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A worm spread model concerning impulsive control strategy is proposed and analyzed. We prove that there exists a globally attractive virus-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 including viruses, worms, and trojan horses. Worms use system vulnerability to search and attack computers. As hardware and software technologies develop and computer networks become an essential tool for daily life, worms start to be a major threat. In June 2010, the Belarusian security firm Virus Block Ada discovered deadly Stuxnet worm. The Stuxnet worm is the first known example of a cyber-weapon that is designed not just to steal and manipulate data but to attack a processing system and cause physical damage. The Stuxnet worm is the first cyber-attack of its kind and has infected thousands of computer systems worldwide.

Consequently, the trial on better understanding the worm propagation dynamics is an important matter for improving the safety and reliability in computer systems and networks. Similar to the biological viruses, there are two ways to study this problem: microscopic and macroscopic. Following a macroscopic approach, since [

In [

Original model.

A population size

One has

In the

As we know, antivirus software is a kind of computer program which can detect and eliminate known worm. There are two common methods to detect worms: using a list of worm signature definition and using a heuristic algorithm to find worm based on common behaviors. It has been observed that it does not always work in detecting a novel worm by using the heuristic algorithm. On the other hand, obviously, it is impossible for antivirus software to find a new worm signature definition on the dated list. So, to keep the antivirus software in high efficiency, it is important to ensure that it is updated. Based on the previous 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 worm.

Based on the previous facts, we propose the following assumptions:

the antivirus software is installed or updated at time

According to the previous assumptions (

Impulse model.

The system (

The initial conditions for (

The organization of this paper is as follows. In Section

To prove our main results, we state three lemmas which will be essential to our proofs.

Consider the following impulsive differential equations:

If

Let

The local stability of virus-free periodic solution may be determined by considering the behaviors of a small amplitude perturbation of the solution. Define

The proof is complete.

If

Because

Let

The proof is complete.

The virus-free periodic solution

Theorem

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

System (

Suppose that

Now, we will prove that there exist

By Lemma

Therefore, it is certain that there exists a

Since

Owing to the randomicity of

The proof of Theorem

Suppose

Let

The proof of Theorem

It follows from Theorem

In this section we have performed some numerical simulations to show the geometric impression of our results.

To demonstrate the global attractivity of virus-free periodic solution of system (

Global attractivity of virus-free periodic solution of system.

To demonstrate the permanence of system (

Permanence of system.

We have analyzed the