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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 8239823, 18 pages
https://doi.org/10.1155/2018/8239823
Research Article

Stability and Hopf Bifurcation of a Delayed Epidemic Model of Computer Virus with Impact of Antivirus Software

1School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
2Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India

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

Received 2 September 2018; Revised 8 October 2018; Accepted 18 October 2018; Published 4 November 2018

Academic Editor: Seenith Sivasundaram

Copyright © 2018 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

In this paper, we investigate an SLBRS computer virus model with time delay and impact of antivirus software. The proposed model considers the entering rates of all computers since every computer can enter or leave the Internet easily. It has been observed that there is a stability switch and the system becomes unstable due to the effect of the time delay. Conditions under which the system remains locally stable and Hopf bifurcation occurs are found. Sufficient conditions for global stability of endemic equilibrium are derived by constructing a Lyapunov function. Formulae for the direction, stability, and period of the bifurcating periodic solutions are conducted with the aid of the normal form theory and center manifold theorem. Numerical simulations are carried out to analyze the effect of some of the parameters in the system on the dynamic behavior of the system.

1. Introduction

Computer viruses are programs created to carry out activities in a computer without consent of its owner. They not only disrupt the normal functionalities of computer system and damage data files in the computer, but also cause heavy economic losses and tremendous social impacts [13]. In recent years, mathematical modeling enjoys popularity in both analyzing and controlling computer viruses based on the similarity between computer viruses and biological viruses. A few works proposing SIR models have appeared in the literatures [46]. In [4], Amador studied a stochastic SIRA epidemic model for computer viruses and analyzed the quasistationary distribution, the extinction time, and the number of infections in order to understand the spreading mechanism of computer viruses. In [5], Ozturk and Gulsu proposed an approximate solution to a modified SIR computer virus model by using the shifted Chebyshev collocation method. In [6], Khanh studied the stability and approximate iterative solutions of an SIR computer virus model with antidotal component.

Considering the latent period of computer viruses, some models with latency are proposed by some scholars [1, 710]. In [7], Yang investigated global stability of a VEISV network worm attack model by using the Li-Muldowney geometric approach. In [8], Keshri et al. proposed a reduced SEIR scale-free network model and studied its stability. In [9], Hosseini et al. formulated a discrete-time SEIRS model of computer virus propagation in scale-free networks and analyzed the local and global stability of the model. In [1], Guillen et al. proposed an improved SEIRS worm model with considering accurate positions for dysfunctional hosts and their replacements in state transition. In [10], Ren and Xu investigated an SEIR-KS computer virus propagation model based on the kill signals. They studied the local and global stability of the model by applying Routh-Hurwitz criterion and Lyapunov functional method. There are also some other models considering the latency of computer viruses with quarantine [1114] and vaccination [1519].

However, as stated in [20], those above models with the exposed compartment neglect the fact that a computer can infect other computers through file copying or file downloading. Therefore, to overcome the above-mentioned defect, computer virus models with infectivity in latent period have received much attention in recent years [2126]. Unfortunately, most of these models still have some defects. On the one hand, they ignore the effect of time delay. As is known, there are some time delays of one type or another in the transmission process of computer viruses due to latent period, temporary immunity period, or other reasons. On the other hand, only the susceptible computers are regarded as the entering computers, but every computer can enter or leave the Internet easily in reality. Finally, they neglect the effect of antivirus software, especially the effect of antivirus software on the susceptible computers. Based on the discussion above, we investigated a delayed SLBRS computer virus model with impact of antivirus software based on the following model proposed in [27]: where , , , and denote the numbers of susceptible, latent, breaking, and recovered computers at time , respectively. More parameters are listed in Table 1 as follows.

Table 1: Parameters and their meanings in this paper.

Considering the temporary immune period of the recovered computers, we incorporate the time delay due to the temporary immunity period into system (1) and obtain the following delayed model:where is the time delay due to the temporary immunity period.

The remainder of the paper is structured as follows. In Section 2, conditions for local stability of the endemic equilibrium and the existence of Hopf bifurcation are performed. Section 3 deals with global stability of the endemic equilibrium. Section 4 is devoted to establishing the formulae to determine the direction, stability, and period of the bifurcating periodic solutions. Some numerical simulations are presented to illustrate the theoretical results in Section 5. We end the paper with a brief conclusion in Section 6.

2. Local Stability and Existence of Hopf Bifurcation

By a direct computation, it can be concluded that system (2) has the endemic equilibrium wherewhere is the positive root of (4)whereandThe Jacobian matrix of system (2) evaluated at iswhereThe corresponding characteristic equations iswithFor , (9) becomes

Clearly, . Hence, it follows from the Hurwitz criterion that all the roots of (11) have negative real parts, if : , and holds.

For , let be the root of (9). Then,Thus,withLet , then (13) becomes

Suppose that (15) has a positive root . Then, (13) has a positive root such that (9) has a pair of purely imaginary roots . For ,whereDifferentiating on both sides of (9) with respect to , one can obtainFurther, we havewhere and .

Therefore, if : , then . Thus, we have the following results based the Hopf bifurcation theorem in [28].

Theorem 1. For system (2), if - hold, then is locally asymptotically stable when ; system (2) undergoes a Hopf bifurcation at when and a family of periodic solutions bifurcate from . is defined as in (16).

3. Global Stability Analysis

Theorem 2. If , withwhere , , , and for , then the endemic equilibrium is globally asymptotically stable.

Proof. LetThen becomes the trivial equilibrium for all , and system (2) can be reduced to the following form:Now, we haveNow, (22) can be rewritten as follows by using above relation,Let . It follows from the above equationWe find that there exists a , such that for all and for , we haveAgain due to form of (29) we consider the following functional:whose derivative along the solution of system (2) is given byAgain let and . Now calculate the derivative of and with the solution of (2), it follows from, respectively, (23) and (24)Now, (25) can be rewritten as follows by using (26),Again let . It follows from the above equationWe find that there exists a , such that for all and for , we haveAgain due to the above form of (36) we consider the following functional:whose right derivative along the solution of the system (2) is given byLet us define a Lyapunov functional as Computing the upper right derivative of along the solution of system (2) and by using (31)-(33) and (38), we obtainwhere , , , and are defined above in (20).
Since the model system (2) is positive invariant, therefore, for all , we have Using the mean value theorem, we havewhere lies between and , lies between and , lies between and , and lies between and . Therefore,Note that . Hence, from theory of global stability and (43), we conclude that the zero solution of the reduced system (22)-(25) is globally asymptotically stable. Therefore, the endemic equilibrium of model system (2) is globally asymptotically stable.

4. Direction and Stability of Hopf Bifurcation

Let , , , , and . System (2) becomeswhere and : and : are defined as follows:and