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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 264874, 16 pages
http://dx.doi.org/10.1155/2012/264874
Research Article

A Stochastic Dynamic Model of Computer Viruses

School of Information Engineering, Guangdong Medical College, Dongguan 523808, China

Received 8 June 2012; Accepted 16 July 2012

Academic Editor: Bimal Kumar Mishra

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.

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 [313]. 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 𝜇𝑁.(𝐻3) Computers are disconnected to the computer network with constant rate 𝜇.(𝐻4)𝑆 computers become 𝐸 computers with rate 𝛼𝑟/𝑁, where 𝑟 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 𝜌𝑆𝑅.(𝐻5)𝐸 computers become 𝐼 computers with constant rate 𝛼; 𝐸 computers become 𝑅 computers with constant rate 𝜌𝑆𝑅; 𝐼 computers become 𝑅 computers with constant rate 𝛾.

According to the above assumptions, the following model (see Figure 1) is derived: ̇𝑆(𝑡)=𝜇𝑁𝛼𝑟𝑁𝐸(𝑡)𝑆(𝑡)𝜌𝑆𝑅𝑆̇(𝑡)𝜇𝑆(𝑡),𝐸(𝑡)=𝛼𝑟𝑁𝐸(𝑡)𝑆(𝑡)𝛼+𝜌𝐸𝑅̇̇𝑅+𝜇𝐸(𝑡),𝐼(𝑡)=𝛼𝐸(𝑡)(𝛾+𝜇)𝐼(𝑡),(𝑡)=𝜌𝑆𝑅𝑆(𝑡)+𝜌𝐸𝑅𝐸(𝑡)𝛾𝐼(𝑡)𝜇𝑅(𝑡).(1.1)

264874.fig.001
Figure 1

Notably, the first three equations in (1.1) do not depend on the fourth equation, since ̇̇̇̇𝑆(𝑡)+𝐸(𝑡)+𝐼(t)+𝑅(𝑡)=1. Therefore, the forth equation can be omitted and the model (1.1) can be rewritten as ̇𝑆(𝑡)=𝜇𝑁𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁𝜌𝑆𝑅̇𝑆(𝑡)𝜇𝑆(𝑡),𝐸(𝑡)=𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁𝛼+𝜌𝐸𝑅̇+𝜇𝐸(𝑡),𝐼(𝑡)=𝛼𝐸(𝑡)(𝛾+𝜇)𝐼(𝑡).(1.2)

In [13], authors have proved the virus-free equilibrium EQvf=((𝜇/(𝜌𝑆𝑅+𝜇))𝑁,0,0) is globally asymptotically stable if 𝑅0=(𝛼𝑟𝜇/(𝛼𝑟𝜇)(𝜌𝑆𝑅+𝜇))1, and the viral equilibrium EQve is globally asymptotically stable if 𝑅0>1, where EQve=𝛼+𝜌𝐸𝑅+𝜇𝛼𝑟𝑁,𝜇𝑁𝛼+𝜌𝐸𝑅𝜌+𝜇𝐸𝑅𝑁+𝜇,𝛼𝛼𝑟𝛾+𝜇𝜇𝑁𝛼+𝜌𝐸𝑅𝜌+𝜇𝐸𝑅𝑁+𝜇.𝛼𝑟(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 𝜇,𝜇 and 𝜇 by 𝜇𝜇+𝜎1̇𝐵1(𝑡),𝜇𝜇+𝜎2̇𝐵2(𝑡), and 𝜇𝜇+𝜎3̇𝐵3(𝑡), where ̇𝐵1̇𝐵(𝑡),2(𝑡), and ̇𝐵3(𝑡) are mutual independent standard Brownian motions with 𝐵1(0)=0,𝐵2(0)=0, and 𝐵3(0)=0, and intensity of white noise 𝜎210,𝜎220 and 𝜎230, respectively. Then the stochastic system is ̇𝑆(𝑡)=𝜇𝑁𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁𝜌𝑆𝑅𝑆(𝑡)𝜇𝑆(𝑡)𝜎1̇𝐵𝑆(𝑡)1̇(𝑡),𝐸(𝑡)=𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁𝛼+𝜌𝐸𝑅+𝜇𝐸(𝑡)𝜎2̇𝐵𝐸(𝑡)1̇(𝑡),𝐼(𝑡)=𝛼𝐸(𝑡)(𝛾+𝜇)𝐼(𝑡)𝜎3̇𝐵𝐼(𝑡)1(𝑡).(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 𝑅01, 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 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑡)𝑑𝑡+𝑔(𝑥(𝑡),𝑡)𝑑𝐵(𝑡),on𝑡𝑡0,(1.5) with the initial value 𝑥(𝑡0)=𝑥0𝑅𝑛. 𝐵(𝑡) denotes 𝑛-dimensional standard Brownian motion defined on the above probability space. Define the differential operator 𝐿 associated with (1.4) by 𝜕𝐿=𝜕𝑥𝑘+12𝑛𝑘,𝑗=1𝑔𝑇𝜕(𝑥,𝑡)𝑔(𝑥,𝑡)2𝜕𝑥𝑘𝜕𝑥𝑗.(1.6) If 𝐿 acts on a function 𝑉, then 𝐿𝑉(𝑥,𝑡)=𝑉𝑡(𝑥,𝑡)+𝑉𝑥1(𝑥,𝑡)𝑓(𝑥,𝑡)+2𝑔trace𝑇(𝑥,𝑡)𝑉𝑥𝑥,(𝑥,𝑡)𝑔(𝑥,𝑡)(1.7) where 𝑉𝑡=𝜕𝑉/𝜕𝑡,𝑉𝑥=(𝜕𝑉/𝜕𝑥1,,𝜕𝑉/𝜕𝑥𝑛),𝑉𝑥𝑥=(𝜕2𝑉/𝜕𝑥𝑘𝜕𝑥𝑘)𝑛𝑛.

By Ito’s formula, if 𝑥(𝑡)𝑅𝑛, then for (1.4), assume that 𝑓(0,𝑡)=0,𝑔(0,𝑡)=0 for all 𝑡𝑡0. So 𝑥(𝑡)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)Δ, then the system (1.2) admits a unique solution (𝑆(𝑡),𝐸(𝑡),𝐼(𝑡)) on 𝑡0, 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(𝑡),𝐸(𝑡),𝐼(𝑡)) on 𝑡[0,𝜏𝑒), where 𝜏𝑒 is the explosion time [2, 13]. To show this solution is global, we need to show that 𝜏𝑒= a. s. Let 𝑘0>0 be sufficiently large so that every component of 𝑥0 lies within the interval [1/𝑘0,𝑘0]. For each integer 𝑘𝑘0, define the stopping time, 𝜏𝑘=inf𝑡0,𝜏𝑒1𝑆(𝑡)𝑘1,𝑘or𝐸(𝑡)𝑘1,𝑘or𝐼(𝑡)𝑘,,𝑘(2.1) where throughout this paper we set inf= (as usual denotes the empty set). Clearly, 𝜏𝑘 is increasing as 𝑘. Set 𝜏=lim𝑘𝜏𝑘, whence 𝜏𝜏𝑒 a. s. If we can show that 𝜏= a. s., then 𝜏𝑒= and (𝑆(𝑡),𝐸(𝑡),𝐼(𝑡)) a. s. for all 𝑡0. 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 𝜀(0,1) such that 𝑃𝜏𝑇>𝜀.(2.2) Hence, there is an integer 𝑘1𝑘0 such that 𝑃𝜏𝑇>𝜀𝑘>𝑘1.(2.3) Define a 𝐶2-function 𝑉 for 𝑋(𝑆,𝐸,𝐼)𝑅3+ by 𝑆𝑉(𝑋)=𝑆𝑎log𝑎+(𝐸1log𝐸)+(𝐼1log𝐼).(2.4) The nonnegativity of this function can be seen from 𝜇+1log𝜇0, for  all 𝜇>0. Using Ito’s formula we get 𝑎𝑑𝑉(𝑋)=𝑎𝑆𝑎𝑑𝑆+2𝑆2(𝑑𝑆)2+11𝐸1𝑑𝐸+2𝐸2(𝑑𝐸)2+11𝐼1𝑑𝐼+2𝐼2(𝑑𝐼)2𝜎𝐿𝑉𝑑𝑡1̇𝐵(𝑆𝑎)1(𝑡)+𝜎2̇𝐵(𝐸1)2(𝑡)+𝜎3̇𝐵(𝐼1)3,(𝑡)(2.5) where 𝑎𝐿𝑉=1𝑆𝜇𝑁𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁𝜌𝑆𝑅+𝑆(𝑡)𝜇𝑆(𝑡)𝑎𝜎212+11𝐸𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁𝛼+𝜌𝑆𝑅+𝜎+𝜇𝐸(𝑡)212+11𝐼[]+𝜎𝛼𝐸(𝑡)(𝛾+𝜇)𝐼(𝑡)232=𝜇𝑁+𝑎𝜌𝑆𝑅+𝜇𝑎+𝛼+𝜌𝑆𝑅+𝜇+𝛾+𝜇+𝑎𝜎212+𝜎222+𝜎232+𝛼𝑟𝑎𝑁𝐸𝜌𝑆𝑅𝑎𝑆𝜇𝑆𝑆𝜇𝑁𝜌𝑆𝑅𝐸𝜇𝐸𝛼𝑟𝑁𝛼𝑆𝛾𝐼𝜇𝐼𝐼𝐸𝜇𝑁+𝑎𝜌𝑆𝑅+𝜇𝑎+𝛼+𝜌𝑆𝑅+𝜇+𝛾+𝜇+𝑎𝜎212+𝜎222+𝜎232+𝛼𝑟𝑎𝑁𝐸𝜌𝐸𝑅𝐸𝜇𝐸.(2.6) By choosing 𝑎=(𝜌𝐸𝑅+𝜇)𝑁/𝛼𝑟, then 𝐿𝑉𝜇𝑁+𝑎𝜌𝑆𝑅+𝜇𝑎+𝛼+𝜌𝑆𝑅+𝜇+𝛾+𝜇+𝑎𝜎212+𝜎222+𝜎232̇𝑀.(2.7) Therefore, 𝜏𝑚0𝑇𝑑𝑉(𝑋)𝜏𝑚0𝑇̇𝑀𝑑𝑡𝜏𝑚0𝑇𝜎1(𝑆𝑎)𝑑𝐵1(𝑡)+𝜎2(𝐸1)𝑑𝐵2(𝑡)+𝜎3(𝐼1)𝑑𝐵3,𝑋𝜏(𝑡)𝐸𝑉𝑚𝑇𝑉(𝑋(0))+𝐸𝜏𝑚0𝑇̇̇𝑀𝑑𝑡𝑉(𝑋(0))+𝑀𝑇.(2.8) Setting Ω𝑚={𝜏𝑚𝑇} for 𝑚𝑚1, then by (2.3), we know that 𝑃(Ω𝑚)𝜀. Note that for every 𝜔Ω𝑚, there is at least one of 𝑆(Ω𝑚,𝜔), 𝐸(Ω𝑚,𝜔), and 𝐼(Ω𝑚,𝜔) that equals either 𝑚 or 1/𝑚. Then 𝑉𝑋𝜏𝑚1(𝑚1log𝑚)𝑚𝑚1+log𝑚𝑚𝑎𝑎log𝑎1𝑚,𝑎+𝑎log𝑎𝑚(2.9) where 1Ω𝑚(𝜔) is the indicator function of Ω𝑚. Let 𝑚 lead to the contradiction that ̇>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=(𝜇𝑁/(𝜌𝑆𝑅+𝜇),0,0) is the virus-free equilibrium of system (1.3), which has been mentioned above, and EQvf is globally stable if 𝑅01, 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 𝜌𝑆𝑅+𝜇>𝜎21,3𝛼2+2𝜌𝑆𝑅+2𝜇>𝜎22,2𝛾+2𝜇𝛼>𝜎23 and 𝑅01, then the solution 𝑋(𝑡) of system (1.3) with initial value 𝑋(0)𝑅3+ has the property lim𝑥1sup𝑡𝐸𝑡0(𝜌1+𝑏)𝑆𝑅+𝜇𝜎21𝜇2(1𝑠)+2𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝜎2(+𝛼𝑠)𝛾+𝜇212𝜎23𝑤2(𝑠)𝑑𝑠(1𝑏)𝜎21𝜇𝜌𝑆𝑅𝑁,+𝜇(3.1) where 𝑏 is positive constants, defined as in the proof.

Proof. For simplicity, let 𝑢(𝑡)=𝑆(𝑡)𝜇𝑁/(𝜌𝑆𝑅+𝜇),𝑣(𝑡)=𝐸(𝑡),𝑤(𝑡)=𝐼(𝑡), system (1.3) can be written as ̇𝑢(𝑡)=𝛼𝑟𝑣(𝑡)𝑁𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁𝜌+𝜇𝑆𝑅+𝜇𝑢(𝑡)𝜎1𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁̇̇+𝜇𝐵(𝑡),𝑣(𝑡)=𝛼𝑟𝑣(𝑡)𝑁𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁+𝜇𝛼+𝜌𝑆𝑅+𝜇𝑣(𝑡)𝜎2̇̇𝑣(𝑡)𝐵(𝑡),𝑤(𝑡)=𝛼𝑣(𝑡)(𝛾+𝜇)𝑤(𝑡)𝜎3̇𝐵𝑤(𝑡)3(𝑡).(3.2) Let 1𝑉(𝑥)=2(𝑢+𝑣)2+12𝑏𝑢2+121𝑏𝑣+2𝑤2=𝑉1+𝑏𝑉2+𝑏𝑉3+𝑉4,(3.3) then 𝑏 is positive constants to be determined later. By Ito’s formula, we compute 𝑑𝑉1𝐿𝑉1𝜎𝑑𝑡(𝑢(𝑡)+𝑣(𝑡))1𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁̇+𝜇𝐵(𝑡)+𝜎2̇𝐵𝑣(𝑡)2,(𝑡)𝐿𝑉1𝜌=(𝑢(𝑡)+𝑣(𝑡))𝑆𝑅+𝜇𝑢(𝑡)𝛼+𝜌𝑆𝑅+1+𝜇𝑣(𝑡)2𝜎21𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁+𝜇2+12𝜎22𝑣2(𝜌𝑡)(𝑢(𝑡)+𝑣(𝑡))𝑆𝑅+𝜇𝑢(𝑡)𝛼+𝜌𝑆𝑅+𝜇𝑣(𝑡)+𝜎21𝑢2(𝑡)+𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2+12𝜎22𝑣2(𝜌𝑡)=𝑆𝑅+𝜇𝜎21𝑢2(𝑡)+𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝑣2(𝑡)+𝛼+2𝜌𝑆𝑅+2𝜇𝑢(𝑡)𝑣(𝑡)𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2,𝑑𝑉2𝐿𝑉2𝑑𝑡𝜎1𝜇𝑢(𝑡)𝑢(𝑡)+𝜌𝑆𝑅̇+𝜇𝐵(𝑡),𝐿𝑉2=𝑢(𝑡)𝛼𝑟𝑣(𝑡)𝑁𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁𝜌+𝜇𝑆𝑅+1+𝜇𝑢(𝑡)2𝜎21𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁+𝜇2𝑢(𝑡)𝛼𝑟𝑣(𝑡)𝑁𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁𝜌+𝜇𝑆𝑅+𝜇𝑢(𝑡)+𝜎21𝑢2(𝑡)+𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2𝜌=𝑆𝑅+𝜇𝜎21𝑢2(𝑡)+𝛼𝑟𝜇𝜌𝑆𝑅+𝜇𝑢(𝑡)𝑣(𝑡)+𝛼𝑟𝑁𝑣(𝑡)𝑢2(𝑡)+𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2𝜌𝑆𝑅+𝜇𝜎21𝑢2(𝑡)+𝛼𝑟𝜇𝜌𝑆𝑅+𝜇𝑢(𝑡)𝑣(𝑡)+𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2,𝑑𝑉3=𝛼𝑟𝑣(𝑡)𝑁𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁+𝜇𝑑𝑡𝛼+𝜌𝑆𝑅+𝜇𝑣(𝑡)𝑑𝑡𝜎2̇=𝑣(𝑡)𝐵(𝑡)𝛼𝑟𝑁𝑣(𝑡)𝑢(𝑡)+𝛼𝑟𝜇𝜌𝑆𝑅+𝜇𝛼+𝜌𝑆𝑅+𝜇𝑣(𝑡)𝑑𝑡𝜎2̇𝑣(𝑡)𝐵(𝑡)𝐿𝑉3𝜎2𝑣̇𝐵(𝑡)(𝑡),𝑑𝑉4=1𝑤(𝑡)(𝛼𝑣(𝑡)(𝛾+𝜇)𝑤(𝑡))+2𝜎23𝑤2(𝑡)𝑑𝑡𝜎3𝑤2̇=(𝑡)𝐵(𝑡)𝛼𝑣(𝑡)𝑤(𝑡)(𝛾+𝜇)𝑤21(𝑡)+2𝜎23𝑤2(𝑡)𝑑𝑡𝜎3𝑤2̇𝐵𝛼(𝑡)(𝑡)2𝑣2(𝑡)+𝑤2(𝑡)(𝛾+𝜇)𝑤21(𝑡)+2𝜎23𝑤2(𝑡)𝑑𝑡𝜎3𝑤2̇=𝛼(𝑡)𝐵(𝑡)21𝛾𝜇+2𝜎23𝑤2(𝛼𝑡)+2𝑣2(𝑡)𝑑𝑡𝜎3𝑤2(̇𝑡)𝐵(𝑡)𝐿𝑉4𝑑𝑡𝜎3𝑤2̇(𝑡)𝐵(𝑡),𝐿𝑉=𝐿𝑉1+𝑏𝐿𝑉2+𝑏𝐿𝑉3+𝐿𝑉4𝜌=𝑆𝑅+𝜇𝜎21𝑢2(𝑡)+𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝑣2+(𝑡)𝛼+2𝜌𝑆𝑅+2𝜇𝑢(𝑡)𝑣(𝑡)𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2𝜌𝑏𝑆𝑅+𝜇𝜎21𝑢2(𝑡)+𝛼𝑟𝜇𝜌𝑆𝑅+𝜇𝑢(𝑡)𝑣(𝑡)+𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2+𝑏𝛼𝑟𝑁𝑣(𝑡)𝑢(𝑡)+𝛼𝑟𝜇𝜌𝑆𝑅+𝜇𝛼+𝜌𝑆𝑅+𝛼+𝜇𝑣(𝑡)21𝛾𝜇+2𝜎23𝑤2𝛼(𝑡)+2𝑣2.(𝑡)(3.4)
Choosing 𝑏=(𝑁(𝛼+2𝜌𝑆𝑅+2𝜇)(𝜌𝑆𝑅+𝜇))/(𝛼𝑟(𝜌𝑆𝑅+𝜇𝑁𝜇)), then we get 𝜌𝐿𝑉=(1+𝑏)𝑆𝑅+𝜇𝜎21𝑢21(𝑡)2𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝑣2𝛼(𝑡)𝛾+𝜇212𝜎23𝑤2(𝑡)𝑏𝛼+𝜌𝑆𝑅+𝜇𝛼𝑟𝜇𝜌𝑆𝑅+𝜇𝑣(𝑡)+(1𝑏)𝜎21𝜇𝜌𝑆𝑅+𝜇2,𝜌𝑑𝑉(1+𝑏)𝑆𝑅+𝜇𝜎21𝑢21(𝑡)2𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝑣2𝛼(𝑡)𝛾+𝜇212𝜎23𝑤2(𝑡)+(1𝑏)𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2𝜎(𝑢(𝑡)+𝑣(𝑡))1𝜇𝑢(𝑡)+𝜌𝑆𝑅𝑁̇𝐵+𝜇1(𝑡)+𝜎2̇𝐵𝑣(𝑡)2(𝑡)𝜎1𝜇𝑢(𝑡)𝑢(𝑡)+𝜌𝑆𝑅𝑁̇+𝜇𝐵(𝑡)𝜎2̇𝐵𝑣(𝑡)2(𝑡)𝜎3𝑤2̇𝐵(𝑡)3(𝑡).(3.5) Integrating this from 0 to 𝑡 and taking the expectation, we have 𝐸[]𝑉(𝑡)𝑉(0)𝐸𝑡0𝜌(1+𝑏)𝑆𝑅+𝜇𝜎21𝑢21(𝑠)+2𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝑣2+𝛼(𝑠)𝛾+𝜇212𝜎23𝑤2(𝑠)(1𝑏)𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2𝑑𝑠.(3.6) Hence, lim𝑥1sup𝑡𝐸𝑡0(𝜌1+𝑏)𝑆𝑅+𝜇𝜎21𝑢2(1𝑠)+2𝛼+𝜌𝑆𝑅1+𝜇2𝜎22𝑣2(+𝛼𝑠)𝛾+𝜇212𝜎23𝑤2(𝑠)𝑑𝑠(1𝑏)𝜎21𝜇𝜌𝑆𝑅𝑁+𝜇2.(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 ̇𝐵1(𝑡). 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 𝐵: there exists a bounded domain 𝑈𝐸𝑙 with regular boundary Γ, having the following properties.(𝐵.1) In the domain 𝑈 and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix 𝐴(𝑥) is bounded away from zero.(𝐵.2) If 𝑥𝐸𝑙/𝑈, the mean time 𝜏 at which a path issuing from 𝑥 reaches the set 𝑈 is finite, and sup𝑥𝐾𝐸𝑥𝜏< for every compact subset 𝐾𝐸𝑙. If (𝐵) holds, then the Markov process 𝑋(𝑡) has a stationary distribution 𝜇(). Let 𝑓() be a function integrable with respect to the measure 𝜇. Then P𝑥lim𝑇1𝑇𝑇0𝑓(𝑋(𝑡))𝑑𝑡=𝐸𝑙𝑓(𝑥)𝜇𝑑(𝑥)=1,𝑥𝐸𝑙.(4.1)

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

Theorem 4.3. If𝜎21<(𝜌𝑆𝑅+𝜇)(1+(𝛼𝑟/𝑆𝑁))(𝑆/(𝑆1)),𝜎22<(𝛼/2)+𝜌𝑆𝑅+𝜇𝜎23<𝛾+𝜇(𝛼/2), and 𝑅0>1, then, for any initial value 𝑋(0)𝑅3+, there is a stationary distribution 𝜇() for system (1.3), and it has an ergodic property, where 𝑎,𝑐 are defined as in the proof, Qve=(𝑆,𝐸,𝐼) is the viral equilibrium of system.

Proof. When 𝑅0>1, there is an viral equilibrium EQve of system (1.3). Then 𝜇𝑁=𝛼𝑟𝐸𝑆𝑁𝜌𝑆𝑅𝑆+𝜇𝑆,𝛼𝑟𝐸𝑆𝑁=𝛼+𝜌𝑆𝑅𝐸+𝜇,𝛼𝐸=(𝛾+𝜇)𝐼.(4.2) Define 𝑉(𝑥)=𝑎𝑆𝑆𝑆𝑆log𝑆+𝐸𝐸𝐸𝐸log𝐸+𝐸𝐸𝐸𝐸log𝐸+12𝑆𝑆+𝐸𝐸2+12𝑐𝑆𝑆2+12𝐼𝐼2=𝑎𝑉1+𝑉2+𝑉3+𝑐𝑉4+𝑉5,(4.3) where 𝑎,𝑐, are positive constants to be determined later. Then 𝑉 is positive definite. By Ito’s formula, we compute 𝑑𝑉1=𝑆1𝑆𝜇𝑁𝛼𝑟𝐸𝑆𝑁𝜌𝑆𝑅𝑆𝜇𝑆𝑑𝑡𝜎1𝑆̇𝐵1+𝐸(𝑡)1𝐸𝛼𝑟𝐸𝑆𝑁𝛼+𝜌𝑆𝑅𝐸+𝜇𝑑𝑡𝜎2𝐸̇𝐵2+1(𝑡)2𝑆𝜎211𝑑𝑡+2𝐸𝜎22𝑑𝑡=𝐿𝑉1𝑆𝑑𝑡1𝑆𝜎1𝑆̇𝐵(𝑡)1𝐸(𝑡)1𝐸𝜎2𝐸̇𝐵(𝑡)2(𝑡),(4.4) where 𝐿𝑉1=𝑆1𝑆𝜇𝑁𝛼𝑟𝐸𝑆𝑁𝜌𝑆𝑅+𝐸𝑆𝜇𝑆𝑑𝑡1𝐸𝛼𝑟𝐸𝑆𝑁𝛼+𝜌𝑆𝑅𝐸+1+𝜇𝑑𝑡2𝑆𝜎211𝑑𝑡+2𝐸𝜎22=𝑆𝑑𝑡1𝑆𝛼𝑟𝑁𝐸𝑆𝜌𝐸𝑆𝑑𝑡+𝑆𝑅𝑆+𝜇+𝐸𝑆𝑑𝑡1𝐸𝛼𝑟𝐸𝑆𝑁𝛼+𝜌𝑆𝑅𝐸+1+𝜇𝑑𝑡2𝑆𝜎211𝑑𝑡+2𝐸𝜎22𝑑𝑡=𝑆𝑆2𝑆𝜌𝑆𝑅++𝜇𝛼𝑟𝑁𝐸𝑆+𝐸𝛼+𝜌𝑆𝑅++𝜇𝛼𝑟𝑁𝑆𝛼𝜌𝑆𝑅𝜇𝐸𝛼𝑟𝑁𝐸𝑆𝛼𝑟𝑁𝐸𝑆2𝑆+12𝑆𝜎21+12𝐸𝜎22𝑆𝑆2𝑆𝜌𝑆𝑅1+𝜇2𝑆𝜎21+12𝐸𝜎22,𝑑𝑉2=𝐸1𝐸𝛼𝑟𝐸𝑆𝑁𝛼+𝜌𝑆𝑅𝐸+𝜇𝑑𝑡𝜎2𝐸̇𝐵2+1(𝑡)2𝐸𝜎22𝑑𝑡=𝐿𝑉2𝐸𝑑𝑡1𝐸𝜎2̇𝐵𝐸(𝑡)2(𝑡).(4.5) Let 𝐵=(𝛼𝑟/𝑁)𝐸𝑆=(𝛼+𝜌𝑆𝑅+𝜇)𝐸 and 𝛼1log𝛼>0, for all 𝛼𝐿𝑉2=𝐸1𝐸𝛼𝑟𝐸𝑆𝑁𝛼+𝜌𝑆𝑅𝐸1+𝜇𝑑𝑡+2𝐸𝜎22=𝐸𝑑𝑡1𝐸𝐵𝐸𝑆𝐸𝑆𝐵𝐸𝐸+12𝐸𝜎22=𝐵𝐸𝑆𝐸𝑆𝐵𝐸𝐸+𝐵𝑆𝑆+𝐵+12𝐸𝜎22𝐵𝐸𝑆𝐸𝑆𝐸𝐸𝑆1+log𝑆+1+12𝐸𝜎22𝐵𝐸𝑆𝐸𝑆𝐸𝐸+𝑆𝑆+12+12𝐸𝜎22=𝐵𝐸𝐸𝑆1𝑆+𝐵𝑆1𝑆+𝑆𝑆+122𝐸𝜎22=𝛼𝑟𝑁𝐸𝐸𝑆𝑆+𝛼𝑟𝑁𝐸𝑆𝑆2𝑆+12𝐸𝜎22,𝑑𝑉3=𝑆𝑆+𝐸𝐸𝜌𝜇𝑁𝑆𝑅+𝜇𝑆𝛼+𝜌𝑆𝑅𝐸+𝜇𝑑𝑡𝑆𝑆+𝐸𝐸𝜎1𝑆̇𝐵1(𝑡)+𝜎2𝐸̇𝐵2+𝜎(𝑡)212𝑆2+𝜎222𝐸2𝑑𝑡,𝐿𝑉3=𝑆𝑆+𝐸𝐸𝜌𝑆𝑅+𝜇𝑆𝑆𝛼+𝜌𝑆𝑅+𝜇𝐸𝐸+𝜎212𝑆2+𝜎222𝐸2𝜌𝑆𝑅+𝜇𝑆𝑆2𝛼+𝜌𝑆𝑅+𝜇𝐸𝐸2𝛼+𝜌𝐸𝑅+𝜌𝑆𝑅+2𝜇𝑆𝑆𝐸𝐸+𝜎21𝑆𝑆+𝑆2+𝜎22𝐸𝐸+𝐸2𝜎21𝜌𝑆𝑅𝜇𝑆𝑆2+𝜎22𝛼𝜌𝐸𝑅𝜇𝐸𝐸2+𝜎21𝑆2+𝜎22𝐸2,𝑑𝑉4=𝑆𝑆𝜇𝑁𝛼𝑟𝑁𝜌𝐸𝑆𝑆𝑅𝑆+𝜇𝑆𝑆𝑆𝜎1̇𝐵1(𝑡)+2𝜎21𝑆2,𝐿𝑉4=𝑆𝑆𝜇𝑁𝛼𝑟𝑁𝜌𝐸𝑆𝑆𝑅𝑆+1+𝜇2𝜎21𝑆2=𝑆𝑆𝛼𝑟𝑁𝐸𝑆𝐸𝑆𝜌𝑆𝑅+𝜇𝑆𝑆+12𝜎21𝑆2=𝛼𝑟𝑁𝑆𝑆𝑆𝐸𝐸𝛼𝑟𝑁𝑆𝑆2𝐸𝛼𝑟𝑁𝜌𝑆𝑅+𝜇𝑆𝑆2+12𝜎21𝑆2𝛼𝑟𝑁𝑆𝑆𝑆𝐸𝐸𝛼𝑟𝑁𝜌𝑆𝑅+𝜇𝜎21𝑆𝑆2+𝜎21𝑆2,𝑑𝑉5=𝐼𝐼[]+𝜎𝛼𝐸(𝛾+𝜇)𝐼232𝐼2=𝐿𝑉5𝑑𝑡𝜎3𝐼𝐼𝐼̇𝐵3,𝐿𝑉5=𝐼𝐼[]+𝜎𝛼𝐸(𝛾+𝜇)𝐼232𝐼2=𝐼𝐼𝛼𝐸𝐸(𝛾+𝜇)𝐼𝐼+𝜎232𝐼2𝛼𝐸𝐸𝐼𝐼𝛾+𝜇𝜎23𝐼𝐼2+𝜎23𝐼2𝛼2𝐸𝐸2𝛾+𝜇𝜎23𝛼2𝐼𝐼2+𝜎23𝐼2.(4.6)
Choosing 𝑎=(𝛼𝑟/(𝜌𝑆𝑅+𝜇)𝑁)𝐸, then 𝑎𝑉1+𝑉2=𝑎𝑆𝑆2𝑆𝜌𝑆𝑅+1+𝜇2𝑆𝜎21+12𝐸𝜎22+𝛼𝑟𝑁𝐸𝐸𝑆𝑆+𝛼𝑟𝑁𝐸𝑆𝑆2𝑆+12𝐸𝜎22=𝛼𝑟𝑁𝐸𝐸𝑆𝑆+12𝑎𝑆𝜎21+12𝐸𝜎22.(4.7)
Choosing 𝑐=1/𝑆, then 𝑎𝑉1+𝑉2+𝑉3+𝑉4+𝑉5𝛼𝑟𝑁𝐸𝐸𝑆𝑆+12𝑎𝑆𝜎21+12(𝑎+1)𝐸𝜎22+𝜎21𝜌𝑆𝑅𝜇𝑆𝑆2+𝜎22𝛼𝜌𝑆𝑅𝜇𝐸𝐸2+𝜎21𝑆2+𝜎22𝐸2+𝑐𝛼𝑟𝑁𝑆𝑆𝑆𝐸𝐸𝛼𝑟𝑁𝜌S𝑅+𝜇𝜎21𝑆𝑆2+𝜎21𝑆2+𝛼2𝐸𝐸2𝛾+𝜇𝜎23𝛼2𝐼𝐼2+𝜎23𝐼2=12𝑎𝜎21𝑆+𝜎21𝑆+𝜎21𝑆2+12(𝑎+1)𝜎22𝐸+𝜎22𝐸2+𝜎23𝐼2+𝜎21𝜌𝑆𝑅𝜇𝑐𝛼𝑟𝑁𝜌𝑆𝑅+𝜇𝜎21𝑆𝑆2+𝜎22𝛼2𝜌𝑆𝑅𝜇𝐸𝐸2𝛾+𝜇𝜎23𝛼2𝐼𝐼2𝛿1+𝛼𝑟𝑆𝑁𝜌𝑆𝑅+𝜇𝜎2111+𝑆𝑆𝑆2𝛼2+𝜌𝑆𝑅+𝜇𝜎22𝐸𝐸2𝛾+𝜇𝜎23𝛼2𝐼𝐼2.(4.8)
Then the ellipsoid 𝛿1+𝛼𝑟𝑆𝑁𝜌𝑆𝑅+𝜇𝜎2111+𝑆𝑆𝑆2𝛼2+𝜌𝑆𝑅+𝜇𝜎22𝐸𝐸2=0(4.9) lies entirely in 𝑅3+. We can take 𝑈 to be a neighborhood of the ellipsoid with 𝑈𝑅3+, so, for 𝑥𝑈/𝑅3+, 𝐿𝑉𝐾 (𝐾 is a positive constant), which implies that condition (𝐵.2) in Lemma 4.1 is satisfied. Hence, the solution 𝑋(𝑡) is recurrent in the domain 𝑈, which, together with Lemma 4.2, implies that 𝑋(𝑡) is recurrent in any bounded domain 𝐷𝑅3+. Besides, for  all 𝐷, there is an 𝜎𝑀=min21𝑆2,𝜎22𝐸2,𝜎23𝐼2𝐷>0,(4.10) such that 3𝑖,𝑗=1𝑎𝑖𝑗𝜉𝑖𝜉𝑗=𝜎21𝑆2𝜉21+𝜎22𝐸2𝜉22+𝜎23𝐼2𝜉23𝑀𝜉2 for all 𝑋𝐷,𝜉𝑅3which implies that condition (𝐵.1) is also satisfied. Therefore, the stochastic system (1.3) has a stationary distribution 𝜇() 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: 𝜇=1/4380,𝑁=100000, 𝛼=1/500, 𝑟=7, 𝜌𝑆𝑅=1/2500, 𝜌𝐸𝑅=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.

fig2
Figure 2: Deterministic and stochastic trajectories around infection-free solution.

To demonstrate the permanence of system (1.4), we take the following set parameter values: 𝜇=1/4380, 𝑁=100000, 𝛼=1/500, 𝑟=30, 𝜌𝑆𝑅=1/2500, 𝜌𝐸𝑅=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.

fig3
Figure 3: Deterministic and stochastic trajectories around virus endemic equilibrium.

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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