Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2012 / Article
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Computer Virus: Theory, Model, and Methods

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Volume 2012 |Article ID 264874 | https://doi.org/10.1155/2012/264874

Chunming Zhang, Yun Zhao, Yingjiang Wu, Shuwen Deng, "A Stochastic Dynamic Model of Computer Viruses", Discrete Dynamics in Nature and Society, vol. 2012, Article ID 264874, 16 pages, 2012. https://doi.org/10.1155/2012/264874

A Stochastic Dynamic Model of Computer Viruses

Academic Editor: Bimal Kumar Mishra
Received08 Jun 2012
Accepted16 Jul 2012
Published26 Aug 2012

Abstract

A stochastic computer virus spread model is proposed and its dynamic behavior is fully investigated. Specifically, we prove the existence and uniqueness of positive solutions, and the stability of the virus-free equilibrium and viral equilibrium by constructing Lyapunov functions and applying Ito's formula. Some numerical simulations are finally given to illustrate our main results.

1. Introduction

A generalized computer virus, including the narrowly defined virus and the worm, is a kind of computer program that can replicate itself and spread from one computer to another. Viruses mainly attack the file system and worms use system vulnerability to search and attack computers. As hardware and software technology developed and computer networks became widespread, computer virus has come to be one major threat to our daily life. Consequently, in order to deal with the threat, the trial on better understanding the computer virus propagation dynamics is an important matter. Similar to the biological virus, there are two ways to study this problem: microscopic and macroscopic. 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 [13], the authors investigated a differential 𝑆𝐸𝐼𝑅 model by making the following assumptions.(𝐻1) The total population of computers is divided into four groups: susceptible, exposed, infected, and recovered computers. Let 𝑆,𝐸,𝐼, and 𝑅 denote the numbers of susceptible, exposed, infected, and recovered computers, respectively. 𝑁 denotes the total number of computers.(𝐻2) New computers are attached to the computer network with rate πœ‡π‘.(𝐻3) Computers are disconnected to the computer network with constant rate πœ‡.(𝐻4)𝑆 computers become 𝐸 computers with rate π›Όπ‘Ÿ/𝑁, where π‘Ÿ denotes the averaged number of neighbor nodes (with various states) that are directly connected; 𝛼 is the transition rate from 𝐸 to 𝐼. 𝑆 computers become 𝑅 computers with rate πœŒπ‘†π‘….(𝐻5)𝐸 computers become 𝐼 computers with constant rate 𝛼; 𝐸 computers become 𝑅 computers with constant rate πœŒπ‘†π‘…; 𝐼 computers become 𝑅 computers with constant rate 𝛾.

According to the above assumptions, the following model (see Figure 1) is derived: ̇𝑆(𝑑)=πœ‡π‘βˆ’π›Όπ‘Ÿπ‘πΈ(𝑑)𝑆(𝑑)βˆ’πœŒπ‘†π‘…π‘†Μ‡(𝑑)βˆ’πœ‡π‘†(𝑑),𝐸(𝑑)=π›Όπ‘Ÿπ‘ξ€·πΈ(𝑑)𝑆(𝑑)βˆ’π›Ό+πœŒπΈπ‘…ξ€ΈΜ‡Μ‡π‘…+πœ‡πΈ(𝑑),𝐼(𝑑)=𝛼𝐸(𝑑)βˆ’(𝛾+πœ‡)𝐼(𝑑),(𝑑)=πœŒπ‘†π‘…π‘†(𝑑)+πœŒπΈπ‘…πΈ(𝑑)βˆ’π›ΎπΌ(𝑑)βˆ’πœ‡π‘…(𝑑).(1.1)

Notably, the first three equations in (1.1) do not depend on the fourth equation, since ̇̇̇̇𝑆(𝑑)+𝐸(𝑑)+𝐼(t)+𝑅(𝑑)=1. Therefore, the forth equation can be omitted and the model (1.1) can be rewritten as ̇𝑆(𝑑)=πœ‡π‘βˆ’π›Όπ‘ŸπΈ(𝑑)𝑆(𝑑)π‘βˆ’πœŒπ‘†π‘…Μ‡π‘†(𝑑)βˆ’πœ‡π‘†(𝑑),𝐸(𝑑)=π›Όπ‘ŸπΈ(𝑑)𝑆(𝑑)π‘βˆ’ξ€·π›Ό+πœŒπΈπ‘…ξ€ΈΜ‡+πœ‡πΈ(𝑑),𝐼(𝑑)=𝛼𝐸(𝑑)βˆ’(𝛾+πœ‡)𝐼(𝑑).(1.2)

In [13], authors have proved the virus-free equilibrium EQvf=((πœ‡/(πœŒπ‘†π‘…+πœ‡))𝑁,0,0) is globally asymptotically stable if 𝑅0=(π›Όπ‘Ÿπœ‡/(π›Όπ‘Ÿπœ‡)(πœŒπ‘†π‘…+πœ‡))≀1, and the viral equilibrium EQve is globally asymptotically stable if 𝑅0>1, where EQve=𝛼+πœŒπΈπ‘…ξ€Έ+πœ‡π›Όπ‘Ÿπ‘,πœ‡π‘ξ€·π›Ό+πœŒπΈπ‘…ξ€Έβˆ’ξ€·πœŒ+πœ‡πΈπ‘…ξ€Έπ‘+πœ‡,π›Όπ›Όπ‘Ÿξƒ¬π›Ύ+πœ‡πœ‡π‘ξ€·π›Ό+πœŒπΈπ‘…ξ€Έβˆ’ξ€·πœŒ+πœ‡πΈπ‘…ξ€Έπ‘+πœ‡.π›Όπ‘Ÿξƒ­ξƒͺ(1.3)

However, in the real world, systems are inevitably affected by environmental noise. Hence the deterministic approach has some limitations in mathematically modeling the transmission of an infectious disease, and it is quite difficult to predict the future dynamics of the system accurately. This happens due to the fact that deterministic models do not incorporate the effect of a fluctuating environment. Stochastic differential equation models play a significant role in various branches of applied sciences, including infectious dynamics, as they provide some additional degree of realism compared to their deterministic counterpart. In this paper, we introduce a noise into (1.2) and we transform the deterministic problem into a corresponding stochastic problem.

In this paper, we introduce randomness into the model by replacing the parameters πœ‡,πœ‡ and πœ‡ by πœ‡β†’πœ‡+𝜎1̇𝐡1(𝑑),πœ‡β†’πœ‡+𝜎2̇𝐡2(𝑑), and πœ‡β†’πœ‡+𝜎3̇𝐡3(𝑑), where ̇𝐡1̇𝐡(𝑑),2(𝑑), and ̇𝐡3(𝑑) are mutual independent standard Brownian motions with 𝐡1(0)=0,𝐡2(0)=0, and 𝐡3(0)=0, and intensity of white noise 𝜎21β‰₯0,𝜎22β‰₯0 and 𝜎23β‰₯0, respectively. Then the stochastic system is ̇𝑆(𝑑)=πœ‡π‘βˆ’π›Όπ‘ŸπΈ(𝑑)𝑆(𝑑)π‘βˆ’πœŒπ‘†π‘…π‘†(𝑑)βˆ’πœ‡π‘†(𝑑)βˆ’πœŽ1̇𝐡𝑆(𝑑)1Μ‡(𝑑),𝐸(𝑑)=π›Όπ‘ŸπΈ(𝑑)𝑆(𝑑)π‘βˆ’ξ€·π›Ό+πœŒπΈπ‘…ξ€Έ+πœ‡πΈ(𝑑)βˆ’πœŽ2̇𝐡𝐸(𝑑)1Μ‡(𝑑),𝐼(𝑑)=𝛼𝐸(𝑑)βˆ’(𝛾+πœ‡)𝐼(𝑑)βˆ’πœŽ3̇𝐡𝐼(𝑑)1(𝑑).(1.4)

The organization of this paper is as follows. In Section 2, we prove the existence and the uniqueness of the nonnegative solution of (1.3). In Section 3, if 𝑅0≀1, we show that the solution is oscillating around the virus-free equilibrium of (1.3). Section 4 focuses on the persistence of the virus. By choosing appropriate Lyapunov function, we show that there is a stationary distribution for (1.3) and that it is persistent if 𝑅0>1. Some numerical simulations are performed in Section 5. In Section 6, a brief conclusion is given.

Throughout this paper, consider the 𝑛-dimensional stochastic differential equation 𝑑π‘₯(𝑑)=𝑓(π‘₯(𝑑),𝑑)𝑑𝑑+𝑔(π‘₯(𝑑),𝑑)𝑑𝐡(𝑑),on𝑑β‰₯𝑑0,(1.5) with the initial value π‘₯(𝑑0)=π‘₯0βˆˆπ‘…π‘›. 𝐡(𝑑) denotes 𝑛-dimensional standard Brownian motion defined on the above probability space. Define the differential operator 𝐿 associated with (1.4) by πœ•πΏ=πœ•π‘₯π‘˜+12π‘›ξ“π‘˜,𝑗=1ξ€Ίπ‘”π‘‡ξ€»πœ•(π‘₯,𝑑)𝑔(π‘₯,𝑑)2πœ•π‘₯π‘˜πœ•π‘₯𝑗.(1.6) If 𝐿 acts on a function 𝑉, then 𝐿𝑉(π‘₯,𝑑)=𝑉𝑑(π‘₯,𝑑)+𝑉π‘₯1(π‘₯,𝑑)𝑓(π‘₯,𝑑)+2𝑔trace𝑇(π‘₯,𝑑)𝑉π‘₯π‘₯ξ€»,(π‘₯,𝑑)𝑔(π‘₯,𝑑)(1.7) where 𝑉𝑑=πœ•π‘‰/πœ•π‘‘,𝑉π‘₯=(πœ•π‘‰/πœ•π‘₯1,…,πœ•π‘‰/πœ•π‘₯𝑛),𝑉π‘₯π‘₯=(πœ•2𝑉/πœ•π‘₯π‘˜πœ•π‘₯π‘˜)π‘›βˆ—π‘›.

By Ito’s formula, if π‘₯(𝑑)βˆˆπ‘…π‘›, then for (1.4), assume that 𝑓(0,𝑑)=0,𝑔(0,𝑑)=0 for all 𝑑β‰₯𝑑0. So π‘₯(𝑑)≑0 is a solution of (1.4), called the trivial solution or equilibrium position.

2. Existence and Uniqueness of the Nonnegative Solution

To investigate the dynamical behavior of a population model, the first concern is whether the solution is positive or not and whether it has the global existence or not. Hence, in this section, we mainly use the Lyapunov analysis method to show that the solution of system (1.3) is positive and global.

Theorem 2.1. Let (𝑆0,𝐸0,𝐼0)βˆˆΞ”, then the system (1.2) admits a unique solution (𝑆(𝑑),𝐸(𝑑),𝐼(𝑑)) on 𝑑β‰₯0, and this solution remains in 𝑅3+ with probability 1.

Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value (𝑆0,𝐸0,𝐼0) there is a unique local solution (𝑆(𝑑),𝐸(𝑑),𝐼(𝑑)) on π‘‘βˆˆ[0,πœπ‘’), where πœπ‘’ is the explosion time [2, 13]. To show this solution is global, we need to show that πœπ‘’=∞ a. s. Let π‘˜0>0 be sufficiently large so that every component of π‘₯0 lies within the interval [1/π‘˜0,π‘˜0]. For each integer π‘˜β‰₯π‘˜0, define the stopping time, πœπ‘˜ξ‚†ξ€Ί=infπ‘‘βˆˆ0,πœπ‘’ξ€Έξ‚€1βˆΆπ‘†(𝑑)βˆ‰π‘˜ξ‚ξ‚€1,π‘˜or𝐸(𝑑)βˆ‰π‘˜ξ‚ξ‚€1,π‘˜or𝐼(𝑑)βˆ‰π‘˜,,π‘˜ξ‚ξ‚‡(2.1) where throughout this paper we set infβˆ…=∞ (as usual βˆ… denotes the empty set). Clearly, πœπ‘˜ is increasing as π‘˜β†’βˆž. Set 𝜏∞=limπ‘˜β†’βˆžπœπ‘˜, whence πœβˆžβ‰€πœπ‘’ a. s. If we can show that 𝜏∞=∞ a. s., then πœπ‘’=∞ and (𝑆(𝑑),𝐸(𝑑),𝐼(𝑑)) a. s. for all 𝑑β‰₯0. In other words, to complete the proof we need to show that 𝜏∞=∞ a. s. For if this statement is false, then there is a pair of constants 𝑇>0 and πœ€βˆˆ(0,1) such that π‘ƒξ€·πœβˆžξ€Έβ‰€π‘‡>πœ€.(2.2) Hence, there is an integer π‘˜1β‰₯π‘˜0 such that π‘ƒξ€½πœβˆžξ€Ύβ‰€π‘‡>πœ€βˆ€π‘˜>π‘˜1.(2.3) Define a 𝐢2-function 𝑉 for 𝑋(𝑆,𝐸,𝐼)βˆˆπ‘…3+ by 𝑆𝑉(𝑋)=π‘†βˆ’π‘Žβˆ’logπ‘Žξ‚+(πΈβˆ’1βˆ’log𝐸)+(πΌβˆ’1βˆ’log𝐼).(2.4) The nonnegativity of this function can be seen from πœ‡+1βˆ’logπœ‡β‰₯0, for  all πœ‡>0. Using Ito’s formula we get ξ‚€π‘Žπ‘‘π‘‰(𝑋)=π‘Žβˆ’π‘†ξ‚π‘Žπ‘‘π‘†+2𝑆2(𝑑𝑆)2+ξ‚€11βˆ’πΈξ‚1𝑑𝐸+2𝐸2(𝑑𝐸)2+ξ‚€11βˆ’πΌξ‚1𝑑𝐼+2𝐼2(𝑑𝐼)2ξ€ΊπœŽβ‰πΏπ‘‰π‘‘π‘‘βˆ’1̇𝐡(π‘†βˆ’π‘Ž)1(𝑑)+𝜎2̇𝐡(πΈβˆ’1)2(𝑑)+𝜎3̇𝐡(πΌβˆ’1)3ξ€»,(𝑑)(2.5) where ξ‚€π‘ŽπΏπ‘‰=1βˆ’π‘†ξ‚ξ‚Έπœ‡π‘βˆ’π›Όπ‘ŸπΈ(𝑑)𝑆(𝑑)π‘βˆ’πœŒπ‘†π‘…ξ‚Ή+𝑆(𝑑)βˆ’πœ‡π‘†(𝑑)π‘ŽπœŽ212+ξ‚€11βˆ’πΈξ‚ξ‚Έπ›Όπ‘ŸπΈ(𝑑)𝑆(𝑑)π‘βˆ’ξ€·π›Ό+πœŒπ‘†π‘…ξ€Έξ‚Ή+𝜎+πœ‡πΈ(𝑑)212+ξ‚€11βˆ’πΌξ‚[]+πœŽπ›ΌπΈ(𝑑)βˆ’(𝛾+πœ‡)𝐼(𝑑)232=ξƒ©πœ‡π‘+π‘ŽπœŒπ‘†π‘…+πœ‡π‘Ž+𝛼+πœŒπ‘†π‘…+πœ‡+𝛾+πœ‡+π‘ŽπœŽ212+𝜎222+𝜎232ξƒͺ+ξ‚ƒπ›Όπ‘Ÿπ‘Žπ‘πΈβˆ’πœŒπ‘†π‘…π‘Žπ‘†βˆ’πœ‡π‘†βˆ’π‘†πœ‡π‘βˆ’πœŒπ‘†π‘…πΈβˆ’πœ‡πΈβˆ’π›Όπ‘Ÿπ‘π›Όπ‘†βˆ’π›ΎπΌβˆ’πœ‡πΌβˆ’πΌπΈξ‚„β‰€πœ‡π‘+π‘ŽπœŒπ‘†π‘…+πœ‡π‘Ž+𝛼+πœŒπ‘†π‘…+πœ‡+𝛾+πœ‡+π‘ŽπœŽ212+𝜎222+𝜎232+π›Όπ‘Ÿπ‘Žπ‘πΈβˆ’πœŒπΈπ‘…πΈβˆ’πœ‡πΈ.(2.6) By choosing π‘Ž=(πœŒπΈπ‘…+πœ‡)𝑁/π›Όπ‘Ÿ, then πΏπ‘‰β‰€πœ‡π‘+π‘ŽπœŒπ‘†π‘…+πœ‡π‘Ž+𝛼+πœŒπ‘†π‘…+πœ‡+𝛾+πœ‡+π‘ŽπœŽ212+𝜎222+𝜎232≐̇𝑀.(2.7) Therefore, ξ€œπœπ‘š0βˆ§π‘‡ξ€œπ‘‘π‘‰(𝑋)β‰€πœπ‘š0βˆ§π‘‡Μ‡ξ€œπ‘€π‘‘π‘‘βˆ’πœπ‘š0βˆ§π‘‡ξ€ΊπœŽ1(π‘†βˆ’π‘Ž)𝑑𝐡1(𝑑)+𝜎2(πΈβˆ’1)𝑑𝐡2(𝑑)+𝜎3(πΌβˆ’1)𝑑𝐡3ξ€»,ξ€·π‘‹ξ€·πœ(𝑑)πΈπ‘‰π‘šξ‚Έξ€œβˆ§π‘‡ξ€Έξ€Έβ‰€π‘‰(𝑋(0))+πΈπœπ‘š0βˆ§π‘‡Μ‡ξ‚ΉΜ‡π‘€π‘‘π‘‘β‰€π‘‰(𝑋(0))+𝑀𝑇.(2.8) Setting Ξ©π‘š={πœπ‘šβ‰€π‘‡} for π‘šβ‰₯π‘š1, then by (2.3), we know that 𝑃(Ξ©π‘š)β‰₯πœ€. Note that for every πœ”βˆˆΞ©π‘š, there is at least one of 𝑆(Ξ©π‘š,πœ”), 𝐸(Ξ©π‘š,πœ”), and 𝐼(Ξ©π‘š,πœ”) that equals either π‘š or 1/π‘š. Then π‘‰ξ€·π‘‹ξ€·πœπ‘šξ‚€1ξ€Έξ€Έβ‰₯(π‘šβˆ’1βˆ’logπ‘š)βˆ§π‘šξ‚βˆ§ξ‚€π‘šβˆ’1+logπ‘šπ‘šβˆ’π‘Žβˆ’π‘Žlogπ‘Žξ‚βˆ§ξ‚€1π‘šξ‚,βˆ’π‘Ž+π‘Žlogπ‘Žπ‘š(2.9) where 1Ξ©π‘š(πœ”) is the indicator function of Ξ©π‘š. Let π‘šβ†’βˆž lead to the contradiction that Μ‡βˆž>𝑉(𝑋(0))+𝑀𝑇=∞. So 𝜏∞=∞ is necessary. The proof of Theorem 2.1 is completed.

3. Stability of Virus-Free Equilibrium

It is clear that EQvf=(πœ‡π‘/(πœŒπ‘†π‘…+πœ‡),0,0) is the virus-free equilibrium of system (1.3), which has been mentioned above, and EQvf is globally stable if 𝑅0≀1, which means that the virus will die out after some period of time. Since there is no virus-free equilibrium of system (1.3), in this section, we show that the solution is oscillating in a small neighborhood of EQvf if the white noise is small.

Theorem 3.1. If πœŒπ‘†π‘…+πœ‡>𝜎21,3𝛼2+2πœŒπ‘†π‘…+2πœ‡>𝜎22,2𝛾+2πœ‡βˆ’π›Ό>𝜎23 and 𝑅0≀1, then the solution 𝑋(𝑑) of system (1.3) with initial value 𝑋(0)βˆˆπ‘…3+ has the property limπ‘₯β†’βˆž1supπ‘‘πΈξ€œπ‘‘0(ξ€·πœŒ1+𝑏)𝑆𝑅+πœ‡βˆ’πœŽ21ξ€Έπœ‡2(ξ‚€1𝑠)+2𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22ξ‚πœŽ2(+𝛼𝑠)𝛾+πœ‡βˆ’2βˆ’12𝜎23𝑀2(𝑠)𝑑𝑠≀(1βˆ’π‘)𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘,+πœ‡ξ‚Άξ‚Ή(3.1) where 𝑏 is positive constants, defined as in the proof.

Proof. For simplicity, let 𝑒(𝑑)=𝑆(𝑑)βˆ’πœ‡π‘/(πœŒπ‘†π‘…+πœ‡),𝑣(𝑑)=𝐸(𝑑),𝑀(𝑑)=𝐼(𝑑), system (1.3) can be written as ̇𝑒(𝑑)=βˆ’π›Όπ‘Ÿπ‘£(𝑑)π‘ξ‚Έπœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ήβˆ’ξ€·πœŒ+πœ‡π‘†π‘…ξ€Έ+πœ‡π‘’(𝑑)βˆ’πœŽ1ξ‚΅πœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚ΆΜ‡Μ‡+πœ‡π΅(𝑑),𝑣(𝑑)=π›Όπ‘Ÿπ‘£(𝑑)π‘ξ‚Έπœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ήβˆ’ξ€·+πœ‡π›Ό+πœŒπ‘†π‘…ξ€Έ+πœ‡π‘£(𝑑)βˆ’πœŽ2̇̇𝑣(𝑑)𝐡(𝑑),𝑀(𝑑)=𝛼𝑣(𝑑)βˆ’(𝛾+πœ‡)𝑀(𝑑)βˆ’πœŽ3̇𝐡𝑀(𝑑)3(𝑑).(3.2) Let 1𝑉(π‘₯)=2(𝑒+𝑣)2+12𝑏𝑒2+121𝑏𝑣+2𝑀2=𝑉1+𝑏𝑉2+𝑏𝑉3+𝑉4,(3.3) then 𝑏 is positive constants to be determined later. By Ito’s formula, we compute 𝑑𝑉1≐𝐿𝑉1ξ‚ΈπœŽπ‘‘π‘‘βˆ’(𝑒(𝑑)+𝑣(𝑑))1ξ‚΅πœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚ΆΜ‡+πœ‡π΅(𝑑)+𝜎2̇𝐡𝑣(𝑑)2ξ‚Ή,(𝑑)𝐿𝑉1ξ€Ίβˆ’ξ€·πœŒ=(𝑒(𝑑)+𝑣(𝑑))𝑆𝑅+πœ‡π‘’(𝑑)βˆ’π›Ό+πœŒπ‘†π‘…ξ€Έξ€»+1+πœ‡π‘£(𝑑)2𝜎21ξ‚΅πœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2+12𝜎22𝑣2(ξ€Ίβˆ’ξ€·πœŒπ‘‘)≀(𝑒(𝑑)+𝑣(𝑑))𝑆𝑅+πœ‡π‘’(𝑑)βˆ’π›Ό+πœŒπ‘†π‘…ξ€Έξ€»+πœ‡π‘£(𝑑)+𝜎21𝑒2(𝑑)+𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2+12𝜎22𝑣2(ξƒ¬ξ€·πœŒπ‘‘)=βˆ’π‘†π‘…+πœ‡βˆ’πœŽ21𝑒2ξ‚€(𝑑)+𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22𝑣2ξ€·(𝑑)+𝛼+2πœŒπ‘†π‘…ξ€Έ+2πœ‡π‘’(𝑑)𝑣(𝑑)βˆ’πœŽ21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξƒ­,𝑑𝑉2≐𝐿𝑉2π‘‘π‘‘βˆ’πœŽ1ξ‚΅πœ‡π‘’(𝑑)𝑒(𝑑)+πœŒπ‘†π‘…ξ‚ΆΜ‡+πœ‡π΅(𝑑),𝐿𝑉2ξ‚»βˆ’=𝑒(𝑑)π›Όπ‘Ÿπ‘£(𝑑)π‘ξ‚Έπœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ήβˆ’ξ€·πœŒ+πœ‡π‘†π‘…ξ€Έξ‚Ό+1+πœ‡π‘’(𝑑)2𝜎21ξ‚΅πœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξ‚»βˆ’β‰€π‘’(𝑑)π›Όπ‘Ÿπ‘£(𝑑)π‘ξ‚Έπœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ήβˆ’ξ€·πœŒ+πœ‡π‘†π‘…ξ€Έξ‚Ό+πœ‡π‘’(𝑑)+𝜎21𝑒2(𝑑)+𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξƒ¬ξ€·πœŒ=βˆ’π‘†π‘…+πœ‡βˆ’πœŽ21𝑒2(𝑑)+π›Όπ‘Ÿπœ‡πœŒπ‘†π‘…+πœ‡π‘’(𝑑)𝑣(𝑑)+π›Όπ‘Ÿπ‘π‘£(𝑑)𝑒2(𝑑)+𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξƒ­ξƒ¬ξ€·πœŒβ‰€βˆ’π‘†π‘…+πœ‡βˆ’πœŽ21𝑒2(𝑑)+π›Όπ‘Ÿπœ‡πœŒπ‘†π‘…+πœ‡π‘’(𝑑)𝑣(𝑑)+𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξƒ­,𝑑𝑉3=π›Όπ‘Ÿπ‘£(𝑑)π‘ξ‚Έπœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚Ήξ€·+πœ‡π‘‘π‘‘βˆ’π›Ό+πœŒπ‘†π‘…ξ€Έ+πœ‡π‘£(𝑑)π‘‘π‘‘βˆ’πœŽ2Μ‡=𝑣(𝑑)𝐡(𝑑)π›Όπ‘Ÿπ‘ξ‚΅π‘£(𝑑)𝑒(𝑑)+π›Όπ‘Ÿπœ‡πœŒπ‘†π‘…βˆ’ξ€·+πœ‡π›Ό+πœŒπ‘†π‘…ξ€Έξ‚Άξ‚Ή+πœ‡π‘£(𝑑)π‘‘π‘‘βˆ’πœŽ2̇𝑣(𝑑)𝐡(𝑑)≐𝐿𝑉3βˆ’πœŽ2𝑣̇𝐡(𝑑)(𝑑),𝑑𝑉4=1𝑀(𝑑)(𝛼𝑣(𝑑)βˆ’(𝛾+πœ‡)𝑀(𝑑))+2𝜎23𝑀2ξ‚„(𝑑)π‘‘π‘‘βˆ’πœŽ3𝑀2Μ‡=(𝑑)𝐡(𝑑)𝛼𝑣(𝑑)𝑀(𝑑)βˆ’(𝛾+πœ‡)𝑀21(𝑑)+2𝜎23𝑀2ξ‚„(𝑑)π‘‘π‘‘βˆ’πœŽ3𝑀2̇𝐡≀𝛼(𝑑)(𝑑)2𝑣2(𝑑)+𝑀2ξ€Έ(𝑑)βˆ’(𝛾+πœ‡)𝑀21(𝑑)+2𝜎23𝑀2ξ‚„(𝑑)π‘‘π‘‘βˆ’πœŽ3𝑀2Μ‡=𝛼(𝑑)𝐡(𝑑)21βˆ’π›Ύβˆ’πœ‡+2𝜎23𝑀2(𝛼𝑑)+2𝑣2(𝑑)π‘‘π‘‘βˆ’πœŽ3𝑀2(̇𝑑)𝐡(𝑑)≐𝐿𝑉4π‘‘π‘‘βˆ’πœŽ3𝑀2Μ‡(𝑑)𝐡(𝑑),𝐿𝑉=𝐿𝑉1+𝑏𝐿𝑉2+𝑏𝐿𝑉3+𝐿𝑉4ξƒ¬ξ€·πœŒ=βˆ’π‘†π‘…+πœ‡βˆ’πœŽ21𝑒2ξ‚€(𝑑)+𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22𝑣2+ξ€·(𝑑)𝛼+2πœŒπ‘†π‘…ξ€Έ+2πœ‡π‘’(𝑑)𝑣(𝑑)βˆ’πœŽ21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξƒ­ξƒ¬ξ€·πœŒβˆ’π‘π‘†π‘…+πœ‡βˆ’πœŽ21𝑒2(𝑑)+π›Όπ‘Ÿπœ‡πœŒπ‘†π‘…+πœ‡π‘’(𝑑)𝑣(𝑑)+𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξƒ­ξ‚Έ+π‘π›Όπ‘Ÿπ‘ξ‚΅π‘£(𝑑)𝑒(𝑑)+π›Όπ‘Ÿπœ‡πœŒπ‘†π‘…βˆ’ξ€·+πœ‡π›Ό+πœŒπ‘†π‘…ξ€Έξ‚Άξ‚Ή+𝛼+πœ‡π‘£(𝑑)21βˆ’π›Ύβˆ’πœ‡+2𝜎23𝑀2𝛼(𝑑)+2𝑣2ξ‚„.(𝑑)(3.4)
Choosing 𝑏=(𝑁(𝛼+2πœŒπ‘†π‘…+2πœ‡)(πœŒπ‘†π‘…+πœ‡))/(π›Όπ‘Ÿ(πœŒπ‘†π‘…+πœ‡βˆ’π‘πœ‡)), then we get ξ€·πœŒπΏπ‘‰=βˆ’(1+𝑏)𝑆𝑅+πœ‡βˆ’πœŽ21𝑒2ξ‚€1(𝑑)βˆ’2𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22𝑣2𝛼(𝑑)βˆ’π›Ύ+πœ‡βˆ’2βˆ’12𝜎23𝑀2ξ‚΅ξ€·(𝑑)βˆ’π‘π›Ό+πœŒπ‘†π‘…ξ€Έβˆ’+πœ‡π›Όπ‘Ÿπœ‡πœŒπ‘†π‘…ξ‚Ά+πœ‡π‘£(𝑑)+(1βˆ’π‘)𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…ξ‚Ά+πœ‡2,ξ€·πœŒπ‘‘π‘‰β‰€βˆ’(1+𝑏)𝑆𝑅+πœ‡βˆ’πœŽ21𝑒2ξ‚€1(𝑑)βˆ’2𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22𝑣2𝛼(𝑑)βˆ’π›Ύ+πœ‡βˆ’2βˆ’12𝜎23𝑀2(𝑑)+(1βˆ’π‘)𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2ξ‚ΈπœŽβˆ’(𝑒(𝑑)+𝑣(𝑑))1ξ‚΅πœ‡π‘’(𝑑)+πœŒπ‘†π‘…π‘ξ‚ΆΜ‡π΅+πœ‡1(𝑑)+𝜎2̇𝐡𝑣(𝑑)2ξ‚Ή(𝑑)βˆ’πœŽ1ξ‚΅πœ‡π‘’(𝑑)𝑒(𝑑)+πœŒπ‘†π‘…π‘ξ‚ΆΜ‡+πœ‡π΅(𝑑)βˆ’πœŽ2̇𝐡𝑣(𝑑)2(𝑑)βˆ’πœŽ3𝑀2̇𝐡(𝑑)3(𝑑).(3.5) Integrating this from 0 to 𝑑 and taking the expectation, we have 𝐸[]ξ€œπ‘‰(𝑑)βˆ’π‘‰(0)β‰€βˆ’πΈπ‘‘0ξƒ¬ξ€·πœŒ(1+𝑏)𝑆𝑅+πœ‡βˆ’πœŽ21𝑒2ξ‚€1(𝑠)+2𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22𝑣2+𝛼(𝑠)𝛾+πœ‡βˆ’2βˆ’12𝜎23𝑀2(𝑠)βˆ’(1βˆ’π‘)𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2𝑑𝑠.(3.6) Hence, limπ‘₯β†’βˆž1supπ‘‘πΈξ€œπ‘‘0(ξ€·πœŒ1+𝑏)𝑆𝑅+πœ‡βˆ’πœŽ21𝑒2(ξ‚€1𝑠)+2𝛼+πœŒπ‘†π‘…1+πœ‡βˆ’2𝜎22𝑣2(+𝛼𝑠)𝛾+πœ‡βˆ’2βˆ’12𝜎23𝑀2ξ‚„(𝑠)𝑑𝑠≀(1βˆ’π‘)𝜎21ξ‚΅πœ‡πœŒπ‘†π‘…π‘ξ‚Ά+πœ‡2.(3.7)

Remark 3.2. Theorem 3.1 shows that the solution of system (1.3) would oscillate around the virus-free equilibrium of system (1.1) if some conditions are satisfied, and the intensity of fluctuation is proportional to 𝜎21, which is the intensity of the white noise ̇𝐡1(𝑑). In a biological interpretation, if the stochastic effect on 𝑆 is small, the solution of system (1.3) will be close to the virus-free equilibrium of system (1.1) most of the time.

4. Permanence

When studying epidemic dynamical systems, we are interested in when the computer viruses will persist in network. For a deterministic model, this is usually solved by showing that the viral equilibrium is a global attractor or is globally asymptotically stable. But, for system (1.3), there is no viral equilibrium. In this section, we show that there is a stationary distribution, which reveals that the computer viruses will persist.

Lemma 4.1 (see [14, 15]). Assumption 𝐡: there exists a bounded domain π‘ˆβŠ‚πΈπ‘™ with regular boundary Ξ“, having the following properties.(𝐡.1) In the domain π‘ˆ and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix 𝐴(π‘₯) is bounded away from zero.(𝐡.2) If π‘₯βˆˆπΈπ‘™/π‘ˆ, the mean time 𝜏 at which a path issuing from π‘₯ reaches the set π‘ˆ is finite, and supπ‘₯∈𝐾𝐸π‘₯𝜏<∞ for every compact subset πΎβŠ‚πΈπ‘™. If (𝐡) holds, then the Markov process 𝑋(𝑑) has a stationary distribution πœ‡(β€’). Let 𝑓(β€’) be a function integrable with respect to the measure πœ‡. Then Pπ‘₯ξ‚»limπ‘‡β†’βˆž1π‘‡ξ€œπ‘‡0ξ€œπ‘“(𝑋(𝑑))𝑑𝑑=𝐸𝑙𝑓(π‘₯)πœ‡π‘‘(π‘₯)=1,βˆ€π‘₯βˆˆπΈπ‘™.(4.1)

Lemma 4.2 (see [14, 15]). Let 𝑋(𝑑) be a regular temporally homogeneous Markov process in 𝐸𝑙. If 𝑋(𝑑) is recurrent relative to some bounded domain π‘ˆ, then it is recurrent relative to any nonempty domain in 𝐸𝑙.

Theorem 4.3. If𝜎21<(πœŒπ‘†π‘…+πœ‡)(1+(π›Όπ‘Ÿ/π‘†βˆ—π‘))(π‘†βˆ—/(π‘†βˆ—βˆ’1)),𝜎22<(𝛼/2)+πœŒπ‘†π‘…+πœ‡πœŽ23<𝛾+πœ‡βˆ’(𝛼/2), and 𝑅0>1, then, for any initial value 𝑋(0)βˆˆπ‘…3+, there is a stationary distribution πœ‡(β€’) for system (1.3), and it has an ergodic property, where π‘Ž,𝑐 are defined as in the proof, Qve=(π‘†βˆ—,πΈβˆ—,πΌβˆ—) is the viral equilibrium of system.

Proof. When 𝑅0>1, there is an viral equilibrium EQve of system (1.3). Then πœ‡π‘=π›Όπ‘ŸπΈβˆ—π‘†βˆ—π‘βˆ’πœŒπ‘†π‘…π‘†βˆ—+πœ‡π‘†βˆ—,π›Όπ‘ŸπΈβˆ—π‘†βˆ—π‘=𝛼+πœŒπ‘†π‘…ξ€ΈπΈ+πœ‡βˆ—,π›ΌπΈβˆ—=(𝛾+πœ‡)πΌβˆ—.(4.2) Define 𝑉(π‘₯)=π‘Žπ‘†βˆ’π‘†βˆ—βˆ’π‘†βˆ—π‘†logπ‘†βˆ—+πΈβˆ’πΈβˆ—βˆ’πΈβˆ—πΈlogπΈβˆ—ξ‚+ξ‚€πΈβˆ’πΈβˆ—βˆ’πΈβˆ—πΈlogπΈβˆ—ξ‚+12ξ€·π‘†βˆ’π‘†βˆ—+πΈβˆ’πΈβˆ—ξ€Έ2+12π‘ξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2+12ξ€·πΌβˆ’πΌβˆ—ξ€Έ2=π‘Žπ‘‰1+𝑉2+𝑉3+𝑐𝑉4+𝑉5,(4.3) where π‘Ž,𝑐, are positive constants to be determined later. Then 𝑉 is positive definite. By Ito’s formula, we compute 𝑑𝑉1=𝑆1βˆ’βˆ—π‘†ξ‚Άξ‚ƒξ‚€πœ‡π‘βˆ’π›Όπ‘ŸπΈπ‘†π‘βˆ’πœŒπ‘†π‘…ξ‚π‘†βˆ’πœ‡π‘†π‘‘π‘‘βˆ’πœŽ1𝑆̇𝐡1ξ‚„+𝐸(𝑑)1βˆ’βˆ—πΈξ‚Άξ‚ƒξ‚€π›Όπ‘ŸπΈπ‘†π‘βˆ’ξ€·π›Ό+πœŒπ‘†π‘…ξ€ΈπΈξ‚+πœ‡π‘‘π‘‘βˆ’πœŽ2𝐸̇𝐡2ξ‚„+1(𝑑)2π‘†βˆ—πœŽ211𝑑𝑑+2πΈβˆ—πœŽ22𝑑𝑑=𝐿𝑉1ξ‚΅π‘†π‘‘π‘‘βˆ’1βˆ’βˆ—π‘†ξ‚ΆπœŽ1𝑆̇𝐡(𝑑)1𝐸(𝑑)βˆ’1βˆ’βˆ—πΈξ‚ΆπœŽ2𝐸̇𝐡(𝑑)2(𝑑),(4.4) where 𝐿𝑉1=𝑆1βˆ’βˆ—π‘†ξ‚Άξ‚ƒξ‚€πœ‡π‘βˆ’π›Όπ‘ŸπΈπ‘†π‘βˆ’πœŒπ‘†π‘…ξ‚ξ‚„+ξ‚΅πΈπ‘†βˆ’πœ‡π‘†π‘‘π‘‘1βˆ’βˆ—πΈξ‚Άξ‚ƒξ‚€π›Όπ‘ŸπΈπ‘†π‘βˆ’ξ€·π›Ό+πœŒπ‘†π‘…ξ€ΈπΈξ‚ξ‚„+1+πœ‡π‘‘π‘‘2π‘†βˆ—πœŽ211𝑑𝑑+2πΈβˆ—πœŽ22=𝑆𝑑𝑑1βˆ’βˆ—π‘†ξ‚Άξ‚ƒπ›Όπ‘Ÿπ‘ξ€·πΈβˆ—π‘†βˆ—ξ€Έξ€·πœŒβˆ’πΈπ‘†π‘‘π‘‘+𝑆𝑅𝑆+πœ‡ξ€Έξ€·βˆ—ξ€Έξ‚„+ξ‚΅πΈβˆ’π‘†π‘‘π‘‘1βˆ’βˆ—πΈξ‚Άξ‚ƒξ‚€π›Όπ‘ŸπΈπ‘†π‘βˆ’ξ€·π›Ό+πœŒπ‘†π‘…ξ€ΈπΈξ‚ξ‚„+1+πœ‡π‘‘π‘‘2π‘†βˆ—πœŽ211𝑑𝑑+2πΈβˆ—πœŽ22𝑑𝑑=βˆ’π‘†βˆ’π‘†βˆ—ξ€Έ2π‘†ξ€·πœŒπ‘†π‘…ξ€Έ++πœ‡π›Όπ‘Ÿπ‘πΈβˆ—π‘†βˆ—+πΈβˆ—ξ€·π›Ό+πœŒπ‘†π‘…ξ€Έ+ξ‚€+πœ‡π›Όπ‘Ÿπ‘π‘†βˆ—βˆ’π›Όβˆ’πœŒπ‘†π‘…ξ‚βˆ’πœ‡πΈβˆ’π›Όπ‘Ÿπ‘πΈβˆ—π‘†βˆ—βˆ’π›Όπ‘Ÿπ‘πΈβˆ—π‘†βˆ—2𝑆+12π‘†βˆ—πœŽ21+12πΈβˆ—πœŽ22ξ€·β‰€βˆ’π‘†βˆ’π‘†βˆ—ξ€Έ2π‘†ξ€·πœŒπ‘†π‘…ξ€Έ1+πœ‡2π‘†βˆ—πœŽ21+12πΈβˆ—πœŽ22,𝑑𝑉2=𝐸1βˆ’βˆ—πΈξ‚Άξ‚ƒξ‚€π›Όπ‘ŸπΈπ‘†π‘βˆ’ξ€·π›Ό+πœŒπ‘†π‘…ξ€ΈπΈξ‚+πœ‡π‘‘π‘‘βˆ’πœŽ2𝐸̇𝐡2ξ‚„+1(𝑑)2πΈβˆ—πœŽ22𝑑𝑑=𝐿𝑉2ξ‚΅πΈπ‘‘π‘‘βˆ’1βˆ’βˆ—πΈξ‚ΆπœŽ2̇𝐡𝐸(𝑑)2(𝑑).(4.5) Let 𝐡=(π›Όπ‘Ÿ/𝑁)πΈβˆ—π‘†βˆ—=(𝛼+πœŒπ‘†π‘…+πœ‡)πΈβˆ— and π›Όβˆ’1βˆ’log𝛼>0, for all 𝛼𝐿𝑉2=𝐸1βˆ’βˆ—πΈξ‚Άξ‚€π›Όπ‘ŸπΈπ‘†π‘βˆ’ξ€·π›Ό+πœŒπ‘†π‘…ξ€ΈπΈξ‚1+πœ‡π‘‘π‘‘+2πΈβˆ—πœŽ22=𝐸𝑑𝑑1βˆ’βˆ—πΈξ‚Άξ‚€ξπ΅πΈπ‘†πΈβˆ—π‘†βˆ—βˆ’ξπ΅πΈπΈβˆ—ξ‚+12πΈβˆ—πœŽ22=ξ‚€ξπ΅πΈπ‘†πΈβˆ—π‘†βˆ—βˆ’ξπ΅πΈπΈβˆ—+ξπ΅π‘†π‘†βˆ—+𝐡+12πΈβˆ—πœŽ22β‰€ξπ΅ξ‚ƒπΈπ‘†πΈβˆ—π‘†βˆ—βˆ’πΈπΈβˆ—βˆ’ξ‚€π‘†1+logπ‘†βˆ—ξ‚ξ‚„+1+12πΈβˆ—πœŽ22β‰€ξπ΅ξ‚ƒπΈπ‘†πΈβˆ—π‘†βˆ—βˆ’πΈπΈβˆ—+ξ‚€π‘†π‘†βˆ—ξ‚ξ‚„+1βˆ’2+12πΈβˆ—πœŽ22=ξπ΅ξ‚€πΈπΈβˆ—π‘†βˆ’1ξ‚ξ‚€π‘†βˆ—ξ‚+ξπ΅ξ‚΅π‘†βˆ’1βˆ—π‘†+π‘†π‘†βˆ—ξ‚Ά+1βˆ’22πΈβˆ—πœŽ22=π›Όπ‘Ÿπ‘ξ€·πΈβˆ’πΈβˆ—ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ+π›Όπ‘Ÿπ‘πΈβˆ—ξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2𝑆+12πΈβˆ—πœŽ22,𝑑𝑉3=ξ€·π‘†βˆ’π‘†βˆ—+πΈβˆ’πΈβˆ—ξ€·πœŒξ€Έξ€Ίπœ‡π‘βˆ’π‘†π‘…ξ€Έξ€·+πœ‡π‘†βˆ’π›Ό+πœŒπ‘†π‘…ξ€ΈπΈξ€»βˆ’ξ€·+πœ‡π‘‘π‘‘π‘†βˆ’π‘†βˆ—+πΈβˆ’πΈβˆ—πœŽξ€Έξ€Ί1𝑆̇𝐡1(𝑑)+𝜎2𝐸̇𝐡2ξ€»+ξƒ©πœŽ(𝑑)212𝑆2+𝜎222𝐸2ξƒͺ𝑑𝑑,𝐿𝑉3=ξ€·π‘†βˆ’π‘†βˆ—+πΈβˆ’πΈβˆ—βˆ’ξ€·πœŒξ€Έξ€Ίπ‘†π‘…+πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έβˆ’ξ€·π›Ό+πœŒπ‘†π‘…+πœ‡ξ€Έξ€·πΈβˆ’πΈβˆ—+πœŽξ€Έξ€»212𝑆2+𝜎222𝐸2ξ€·πœŒβ‰€βˆ’π‘†π‘…+πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2βˆ’ξ€·π›Ό+πœŒπ‘†π‘…+πœ‡ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έ2βˆ’ξ€·π›Ό+πœŒπΈπ‘…+πœŒπ‘†π‘…+2πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έ+𝜎21ξ€Ίξ€·π‘†βˆ’π‘†βˆ—ξ€Έ+π‘†βˆ—2ξ€»+𝜎22ξ€Ίξ€·πΈβˆ’πΈβˆ—ξ€Έ+πΈβˆ—2ξ€»β‰€ξ€·πœŽ21βˆ’πœŒπ‘†π‘…βˆ’πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2+ξ€·πœŽ22βˆ’π›Όβˆ’πœŒπΈπ‘…βˆ’πœ‡ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έ2+𝜎21π‘†βˆ—2+𝜎22πΈβˆ—2,𝑑𝑉4=ξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ‚€πœ‡π‘βˆ’π›Όπ‘Ÿπ‘ξ€·πœŒπΈπ‘†βˆ’π‘†π‘…ξ€Έπ‘†ξ‚ξ€·+πœ‡βˆ’π‘†π‘†βˆ’π‘†βˆ—ξ€ΈπœŽ1̇𝐡1(𝑑)+2𝜎21𝑆2,𝐿𝑉4=ξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ‚€πœ‡π‘βˆ’π›Όπ‘Ÿπ‘ξ€·πœŒπΈπ‘†βˆ’π‘†π‘…ξ€Έπ‘†ξ‚+1+πœ‡2𝜎21𝑆2=ξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ‚ƒβˆ’π›Όπ‘Ÿπ‘ξ€·πΈπ‘†βˆ’πΈβˆ—π‘†βˆ—ξ€Έβˆ’ξ€·πœŒπ‘†π‘…+πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ‚„+12𝜎21𝑆2=βˆ’π›Όπ‘Ÿπ‘π‘†βˆ—ξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έβˆ’π›Όπ‘Ÿπ‘ξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2πΈβˆ’π›Όπ‘Ÿπ‘ξ€·πœŒπ‘†π‘…+πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2+12𝜎21𝑆2β‰€βˆ’π›Όπ‘Ÿπ‘π‘†βˆ—ξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έβˆ’ξ‚€π›Όπ‘Ÿπ‘ξ€·πœŒπ‘†π‘…ξ€Έ+πœ‡βˆ’πœŽ21ξ‚ξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2+𝜎21π‘†βˆ—2,𝑑𝑉5=ξ€·πΌβˆ’πΌβˆ—ξ€Έ[]+πœŽπ›ΌπΈβˆ’(𝛾+πœ‡)𝐼232𝐼2=𝐿𝑉5π‘‘π‘‘βˆ’πœŽ3πΌξ€·πΌβˆ’πΌβˆ—ξ€ΈΜ‡π΅3,𝐿𝑉5=ξ€·πΌβˆ’πΌβˆ—ξ€Έ[]+πœŽπ›ΌπΈβˆ’(𝛾+πœ‡)𝐼232𝐼2=ξ€·πΌβˆ’πΌβˆ—π›Όξ€·ξ€Έξ€ΊπΈβˆ’πΈβˆ—ξ€Έξ€·βˆ’(𝛾+πœ‡)πΌβˆ’πΌβˆ—+πœŽξ€Έξ€»232𝐼2ξ€·β‰€π›ΌπΈβˆ’πΈβˆ—ξ€Έξ€·πΌβˆ’πΌβˆ—ξ€Έβˆ’ξ€·π›Ύ+πœ‡βˆ’πœŽ23ξ€Έξ€·πΌβˆ’πΌβˆ—ξ€Έ2+𝜎23πΌβˆ—2≀𝛼2ξ€·πΈβˆ’πΈβˆ—ξ€Έ2βˆ’ξ‚€π›Ύ+πœ‡βˆ’πœŽ23βˆ’π›Ό2ξ‚ξ€·πΌβˆ’πΌβˆ—ξ€Έ2+𝜎23πΌβˆ—2.(4.6)
Choosing π‘Ž=(π›Όπ‘Ÿ/(πœŒπ‘†π‘…+πœ‡)𝑁)πΈβˆ—, then π‘Žπ‘‰1+𝑉2ξƒ¬βˆ’ξ€·=π‘Žπ‘†βˆ’π‘†βˆ—ξ€Έ2π‘†ξ€·πœŒπ‘†π‘…ξ€Έ+1+πœ‡2π‘†βˆ—πœŽ21+12πΈβˆ—πœŽ22ξƒ­+ξƒ¬π›Όπ‘Ÿπ‘ξ€·πΈβˆ’πΈβˆ—ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ+π›Όπ‘Ÿπ‘πΈβˆ—ξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2𝑆+12πΈβˆ—πœŽ22ξƒ­=π›Όπ‘Ÿπ‘ξ€·πΈβˆ’πΈβˆ—ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ+12π‘Žπ‘†βˆ—πœŽ21+12πΈβˆ—πœŽ22.(4.7)
Choosing 𝑐=1/π‘†βˆ—, then π‘Žπ‘‰1+𝑉2+𝑉3+𝑉4+𝑉5β‰€π›Όπ‘Ÿπ‘ξ€·πΈβˆ’πΈβˆ—ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ+12π‘Žπ‘†βˆ—πœŽ21+12(π‘Ž+1)πΈβˆ—πœŽ22+ξ‚ƒξ€·πœŽ21βˆ’πœŒπ‘†π‘…βˆ’πœ‡ξ€Έξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2+ξ€·πœŽ22βˆ’π›Όβˆ’πœŒπ‘†π‘…βˆ’πœ‡ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έ2+𝜎21π‘†βˆ—2+𝜎22πΈβˆ—2ξ‚„ξ‚ƒβˆ’+π‘π›Όπ‘Ÿπ‘π‘†βˆ—ξ€·π‘†βˆ’π‘†βˆ—ξ€Έξ€·πΈβˆ’πΈβˆ—ξ€Έβˆ’ξ‚€π›Όπ‘Ÿπ‘ξ€·πœŒS𝑅+πœ‡βˆ’πœŽ21ξ‚ξ€·π‘†βˆ’π‘†βˆ—ξ€Έ2+𝜎21π‘†βˆ—2ξ‚„+𝛼2ξ€·πΈβˆ’πΈβˆ—ξ€Έ2βˆ’ξ‚€π›Ύ+πœ‡βˆ’πœŽ23βˆ’π›Ό2ξ‚ξ€·πΌβˆ’πΌβˆ—ξ€Έ2+𝜎23πΌβˆ—2ξ‚„=12π‘ŽπœŽ21π‘†βˆ—+𝜎21π‘†βˆ—+𝜎21π‘†βˆ—2+12(π‘Ž+1)𝜎22πΈβˆ—+𝜎22πΈβˆ—2+𝜎23πΌβˆ—2+ξ‚ƒξ€·πœŽ21βˆ’πœŒπ‘†π‘…ξ€Έξ‚€βˆ’πœ‡βˆ’π‘π›Όπ‘Ÿπ‘ξ€·πœŒπ‘†π‘…ξ€Έ+πœ‡βˆ’πœŽ21ξ€·ξ‚ξ‚„π‘†βˆ’π‘†βˆ—ξ€Έ2+ξ‚€πœŽ22βˆ’π›Ό2βˆ’πœŒπ‘†π‘…ξ‚ξ€·βˆ’πœ‡πΈβˆ’πΈβˆ—ξ€Έ2βˆ’ξ‚€π›Ύ+πœ‡βˆ’πœŽ23βˆ’π›Ό2ξ‚ξ€·πΌβˆ’πΌβˆ—ξ€Έ2β‰π›Ώβˆ’ξ‚ƒξ‚€1+π›Όπ‘Ÿπ‘†βˆ—π‘ξ‚ξ€·πœŒπ‘†π‘…ξ€Έ+πœ‡βˆ’πœŽ21ξ‚€11+π‘†βˆ—ξ€·ξ‚ξ‚„π‘†βˆ’π‘†βˆ—ξ€Έ2βˆ’ξ‚€βˆ’π›Ό2+πœŒπ‘†π‘…+πœ‡βˆ’πœŽ22ξ‚ξ€·πΈβˆ’πΈβˆ—ξ€Έ2βˆ’ξ‚€π›Ύ+πœ‡βˆ’πœŽ23βˆ’π›Ό2ξ‚ξ€·πΌβˆ’πΌβˆ—ξ€Έ2.(4.8)
Then the ellipsoid π›Ώβˆ’ξ‚ƒξ‚€1+π›Όπ‘Ÿπ‘†βˆ—π‘ξ‚ξ€·πœŒπ‘†π‘…ξ€Έ+πœ‡βˆ’πœŽ21ξ‚€11+π‘†βˆ—ξ€·ξ‚ξ‚„π‘†βˆ’π‘†βˆ—ξ€Έ2βˆ’ξ‚€βˆ’π›Ό2+πœŒπ‘†π‘…+πœ‡βˆ’πœŽ22ξ‚ξ€·πΈβˆ’πΈβˆ—ξ€Έ2=0(4.9) lies entirely in 𝑅3+. We can take π‘ˆ to be a neighborhood of the ellipsoid with π‘ˆβŠ‚π‘…3+, so, for π‘₯βˆˆπ‘ˆ/𝑅3+, πΏπ‘‰β‰€βˆ’πΎ (𝐾 is a positive constant), which implies that condition (𝐡.2) in Lemma 4.1 is satisfied. Hence, the solution 𝑋(𝑑) is recurrent in the domain π‘ˆ, which, together with Lemma 4.2, implies that 𝑋(𝑑) is recurrent in any bounded domain π·βŠ‚π‘…3+. Besides, for  all 𝐷, there is an ξ€½πœŽπ‘€=min21𝑆2,𝜎22𝐸2,𝜎23𝐼2ξ€Ύβˆˆπ·>0,(4.10) such that βˆ‘3𝑖,𝑗=1π‘Žπ‘–π‘—πœ‰π‘–πœ‰π‘—=𝜎21𝑆2πœ‰21+𝜎22𝐸2πœ‰22+𝜎23𝐼2πœ‰23β‰₯π‘€β€–πœ‰2β€– for all π‘‹βˆˆπ·,πœ‰βˆˆπ‘…3which implies that condition (𝐡.1) is also satisfied. Therefore, the stochastic system (1.3) has a stationary distribution πœ‡(βˆ—) and it is ergodic. This completes the proof.

5. Numerical Simulations

In this section, we have performed some numerical simulations to show the geometric impression of our results. To demonstrate the global stability of infection-free solution of system (1.3) we take following set parameter values: πœ‡=1/4380,𝑁=100000, 𝛼=1/500, π‘Ÿ=7, πœŒπ‘†π‘…=1/2500, πœŒπΈπ‘…=1/300, 𝛾=1/500,𝜎21=0.0006, 𝜎22=0.001, 𝜎23=0.002. In this case, we have 𝑅0=0.9147<1. In Figures 2(a), 2(b), and 2(c), we have displayed, respectively, the susceptible, infected and recovered computer of system (1.4) with initial conditions: 𝑆(0)=3,𝐸(0)=0.1 and 𝐼(0)=0.1.

To demonstrate the permanence of system (1.4), we take the following set parameter values: πœ‡=1/4380, 𝑁=100000, 𝛼=1/500, π‘Ÿ=30, πœŒπ‘†π‘…=1/2500, πœŒπΈπ‘…=1/300, 𝛾=1/500, 𝜎21=0.0006, 𝜎22=0.001, 𝜎23=0.002. In this case, we have 𝑅1=3.9201>1. In Figures 3(a), 3(b), and 3(c), we have displayed, respectively, the susceptible and infected population of system (1.4) with initial conditions: 𝑆(0)=15000,𝐸(0)=2000 and 𝐼(0)=2000.

6. Conclusion

In this paper, a stochastic computer virus spread model has been proposed and analyzed. First, we prove the existence and uniqueness of positive solutions. Then, by constructing Lyapunov functions and applying Ito’s formula, the stability of the virus-free equilibrium and viral equilibrium is studied.

Acknowledgments

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).

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Copyright © 2012 Chunming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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