Computer virus spread model concerning impulsive control strategy is proposed and analyzed. We prove that there exists a globally attractive infection-free periodic solution when the vaccination rate is larger than

Computer virus is a kind of computer program that can replicate itself and spread from one computer to others. Viruses mainly attack the file system and worms use system vulnerability to search and attack computers. As hardware and software technology develop and computer networks become an essential tool for daily life, the computer virus starts to be a major threat. Consequently, the trial on better understanding of the computer virus propagation dynamics is an important matter for improving the safety and reliability in computer systems and networks. Similar to the biological virus, there are two ways to study this problem: microscopic and macroscopic models. Following a macroscopic approach, since [

In [

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

New computers are attached to the computer network with constant rate

Computers are disconnected to the computer network with the constant rate

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

Original model.

As we know, antivirus software is a kind of computer program which can detect and eliminate known viruses. There are two common methods that an antivirus software application uses to detect viruses: using a list of virus signature definitions and using a heuristic algorithm to find viruses based on common behaviors. It has been observed that it does not always work in detecting a novel computer virus by using the heuristic algorithm. On the other hand, obviously, it is impossible for antivirus software to find new computer viruss signature definitions on the dated list. So, to keep the antivirus soft in high efficiency, it is important to ensure that it is updated. Based on the above facts, we propose an impulsive system to model the process of periodic installing or updating antivirus software on susceptible computers at fixed time for controlling the spread of computer virus.

Based on above facts, we propose the following assumptions.

The antivirus software is installed or updated at time

According to the above assumptions (H1)–(H6), and for the reason of simplicity we propose the following model with one time delay (see Figure

Impulse Model.

The total population size

Before going into any details, we simplify model (

The organization of this paper is as follows. In Section

In this section, we prove that the infection-free periodic solution is globally attractive under some conditions. To prove the main results, two lemmas (given in [

Consider the following impulsive system:

Consider the following linear neutral delay equation:

When

Consider system (

assume that

assume that

From the third and sixth equations of system (

The infection-free periodic solution

Since

From the first equation of system (

The infection-free periodic solution

Theorem

In this section, we say the computer virus is local if the infectious population persists above a certain positive level for sufficiently large time. The locality viruses can be well captured and studied through the notion of permanence.

System (

System (

Suppose that

Now, we will prove there exist

Consider the following comparison system:

Corollary

Therefore, it is certain that there exists

Owing to the randomicity of

Suppose

Let

By Lemmas

It follows from Theorem

In this section, we perform some numerical simulations to show the geometric impression of our results. To demonstrate the global attractivity of infection-free periodic solution to system (

Global attractivity of infection-free periodic solution.

To demonstrate the permanence of system (

permanence of system.

We have analyzed the delayed

In this paper, we have only discussed two cases: (i)

This paper is supported by the National Natural Science Foundation of China (no. 61170320), the Natural Science Foundation of Guangdong Province (no. S2011040002981) and the Scientific research Foundation of Guangdong Medical College (no. KY1048).