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

Zizhen Zhang1,2 and Huizhong Yang¹

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 [3–5], SIRS [6–8], 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 [14–23]. 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 , . By Theorem 2, we know that when the positive equilibrium is asymptotically stable which can be illustrated by Figures 1 and 2. However, if we let , the positive equilibrium becomes unstable and a Hopf bifurcation occurs and a branch of periodic solutions bifurcates from the positive equilibrium . This property can be shown in Figures 3 and 4. Similarly, we have , for , . The corresponding wave forms and plots are shown in Figures 5, 6, 7, and 8.

(a)

(b)

(c)

(d)

Figure 1: The track of the states , , , and for .

Figure 2: The phase plot of the states , , and for .

(a)

(b)

(c)

(d)

Figure 3: The track of the states , , , and for .

Figure 4: The phase plot of the states , , and for .

(a)

(b)

(c)

(d)

Figure 5: The track of the states , , , and for .

Figure 6: The phase plot of the states , , and for .

(a)

(b)

(c)

(d)

Figure 7: The track of the states , , , and for .

Figure 8: The phase plot of the states , , and for .

For , we get , . According to Theorem 4, we can conclude that when , the positive equilibrium is asymptotically stable, which can be seen from Figures 9 and 10. If ; then, the positive equilibrium becomes unstable, and a branch of bifurcating periodic solutions occurs, which can be shown in Figures 11 and 12.

(a)

(b)

(c)

(d)

Figure 9: The track of the states , , , and for .

Figure 10: The phase plot of the states , , and for .

(a)

(b)

(c)

(d)

Figure 11: The track of the states , , , and for .

Figure 12: The phase plot of the states , , and for .

Lastly, we have , for , . The corresponding wave forms and phase plots are shown in Figures 13, 14, 15, and 16. In addition, we have , , , , . Thus, by Theorem 6, we can conclude that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the periodic solutions increases.

(a)

(b)

(c)

(d)

Figure 13: The track of the states , , , and for and .

Figure 14: The phase plot of the states , , and for and .

(a)

(b)

(c)

(d)

Figure 15: The track of the states , , , and for and .

Figure 16: The phase plot of the states , , and for and .

5. Conclusions

Based on the SEIR model considered in the literature [10], we propose an SEIRS model with two delays for the propagation of computer viruses in networks. The effects of the two delays on the dynamics of the model are investigated. It is found that the two delays may lead to local Hopf bifurcation and make the model unstable under some certain conditions. When the corresponding delay is suitable and small, the positive equilibrium is asymptotically stable. In this case, the propagation of computer viruses can be predicted and controlled. However, a local Hopf bifurcation occurs and a branch of periodic solutions bifurcates from the positive equilibrium when the corresponding delay passes though a critical value. In such conditions, propagation of the computer viruses is out of control. This phenomenon is not welcome in networks. Therefore, In order to control and even eliminate the propagation of computer viruses, the two delays and in the model should remain less than the corresponding critical value. Furthermore, by using the normal form theory and center manifold theorem, the properties of the Hopf bifurcation such as direction and stability are determined. Some numerical simulations are also included to testify the theoretical analysis.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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