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

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.