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

Zizhen Zhang,Luca Guerrini,Dianjie Bi,Yougang Wang

doi:10.1155/2017/3536125

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.

Highlights

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]
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 [3,4,5,6,7,8], stochastic models [9,10,11,12], and delayed models [13,14,15,16,17,18]
A stability switch occurs even when an ignored delay is small for a dynamical system

Summary

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]. 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. + ηI (t − τ1) , χV (t τ2) , where τ1 is the time delay due to the period that antivirus software uses to clean the viruses in the infected and quarantined computers and τ2 is the time delay due to the temporary immunity period of the recovered and the vaccinated computers.

Analysis of Hopf Bifurcation

Properties of the Hopf Bifurcation

Numerical Simulation

Conclusions