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By considering the varying latency period of computer virus, we propose a novel model for computer virus propagation in network. Under this model, we give the threshold value determining whether or not the virus finally dies out, and study the local stability of the virus-free and virus equilibrium. It is found that the model may undergo a Hopf bifurcation. Next, we use different methods to prove the global asymptotic stability of the equilibria: the virus-free equilibrium by using the direct Lyapunov method and virus equilibrium by using a geometric approach. Finally, some numerical examples are given to support our conclusions.

With the advance of computer software and hardware and communication technologies, the number and sort of computer viruses have increased dramatically, which causes huge losses to the human society. Therefore, establishing reasonable computer-virus-propagation models by considering the characteristics of computer virus and, by model analysis, understanding the spread law of the virus over the network, are a currently hot topic of research.

Towards this goal, the classical SIR (susceptible-infected-recovered) model [

Motivated by the previous work, this paper proposes and studies a computer-virus-propagation model with varying latency period, known as the SIRC model. We obtain the threshold value determining whether the virus dies out completely, study the local asymptotic stabilities of the equilibria of the model and it is found that, model may undergo a Hopf bifurcation. Next, we prove the global asymptotic stability of the virus-free equilibrium by using the direct Lyapunov method, prove the global asymptotic stability of the virus equilibrium by using a geometric approach. By introducing varying time delay, the model may truly reflect the virus propagation and hence, the corresponding results may help understand and prevent the spread of computer virus over a computer network.

The remaining materials of this paper are organized this way: Section

Consider the classical SIR computer virus model proposed in [

The computer virus has latent and unpredictable characteristics [

The virus in susceptible computer has a latency period. Moreover, this latency period is varying, which can be reflected by the following expression:

Only when the virus breaks out can the susceptible computers become the infected ones.

We choose a typical class of kernels

By incorporating these factors into model (

This section investigates the equilibria of model (

First, model (

Consider model (

The virus-free equilibrium

Next, when

Consider model (

The virus equilibrium

From the above analysis, we can see that that there exists a stability switch for

Indeed, when

If

In this section, we will discuss the global stability of the model.

when

Define

In the following, we use the geometrical approach [

Let

Denote by

There exists a compact absorbing set

Equation (

Let

From the above outline, a theorem can be given as follows:

Assume that _{1}) and (H_{2}) hold, if

Now, we discuss the global stability of the virus equilibrium

Model (_{1}). If _{2}) is met.

The Jacobian matrix of model (

This leads to

From the above discussions, we can obtain the following theorem:

When

In this section, we make some numerical simulations to understand the obtained theorems. Let

Distribution of computers versus time when

Distribution of computers versus time when

In this paper, by considering varying latency period of computer virus, we propose a model for computer virus propagation in network. First, we give the threshold value

The authors wish to thank the anonymous editors and reviewers.