^{1, 2}

^{2}

^{1}

^{2}

A delayed SIQR computer virus model is considered. It has been observed that there exists a critical value of delay for the stability of virus prevalence by choosing the delay as a bifurcation parameter. Furthermore, the properties of the Hopf bifurcation such as direction and stability are investigated by using the normal form method and center manifold theory. Finally, some numerical simulations for supporting our theoretical results are also performed.

Recently, many scholars have been studying the prevalence of computer viruses by establishing reasonable mathematics models [

As is known, many computer viruses have different kinds of delays when they spread, such as latent period delay [

Gan et al. investigated global attractivity and sustainability of system (

This paper is organized as follows. In Section

In this section, we mainly focus on the local stability of positive equilibrium and existence of local Hopf bifurcation. It is not difficult to verify that if the basic reproduction number

If condition

Thus, if condition

If the conditions

In this section, we investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using the normal form theory and the center manifold theorem in [

Let

For

Let

Following the algorithms given in [

For system (

In this section, a numerical example is given to support the theoretical results in Sections

The track of the states

The phase plot of the states

The phase plot of the states

The track of the states

The phase plot of the states

The phase plot of the states

In addition, according to the numerical simulation, we find that the onset of the Hopf bifurcation can be delayed by decreasing the number of new nodes connected to a network or increasing the immunization rate of the new nodes. Therefore, the managers of a real network should control the number of the new nodes connected to network and strengthen the immunization of the new nodes in order to delay and control the onset of the Hopf bifurcation, so as to make the propagation of computer viruses be predicted and controlled easily.

In this paper, the problem of Hopf bifurcation for a delayed SIQR computer virus model has been studied. The stability of the positive equilibrium and the existence of Hopf bifurcation under this model are analyzed. It has been found that when the delay is suitable small (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the anonymous referees and the editor for their valuable comments and suggestions on the paper. The research was supported by the National Nature Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2014A005).