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- Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 371792, 12 pages
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.
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.
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 [3–5], 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 , 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] 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 , , and denote their corresponding numbers at time . This model involves four positive parameters: denotes the rate at which external computers are connected to the network, denotes the recovery rate of infected computers due to the antivirus ability of the network, denotes the rate at which one computer is removed from the network, 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 [1–5, 9, 10].
The computer virus has latent and unpredictable characteristics . 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: where is the delay kernel , is the distributed delay, indicates how 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 where 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 which implies that the effect of previous events decreases exponentially.
By incorporating these factors into model (2.1), we get the following model: We define a new variable which indicates in susceptible computer, the effect of latent virus on infection. Then model (2.5) becomes Because , , and are independent of variable , this paper focuses on the following model: Adding the first two equations of model (2.8), we can obtain Therefore, and are bounded, that is, . From the third equation of model (2.8), we can obtain It is easy to see that is bounded, that is, . Thus, the set 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 .
First, model (2.8) has a virus-free equilibrium . The characteristic equation of the corresponding linearized system with respect to is The three eigenvalues are . Thus, we immediately get
Theorem 3.1. Consider model (2.8).(a)The virus-free equilibrium is locally asymptotically stable if .(b) is unstable if .
Next, when , model (2.8) has a positive virus equilibrium , where The characteristic equation of the corresponding linearized system near is which equals where A simple calculation gives If , that is, , is locally asymptotically stable, where , and is equivalent to . From the above analysis, we obtain the following Theorem:
Theorem 3.2. Consider model (2.8). Suppose .(a)The virus equilibrium is locally asymptotically stable if .(b) is unstable if .
Remark 3.3. From the above analysis, we can see that that there exists a stability switch for : changes its stability when goes across the critical value , 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 , (3.4) has two complex conjugate roots, . It is noted that , , and Let , which leads to . Thus, we obtain.
Theorem 3.4. If , model (2.8) undergoes a Hopf bifurcation with respect to the virus equilibrium when goes across the value of .
4. Global Stability
In this section, we will discuss the global stability of the model.
Theorem 4.1. when , the virus-free equilibrium is global stability.
If , then
Since all the model parameters are positive, it follows that for with if and only if or . Hence, is a Lyapunov function on . Thus, as . Using in the first equation of (2.8) shows that as . Therefore, it follows from the Lasalle’s invariance principle, that every solution of the model, starting from within , approaches as .
In the following, we use the geometrical approach [13, 14] to discuss the global stability of virus equilibrium . First, we give a brief outline of this approach.
Let be a function for in an open set . Consider the following equation:
Denote by the solution with . Then, the following assumptions are made:(H1) There exists a compact absorbing set .(H2) Equation (4.3) has a unique equilibrium in .
Let be an matrix-valued function that is for . Assume that exists and is continuous for , the compact absorbing set. A quantity is defined as where The matrix is obtained by replacing each entry of by its derivative in the direction of , and is defined by which is the Lozinskil measure of with respect to a vector norm in .
From the above outline, a theorem can be given as follows:
Now, we discuss the global stability of the virus equilibrium of model (2.8).
Model (2.8) has a unique virus equilibrium in , hence it satisfies the assumption (H1). If , 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 . Therefore, the assumption (H2) is met.
The Jacobian matrix of model (2.8) is and its second additive compound matrix is Set the function Then and the matrix can be written as where , and Select the norm in as the following: where denotes the vector in , let denote the Lozinskii measure with respect to this norm, then where , are matrix norms with respect to the vector norm. Thus, Under the condition of , Therefore According to the second equation of model (2.8), we can obtain Hence, Therefore, For , where .
This leads to which implies that .
From the above discussions, we can obtain the following theorem:
Theorem 4.3. When , if then the unique equilibrium is globally asymptotically stable.
5. Numerical Simulations
In this section, we make some numerical simulations to understand the obtained theorems. Let , then . Hence, the virus-free equilibrium is asymptotically stable (see Figure 1), that is, the virus would extinguish after a period of time. In contrast, let yield . In this case, when and , the virus equilibrium would become stable (see Figure 2) and unstable (see Figure 3), respectively.
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 determining whether the virus extinguishes, and study the local stabilities of the virus-free equilibrium and virus equilibrium under this model. It is found that changes the stability of and time delay parameter changes the stability of , 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.
The authors wish to thank the anonymous editors and reviewers.
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