Abstract
A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.
Highlights
In the wake of developments in computer technology and communication technology, there is a rapid increase in computer viruses which has brought about huge financial losses [1,2,3]
The object of this paper is to study the existence and properties of the Hopf bifurcation of system (2)
For system (2), if conditions (H1), (H2), and (H3) hold, the endemic equilibrium D∗(S∗, E∗, I∗, Q∗, R∗) is locally asymptotically stable when τ ∈ [0, τ0); a Hopf bifurcation occurs at the endemic equilibrium D∗(S∗, E∗, I∗, Q∗, R∗) when τ = τ0 and a family of periodic solutions bifurcate from the endemic equilibrium D∗(S∗, E∗, I∗, Q∗, R∗) near τ = τ0
Summary
In the wake of developments in computer technology and communication technology, there is a rapid increase in computer viruses which has brought about huge financial losses [1,2,3]. Dong et al [9] proposed a delayed SEIR computer virus model with multistate antivirus and studied the Hopf bifurcation of the model by choosing the delay where the infectious nodes use antivirus software to clean the viruses as the bifurcation parameter. Zhang et al [24] studied the existence and properties of the Hopf bifurcation of a computer virus model with antidote in vulnerable system by regarding the time delay due to the period that the infected computers use to reinstall system as a bifurcation parameter. − (d + γ) R (t) , where τ is the time delay due to the period that the antivirus software uses to clean the worms in the exposed, the infectious, and the quarantined computers. An example together with its numerical simulations is presented in order to illustrate the effectiveness of our obtained theoretical results
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