Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

Abstract

This paper is devoted to the study of an SIRS computer virus propagation model with two delays and multistate antivirus measures. We demonstrate that the system loses its stability and a Hopf bifurcation occurs when the delay passes through the corresponding critical value by choosing the possible combination of the two delays as the bifurcation parameter. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by means of the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate the obtained results.

1. Introduction

With the rapid development of computer technologies and network applications, the threat of computer viruses to the world would become increasingly serious. It is of vital importance to understand how computer viruses spread over computer network and to control the computer viruses’ propagation in computer networks. To this end, many mathematical models have been studied to illustrate the dynamical behavior of computer viruses spreading since Murray [1] suggested that computer viruses share some traits of biological viruses. In [2, 3], Kephart and White used the SIS model to describe the propagation of computer viruses. In [4], Zou et al. investigated how the spread of red worms is affected by the worm characteristics based on the SIR model. In [5, 6], Yuan et al. proposed the SEIR computer virus model and studied the dynamics of the model, respectively. In [7], Mishra and Pandey formulated an SEIRS model for the transmission of worms in computer network through vertical transmission. In addition, there are also some researchers who proposed the computer virus models with vaccination and quarantine strategy [8–10].

In fact, many computer viruses have different kinds of delays when the viruses spread, such as latent period delay [11, 12], immunity period delay [12–15], and the delay due to the period that the anti-virus software needs to clean the viruses [6]. In [12], Feng et al. proposed the following computer virus propagation model with dual delays and multistate antivirus measures based on the classical SIR epidemic model in [16]: where , , and represent the numbers of susceptible, infected, and recovered hosts in computer networks at time , respectively. is the number of the hosts which are attached to the computer networks and is the proportion of the new hosts which are susceptible. is the death rate of the hosts. , , , and are the state transition rates between the classes , , and . is the latent period of the computer viruses and is the temporary immune period of the recovered hosts. For the convenience of analysis, Feng et al. [12] let ; then, system (1) becomes the following form:

By regarding the time delay as the bifurcation parameter, Feng et al. [12] studied the existence and properties of Hopf bifurcation of system (2). As is known, it needs some time to clean the viruses in the infected hosts for the antivirus software. Therefore, it is reasonable to take into account the time delay due to the period that the antivirus software uses to clean the viruses in the infected hosts in system (2). To this end, we consider the following system with two delays: where is the time delay due to the latent period of the computer viruses and the temporary immune period of the recovered hosts. is the time delay due to the period that the antivirus software uses to clean the viruses in the infected hosts.

The remaining materials of this paper are organized in this fashion: local stability and existence of local Hopf bifurcation are discussed in Section 2. Properties of the Hopf bifurcation such as the direction and stability are investigated in Section 3. Some numerical simulations are carried out to verify the theoretical results in Section 4 and, finally, this work is summarized in Section 5.

2. Local Stability and Existence of Local Hopf Bifurcation

By direct computation, it can be concluded that if , then system (3) has a unique positive equilibrium , where The characteristic equation of system (3) at is from which one can obtain where with

From the expressions of , , , , , and , one can obtain . Therefore, (6) can be transformed into the following form:

Case 1 (). When , (9) is equivalent to where
It is easy to get that . Therefore, according to the Routh-Hurwitz criterion, we can conclude that if , then the positive equilibrium of system (3) is locally asymptotically stable when .

Case 2 (, ). When and , (9) becomes the following: where Multiplying on both sides of (12), it is easy to get Let be the root of (14). Then, Then, one can obtain where Since , we have where Let ; then, (18) becomes
In order to give the main results in the present paper, we make the following assumption.Equation (20) has at least one positive real root.
If the condition holds, then there exists a positive root of (20) which can make (14) have a pair of purely imaginary roots . For , the corresponding critical value of delay is Differentiating (14) with respect to , we get Thus, where
It is obvious that if the condition holds, then . According to the Hopf bifurcation theorem in [17], the following results hold.

Theorem 1. If the conditions - hold, the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 3 (, ). When and , (9) becomes where Let be the root of (25). Then, which follows that with Let ; then, (28) becomes Let Discussion about the roots of (30) is similar to that in [18].

Lemma 2. (i) If , then (30) has at least one positive root.
(ii) If and , then (30) has no positive root.
(iii) If and , then (30) has positive root if and only if and .

In what follows, we suppose that the coefficients in (30) satisfy the following condition:(a) or (b) , , , and .

If the condition holds, we know that there exists a positive root of (30) such that (25) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is Differentiating two sides of (25) with respect to , we have Thus, where and .

Obviously, if the condition holds, then . According to the Hopf bifurcation theorem in [17], the following results hold.

Theorem 3. If the conditions - hold, the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at when .

Case 4 (, , ). We consider (9) with in its stable interval and choose as a bifurcation parameter. Multiplying by , (9) becomes Let be the root of (35). Then, where Then, we can obtain where Then, we can get a function with respect to : Next, we suppose that :   (40) has at least one positive real root.
If the condition holds, then there exists a such that (35) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is Taking the derivative with respect to in (35), we get where Thus, where
Thus, if the condition holds, then , which implies that the transversality condition is satisfied. According to the Hopf bifurcation theorem in [17], we can conclude the discussions above as follows.

Theorem 4. If the conditions - hold and , the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at when .

Case 5 (, and ). We consider (9) with in its stable interval and is considered as a bifurcation parameter.
Let be the root of (9). Then, where Then, we have where
Similar to Case 4, we suppose that :   (48) has at least one positive real root. If the condition holds, then there exists a such that (9) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is Differentiating (9) with respect to , we have where Define
If the condition holds, then . Therefore, according to the Hopf bifurcation theorem in [17], we can conclude the discussions above as follows.

Theorem 5. If the conditions - hold and , the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at when .

3. Direction and Stability of the Hopf Bifurcation

In this section, we determine the properties of the Hopf bifurcation of system (3) with respect to for . Throughout this section, we assume that , where .

Let , ; then, is the Hopf bifurcation value of system (3). Rescale the time delay . Let , let , and let ; then, system (3) can be transformed into an FDE in : where and and are given, respectively, by with

By the Riesz representation theorem, there exists a function of bounded variation for such that In fact, we can choose For , we define Then, system (54) is equivalent to where for .

For , define and the bilinear form where .

Let and be the eigenvectors of and corresponding to and , respectively. By a direction computation, we get From (62), we obtain Then, one can see that and .

Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [17] and using a computation process similar to that in [19, 20]: with where and can be calculated by the following two equations: