− 𝜔 (𝑒𝛾 + 𝜇) (𝐿 − 𝐿

∗)2

− 𝜔𝛼 (𝐿 − 𝐿∗) (𝐿𝜏 − 𝐿

∗)

+ [𝜔𝛽 (1 − 𝑆 − 𝑆∗) − 𝜔𝑒𝛾 − (1 − 𝑒) 𝛾]

× (𝑆 − 𝑆∗) (𝐿 − 𝐿

∗)

= − [(𝜇 + 𝛾) (1 − 𝑒) 𝛾

𝛼 + 𝜇 + 𝛾(𝑆∗− 𝑆)2

+ 𝜔 (𝑒𝛾 + 𝜇) (𝐿 − 𝐿∗)2

− [𝜔𝛽 (𝑆 − 1) + 𝜔𝛽𝑆∗+ 𝜔𝑒𝛾 + (1 − 𝑒) 𝛾]

× (𝑆∗− 𝑆) (𝐿 − 𝐿

∗)]

+ 𝛽 (𝑆 − 1) (𝑆 − 𝑆∗)2

− 𝜔𝛼 (𝐿 − 𝐿∗) (𝐿𝜏 − 𝐿

∗) .

(24)

Let(𝜇 + 𝛾) (1 − 𝑒) 𝛾

𝛼 + 𝜇 + 𝛾× 𝜔 (𝑒𝛾 + 𝜇) ≥ 2𝜔𝛽 (𝑆 − 1)

+ 2𝜔𝛽𝑆∗+ 2𝜔𝑒𝛾 + 2 (1 − 𝑒) 𝛾.

(25)

That is,

(𝜇 + 𝛾) (1 − 𝑒) 𝛾

𝛼 + 𝜇 + 𝛾× 𝜔 (𝑒𝛾 + 𝜇) − 2𝜔 (𝛽𝑆

∗+ 𝑒𝛾) ≥ (1 − 𝑒) 𝛾,

(26)

Mathematical Problems in Engineering 5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

The number of latent nodes L

The n

umbe

r of b

reak

ing

node

s B

−0.05−0.05

E0

Figure 1: Phase diagram of 𝐿(𝑡) and 𝐵(𝑡) in the case 𝛽 = 0.02, 𝛼 =

0.23, 𝜇 = 0.4, 𝛾 = 0.58, 𝑒 = 0.35, and 𝜏 = 1 under the different valuesof 𝐿(0) and 𝐵(0).

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

Time t

−0.05

L+B

Figure 2: Evolutions of 𝐿(𝑡) + 𝐵(𝑡) in the case 𝛽 = 0.05, 𝛼 = 0.3,𝜇 = 0.4, 𝛾 = 0.58, 𝑒 = 0.18, and 𝜏 = 2 under the values of 𝐿(0) = 0.1

and 𝐵(0) = 0.1.

under the condition of (𝜇 + 𝛾)(1 − 𝑒)𝜆(𝑒𝛾 + 𝜇) − 2(𝛽𝑆∗

+

𝑒𝛾)(𝛼 + 𝜇 + 𝛾) > 0, from which we can conclude that 𝜔 >

(1−𝑒)𝛾(𝛼+𝜇+𝛾)/((𝜇+𝛾)(1−𝑒)𝜆(𝑒𝛾+𝜇)−2(𝛽𝑆∗+𝑒𝛾)(𝛼+𝜇+𝛾)).

In addition, suppose that 𝜏 is small enough, then𝐿𝜏 ≈ 𝐿.Theylead to 𝑉 < 0. Applying the Lyapunov-LaSalle type theorem,it shows that lim

𝑡→∞𝐿(𝑡) = 𝐿

∗and lim

𝑡→∞𝐵(𝑡) = 𝐵

∗.

Hence, when 𝑅0

> 1, the virus equilibrium 𝐸∗is globally

asymptotically stable.

4. Numerical Simulations and Discussion

In this section, numerical simulations are carried out tosupport the analytical conclusion and to illustrate possiblebehavioral scenarios of the model. Figure 1 exhibits theevolutions of 𝐿(𝑡) and 𝐵(𝑡) with time, where the virus-free

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

L

B

E∗

Figure 3: Phase diagramof𝐿(𝑡) and𝐵(𝑡) in the case𝛽 = 0.4,𝛼 = 0.1,𝜇 = 0.05, 𝛾 = 0.15, 𝑒 = 0.5, 𝜏 = 1, 𝑅

0= 5.33 > 1, and 𝐸

∗=

(0.542, 0.271) under the different values of 𝐿(0) and 𝐵(0).

0 2 4 6 8 10 12 14 16 18 200.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9L+B

Time t

Figure 4: Evolutions of 𝐿(𝑡) + 𝐵(𝑡) in the case 𝛽 = 0.56, 𝛼 = 0.15,𝜇 = 0.05, 𝛾 = 0.15, 𝑒 = 0.5, 𝜏 = 3, and 𝑅

0= 6.817 > 1 under the

values of 𝐿(0) = 0.3 and 𝐵(0) = 0.2.

equilibrium is globally asymptotically stable, consistent withTheorem 2. Furthermore, an equilibrium is virus-free if andonly if 𝐿(𝑡) + 𝐵(𝑡) = 0, which means that the virus would beextinct in the network, as shown inFigure 2. Figure 3 plots theevolutions of 𝐿(𝑡) and 𝐵(𝑡) with time. One can observe that,for any initial state, the solution would approach a fixed level;that is, the virus equilibrium is globally asymptotically stable.Besides, an equilibrium is viral if and only if 𝐿(𝑡) + 𝐵(𝑡) =

0, and its global asymptotical stability means that the virusspreads in the network continuously and stably, as shownin Figure 4. Figure 5 illustrates the complex impacts of delay𝜏 on the spreading behavior of the virus. The evolutions ofcomparing 𝜏 < 𝜏

0with 𝜏 > 𝜏

0between 𝐿(𝑡) and 𝐵(𝑡) are

carried out. It can be seen that, virus equilibrium 𝐸∗is stable

when 𝜏 < 𝜏0and then when delay 𝜏 increases to the critical

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

Time t0 50 100 150 200 250 300 350 400 450 500

Time t

Lant

ent a

nd b

reak

ing

com

pute

rs

Lant

ent a

nd b

reak

ing

com

pute

rs

L

B

L

B

0

0.5

1𝜏 < 𝜏0 𝜏 > 𝜏0

−0.5

Figure 5: Evolutions of comparison of 𝜏 < 𝜏0with 𝜏 < 𝜏

0between 𝐿(𝑡) and 𝐵(𝑡) in the case 𝛽 = 0.85, 𝛼 = 0.85, 𝜇 = 0.2, 𝛾 = 0.095, 𝑒 = 0.85,

𝜏 = 2.3 (left) and 𝜏 = 2.4 (right), 𝑅0= 4.05 > 1, and 𝐸

∗= (0.193, 0.559) under the values of 𝐿(0) = 0.006 and 𝐵(0) = 0.001.

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

Late

nt an

d br

eaki

ng co

mpu

ters

0

0.2

0.4

0.6

0.8

Late

nt an

d br

eaki

ng co

mpu

ters

0

0.2

0.4

0.6

0.8

Late

nt an

d br

eaki

ng co

mpu

ters

0

0.2

0.4

0.6

0.8La

tent

and

brea

king

com

pute

rs

B

L

B

L

B

L

B

L

0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700

0 100 200 300 400 500 600 700

𝜏 = 5𝜏 = 10

𝜏 = 15𝜏 = 25

Time t

Time t Time t

Time t

Figure 6: Evolutions of 𝐿(𝑡) and 𝐵(𝑡) with the different values of time delay 𝜏 in the case 𝛽 = 0.85, 𝛼 = 0.85, 𝜇 = 0.2, 𝛾 = 0.95, 𝑒 = 0.85,𝑅0= 3.26 > 1, and 𝐸

∗= (0.399, 0.295) under the values of 𝐿(0) = 0.006 and 𝐵(0) = 0.001.

Mathematical Problems in Engineering 7

0 10 20 30 40 50 60 70 80 90 1000.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 10 20 30 40 50 60 70 80 90 100

L

0.1

0.15

0.2

0.25

0.3

0.35

0.4

B

𝛾 = 0.15

𝛾 = 0.55𝛾 = 0.15

𝛾 = 0.55

Time t Time t

Figure 7: Evolutions of 𝐿(𝑡) and 𝐵(𝑡) with the different scan rates in the case 𝛽 = 0.85, 𝛼 = 0.25, 𝜇 = 0.2, 𝑒 = 0.15, and 𝜏 = 2.5 under thevalues of 𝐿(0) = 0.6 and 𝐵(0) = 0.2.

0 10 20 30 40 50 60 70 80 90 1000.35

0.4

0.45

0.5

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

Time t0 10 20 30 40 50 60 70 80 90 100

Time t

L

e = 0.35

e = 0.65

e = 0.35

e = 0.65

B

Figure 8: Evolutions of 𝐿(𝑡) and 𝐵(𝑡) with the different parameter 𝑒 in the case 𝛽 = 0.85, 𝛼 = 0.25, 𝜇 = 0.2, 𝛾 = 0.35, and 𝜏 = 2.5 under thevalues of 𝐿(0) = 0.6 and 𝐵(0) = 0.2.

value 𝜏0, it loses its stability and aHopf bifurcation arises; then

it exceeds the value of 𝜏0beyond which the virus propagation

will become unstable, in agreement with Theorem 3. InFigure 6, the effect of delay with 𝜏 ∈ (5, 10, 15, 25) on thenumber of latent and breaking-out computers is illustrated.The role of key parameters 𝛾 and 𝑒 in the variation of the latentand breaking-out compartments is shown in Figures 7-8. Asexpected, one can observe that, for higher value of scan rates𝛾, the percentage of latent computers increases, in contrastto that of breaking-out ones. However, the percentages ofboth latent and breaking-out computers rise as 𝑒 increases.Figure 9 shows the appearance of periodic solutions with thetransmission from the stable state to the unstable one.

5. Conclusions

In real networks, the outbreak of computer virus usually lagsand the antivirus ability of network is not fully complete.Aiming at characterizing these situations, a new computervirus propagation model is established. By theoretical anal-ysis, the following conclusions can be obtained.

(1) If 𝑅0< 1 hold, the virus-free equilibrium 𝐸

0is glob-

ally asymptotically stable under certain conditions forall 𝜏 > 0, which implies that the viruswould be extinctin the network. In such conditions, it is unnecessaryfor us to take practices in a real network. Say, the virusshould be left alone.

8 Mathematical Problems in Engineering

0

0.2

0.4

0.6

0.8

1

Late

nt an

d br

eaki

ng co

mpu

ters

L

B

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

Figure 9: Appearance of periodic solutions in the case 𝛽 = 0.85, 𝛼 =

0.85, 𝜇 = 0.2, 𝛾 = 0.095, 𝑒 = 0.85, 𝜏 = 2.375, and 𝐸∗= (0.193, 0.559)

under the values of 𝐿(0) = 0.35 and 𝐵(0) = 0.45.

(2) If 𝑅0

> 1 hold, the virus equilibrium 𝐸0is globally

asymptotically stable, which means that the virusesspread in the network continuously and stably. Inthis case, some efforts can be made to keep the virusprevalence to below a proper level.

(3) The critical delay 𝜏0where the Hopf bifurcation

occurs is obtained, where

𝜏0=

1

𝜂arccos[

𝜂2(1 − 𝛼𝑝

1) − 𝑝0

𝜂2𝛼2 + 𝑞0

] +2𝑘𝜋

𝜂,

𝑘 = 0, 1, 2, 3, . . . .

(27)

(4) When the delay 𝜏 < 𝜏0, the virus propagation is stable.

In such conditions, the spreading behavior of viruswould be divinable.

(5) When the delay 𝜏 > 𝜏0, the virus propagation is unsta-

ble. In such conditions, the virus spreading would beout of control.

Moreover, numerical simulations are presented to demon-strate the analytical results and to illustrate possible behav-ioral scenarios of the model. It is shown that

(1) For virus equilibrium, the larger the delay is, thelonger it takes to settle down towards its steady states.

(2) As expected, the increase of the scan rate can reducethe percentage of the breaking-out computers butincrease the percentage of the latent ones, which sug-gests that we run the antivirus software as often aspossible.

(3) As expected, the increase of the antivirus ability ofthe software can reduce the percentage of the infected

(latent and breaking-out) computers in the network,which suggests that we invest more in their develop-ments.

Our results may provide some understanding of the spread-ing behaviors of computer viruses.

Conflict of Interests

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

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China underGrant no. 61304117, theNatural ScienceFoundation of the Jiangsu Higher Education Institutions ofChina under Grant no. 13KJB520008, and the DoctorateTeacher Support Project of Jiangsu Normal University underGrant no. 12XLR021.

References

[1] J. O. Kephart, T. Hogg, and B. A. Huberman, “Dynamics ofcomputational ecosystems,” Physical Review A, vol. 40, no. 1, pp.404–421, 1989.

[2] J. C. Wierman and D. J. Marchette, “Modeling computer virusprevalence with a susceptible-infected-susceptible model withreintroduction,”Computational Statistics andDataAnalysis, vol.45, no. 1, pp. 3–23, 2004.

[3] J. R. C. Piqueira and V. O. Araujo, “A modified epidemiologicalmodel for computer viruses,”Applied Mathematics and Compu-tation, vol. 213, no. 2, pp. 355–360, 2009.

[4] J. G. Ren, X. F. Yang, Q. Y. Zhu, L. X. Yang, and C. Zhang,“A novel computer virus model and its dynamics,” NonlinearAnalysis: Real World Applications, vol. 13, no. 1, pp. 376–384,2012.

[5] B. K.Mishra andD. K. Saini, “SEIRS epidemicmodel with delayfor transmission of malicious objects in computer network,”AppliedMathematics and Computation, vol. 188, no. 2, pp. 1476–1482, 2007.

[6] M. Draief, A. Ganesh, and L. Massoulie, “Thresholds for virusspread on networks,” The Annals of Applied Probability, vol. 18,no. 2, pp. 359–378, 2008.

[7] B. K.Mishra andN. Jha, “SEIQRSmodel for the transmission ofmalicious objects in computer network,” Applied MathematicalModelling, vol. 34, no. 3, pp. 710–715, 2010.

[8] B. K. Mishra and N. Jha, “Fixed period of temporary immu-nity after run of anti-malicious software on computer nodes,”AppliedMathematics and Computation, vol. 190, no. 2, pp. 1207–1212, 2007.

[9] Q. Zhu, X. Yang, and J. Ren, “Modeling and analysis ofthe spread of computer virus,” Communications in NonlinearScience and Numerical Simulation, vol. 17, no. 12, pp. 5117–5124,2012.

[10] C. Gan, X. Yang, W. Liu, Q. Zhu, and X. Zhang, “An epidemicmodel of computer viruses with vaccination and generalizednonlinear incidence rate,” Applied Mathematics and Computa-tion, vol. 222, pp. 265–274, 2013.

Mathematical Problems in Engineering 9

[11] C. Zhang, Y. Zhao, and Y.Wu, “An impulse model for computerviruses,” Discrete Dynamics in Nature and Society, vol. 2012,Article ID 260962, 13 pages, 2012.

[12] S. Kondakci, “Epidemic state analysis of computers undermalware attacks,” SimulationModelling Practice andTheory, vol.16, no. 5, pp. 571–584, 2008.

[13] F. G. Wang, Y. K. Zhang, C. G. Wang, J. F. Ma, and S. J.Moon, “Stability analysis of a SEIQV epidemic model for rapidspreading worms,” Computers and Security, vol. 29, no. 4, pp.410–418, 2010.

[14] L. P. Song, Z. Jin, G. Q. Sun, J. Zhang, and X. Han, “Influence ofremovable devices on computer worms: dynamic analysis andcontrol strategies,”Computers &Mathematics with Applications,vol. 61, no. 7, pp. 1823–1829, 2011.

[15] J. Amador, “The stochastic SIRA model for computer viruses,”Applied Mathematics and Computation, vol. 232, pp. 1112–1124,2014.

[16] K. M. Fuhrman, G. A. Pinter, and J. A. Berges, “Dynamics ofa virus-host model with an intrinsic quota,” Mathematical andComputer Modelling, vol. 53, no. 5-6, pp. 716–730, 2011.

[17] L. Yang, X. Yang, L. Wen, and J. Liu, “A novel computer viruspropagation model and its dynamics,” International Journal ofComputer Mathematics, vol. 89, no. 17, pp. 2307–2314, 2012.

[18] X. Yang and L.-X. Yang, “Towards the epidemiological mod-eling of computer viruses,” Discrete Dynamics in Nature andSociety, vol. 2012, Article ID 259671, 2012.

[19] L. Yang, X. Yang, Q. Zhu, and L. Wen, “A computer virusmodel with graded cure rates,” Nonlinear Analysis: Real WorldApplications, vol. 14, no. 1, pp. 414–422, 2013.

[20] Y. B. Kafai, “Understanding virtual epidemics: children’s folkconceptions of a computer virus,” Journal of Science Educationand Technology, vol. 17, no. 6, pp. 523–529, 2008.

[21] Z. H. Zuo, Q. X. Zhu, and M. T. Zhou, “On the time complexityof computer viruses,” IEEE Transactions on InformationTheory,vol. 51, no. 8, pp. 2962–2966, 2005.

[22] D. Li and W. Ma, “Asymptotic properties of a HIV-1 infectionmodel with time delay,” Journal of Mathematical Analysis andApplications, vol. 335, no. 1, pp. 683–691, 2007.

[23] R. Naresh, A. Tripathi, and D. Sharma, “A nonlinear AIDSepidemic model with screening and time delay,” Applied Math-ematics and Computation, vol. 217, no. 9, pp. 4416–4426, 2011.

[24] J. Wang, J. Zhang, and Z. Jin, “Analysis of an SIR modelwith bilinear incidence rate,” Nonlinear Analysis: Real WorldApplications, vol. 11, no. 4, pp. 2390–2402, 2010.

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