A delayed SEIQRS model for the transmission of malicious objects in computer network is considered in this paper. Local stability of the positive equilibrium of the model and existence of local Hopf bifurcation are investigated by regarding the time delay due to the temporary immunity period after which a recovered computer may be infected again. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Numerical simulations are also presented to support the theoretical results.

The action of malicious objects throughout a network can be studied by using epidemic models due to the high similarity between malicious objects and biological viruses [

It is well known that time delays can cause a stable equilibrium to become unstable and make a system bifurcate periodic solutions and dynamical systems with delay have been studied by many scholars [

The main purpose is to investigate the effects of the delay on system (

According to the analysis in [

Let

For system (

For

Suppose that condition

In the previous section, we have obtained the conditions under which system (

Define

For system (

Let

By the Riesz representation theorem, there exists a

For

The adjoint operator

Let

Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [

Then, we can get the expression of

In this section, we use a numerical example to support the theoretical analysis above in this paper. We take the following particular case of system (

It is easy to verify that

The phase plot of the states

The phase plot of the states

The phase plot of the states

The phase plot of the states

This paper is concerned with a delayed SEIQRS model for the transmission of malicious objects in computer network. The theoretical analysis for the delayed model is given and the main results are given in terms of local stability and local Hopf bifurcation. By regarding the delay due to the temporary immunity period after which a recovered computer may be infected again, we have proven that when the delay passes through the critical value, the model undergoes a Hopf bifurcation. The occurrence of Hopf bifurcation means that the state of virus prevalence changes from a positive equilibrium to a limit cycle, which is not welcomed in networks. Hence, we should control the phenomenon by combining some bifurcation control strategies and other relative features of virus prevalence. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Finally, some numerical simulations are presented to clarify our theoretical results.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is grateful to the referees and the editor for their valuable comments and suggestions on the paper.