Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 5649584, 21 pages
http://dx.doi.org/10.1155/2016/5649584
Research Article

Dynamical Analysis of a Computer Virus Model with Delays

1Department of Mathematics and Physics, Bengbu University, Bengbu 233030, China
2Laboratoire de Physique Statistique, Ecole Normale Supérieure, PSL Research University, Université Paris Diderot Sorbonne Paris-Cité, Sorbonne Universités, UPMC Univ Paris 06, CNRS, 24 rue Lhomond, 75005 Paris, France
3Department of Management, Polytechnic University of Marche, 60121 Ancona, Italy

Received 5 August 2016; Accepted 29 September 2016

Academic Editor: Vincenzo Scalzo

Copyright © 2016 Juan Liu 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

An SIQR computer virus model with two delays is investigated in the present paper. The linear stability conditions are obtained by using characteristic root method and the developed asymptotic analysis shows the onset of a Hopf bifurcation occurs when the delay parameter reaches a critical value. Moreover the direction of the Hopf bifurcation and stability of the bifurcating period solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical investigations are carried out to show the feasibility of the theoretical results.

1. Introduction

The advances of information technology and the wide-spread popularity of computer networks have increased the interest in the computer viruses. In the past decades, many epidemic models such as SIR model [1, 2], SIRS model [3, 4], SEIR model [57], SEIRS model [8] and SEIQRS model [9] characterizing the spread of computer viruses in networks were investigated by many scholars. It is worthwhile to note that the dynamical models above neglect the time delay in the spreading process of computer viruses. To our knowledge, there have been some computer virus models with time delay proposed to depict the spread of a computer virus. In [10, 11], Muroya et al. studied the global stability of a delayed SIRS computer virus propagation model, respectively. In [12], Feng et al. investigated the Hopf bifurcation of a delayed SIRS viral infection model in computer networks by regarding the delay due to temporary immune period of the recovered computers as a bifurcation parameter. In [13], Dong et al. analyzed the Hopf bifurcation of a delayed SEIR computer virus model with multistate antivirus by regarding the time delay due to the period that the computers use antivirus software to clean the viruses as the bifurcation parameter. In [14], Zhang and Yang studied the Hopf bifurcation of the following SIQR computer virus model with time delay: where , , , and denote the numbers of nodes in states susceptible, infectious, quarantined, and recovered at time , respectively. , , , , , , , and are the parameters of system (1) and is the time delay due to the latent period of the computer virus. Zhang and Yang obtained the sufficient conditions for the local stability and existence of local Hopf bifurcation by regarding the delay as a bifurcation parameter and investigated the properties of the Hopf bifurcation by using the normal form method and center manifold theory. As stated in [13], it usually takes a period to clean the viruses in the computers infected by viruses for antivirus software. Therefore, it is reasonable to incorporate the time delay due to the period that antivirus software uses to clean viruses in the infectious and the quarantined computers into system (1). Bearing all above in mind, this paper deals with the analysis of the Hopf bifurcation of the following system with two delays: where is the time delay due to the latent period of the computer virus and is the time delay due to the period that the antivirus software uses to clean the computer viruses in the infectious and the quarantined nodes.

This paper mainly investigates the effect of the two delays on system (2). The remainder of this paper is organized as follows. The local stability of the positive equilibrium and existence of local Hopf bifurcation are analyzed in Section 2 by choosing different combination of the two delays as a bifurcation parameter. Direction of the Hopf bifurcation, stability, and period of the bifurcating periodic solutions on the center manifold are determined in Section 3. Some numerical simulations are presented to illustrate the validity of the main results in Section 4. Finally a critical analysis and further research directions are contained in Section 5.

2. Local Stability of the Positive Equilibrium and Existence of Local Hopf Bifurcation

According to the analysis in [14] we know that if , system (2) has a unique positive equilibrium , where The characteristic equation of system (2) at is from which we obtain where with

Case 1 (). When , (5) becomes Thus, if the condition (9) holds, then the positive equilibrium of system (2) is locally asymptotically stable when .

Case 2 (, ). When , , (5) becomes whereLet be the root of (10); we getfrom which one can obtain where Let ; then (13) becomes Obviously, if all the coefficients of system (2) are given, then we can obtain all the roots of (15) by Matlab software package easily. Thus, we make the following assumption.
(15) has at least one positive root.
If the condition holds, then there exists a positive root for (15) such that (10) has a pair of purely imaginary roots . For , where Substituting into the left side of (10) and taking the derivative with respect to , one can obtain Thus, where and .
Thus, if the condition holds, then . According to the Hopf bifurcation theorem in [15], we have the following results.

Theorem 1. If the conditions - hold, then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 3 (, ). When , , (5) becomes where Multiplying by , (20) becomes Let () be the root of (22); then It follows that where Then, we can get with Let ; (26) becomes Similar as in Case 2, we make the following assumption.
(28) has at least one positive root.

If condition holds, there exists a positive root for (22) such that (22) has a pair of purely imaginary root . For , Differentiating the two sides of (22) regarding , we obtain where where

Therefore, if condition () holds, then Re. Thus, we have the following results according to the Hopf bifurcation theorem in [15].

Theorem 2. If conditions - hold, then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 4 (, and ). Let be the root of (5); then we obtain where Then, we can obtain the following equation with respect to : where In order to give the main results in this paper, we make the following assumption.
(35) has at least one positive root.
If the condition holds, then there exists a positive root for (35) such that (5) has a pair of purely imaginary root . For , Substituting into (5) and differentiating both sides of it with respect to , then where We therefore derive that where Therefore, if condition holds, then . Thus, we have the following results according to the Hopf bifurcation theorem in [15].

Theorem 3. For system (2), if conditions - hold and , then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from near .

Case 5 (, and ). Multiplying by , (5) becomes Let be the root of (42); then we get where