Hopf bifurcation analysis for an epidemic model over the Internet with two delays

Tao Zhao,Dianjie Bi

doi:10.1186/s13662-018-1541-y

Tao Zhao, Dianjie Bi

Open Access

https://doi.org/10.1186/s13662-018-1541-y

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Abstract

Taking the delay due to the latent period of computer viruses and the delay due to the period that the anti-virus software uses to clean the computer viruses as the bifurcation parameters, local Hopf bifurcation of an epidemic model over the Internet is studied. We discuss the existence of the Hopf bifurcation under four conditions: (1) tau _{1}>0, tau_{2}=0, (2) tau_{1}=0, tau_{2}>0, (3) tau_{1}=tau _{2}=tau>0, and (4) tau_{1}>0, tau_{2}in(0, tau_{20}). Properties of the Hopf bifurcation about condition (4) are investigated by means of the center manifold theorem and the normal form theory. Finally, some simulations are presented to support our obtained results.

Highlights

The Internet is an indispensable part of our everyday life and it offers us more and more functionalities and facilities
In [5, 6], Ren et al investigated the Hopf bifurcation of a delayed SIRS computer virus propagation model
Compared with the model considered in the literature [24], we investigate the effects of the time delay due to the latent period of the latent computers, and the effects of the time delay due to the period that the anti-virus software uses to clean computer viruses in the breaking and the quarantined computers

Summary

Introduction

The Internet is an indispensable part of our everyday life and it offers us more and more functionalities and facilities. In [5, 6], Ren et al investigated the Hopf bifurcation of a delayed SIRS computer virus propagation model. In [18], Dong et al proposed a delayed SEIR (Susceptible–Exposed–Infectious–Recovered) computer virus model with multistate antivirus and studied the dynamical behaviors, which include local asymptotical stability and local Hopf bifurcation, by regarding the time delay as bifurcating parameter. Considering the fact that the recovered computers may be infected again after a temporary immunity period, Zhang and Yang [19] proposed a computer virus model with two delays based on the work in [18] and studied the Hopf bifurcation by regarding the possible combinations. Τ is the time delay due to the period that the anti-virus software uses to clean the computer viruses in the breaking and the quarantined computers.

Local stability of viral equilibrium and existence of local Hopf bifurcation

Properties of the Hopf bifurcation

Conclusions