^{1}

^{2}

^{1}

^{1}

^{2}

A computer virus model with infection delay and recovery delay is considered. The sufficient conditions for the global stability of the virus infection equilibrium are established. We show that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits. By the normal form and center manifold theory, the direction of the Hopf bifurcation and stability of the bifurcating periodic orbits are determined. Numerical simulations are provided to support our theoretical conclusions.

In recent years, the computer networks have become more and more popular, and people can find many useful things through computer networks. However, computer virus flows and spoils the correct operation of computer. As the computer networks become necessary tools in our daily life, computer virus becomes a major threat [

Cohen [

Dong et al. [

Since there is a period of time from virus entering a host to active state [

Ren et al. [

Then we can control the computer virus propagation using the epidemiological threshold value. There are many research works about epidemiological models; see [

The paper is organized as follows. In Section

The initial conditions are

From the standard theory of functional differential equation [

System (

Let

Consider

Consider system (

For

We introduce a set of parameter values:

In the case of

Consider

if

if

if

if

Let

Let

Consider system (

assume that one of the following two conditions holds:

Then the virus infection equilibrium

in either case,

the virus infection equilibrium

From the above discussions, the sufficient conditions for the existence of Hopf bifurcation were given for

Fix

Following the procedure in [

From (

Assume that the conditions of Theorem

We introduce a set of parameter values:

Behavior and phase portrait of system (

Behavior and phase portrait of system (

The following result shows the global stability of the virus-free equilibrium.

Consider the following.

If

Next we will prove the global stability of the virus infection equilibrium of system (

Consider system (

Define a Lyapunov function

For

We choose a set of parameter values:

Numerical simulations of system (

We consider a computer virus model with infection delay and recovery delay. When

Our results show that we can take measures to make the basic reproduction number

However, our results should be viewed carefully because the model used in this paper is simplified and possibly does not explain all relevant dynamics of computer virus. Then we need more realistic computer virus models to study the real network.

The authors are grateful to the editor and the anonymous reviewers for their valuable comments and suggestions which greatly improved the paper. This research is supported by the National Natural Science Foundation of China (no. 11371112) and in part by the National Natural Science Foundation of China (no. 11031002) and by the Heilongjiang Provincial Natural Science Foundation (no. A201208).

The authors declared that they have no conflict of interests to this work.