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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 371792, 12 pages
http://dx.doi.org/10.1155/2012/371792
Research Article

Dynamics of a Delay-Varying Computer Virus Propagation Model

1College of Computer, Jiangsu Normal University, Xuzhou 221116, China
2College of Bioengineering, Chongqing University, Chongqing 400044, China

Received 4 June 2012; Accepted 19 July 2012

Academic Editor: Xiaofan Yang

Copyright © 2012 Jianguo Ren et al. 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.

Abstract

By considering the varying latency period of computer virus, we propose a novel model for computer virus propagation in network. Under this model, we give the threshold value determining whether or not the virus finally dies out, and study the local stability of the virus-free and virus equilibrium. It is found that the model may undergo a Hopf bifurcation. Next, we use different methods to prove the global asymptotic stability of the equilibria: the virus-free equilibrium by using the direct Lyapunov method and virus equilibrium by using a geometric approach. Finally, some numerical examples are given to support our conclusions.

1. Introduction

With the advance of computer software and hardware and communication technologies, the number and sort of computer viruses have increased dramatically, which causes huge losses to the human society. Therefore, establishing reasonable computer-virus-propagation models by considering the characteristics of computer virus and, by model analysis, understanding the spread law of the virus over the network, are a currently hot topic of research.

Towards this goal, the classical SIR (susceptible-infected-recovered) model [1, 2], as well as its extensions [35], is extended to explore the behavior of computer virus propagation in network. Based on these classical models and by considering the computer virus fixed latent period, Mishra et al. [6, 7] proposed delayed SIRS, SEIR computer virus models with a fixed period of temporary immunity, which accounts for the temporary recovery from the infection of virus. In [8], Tan and Han proposed an SIRS computer virus model with fixed latency and temporal immune periods, studied the effect of time delays on the stability of the equilibria, and gave some conditions for the equilibria to be locally asymptotically stable for all delays.

Motivated by the previous work, this paper proposes and studies a computer-virus-propagation model with varying latency period, known as the SIRC model. We obtain the threshold value determining whether the virus dies out completely, study the local asymptotic stabilities of the equilibria of the model and it is found that, model may undergo a Hopf bifurcation. Next, we prove the global asymptotic stability of the virus-free equilibrium by using the direct Lyapunov method, prove the global asymptotic stability of the virus equilibrium by using a geometric approach. By introducing varying time delay, the model may truly reflect the virus propagation and hence, the corresponding results may help understand and prevent the spread of computer virus over a computer network.

The remaining materials of this paper are organized this way: Section 2 introduces the mathematical model to be discussed; Section 3 studies the local stability of the virus-free and virus equilibrium of model, respectively, examines the stability switch for a virus equilibrium, and shows that our model may admit a Hopf bifurcation; Section 4 uses different methods to prove the global asymptotic stability of the equilibria. In Section 5, some numerical examples are given to support our conclusions. We end the paper with a brief discussion in Section 6.

2. Mathematical Model

Consider the classical SIR computer virus model proposed in [1, 2] 𝑑𝑆𝑑𝑡=𝑏𝛽𝑆(𝑡)𝐼(𝑡)𝜇𝑆(𝑡),𝑑𝐼𝑑𝑡=𝛽𝑆(𝑡)𝐼(𝑡)(𝛾+𝜇)𝐼(𝑡),𝑑𝑅𝑑𝑡=𝛾𝐼(𝑡)𝜇𝑅(𝑡).(2.1) Here it is assumed that all the computers connected to the network in concern are classified into three categories: susceptible, infected, and recovered computers. Let 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) denote their corresponding numbers at time 𝑡. This model involves four positive parameters: 𝑏 denotes the rate at which external computers are connected to the network, 𝛾 denotes the recovery rate of infected computers due to the antivirus ability of the network, 𝜇 denotes the rate at which one computer is removed from the network, 𝛽 denotes the rate at which, when having connection to one infected computer, one susceptible computer can become infected. For some variants of this model, see [15, 9, 10].

The computer virus has latent and unpredictable characteristics [11]. A sophisticated computer virus program, when entering into the computer system, does not immediately break out. The longer the latency of a computer virus, the wider its spreading scope will be. On one hand, the computer virus program can not be detected without use of the specialized programs. The virus can stay quietly in the disk or CD a few days, even years, and when the time comes, it will break out to reproduce, spread, and continue to harm. On the other hand, there is a trigger mechanism within the computer virus, if the trigger conditions are not met, the computer virus does not do any other damage. Only when the trigger conditions are met, can the virus be activated to do some damages. Without loss of reality, the following assumptions are made:(1)The virus in susceptible computer has a latency period. Moreover, this latency period is varying, which can be reflected by the following expression: 𝑡𝑆(𝜏)𝐷(𝑡𝜏)𝑑𝜏,(2.2) where 𝐷 is the delay kernel [12], 𝜏 is the distributed delay, 𝑆(𝜏) indicates how 𝑆(𝑡) is affected by their previous values.(2)Only when the virus breaks out can the susceptible computers become the infected ones.

We choose a typical class of kernels 𝐷𝜎(𝑡𝜏)=𝑛+1(𝑡𝜏)𝑛[]exp𝜎(𝑡𝜏)𝑛!,𝑡0,𝑛=0,1,2,,(2.3) where 𝜎 is a positive constant indicating the average delay of the collected information on the virus infection. In this paper, we simply take the weak kernel 𝐷(𝑡𝜏)=𝜎𝑒𝜎(𝑡𝜏),𝜎>0,(2.4) which implies that the effect of previous events decreases exponentially.

By incorporating these factors into model (2.1), we get the following model: 𝑑𝑆𝑑𝑡=𝑏𝛽𝑡𝑆(𝜏)𝐷(𝑡𝜏)𝑑𝜏𝐼(𝑡)𝜇𝑆(𝑡),𝑑𝐼𝑑𝑡=𝛽𝑡𝑆(𝜏)𝐷(𝑡𝜏)𝑑𝜏𝐼(𝑡)(𝜇+𝛾)𝐼(𝑡),𝑑𝑅𝑑𝑡=𝛾𝐼(𝑡)𝜇𝑅(𝑡).(2.5) We define a new variable 𝐶(𝑡)=𝑡𝑆(𝜏)𝐷(𝑡𝜏)𝑑𝜏,(2.6) which indicates in susceptible computer, the effect of latent virus on infection. Then model (2.5) becomes 𝑑𝑆𝑑𝑡=𝑏𝛽𝐶(𝑡)𝐼(𝑡)𝜇𝑆(𝑡),𝑑𝐼𝑑𝑡=𝛽𝐶(𝑡)𝐼(𝑡)(𝜇+𝛾)𝐼(𝑡),𝑑𝑅𝑑𝑡=𝛾𝐼(𝑡)𝜇𝑅(𝑡),𝑑𝐶=1𝑑𝑡𝜎1𝑆(𝑡)𝜎𝐶(𝑡).(2.7) Because 𝑆, 𝐼, and 𝐶 are independent of variable 𝑅, this paper focuses on the following model: 𝑑𝑆𝑑𝑡=𝑏𝛽𝐶(𝑡)𝐼(𝑡)𝜇𝑆(𝑡),𝑑𝐼𝑑𝑡=𝛽𝐶(𝑡)𝐼(𝑡)(𝜇+𝛾)𝐼(𝑡),𝑑𝐶=1𝑑𝑡𝜎1𝑆(𝑡)𝜎𝐶(𝑡).(2.8) Adding the first two equations of model (2.8), we can obtain 𝑑(𝑆+𝐼)𝑑𝑡=𝑏𝜇(𝑆+𝐼)𝛾𝐼,(2.9) Therefore, 𝑆(𝑡) and 𝐼(𝑡) are bounded, that is, 𝑆+𝐼𝑏/𝜇. From the third equation of model (2.8), we can obtain 𝑑𝐶=1𝑑𝑡𝜎1𝑆𝜎𝑏𝐶1𝜎𝜇𝜎𝐶.(2.10) It is easy to see that 𝐶(𝑡) is bounded, that is, 𝐶𝑏/𝑢. Thus, the set Ω=(𝑆,𝐼,𝐶)𝑅3+𝑏,𝑆+𝐼𝜇𝑏,𝐶𝜇(2.11) is the positively invariant set of model (2.8).

3. The Equilibria and Local Stability

This section investigates the equilibria of model (2.8) and their stability. For that purpose, let us introduce the basic reproduction number, which is defined as 𝑅0=𝑏𝛽/𝜇(𝜇+𝛾).

First, model (2.8) has a virus-free equilibrium 𝐸0=(𝑏/𝜇,0,𝑏/𝜇). The characteristic equation of the corresponding linearized system with respect to 𝐸0 is det𝜇𝜆𝛽𝑏𝜇00𝛽𝑏𝜇1(𝜇+𝛾)𝜆0𝜎10𝜎𝜆=0.(3.1) The three eigenvalues are 𝜇,1/𝜎,and𝑏𝛽/𝜇(𝛾+𝜇). Thus, we immediately get

Theorem 3.1. Consider model (2.8).(a)The virus-free equilibrium 𝐸0 is locally asymptotically stable if 𝑅0<1.(b)𝐸0 is unstable if 𝑅0>1.

Next, when 𝑅0>1, model (2.8) has a positive virus equilibrium 𝐸=(𝑆,𝐼,𝐶), where 𝑆=𝑏𝜇𝑅0,𝐼=𝜇𝑅01𝛽,𝐶=𝑏𝜇𝑅0.(3.2) The characteristic equation of the corresponding linearized system near 𝐸 is det𝜇𝜆𝛽𝐶𝛽𝐼0𝜆𝛽𝐼1𝜎10𝜎𝜆=0,(3.3) which equals 𝜆3+𝑝0𝜆2+𝑝1𝜆+𝑝2=0,(3.4) where 𝑝01=𝜇+𝜎,𝑝1=𝛽𝐼1+𝜇𝜎,𝑝2=1𝜎𝛽2𝐶𝐼.(3.5) A simple calculation gives 𝑝0𝑝1𝑝2=1𝜎+𝜇𝛽𝐼1+𝜇𝜎𝛽2𝐶𝐼1𝜎=1𝜎𝜇𝛽𝐼+𝜇𝛽2𝐶𝐼+𝛽𝐼1+𝜇𝜎.(3.6) If 𝑝0𝑝1𝑝2>0, that is, 𝜎<𝜎, 𝐸 is locally asymptotically stable, where 𝜎=𝑅0/(𝑅01)(𝛾+𝜇)𝜇𝑅0, and 𝜎>0 is equivalent to 𝑅0>1+𝜇/𝛾. From the above analysis, we obtain the following Theorem:

Theorem 3.2. Consider model (2.8). Suppose 𝑅0>1+𝜇/𝛾.(a)The virus equilibrium 𝐸 is locally asymptotically stable if 𝜎<𝜎.(b)𝐸 is unstable if 𝜎>𝜎.

Remark 3.3. From the above analysis, we can see that that there exists a stability switch for 𝐸: 𝐸 changes its stability when 𝜎 goes across the critical value 𝜎, which may result in a Hopf bifurcation and, hence, can be exploited to find an effective strategy for preventing the spread of computer virus.
Indeed, when 𝜎=𝜎, (3.4) has two complex conjugate roots, 𝜆1,2=𝛼(𝑇)±𝑖𝜔(𝑇). It is noted that 𝛼(𝜎)=0, 𝜔(𝜎)=𝑝1>0, and Δ=𝑑𝛼|||𝑑𝜎𝜎=𝜎𝑅𝜇(𝛾+𝜇)01𝜎𝜇2𝑅02𝜇𝑅02𝜎3(𝜇+1/𝜎)2+2𝜎2𝜇𝑅0.(3.7) Let 𝑓(𝑅0)=𝜎𝜇(𝛾+𝜇)(𝑅01)𝜎𝜇2𝑅02𝜇𝑅00, which leads to Δ0. Thus, we obtain.

Theorem 3.4. If 𝑅0>1+𝜇/𝛾, model (2.8) undergoes a Hopf bifurcation with respect to the virus equilibrium when 𝜎 goes across the value of 𝜎.

4. Global Stability

In this section, we will discuss the global stability of the model.

Theorem 4.1. when 𝑅0<1, the virus-free equilibrium 𝐸0 is global stability.

Proof. Define 1𝑉(𝑆,𝐼,𝐶)=2𝐼2.(4.1) If 𝑅0<1, then ̇[]𝐼𝑉(𝑆,𝐼,𝐶)=𝛽𝐶(𝜇+𝛾)2𝛽𝑏𝜇𝐼(𝜇+𝛾)2𝑅(𝜇+𝛾)0𝐼120.(4.2) Since all the model parameters are positive, it follows that ̇𝑉(𝑆,𝐼,𝐶)<0 for 𝑅0<1 with ̇𝑉(𝑆,𝐼,𝐶)=0 if and only if 𝐼=0 or 𝑅0=1. Hence, 𝑉 is a Lyapunov function on Ω. Thus, 𝐼0 as 𝑡. Using 𝐼=0 in the first equation of (2.8) shows that 𝑆𝑏/𝜇 as 𝑡. Therefore, it follows from the Lasalle’s invariance principle, that every solution of the model, starting from within Ω, approaches 𝐸0 as 𝑡.
In the following, we use the geometrical approach [13, 14] to discuss the global stability of virus equilibrium 𝐸. First, we give a brief outline of this approach.
Let 𝑥𝑓(𝑥)𝑅𝑛 be a 𝐶1 function for 𝑥 in an open set 𝐷𝑅𝑛. Consider the following equation: ̇𝑥=𝑓(𝑥).(4.3)
Denote by 𝑥(𝑡,𝑥0) the solution with 𝑥(𝑡,𝑥0)=𝑥0. Then, the following assumptions are made:(H1) There exists a compact absorbing set 𝐾𝐷.(H2) Equation (4.3) has a unique equilibrium 𝑥0 in 𝐷.
Let 𝑥𝑝(𝑥) be an (𝑛2)×(𝑛2) matrix-valued function that is 𝐶1 for 𝑥𝐷. Assume that 𝑝1(𝑥) exists and is continuous for 𝑥𝐾, the compact absorbing set. A quantity 𝑞2 is defined as 𝑞2=limsup𝑥sup𝑥0𝐾1𝑡𝑡0𝜇𝐵𝑥𝑠,𝑥0𝑑𝑠,(4.4) where 𝐵=𝑝𝑓𝑝1+𝑝𝜕𝑓[2]𝑝𝜕𝑥1.(4.5) The matrix 𝑝𝑓 is obtained by replacing each entry of 𝑝 by its derivative in the direction of 𝑓, and 𝜇(𝐵) is defined by 𝜇(𝐵)=lim0+||||𝐼+𝐵1,(4.6) which is the Lozinskil measure of 𝐵 with respect to a vector norm || in 𝑅𝑁.

From the above outline, a theorem can be given as follows:

Theorem 4.2 (see [13]). Assume that 𝐷 is simply connected, and that the assumptions (H1) and (H2) hold, if  𝑞2<0, then the unique equilibrium 𝑥0 of (4.3) is globally asymptotically stable.

Now, we discuss the global stability of the virus equilibrium 𝐸 of model (2.8).

Model (2.8) has a unique virus equilibrium 𝐸 in Ω, hence it satisfies the assumption (H1). If 𝑅0>1, then virus-free equilibrium is not stable, and the solutions of model (2.8) are bounded, which ensure model (2.8) has a compact set in Ω. Therefore, the assumption (H2) is met.

The Jacobian matrix of model (2.8) is 1𝐽=𝜇𝛽𝐶𝛽𝐼0𝛽𝐶𝜇𝛾𝛽𝐼𝜎10𝜎,(4.7) and its second additive compound matrix is 𝐽[2]=1𝛽𝐶2𝜇𝛾𝛽𝐼𝛽𝐼0𝜇𝜎1𝛽𝐶𝜎10𝛽𝐶(𝜇+𝛾)𝜎.(4.8) Set the function 𝑆𝑃(𝑋)=𝑃(𝑆,𝐼,𝐶)=diag𝐼,𝑆𝐼,𝑆𝐼.(4.9) Then 𝑃𝑓𝑃1𝑆=diag𝑆𝐼𝐼,𝑆𝑆𝐼𝐼,𝑆𝑆𝐼𝐼,(4.10) and the matrix 𝐵=𝑃𝑓𝑃1+𝑃𝐽[2]𝑃1 can be written as 𝐵𝐵=11𝐵12𝐵21𝐵22,(4.11) where 𝐵11=𝑆/𝑆𝐼/𝐼+𝛽𝐶2𝜇𝑟,𝐵12=(𝛽𝐼,𝛽𝐼),𝐵12=(0,1/𝜎)𝑇, and 𝐵22=𝑆𝑆𝐼𝐼1𝜇𝜎0𝑆𝛽𝐶𝑆𝐼𝐼1𝜇𝛾𝜎+𝛽𝐶.(4.12) Select the norm in 𝑅3 as the following: ||||𝑢,𝑣,𝜔=max{|𝑢|,|𝑣|+|𝜔|},(4.13) where (𝑢,𝑣,𝜔) denotes the vector in 𝑅3, let 𝜇 denote the Lozinskii measure with respect to this norm, then 𝜇𝑔(𝐵)sup1,𝑔2𝜇𝐵=sup11+||𝐵12||𝐵,𝜇22+||𝐵21||,(4.14) where |𝐵12|, |𝐵21| are matrix norms with respect to the 𝐿1 vector norm. Thus, 𝜇𝐵11=𝑆𝑆𝐼𝐼||𝐵+𝛽𝐶2𝜇𝑟,12||||𝐵=𝛽𝐼,21||=1𝜎.(4.15) Under the condition of 𝐶𝛾/2𝛽, 𝜇𝐵22=𝑆𝑆𝐼𝐼1𝜇𝜎.(4.16) Therefore 𝑔1=𝑆𝑆𝐼𝐼𝑔+𝛽𝐶+𝛽𝐼2𝜇𝑟,2=𝑆𝑆𝐼𝐼𝜇.(4.17) According to the second equation of model (2.8), we can obtain 𝐼𝐼=𝛽𝐶𝜇𝑟.(4.18) Hence, 𝑔1=𝑆𝑆𝐼𝐼𝑆+𝛽𝐼+𝛽𝐶2𝜇𝛾𝑆2𝜇+𝛾𝑏𝛽𝜇,𝑔2=𝑆𝑆𝐼𝐼𝑆𝜇𝑆𝑅𝜇(𝜇+𝛾)0𝑆1𝑆𝜇.(4.19) Therefore, 𝑆𝜇(𝐵)𝑆𝑏.(4.20) For 𝑡𝑡1, where 𝑏=min{𝜇,2𝜇+𝛾𝑏𝛽/𝜇}.

This leads to 1𝑡𝑡01𝜇(𝐵)𝑑𝑠=𝑡𝑡101𝜇(𝐵)𝑑𝑠+𝑡𝑡𝑡11𝜇(𝐵)𝑑𝑠𝑡𝑡101𝜇(𝐵)𝑑𝑠+𝑡log𝑆(𝑡)𝑆𝑡1𝑏,(4.21) which implies that 𝑞2𝑏/2<0.

From the above discussions, we can obtain the following theorem:

Theorem 4.3. When 𝑅0>1, if then the unique equilibrium 𝐸 is globally asymptotically stable.

5. Numerical Simulations

In this section, we make some numerical simulations to understand the obtained theorems. Let 𝑏=20,𝛽=0.01,𝛾=0.4,𝜇=0.2,and𝜎=5, then 𝑅00.83<1. Hence, the virus-free equilibrium 𝐸0(50.00,0) is asymptotically stable (see Figure 1), that is, the virus would extinguish after a period of time. In contrast, let 𝑏=20,𝛽=0.02,𝛾=0.6,and𝜇=0.1 yield 𝜎5.91. In this case, when 𝜎=4<𝜎 and 𝜎=7.5>𝜎, the virus equilibrium 𝐸(35.00,9.29) would become stable (see Figure 2) and unstable (see Figure 3), respectively.

371792.fig.001
Figure 1: 𝐼 versus 𝑆, where 𝑏=20,𝛽=0.01,𝛾=0.4,𝜇=0.2,𝜎=5, and 𝑅00.83 satisfy stable condition for the virus-free equilibrium 𝐸0(50.00, 0).
371792.fig.002
Figure 2: Distribution of computers versus time when 𝑏=10,𝛽=0.02,𝛾=0.6,𝜇=0.1, and 𝜎=4, where 𝑅0=2.86>1 and 𝜎5.91>𝜎, satisfy the stable condition for the virus equilibrium 𝐸(35.00,9.29).
371792.fig.003
Figure 3: Distribution of computers versus time when 𝑏=20,𝛽=0.02,𝛾=0.6,𝜇=0.1,and𝜎=7.5, where 𝑅0=2.86>1 and 𝜎5.91<𝜎 satisfies the unstable condition for the virus equilibrium 𝐸.

6. Discussions

In this paper, by considering varying latency period of computer virus, we propose a model for computer virus propagation in network. First, we give the threshold value 𝑅0 determining whether the virus extinguishes, and study the local stabilities of the virus-free equilibrium 𝐸0 and virus equilibrium 𝐸 under this model. It is found that 𝑅0 changes the stability of 𝐸0 and time delay parameter 𝜎 changes the stability of 𝐸, and that the model may undergo a Hopf bifurcation. Next, we use two different methods to prove the global asymptotic stabilities of the equilibria: the virus-free equilibrium by using the direct Lyapunov method and virus equilibrium by using a geometric approach. Finally, some numerical examples are given to support our conclusions.

Acknowledgments

The authors wish to thank the anonymous editors and reviewers.

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