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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 560804, 18 pages
http://dx.doi.org/10.1155/2013/560804
Research Article

Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays

1Key Laboratory of Advanced Process Control for Light Industry, Jiangnan University, Wuxi 214122, China
2School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China

Received 18 August 2013; Accepted 26 August 2013

Academic Editor: Luca Guerrini

Copyright © 2013 Zizhen Zhang and Huizhong Yang. 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

This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.

1. Introduction

Since the pioneering work of Kephart and White [1, 2], many classical epidemic models such as SIR [35], SIRS [68], SEIR [9, 10], and SEIRS [11, 12], SEIQV [13] have been used to describe the spread of a computer virus in computer network due to the high similarity between computer viruses and biological viruses. In [9], Yuan and Chen proposed the following SEIR model: where , , , and denote the numbers of nodes at time in states susceptible, exposed, infectious, and recovered, respectively. Yuan and Chen [9] studied the behaviors of virus propagation with the presence of antivirus countermeasures by analyzing the equilibrium stability of system (1).

It is well known that time delays of one or other reasons can cause a stable equilibrium to become unstable and make a system bifurcate periodic solutions and dynamical systems with delay have been studied by many scholars [1423]. Starting from this point and considering that the antivirus software may use a period to clean the viruses in a computer, Dong et al. [10] proposed the following model with delay: where is the total number of computers in a network. describes the impact of quarantine or replacement. describes the impact of implementing real-time immunization. describes the impact of cleaning the virus and immunizing the computers. is the transmission coefficient. and are the state transition rates. And is the period which a computer uses antivirus software to clean viruses. Dong et al. [10] discussed the local stability and existence of local Hopf bifurcation of system (2). Properties of the Hopf bifurcation were also investigated in [10].

However, Yuan and Chen [9] Dong et al. [10] supposed that the recovered computers have a permanent immunization period and can no longer be infected. This is not consistent with real situation. In order to overcome limitation and considering that the recovered computers may be infected again after a temporary immunity period, we investigate the following system with two delays in this paper: where is the period that a computer uses antivirus software to clean viruses and is the temporary immunity period after which a recovered computer may be infected again. is the transition rate from to .

This paper is organized as follows. In Section 2, local stability of the positive equilibrium and the existence of local Hopf bifurcation are discussed. In Section 3, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. In order to illustrate the validity of the theoretical analysis, some numerical simulations are presented in Section 4. Some main conclusions are drawn in Section 5.

2. Stability and Existence of Local Hopf Bifurcation

In this section, we will study the stability of positive equilibrium and the existence of Hopf bifurcation. It is not difficult to verify if system (3) has a unique positive equilibrium , where is called the basic reproduction number.

Linearizing system (3) at the positive equilibrium yields the following linear system: where Thus, the characteristic equation of system (3) at is where

Case 1 (). Equation (8) reduces to where
By the Routh-Hurwitz criterion, sufficient conditions for all roots of (10) to have a negative real part are given in the following form:
Thus, if condition (12) holds, then the positive equilibrium is locally asymptotically stable in the absence of delay.

Case 2 (). When , , (8) becomes the following form: where Let () be the root of (13). Then, we can get which follows that where Let ; then, (16) becomes Discussion about the roots of (18) is similar to that in [15]. Therefore, we have the following lemma.

Lemma 1. Let Then, for (18), one has the following:(i)if , (18) has at least one positive root;(ii)if and , (18) has positive roots if and only if and ;(iii)if and , (18) has positive roots if and only if there exists at least one such that and .
In what follows, we assume that : the coefficients in satisfy one of the following conditions in (a)–(c):(a);(b), , , and ;(c), , and there exists at least one such that and .
If holds, we know that (16) has at least a positive root such that (13) has a pair of purely imaginary root . The corresponding critical value of the delay is where Differentiating both sides of (13) regarding , we can obtain Thus, From (16), we can get where .
Thus, if condition holds, then the transversality condition is satisfied. According to the Hopf bifurcation theorem in [24], we have the following results.

Theorem 2. Suppose that the conditions - hold. Consider(i)the positive equilibrium of system (3) is asymptotically stable for ;(ii)system (3) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from the positive equilibrium near .

Case 3 (). When , , (8) becomes the following form: where Let () be the root of (25), and we have from which, we can obtain where Let , then (28) becomes Define Next, we assume that : the coefficients in satisfy one of the following conditions in (a′)–(c′):(a′);(b′), , , and ;(c′) , , and there exists at least one such that and .According to Lemma 1, we know that if the condition holds, then (28) has at least a positive root such that (25) has a pair of purely imaginary roots . The corresponding critical value of the delay is where As in Case 2, we know that if where , then . That is, the transversality condition is satisfied if the condition holds. Thus, according to the Hopf bifurcation theorem in [24], we have the following results.

Theorem 3. Suppose the conditions - hold.(i)The positive equilibrium of system (3) is asymptotically stable for .(ii)System (3) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from the positive equilibrium near .

Case 4 (). Substituting into (8), then (8) becomes where Multiplying by , (34) becomes Let () be the root of (36); consider which follows that where As is known, . Thus, we have where Let , then (40) becomes In order to give the main results in this paper, we make the following assumption.

(42) has at least one positive real root.

Suppose that condition holds. Without loss of generality, we assume that (42) has eight positive real roots, which are denoted as , respectively. Then (40) has eight positive real roots . For every fixed , the corresponding critical value of time delay is Define Taking the derivative of with respect to (36), it is easy to obtain Thus, we have where Obviously, if condition holds, then . Thus, by the Hopf bifurcation theorem in [24], we have the following results.

Theorem 4. Suppose that conditions - hold.(i)The positive equilibrium of system (3) is asymptotically stable for .(ii)System (3) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from the positive equilibrium near .

Case 5 (, ). We consider (8) with in its stable interval, regarding as a parameter.
Let () be the root of (8); we obtain where We assume that (48) has at least finite positive roots. And we denote the positive roots of (48) as , , and . For every fixed , the corresponding critical value of time delay is where Let Next, we make the following assumption: . Thus, by the Hopf bifurcation theorem in [24], we have the following results.

Theorem 5. Suppose that conditions - hold and .(i)The positive equilibrium of system (3) is asymptotically stable for .(ii)System (3) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from the positive equilibrium near .

3. Direction and Stability of the Hopf Bifurcation

In this section, we will investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions w.r.t. for by using the normal form theory and the center manifold argument in [24]. We assume that where .

Let , , , , , and normalize . Then system (3) can be transformed into the following form: where and where

By the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation such that In fact, we choose For , we define Then system (53) can be transformed into the following operator equation: The adjoint operator of is defined by associated with a bilinear form where .

Let be the eigenvector of corresponding to , and let be the eigenvector of corresponding to . From the definition of and , we can get From (61), we can get Then, we choose such that , .

Next, we can get the coefficients which can be used to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by following the algorithms introduced in [24]: with where and can be determined by the following equations respectively

with

Therefore, we can calculate the following values: Based on the discussion above, we can obtain the following results.

Theorem 6. For system (3), if (), the Hopf bifurcation is supercritical (subcritical). If () the bifurcating periodic solutions are stable (unstable). If (), the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Simulation

In this section, we present some numerical simulations to verify the theoretical analysis in Sections 2 and 3. Let , , , , , , , . Then, we can get a particular case of system (3):

Then, we can get and the unique positive equilibrium of system (70). Further, we have , , , . That is, condition holds.

For , , by some complicated computation, we obtain <