ZENG Jijin, LIU Yujing

(a.School of Business; b.Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques,Gannan Normal university, Ganzhou 341000, China)

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A Propagation Model of Computer Virus with Vaccination Strategy* 1

ZENG Jiajiana, LIU Yujiangb

(a.SchoolofBusiness;b.KeyLaboratoryofJiangxiProvinceforNumericalSimulationandEmulationTechniques,GannanNormaluniversity,Ganzhou341000,China)

Abstract:A mathematical model describing the propagation of computer virus on the Internet is presented, in which continuous control strategy of and impulsive vaccination are taken into consideration. We obtain the reproduction number R0, and further prove that a computer virus cannot invade if R0 is less than unit and can invade if R0 is larger than unit. Several numerical simulations are given to illustrate our main results. Furthermore we find that the combination of continuous and impulsive vaccination is helpful to eliminate the virus.

Key words:computer virus; impulsive vaccination; extinction

1Introduction

More and more applications depend on computer network. Computer viruses attempt to parasitize themselves on a host and spread to other computer through computer network, particularly Internet, damage all kinds of network resources[1].

Due to the high similarity between computer viruses and biological viruses[2], the classical SIR computer virus propagation model was proposed[3]. Since then, more computer virus propagation compartment models have been presented[4-6]. Vaccination is widely regarded as one of the most effective measures of repressing computer virus[4].

This paper aim to investigate the effects of continuous vaccination and impulsive vaccination strategies on the spread of computer viruses and it organize as follows. In Section 2, we formulate the computer virus propagation model. The basic reproduction number is given, followed by an analysis of the model in Sections 3 and 4. The results are illustrated by numerical simulations in Section 5, followed by a discussion of potential further research.

2Model formulation and preliminaries

LetS(t),I(t) andR(t) denote the numbers of susceptible, infected, and recovered nodes in the Internet at timet, respectively. In this section, we propose a propagation model with hybrid vaccination strategy based on [6].

(1)

where

(2)

with initial conditionsS(0+)=S0>0,I(0+)=I0>0.

Lemma 1 IfS(0+)=S0>0 andI(0+)=I0>0, thenS(t)>0 andI(t)>0 for eacht>0.

Lemma 2[7]Let us consider the following linear impulsive system:

(3)

wherea,b>0,0<θ<1. Then there exists a unique positive periodic solution of system (3)

which is globally asymptotically stable.

3The global stability of the virus-free periodic solution

Let us firstly demonstrate the existence of the virus-free periodic solution of system (2). IfI(t)=0 for allt≥0, the dynamic of susceptible individuals must satisfy:

(4)

Theorem 1System (2) always has a virus-free periodic solution (S*(t),0).

In the following, we determine the global attractivity condition of the virus-free periodic solution (S*(t),0). According to[8], we haveF(t)=βS*(t),V(t)=d.LetR0be the basic reproduction number of system (2). Then

(5)

Theorem 2IfR0<1, then the virus-free periodic solution (S*(t),0) of system (2) is globally asymptotically stable.

ProofFirst, we demonstrate that the local stability of the virus-free periodic solution (S*(t),0). LetS(t)=u(t)+S*(t),I(t)=v(t), we get the following linear approximation system of (2) relative to the solution (S*(t),0).

(6)

We can easy obtain the fundamental solution matrix of system (6)

According to the Floquet theory, if |λ2|<1, i.e.R0<1 holds, then (S*(t),0) is locally stable.

In the following, we will prove the global attractivity of the virus-free periodic solution. SinceR0<1, we can chooseε>0 sufficiently small such that

(7)

(8)

Remark 1Theorem 2 shows that computer viruses will die out ifR0<1.

4Permanence

Lemma 3IfR0>1, then for anyt1>0, it is impossible thatI(t)t1.

ProofSuppose that the claim is not valid, thenI(t)t1. SinceR0>1. we can chooseε>0 sufficiently small such that

(9)

(10)

(11)

Integrating the second equation of system (2), by (11) we have

Theorem 3Suppose thatR0>1. Then there is a positive constantηsuch that each positive solution (S(t),I(t)) of system (2) satisfiesI(t)≥ηfor sufficient larget, whereη=Ilexp(-dT).

ProofAccording to Lemma 3, we know that there are left two cases to be discussed.

Sett′(>t1) andρ>0 satisfyI(t′)=I(t′+ρ)=IlandI(t)

Case iiIfρ>T, we consider the two cases: A) ift∈(t′,t′+T], the conclusion is evident by Case i;

Since this kind of interval [t′,t′+ρ] is chosen in an arbitrary way (we only needt′to be sufficiently large), we conclude thatI(t)≥ηfor all largetin the second case. In view of our above discussions, the choice ofηis independent of the positive solution of system (2), and we have proved that any solution of system of (2) satisfiesI(t)≥ηfor sufficiently larget.

Theorem 4System (2) is permanent ifR0>1.

By the similar arguments as those in the proof of Theorem 2, we have that there is aε0>0(xsufficient small) such that

(12)

Let Δ0={(S,I)∶Sl≤S,Il≤I,S+I≤N}. From Theorem 3 and inequality (12), we know that the set Δ0is a global attractor in Δ, and then, for every solution of system (2) will eventually enter and remain in region Δ. Therefore, system (2) is permanent. The proof of Theorem 4 is complete.

5Numerical Results

We shall verify the validity of the proposed model through numerical simulation. Consider system (2) with Λ=1 000,γ2=0.2,γ1=0.13,β=0.000 3,α=0.5,μ=0.2,σ=0.3,θ=0.5 andx0=(2 500,2 500,0). AsR0=0.908 3<1, it follows from Theorem 2 that virus-free periodic solution is globally asymptotically stable. Otherwise, change only the parameterαof system (2)α=0.3, thenR0=1.180 8>1. It follows from Theorem 4, that computer virus will persist on the Internet. Fig.1 and Fig.2 show that the computer virus will die out whenR0<1 and will be epidemic whenR0>1, respectively.

Fig.1　Time series of solution for system(2).The virus dies out.　Fig.2　Time series of solution for system(2).The virus is permanent.

To assess the effect of continuous vaccination on controlling the speed of computer virus, we numerically compute the infectious nodes as shown in Fig.3. The increasing of the continuous vaccination rate leads to decreasing the number of infectious nodes, as well asR0. Whenα=0,0.2,0.4,R0>1, the computer virus will be epidemic, meanwhile,α=0.6,0.8,R0>1, the computer virus will die out.

It follows from Fig.4 that, if we fixT=12, continuous vaccination rateα=0.5 and impulsive vaccination rateθ=0.3, virus will die out (R0=0.925 9). Otherwise, continuous vaccination rateα=0.3 and impulsive vaccination rateθ=0.5, virus will be epidemic (R0=1.151 6). But if the impulse-time interval decreases, the final size of virus decreases correspondingly. When the impulse-time intervalT=2, virus will die out (R0=0.985 6).

Fig.3　Time series of I(t) for system (2), 　　　　　Fig.4　Time series of I(t) for system (2), which have different continuous which have different continuous vaccination rate α.　　　　　α and impulsive vaccination rate θ and the impulse-time interval T.

Fig.5 indicates that theR0increases with the increasing of the impulsive vaccination rateθ. Decreasing the impulse-time intervalTleads to a decrease the value ofR0. Especially, if we fix the impulse-time intervalTbelow a threshold value, improving the impulsive vaccination rate simultaneously has effect on the eradication of computer viruses.

Fig.5　The basic reproduction number for system (2), 　which have different impulsive vaccination rate θ and the pulse interval T.

6Conclusion

This paper has studied the long-term behavior of computer virus propagation. An elaborate analysis of the model including the basic reproduction number, the existence of virus-free periodic solution, its global stability and the permanent of the virus has been conducted. It is found that computer virus on the Internet would tend to extinction or permanence according to the value of the basic reproduction number. To illustrate the obtained main results, some numerical simulations have been examined. Our results imply that the combination of continuous and impulsive vaccination is benefit for the inhibition of computer virus spreading.

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* 收稿日期：2016-03-16

DOI：10.13698/j.cnki.cn36-1037/c.2016.03.002

基金項目：國家自然科學(xué)基金項目(11261004)；江西省社會科學(xué)規(guī)劃項目(14XW08)；江西省十二五教育科學(xué)規(guī)劃項目(15ZD3LYB031)

作者簡介：曾家健(1975-)，女，廣東潮洲人，贛南師范學(xué)院商學(xué)院副教授，研究方向：區(qū)域經(jīng)濟(jì).

中圖分類號：O175.1

文獻(xiàn)標(biāo)志碼：A

文章編號：1004-8332(2016)03-0006-05

一類具有脈沖控制的計算機(jī)病毒傳播模型研究

曾家健a，劉于江b

(贛南師范大學(xué) a.商學(xué)院；b.江西省數(shù)值模擬與仿真技術(shù)重點實驗室，江西 贛州341000)

摘要：建立一類具有連續(xù)和脈沖控制的計算機(jī)網(wǎng)絡(luò)病毒傳播模型，得到基本再生數(shù)R0，并證明當(dāng)R0<1時，病毒將消除；當(dāng)R0>1時，病毒會擴(kuò)散；數(shù)據(jù)模擬驗證了理論結(jié)果.研究結(jié)果表明連續(xù)和脈沖相結(jié)合的控制策略有利于病毒的消除.

關(guān)鍵詞：計算機(jī)病毒；脈沖接種；滅絕

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