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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 3536125, 17 pages
https://doi.org/10.1155/2017/3536125
Research Article

Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination

1School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
2Department of Management, Marche Polytechnic University, Piazza Martelli 8, 60121 Ancona, Italy

Correspondence should be addressed to Zizhen Zhang; moc.361@adiahzzz

Received 5 January 2017; Accepted 23 February 2017; Published 20 March 2017

Academic Editor: Lu-Xing Yang

Copyright © 2017 Zizhen Zhang 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.

Abstract

A further generalization of an SEIQRS-V (susceptible-exposed-infectious-quarantined-recovered-susceptible with vaccination) computer virus propagation model is the main topic of the present paper. This paper specifically analyzes effects on the asymptotic dynamics of the computer virus propagation model when two time delays are introduced. Sufficient conditions for the asymptotic stability and existence of the Hopf bifurcation are established by regarding different combination of the two delays as the bifurcation parameter. Moreover, explicit formulas that determine the stability, direction, and period of the bifurcating periodic solutions are obtained with the help of the normal form theory and center manifold theorem. Finally, numerical simulations are employed for supporting the obtained analytical results.

1. Introduction

Computer viruses, including conventional viruses and network worms, can propagate among computers with no human awareness and popularization of Internet has been the major propagation channel of viruses [1, 2]. The past few decades have witnessed the great financial losses caused by computer viruses. Therefore, it is of considerable importance to investigate the laws describing propagation of computer viruses in order to provide some help with preventing computer viruses. For that purpose and in view of the fact that propagation of computer viruses among computers resembles that of biological viruses among a population, many dynamical models describing propagation of computer viruses across the Internet have been established by the scholars at home and abroad, such as conventional models [38], stochastic models [912], and delayed models [1318]. There are also some other computer virus models [1921] combined with network theory to investigate the impact of the network topology, the patch forwarding, and the network eigenvalue on the viral prevalence.

As is known, vaccination is regarded as one of the most effective measures of preventing computer viruses and the awareness that there exist many infected computers would enhance the probability that the user of a susceptible computer will make his computer vaccinated [22, 23]. However, the mentioned models above neglect the influence of vaccination strategy on the propagation of computer viruses. Recently, considering the importance of vaccination, Kumar et al. [24] proposed the following SEIQRS-V computer virus propagation model:where , , , , , and denote the numbers of the uninfected computers, the exposed computers, the infected computers, the quarantined computers, recovered computers, and vaccinated computers at time , respectively. is the birth rate of new computers in the network; is the death rate of the computers due to the reason other than the attack of viruses; is the death rate of computers due to the attack of viruses; is the contact rate of the uninfected computers; , , , , , , and are the transition rates between the states in system (1).

Obviously, system (1) neglects the delays in the procedure of viruses’ propagation and it is investigated under the assumption that the transition between the states is instantaneous. This is not reasonable with reality. For example, it needs a period to clean the viruses in the infected and quarantined computers for antivirus software and there is usually a temporary immunity period for the recovered and the vaccinated computers because of the effect of the antivirus software. In addition, a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this, we introduce two delays into system (1) and get the following delayed system: where is the time delay due to the period that antivirus software uses to clean the viruses in the infected and quarantined computers and is the time delay due to the temporary immunity period of the recovered and the vaccinated computers.

To the best of our knowledge, until now, there is no good analysis on system (2). Therefore, it is meaningful to analyze the proposed system with two delays.

The rest of this paper is organized as follows. In the next section, we analyze the threshold of Hopf bifurcation of system (2) by regarding different combination of the two delays as the bifurcation parameter. In Section 3, by means of the normal form theory and center manifold theorem, direction and stability of the Hopf bifurcation for and are investigated. Simulation results of system (2) are shown in Section 4. Finally, we finish the paper with conclusions in Section 5.

2. Analysis of Hopf Bifurcation

By direct computation, we know that if and , then system (2) has a unique viral equilibrium , where The linearized section of system (2) at is as follows: where Then, the characteristic equation for system (4) can be obtained: with

Case 1 (). When , (6) becomes where Clearly, . Thus, if condition (see (10)) holds, then system (2) without delay is locally asymptotically stable:

Case 2 (; ). Equation (6) equals where Multiplying on left and right of (11), one has Assume that is the root of (13): with Thus, one can obtain the expressions of and as follows: Then, we can get Suppose that (see (17)) has at least one positive root.
If condition holds, then there exists such that (13) has a pair of purely imaginary roots . For , Differentiating (13) with respect to , one has where Thus, with Thus, if condition holds, then . Based on the Hopf bifurcation theorem in [25], we have the following results.

Theorem 1. Suppose that conditions , , and hold for system (2). The viral equilibrium is locally asymptotically stable when and a Hopf bifurcation occurs at the viral equilibrium when .

Case 3 (; ). Equation (6) becomeswhereMultiplying on left and right of (23), one has Let be the root of (25):withThen,And the equation following equation regarding can be obtained: Suppose that (see (29)) has at least one positive root.
If condition holds, then there exists such that (25) has a pair of purely imaginary roots . For , Differentiate both sides of (25) with respect to . Then, whereThus,withSimilar to Case 2, we know that if condition holds, then . In conclusion, we have the following results.

Theorem 2. Suppose that conditions , , and hold for system (2). The viral equilibrium is locally asymptotically stable when and a Hopf bifurcation occurs at the viral equilibrium when .

Case 4 (; ). Regarding as the bifurcation parameter when , multiplying by , (6) becomes Let be the root of (35), and for the convenience we still denote as ; then, where