Journal of Applied Mathematics

Volume 2014 (2014), Article ID 492198, 8 pages

http://dx.doi.org/10.1155/2014/492198

Research Article

Hopf Bifurcation in a Delayed SEIQRS Model for the Transmission of Malicious Objects in Computer Network

Juan Liu

Department of Mathematics and Physics, Bengbu College, Bengbu 233030, China

Received 26 December 2013; Revised 19 February 2014; Accepted 19 February 2014; Published 23 March 2014

Academic Editor: Sabri Arik

Copyright © 2014 Juan Liu. 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.

A delayed SEIQRS model for the transmission of malicious objects in computer network is considered in this paper. Local stability of the positive equilibrium of the model and existence of local Hopf bifurcation are investigated by regarding the time delay due to the temporary immunity period after which a recovered computer may be infected again. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Numerical simulations are also presented to support the theoretical results.

1. Introduction

The action of malicious objects throughout a network can be studied by using epidemic models due to the high similarity between malicious objects and biological viruses [1–7]. In [2], Thommes and Coates proposed a modified version of the SEI model to predict virus propagation in a network. In [3], Mishra and Pandey proposed a SEIRS epidemic model for the transmission of worms with vertical transmission. Recently, antivirus counter measures such as virus immunization and quarantine strategy have been introduced into some epidemic models in order to study the prevalence of virus. In [7], Mishra and Jha proposed the following SEIQRS model for the transmission of malicious objects in computer network:

where , , , , and denote the sizes of nodes at time in the states susceptible, exposed, infectious, quarantined, and recovered, respectively. The parameters , , and are positive constants. is the recruitment rate of susceptible nodes to the computer network. is the crashing rate of nodes due to the reason other than the attack of malicious objects. is the crashing rate of nodes due to the attack of malicious objects. is the transmission rate. , , , , and are the state transition rates. Mishra and Jha [7] investigated the global stability of the unique endemic equilibrium for the system (1).

It is well known that time delays 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 [8–15]. In [9], Feng et al. investigated the Hopf bifurcation of a delayed viral infection model in computer networks by using theories of stability and bifurcation. In [11], Dong et al. proposed a computer virus model with time delay based on an SEIR model and studied the dynamical behaviors such as local stability and local Hopf bifurcation by regarding the time delay as a bifurcating parameter. Motivated by the work above and considering the temporary immunity period after which a recovered computer may be infected again, we introduce the time delay due to the temporary immunity period into system (1) and get the following system with time delay:

where is the time delay due to the temporary immunity period.

The main purpose is to investigate the effects of the delay on system (2) and this paper is organized as follows. Sufficient conditions for the local stability and existence of local Hopf bifurcation are obtained by regarding the delay due to the temporary immunity period after which a recovered computer may be infected again as a bifurcating parameter in Section 2. Direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are determined by the normal form method and center manifold theorem in Section 3. In Section 4, we give a numerical example to support the theoretical results in the paper.

2. Stability and Existence of Local Hopf Bifurcation

According to the analysis in [7], we can get that if the condition : holds, then system (2) has a unique positive equilibrium , where

Let , , , , and . Dropping the bars for the sake of simplicity, system (2) can be rewritten as the following system:

where

Then, we can get the linearized system of system (4) as follows:

Thus, the characteristic equation of system (6) at the positive equilibrium is

where

For the existence of local Hopf bifurcation of system (2), we give the following result.

Theorem 1. For system (2), if conditions hold, then the positive equilibrium of system (2) is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when , where the conditions and the expression of are defined in the following analysis.

Proof. For , (7) becomes

where

Obviously, . Therefore, if condition : (11) holds, then the positive equilibrium is locally asymptotically stable without delay. Consider

For , let be a root of (7). Then, we can get

from which we obtain

with

Let , then (14) becomes

In order to give the main results in this paper, we make the following assumption.

Equation (16) has at least one positive real root.

Suppose that condition holds. Without loss of generality, we suppose that (16) has five positive roots, which are denoted as , respectively. Then, (14) has five positive roots . For every fixed , the corresponding critical value of time delay is

where

Let

Taking the derivative of with respect to in (7), we obtain

Then, we have

where and . Obviously, if condition    holds, then . That is, if condition   holds, then the transversality condition is satisfied. The proof of Theorem 1 is completed.

3. Properties of the Hopf Bifurcation

In the previous section, we have obtained the conditions under which system (2) undergoes Hopf bifurcation at the positive equilibrium when the delay crosses though the critical value . In this section, we give the formula that determines the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (2).

Define

where is defined in the following analysis. Further, we give the following result with respect to the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (2).

Theorem 2. For system (2), 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).

Proof. Let , , so that is the Hopf bifurcation value of system (2). Rescaling the time delay by .  , , , , and , then system (2) can be transformed into the following form:

where and , are given, respectively, by

By the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation such that

In fact, we choose

where is the Dirac delta function.

For , we define

Then system (23) can be transformed into the following operator equation:

where for .

The adjoint operator of is defined by

associated with a bilinear form

where .

Let be the eigenvector of corresponding to and the eigenvector of corresponding to . From the definition of and and by a simple computation, we obtain

From (31), we have

Let

Then, , .

Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [16] as follows:

with

where and can be determined by the following equations, respectively:

with

Then, we can get the expression of as follows:

Further we have

where the sign determines the direction of the Hopf bifurcation, the sign determines the stability of the bifurcating periodic solutions, and the sign of determines the period of the bifurcating periodic solutions. The proof of Theorem 2 is completed.

4. Numerical Simulation

In this section, we use a numerical example to support the theoretical analysis above in this paper. We take the following particular case of system (2) in which , , , , , , , , and . Consider

It is easy to verify that . Thus, condition holds. The positive equilibrium can be obtained by solving the equations in system (41). By some complex computations, we obtain that (16) has one positive root , and further we have , . First, we choose ; the corresponding phase plots are shown in Figures 1 and 2; it is easy to see from Figures 1 and 2 that system (41) is asymptotically stable. Then, we choose . The corresponding phase plots are shown in Figures 3 and 4. It is easy to see that system (41) undergoes Hopf bifurcation. In addition, we have , . Thus, we have , , and . From Theorem 2, we can conclude that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable, and the period of the periodic solutions increases.

Figure 1: The phase plot of the states , , and for .

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

Figure 3: The phase plot of the states , , and for .

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

5. Conclusions

This paper is concerned with a delayed SEIQRS model for the transmission of malicious objects in computer network. The theoretical analysis for the delayed model is given and the main results are given in terms of local stability and local Hopf bifurcation. By regarding the delay due to the temporary immunity period after which a recovered computer may be infected again, we have proven that when the delay passes through the critical value, the model undergoes a Hopf bifurcation. The occurrence of Hopf bifurcation means that the state of virus prevalence changes from a positive equilibrium to a limit cycle, which is not welcomed in networks. Hence, we should control the phenomenon by combining some bifurcation control strategies and other relative features of virus prevalence. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Finally, some numerical simulations are presented to clarify our theoretical results.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the referees and the editor for their valuable comments and suggestions on the paper.

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