An Impulse Model for Computer Viruses

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 andworms 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 1, 2 took the first step towards modeling the spread behavior of computer virus, much effort has been done in the area of developing a mathematical model for the computer virus propagation 3– 13 . These models provide a reasonable qualitative understanding of the conditions under which viruses spread much faster than others. In 4 , the authors investigated a differential SIRS model by making the following assumptions.


Introduction
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 1, 2 took the first step towards modeling the spread behavior of computer virus, much effort has been done in the area of developing a mathematical model for the computer virus propagation 3-13 . These models provide a reasonable qualitative understanding of the conditions under which viruses spread much faster than others.
In 4 , the authors investigated a differential SIRS model by making the following assumptions.
H1 The total population of computers is divided into three groups: susceptible, infected, and recovered computers. Let S, I, and R denote the numbers of susceptible, infected and recovered computers, respectively.
2 Discrete Dynamics in Nature and Society According to the above assumptions, the following model see Figure 1 is derived :

1.1
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.
H5 The antivirus software is installed or updated at time t kT k ∈ N , where T is the period of the impulsive effect.
H6 S computers are successfully vaccinated from S class to R class with rate θ 0 < θ < 1 .
Discrete Dynamics in Nature and Society 3 According to the above assumptions H1 -H6 , and for the reason of simplicity we propose the following model with one time delay see Figure 2 : The total population size N t can be determined by N t S t I t R t to form the differential equationṄ which is derived by adding the equations in system 1.1 . Thus the total population size N may vary in time.
Before going into any details, we simplify model 1.1 and restrict our attention to the following model:

1.5
The initial conditions for 1.5 are Discrete Dynamics in Nature and Society Figure 2: Impulse Model.
From physical considerations, we discuss system 1.5 in the closed set where R 3 denotes the nonnegative cone of R 3 including its lower-dimensional faces. Note that it is positively invariant with respect to 1.7 . The organization of this paper is as follows. In Section 2, we first state three lemmas which are essential to our proofs and establish sufficient condition for the global attractivity of infection-free periodic solution. The sufficient condition for the permanence of the model is obtained in Section 3. Some numerical simulations are performed in Section 4. In the final section, a brief conclusion is given and some future research directions are also pointed out.

Global Attractivity of Infection-Free Periodic Solution
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 14 which are essential to the proofs are stated here.
Lemma 2.1 see 14 , Lemma 1 . Consider the following impulsive system: Then there exists a unique positive periodic solution of system 2.1 Discrete Dynamics in Nature and Society Lemma 2.2 see 14 , Lemma 2 . Consider the following linear neutral delay equation: If |λ| < 1, a 2 1 < a 2 2 or −a 1 a 2 / 0, then increasing τ does not change the stability of 2.3 .
Corollary 2.3. Consider system 2.4 and assume that a 1 , a 2 , ω > 0; x t > 0 for −ω ≤ t ≤ 0. Then we have the following statements: From the third and sixth equations of system

2.5
From the second and fourth equations of system 3.5 , we have lim t → ∞ N t b/μ and have the following limit systems of 3.5 :
Proof. Since R 0 < 1, we can choose ε 1 > 0 sufficiently small such that It follows from the third equation of system 1.5 thaṫ There exists an integer k 1 > 0 such that N t ≤ b/μ, t > k 1 τ.
From the first equation of system 1.5 , we havė For t > k 1 τ, k > k 1 , we consider the following comparison differential system:

2.12
In view of Lemma 2.1, we know that the unique periodic solution of system 2.12 is of the form

2.13
and it is globally asymptotically stable. From 1.5 , we havė Therefore, for any ε 1 > 0 sufficiently small , there exists an integer k 3 > k 2 such that I t < ε for all t > k 3 τ. From the third equation of system 1.5 , we have Consider the comparison equatioṅ It is easy to see that lim t → ∞ z t b −αε 1 /μ. It follows by the comparison theorem that there exists an integer k 4 > k 3 such that

2.18
Since ε 1 is arbitrarily small, from lim t → ∞ sup N t ≤ b/μ and 2.18 we have It follows from 2.15 and 2.19 that there exists k 5 > k 4 such that

2.20
Hence, from the first equation of system 1.5 we have thaṫ for t > k 5 τ.

2.21
Consider the following comparison impulsive differential equations for t > k 5 τ and k > k 5 ,

2.22
In view of Lemma 2.1, we periodic solution of system According to the comparison theorem for impulsive differential equation, there exists an integer k 6 > k 5 such that S t > u e t − ε 1 , kτ < t ≤ k 1 τ, k > k 6 .

2.25
Because ε 1 is arbitrarily small, it follows from 2.25 that is globally attractive, that is,

2.27
It follows from 2.15 , 2.19 , 2.27 , and the restriction N t S t I t R t that lim x → ∞ R t b/μ − S e t . Hence, the infection-free periodic solution S e t , 0, b/μ of system 1.5 is globally attractive. The proof is completed. Theorem 2.4 determines the global attractivity of 1.5 in Ω for the case R 0 < 1. Its realistic implication is that the infected computers vanish so the computer virus removed from the network. Corollary 2.5 implies that the computer virus will disappear if the vaccination rate is larger than θ 0 .

Permanence
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.
Definition 3.1. System 1.5 is said to be uniformly persistent if there is an ϕ > 0 independent of the initial data such that every solution S t , I t , R t , N t with initial conditions 1.7 of system 1.5 satisfies Discrete Dynamics in Nature and Society 9 Definition 3.2. System 1.5 is said to be permanent if there exists a compact region Ω 0 ∈ int Ω such that every solution of system 1.5 with initial data 1.7 will eventually enter and remain in region Ω 0 . Denote Proof. Now, we will prove there exist m I > 0 and a sufficiently large t p such that I t ≥ m I holds for all t > t p . Suppose that I t < m * I for all t > t 0 . From the first equation of 1.5 , we haveṠ Consider the following comparison system: 3.4 By Lemma 2.1, we know that, there exists t 1 such that It follows from the second equation of 1.5 thatİ t > βSI t − τ − μ γ α I t . Consider the comparison systemż 3 t βSI t − τ − μ γ α z 3 t . Noting that R 1 > 1 and ε is sufficiently small, we have βS > μ γ α . Corollary 2.3 implies that t → ∞, I t > z 3 t → ∞. This contradicts I t ≤ b. Hence, we can claim that, for any t 0 > 0, it is impossible that 3.6 By the claim, we are left to consider two cases. First, I t ≤ m * I for t large enough. Second, I t oscillates about m * I for t large enough. Obviously, there is nothing to prove for the first case. For the second case, we can choose t 2 > t 1 and ξ > 0 satisfy I t 2 I t 2 ξ m * I , I t < m * I , for t 2 < t < t 2 ξ 3.7 I t is uniformly continuous since the positive solutions to 1.5 are ultimately bounded and I t is not effected by impulses.

Numerical Simulations
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 1.5 , we take the following parameter values: b 1, μ 0.5, γ 0.3, α 0.02, β 0.3, ν 0.7, θ 0.4, and τ 1. In this case, we have R 0 0.5894 < 1. In Figures 3 a , 3 b , and 3 c display respectively the susceptible, infected and recovered population of system 1.5 with initial conditions: S 0 3, I 0 4 and R 0 5. Figure 3 d shows their corresponding phase-portrait.

Conclusion
We have analyzed the delayed SIRS model with pulse vaccination and varying total population size. We have shown that R 1 > 1 or θ < θ 1 implies that the disease will be endemic, whereas R 0 < 1 or θ > θ 0 implies that the disease will fade out. We have also established sufficient condition for the permanence of the model. Our results indicate that a short interpulse time or a large pulse vaccination rate will lead to eradication of the computer virus.
In this paper, we have only discussed two cases: i R 0 < 1 or θ > θ 0 and ii R 1 > 1 or θ < θ 1 . But for closed interval R 0 , R 1 or θ 1 ,θ 0 , the dynamical behavior of model 3 have not been studied, and the threshold parameter for the reproducing number or the pulse vaccination rate between the extinction of the computer viruses and the uniform persistence of the viruses have not been obtained. These issues would be left to our future consideration.