Based on complex network, this paper proposes a novel computer virus propagation model which is motivated by the traditional SEIRQ model. A systematic analysis of this new model shows that the virus-free equilibrium is globally asymptotically stable when its basic reproduction is less than one, and the viral equilibrium is globally attractive when the basic reproduction is greater than one. Some numerical simulations are finally given to illustrate the main results, implying that these results are applicable to depict the dynamics of virus propagation.

Computer viruses, including the narrowly defined viruses and network worms, are loosely defined as malicious codes that can replicate themselves and spread among computers. Usually, computer viruses attack computer systems directly, while worms mainly attack computers by searching for system or software vulnerabilities. With the rapid popularization of the Internet and mobile wireless networks, network viruses have posed a major threat to our work and life. To thwart the fast spread of computer viruses, it is critical to have a comprehensive understanding of the way that computer viruses propagate. Kephart and White [

Recently, Mishra and Jha [

The population has a homogeneous degree distribution.

The total population of computers is divided into five groups: susceptible, exposed, infected, quarantine and recovered computers. Let

New computers are attached to the Internet at rate

Computers are disconnected from the Internet naturally at a constant rate

According to the above assumptions, the following model is derived (see Figure

Original model.

In view of the fact that the Internet topology is scale-free rather than exponential in its degree distribution [

For convenience, computers on the Internet are called as nodes in the sequel. For our purpose, the following additional assumptions are imposed on the previous SEIQRS model.

The node degrees of the network asymptotically follow a power-law distribution,

The total number of nodes does not change or, equivalently,

The probability that a link has an

By applying the mean-field technique to the above assumptions, we get a new epidemic model of computer virus, which is formulated as (see Figure

Our model.

with initial conditions

Note that, for every

with initial conditions

The organization of this paper is as follows. Section

The basic reproduction number

where

Consider system (

if

After imposing the stationarity condition, we have

It is easily verified that

where

If

It can be seen from Theorem

It is clear that

For convenience, let

Let

with initial condition

Consider system (

The characteristic equation with respect to

We obtain

This equation has negative roots

Suppose

Consider a system

The set

which have

as their respective outer normal vectors. For

Thus, the claimed result follows from Lemma

Consider an

where

the origin forms the largest positively invariant set included in

Then, one has that

We are ready to prove.

Consider system (

Let

Moreover,

We will ascertain the global attractivity of the viral equilibrium.

If

Theorem

where

After equivalent deformation, it follows that

But

Thus, from the above inequality, we get

This is a contradiction. Similarly, we can also get contradictions when

where

If

And we obtain

According to the definition of

Then if

or

Since

Similarly, we can testify that

Both

then we have

According to the LaSalle invariant set principle, any solution of (

Consider system (

In this section, some numerical simulations are given to support our results. To demonstrate the global stability of the infection-free solution of system (

Global stability of infection-free solution.

To demonstrate the global attractivity of the viral equilibrium of system (

Global attractivity of infection solution.

Consider system (

Evolution of

To clearly understand how the Internet topology affects the spread of computer viruses, a new model capturing the epidemics of computer viruses on scale-free networks has been proposed. The basic reproduction number

The work is supported by the National Natural Science Foundation of China under Grant no. 61304117, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant no. 13KJB520008, the doctorate teacher support project of JiangSu Normal University under Grant no. 12XLR021.