# A Stochastic Dynamic Model of Computer Viruses

Authors: Chunming Zhang; Yun Zhao; Yingjiang Wu; Shuwen Deng
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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.


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## Body

## 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. LetS,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 rateμN.(H3)
Computers are disconnected to the computer network with constant rateμ.(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 Figure1) is derived:
(1.1)S˙(t)=μN-αrNE(t)S(t)-ρSRS(t)-μS(t),E˙(t)=αrNE(t)S(t)-(α+ρER+μ)E(t),I˙(t)=αE(t)-(γ+μ)I(t),R˙(t)=ρSRS(t)+ρERE(t)-γI(t)-μR(t).Figure 1Notably, 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
(1.2)S˙(t)=μN-αrE(t)S(t)N-ρSRS(t)-μS(t),E˙(t)=αrE(t)S(t)N-(α+ρER+μ)E(t),I˙(t)=αE(t)-(γ+μ)I(t).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
(1.3)EQve=((α+ρER+μ)αrN,μN(α+ρER+μ)-(ρER+μ)Nαr,αγ+μ[μN(α+ρER+μ)-(ρER+μ)Nαr]).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 parametersμ,μ and μ by μ→μ+σ1B˙1(t),μ→μ+σ2B˙2(t), and μ→μ+σ3B˙3(t), where B˙1(t),B˙2(t), and B˙3(t) are mutual independent standard Brownian motions with B1(0)=0,B2(0)=0, and B3(0)=0, and intensity of white noise σ12≥0,σ22≥0 and σ32≥0, respectively. Then the stochastic system is
(1.4)S˙(t)=μN-αrE(t)S(t)N-ρSRS(t)-μS(t)-σ1S(t)B˙1(t),E˙(t)=αrE(t)S(t)N-(α+ρER+μ)E(t)-σ2E(t)B˙1(t),I˙(t)=αE(t)-(γ+μ)I(t)-σ3I(t)B˙1(t).The organization of this paper is as follows. In Section2, 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 then-dimensional stochastic differential equation
(1.5)dx(t)=f(x(t),t)dt+g(x(t),t)dB(t),on⁡t≥t0,
with the initial value x(t0)=x0∈Rn. B(t) denotes n-dimensional standard Brownian motion defined on the above probability space. Define the differential operator L associated with (1.4) by
(1.6)L=∂∂xk+12∑k,j=1n[gT(x,t)g(x,t)]∂2∂xk∂xj.
If L acts on a function V, then
(1.7)LV(x,t)=Vt(x,t)+Vx(x,t)f(x,t)+12trace[gT(x,t)Vxx(x,t)g(x,t)],
where Vt=∂V/∂t,Vx=(∂V/∂x1,…,∂V/∂xn),Vxx=(∂2V/∂xk∂xk)n*n.By Ito’s formula, ifx(t)∈Rn, then for (1.4), assume that f(0,t)=0,g(0,t)=0 for all t≥t0. 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 t≥0, and this solution remains in R+3 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 k≥k0, define the stopping time,
(2.1)τk=inf⁡{t∈[0,τe)∶S(t)∉(1k,k)orE(t)∉(1k,k)orI(t)∉(1k,k)},
where throughout this paper we set inf∅=∞ (as usual ∅ denotes the empty set). Clearly, τk is increasing as k→∞. Set τ∞=limk→∞τk, whence τ∞≤τe a. s. If we can show that τ∞=∞ a. s., then τe=∞ and (S(t),E(t),I(t)) a. s. for all t≥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 ɛ∈(0,1) such that
(2.2)P(τ∞≤T)>ɛ.
Hence, there is an integer k1≥k0 such that
(2.3)P{τ∞≤T}>ɛ∀k>k1.
Define a C2-function V for X(S,E,I)∈R+3 by
(2.4)V(X)=(S-a-log⁡Sa)+(E-1-log⁡E)+(I-1-log⁡I).
The nonnegativity of this function can be seen from μ+1-logμ≥0, for  all μ>0. Using Ito’s formula we get
(2.5)dV(X)=(a-aS)dS+a2S2(dS)2+(1-1E)dE+12E2(dE)2+(1-1I)dI+12I2(dI)2≐LVdt-[σ1(S-a)B˙1(t)+σ2(E-1)B˙2(t)+σ3(I-1)B˙3(t)],
where
(2.6)LV=(1-aS)[μN-αrE(t)S(t)N-ρSRS(t)-μS(t)]+aσ122+(1-1E)[αrE(t)S(t)N-(α+ρSR+μ)E(t)]+σ122+(1-1I)[αE(t)-(γ+μ)I(t)]+σ322=(μN+aρSR+μa+α+ρSR+μ+γ+μ+aσ122+σ222+σ322)+[αraNE-ρSRS-μS-aSμN-ρSRE-μE-αrNS-γI-μI-αIE]≤μN+aρSR+μa+α+ρSR+μ+γ+μ+aσ122+σ222+σ322+αraNE-ρERE-μE.
By choosing a=(ρER+μ)N/αr, then
(2.7)LV≤μN+aρSR+μa+α+ρSR+μ+γ+μ+aσ122+σ222+σ322≐M˙.
Therefore,
(2.8)∫0τm∧TdV(X)≤∫0τm∧TM˙dt-∫0τm∧T[σ1(S-a)dB1(t)+σ2(E-1)dB2(t)+σ3(I-1)dB3(t)],EV(X(τm∧T))≤V(X(0))+E[∫0τm∧TM˙dt]≤V(X(0))+M˙T.
Setting Ωm={τm≤T} for m≥m1, then by (2.3), we know that P(Ωm)≥ɛ. Note that for every ω∈Ωm, there is at least one of S(Ωm,ω), E(Ωm,ω), and I(Ωm,ω) that equals either m or 1/m. Then
(2.9)V(X(τm))≥(m-1-log⁡m)∧(1m-1+log⁡m)∧(m-a-alog⁡ma)∧(1m-a+alog⁡am),
where 1Ωm(ω) is the indicator function of Ωm. Let m→∞ lead to the contradiction that ∞>V(X(0))+M˙T=∞. So τ∞=∞ is necessary. The proof of Theorem 2.1 is completed.


## 3. Stability of Virus-Free Equilibrium

It is clear thatEQvf=(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 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.
IfρSR+μ>σ12,3α2+2ρSR+2μ>σ22,2γ+2μ-α>σ32 and R0≤1, then the solution X(t) of system (1.3) with initial value X(0)∈R+3 has the property
(3.1)lim⁡x→∞sup⁡1tE∫0t[(1+b)(ρSR+μ-σ12)μ2(s)+(12α+ρSR+μ-12σ22)σ2(s)sup⁡+(γ+μ-α2-12σ32w2(s))ds≤(1-b)σ12(μρSR+μN)],
where b is positive constants, defined as in the proof.Proof.
For simplicity, letu(t)=S(t)-μN/(ρSR+μ),v(t)=E(t),w(t)=I(t), system (1.3) can be written as
(3.2)u˙(t)=-αrv(t)N[u(t)+μρSR+μN]-(ρSR+μ)u(t)-σ1(u(t)+μρSR+μN)B˙(t),v˙(t)=αrv(t)N[u(t)+μρSR+μN]-(α+ρSR+μ)v(t)-σ2v(t)B˙(t),w˙(t)=αv(t)-(γ+μ)w(t)-σ3w(t)B˙3(t).
Let
(3.3)V(x)=12(u+v)2+12bu2+12bv+12w2=V1+bV2+bV3+V4,
then b is positive constants to be determined later. By Ito’s formula, we compute
(3.4)dV1≐LV1dt-(u(t)+v(t))[σ1(u(t)+μρSR+μN)B˙(t)+σ2v(t)B˙2(t)],LV1=(u(t)+v(t))[-(ρSR+μ)u(t)-(α+ρSR+μ)v(t)]+12σ12(u(t)+μρSR+μN)2LV1=+12σ22v2(t)LV1≤(u(t)+v(t))[-(ρSR+μ)u(t)-(α+ρSR+μ)v(t)]+σ12u2(t)+σ12(μρSR+μN)2LV1=+12σ22v2(t)LV1=-[(μρSR+μN)2(ρSR+μ-σ12)u2(t)+(α+ρSR+μ-12σ22)v2(t)+(α+2ρSR+2μ)u(t)v(t)+LV1=--σ12(μρSR+μN)2],dV2≐LV2dt-σ1u(t)(u(t)+μρSR+μ)B˙(t),LV2=u(t){-αrv(t)N[u(t)+μρSR+μN]-(ρSR+μ)u(t)}LV2=+12σ12(u(t)+μρSR+μN)2LV2≤u(t){-αrv(t)N[u(t)+μρSR+μN]-(ρSR+μ)u(t)}+σ12u2(t)LV2=+σ12(μρSR+μN)2LV2=-[(μρSR+μN)2(ρSR+μ-σ12)u2(t)+αrμρSR+μu(t)v(t)+αrNv(t)u2(t)+LV1=-+σ12(μρSR+μN)2]LV2≤-[(ρSR+μ-σ12)u2(t)+αrμρSR+μu(t)v(t)+σ12(μρSR+μN)2],dV3=αrv(t)N[u(t)+μρSR+μN]dt-(α+ρSR+μ)v(t)dt-σ2v(t)B˙(t)dV3=[αrNv(t)u(t)+(αrμρSR+μ-(α+ρSR+μ))v(t)]dt-σ2v(t)B˙(t)dV3≐LV3-σ2v(t)B˙(t),dV4=[w(t)(αv(t)-(γ+μ)w(t))+12σ32w2(t)]dt-σ3w2(t)B˙(t)dV3=[αv(t)w(t)-(γ+μ)w2(t)+12σ32w2(t)]dt-σ3w2(t)B˙(t)dV3≤[α2(v2(t)+w2(t))-(γ+μ)w2(t)+12σ32w2(t)]dt-σ3w2(t)B˙(t)dV3=[(α2-γ-μ+12σ32)w2(t)+α2v2(t)]dt-σ3w2(t)B˙(t)dV3≐LV4dt-σ3w2(t)B˙(t),LV=LV1+bLV2+bLV3+LV4LV=-[(μρSR+μN)2(ρSR+μ-σ12)u2(t)+(α+ρSR+μ-12σ22)v2(t)+LV1=-+(α+2ρSR+2μ)u(t)v(t)-σ12(μρSR+μN)2]LV=-b[(ρSR+μ-σ12)u2(t)+αrμρSR+μu(t)v(t)+σ12(μρSR+μN)2]LV=+b[αrNv(t)u(t)+(αrμρSR+μ-(α+ρSR+μ))v(t)]LV=+[(α2-γ-μ+12σ32)w2(t)+α2v2(t)].
Choosingb=(N(α+2ρSR+2μ)(ρSR+μ))/(αr(ρSR+μ-Nμ)), then we get
(3.5)LV=-(1+b)(ρSR+μ-σ12)u2(t)-(12α+ρSR+μ-12σ22)v2(t)-(γ+μ-α2-12σ32)w2(t)-b((α+ρSR+μ)-αrμρSR+μ)v(t)+(1-b)σ12(μρSR+μ)2,dV≤-(1+b)(ρSR+μ-σ12)u2(t)-(12α+ρSR+μ-12σ22)v2(t)-(γ+μ-α2-12σ32)w2(t)+(1-b)σ12(μρSR+μN)2-(u(t)+v(t))[σ1(u(t)+μρSR+μN)B˙1(t)+σ2v(t)B˙2(t)]-σ1u(t)(u(t)+μρSR+μN)B˙(t)-σ2v(t)B˙2(t)-σ3w2(t)B˙3(t).
Integrating this from 0 to t and taking the expectation, we have
(3.6)E[V(t)]-V(0)≤-E∫0t[(μρSR+μN)2(1+b)(ρSR+μ-σ12)u2(s)+(12α+ρSR+μ-12σ22)v2(s)LV1=-++++(γ+μ-α2-12σ32)w2(s)-(1-b)σ12(μρSR+μN)2]ds.
Hence,
(3.7)lim⁡x→∞sup⁡1tE∫0t[(1+b)(ρSR+μ-σ12)u2(s)+(12α+ρSR+μ-12σ22)v2(s)++∫+(γ+μ-α2-12σ32)w2(s)]ds≤(1-b)σ12(μρSR+μN)2.Remark 3.2.
Theorem3.1 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 σ12, which is the intensity of the white noise B˙1(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]).
AssumptionB: there exists a bounded domain U⊂El with regular boundary Γ, having the following properties.(B.1)
In the domainU and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero.(B.2)
Ifx∈El/U, the mean time τ at which a path issuing from x reaches the set U is finite, and supx∈KExτ<∞ for every compact subset K⊂El. 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
(4.1)Px{lim⁡T→∞1T∫0T‍f(X(t))dt=∫El‍f(x)μd(x)}=1,∀x∈El.Lemma 4.2 (see [14, 15]).
LetX(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.
Ifσ12<(ρSR+μ)(1+(αr/S*N))(S*/(S*-1)),σ22<(α/2)+ρSR+μσ32<γ+μ-(α/2), and R0>1, then, for any initial value X(0)∈R+3, 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.
WhenR0>1, there is an viral equilibrium EQve of system (1.3). Then
(4.2)μN=αrE*S*N-ρSRS*+μS*,αrE*S*N=(α+ρSR+μ)E*,αE*=(γ+μ)I*.
Define
(4.3)V(x)=a(S-S*-S*log⁡SS*+E-E*-E*log⁡EE*)+(E-E*-E*log⁡EE*)+12(S-S*+E-E*)2+12c(S-S*)2+12(I-I*)2=aV1+V2+V3+cV4+V5,
where a,c, are positive constants to be determined later. Then V is positive definite. By Ito’s formula, we compute
(4.4)dV1=(1-S*S)[(μN-αrESN-ρSRS-μS)dt-σ1SB˙1(t)]+(1-E*E)[(αrESN-(α+ρSR+μ)E)dt-σ2EB˙2(t)]+12S*σ12dt+12E*σ22dt=LV1dt-(1-S*S)σ1S(t)B˙1(t)-(1-E*E)σ2E(t)B˙2(t),
where
(4.5)LV1=(1-S*S)[(μN-αrESN-ρSRS-μS)dt]=+(1-E*E)[(αrESN-(α+ρSR+μ)E)dt]=+12S*σ12dt+12E*σ22dt=(1-S*S)[αrN(E*S*-ES)dt+(ρSR+μ)(S*-S)dt]=+(1-E*E)[(αrESN-(α+ρSR+μ)E)dt]=+12S*σ12dt+12E*σ22dt=-(S-S*)2S(ρSR+μ)+αrNE*S*+E*(α+ρSR+μ)=+(αrNS*-α-ρSR-μ)E-αrNE*S*-αrNE*S*2S=+12S*σ12+12E*σ22≤-(S-S*)2S(ρSR+μ)12S*σ12+12E*σ22,dV2=(1-E*E)[(αrESN-(α+ρSR+μ)E)dt-σ2EB˙2(t)]+12E*σ22dt=LV2dt-(1-E*E)σ2E(t)B˙2(t).
Let B^=(αr/N)E*S*=(α+ρSR+μ)E* and α-1-logα>0, for all α(4.6)LV2=(1-E*E)(αrESN-(α+ρSR+μ)E)dt+12E*σ22dtLV2=(1-E*E)(B^ESE*S*-B^EE*)+12E*σ22LV2=(B^ESE*S*-B^EE*+B^SS*+B^)+12E*σ22LV2≤B^[ESE*S*-EE*-(1+log⁡SS*)+1]+12E*σ22LV2≤B^[ESE*S*-EE*+(SS*-2)+1]+12E*σ22LV2=B^(EE*-1)(SS*-1)+B^(S*S+SS*-2)+12E*σ22LV2=αrN(E-E*)(S-S*)+αrNE*(S-S*)2S+12E*σ22,dV3=(S-S*+E-E*)[μN-(ρSR+μ)S-(α+ρSR+μ)E]dtdV3=-(S-S*+E-E*)[σ1SB˙1(t)+σ2EB˙2(t)]+(σ122S2+σ222E2)dt,LV3=(S-S*+E-E*)[-(ρSR+μ)(S-S*)-(α+ρSR+μ)(E-E*)]LV3=+σ122S2+σ222E2LV3≤-(ρSR+μ)(S-S*)2-(α+ρSR+μ)(E-E*)2LV3≤-(α+ρER+ρSR+2μ)(S-S*)(E-E*)LV3≤+σ12[(S-S*)+S*2]+σ22[(E-E*)+E*2]LV3≤(σ12-ρSR-μ)(S-S*)2+(σ22-α-ρER-μ)(E-E*)2+σ12S*2+σ22E*2,dV4=(S-S*)(μN-αrNES-(ρSR+μ)S)-S(S-S*)σ1B˙(t)+12σ12S2,LV4=(S-S*)(μN-αrNES-(ρSR+μ)S)+12σ12S2LV4=(S-S*)[-αrN(ES-E*S*)-(ρSR+μ)(S-S*)]+12σ12S2LV4=-αrNS*(S-S*)(E-E*)-αrN(S-S*)2E-αrN(ρSR+μ)(S-S*)2+12σ12S2LV4≤-αrNS*(S-S*)(E-E*)-(αrN(ρSR+μ)-σ12)(S-S*)2+σ12S*2,dV5=(I-I*)[αE-(γ+μ)I]+σ322I2=LV5dt-σ3I(I-I*)B˙3,LV5=(I-I*)[αE-(γ+μ)I]+σ322I2LV4=(I-I*)[α(E-E*)-(γ+μ)(I-I*)]+σ322I2LV4≤α(E-E*)(I-I*)-(γ+μ-σ32)(I-I*)2+σ32I*2LV4≤α2(E-E*)2-(γ+μ-σ32-α2)(I-I*)2+σ32I*2.
Choosinga=(αr/(ρSR+μ)N)E*, then
(4.7)aV1+V2=a[-(S-S*)2S(ρSR+μ)+12S*σ12+12E*σ22]+[αrN(E-E*)(S-S*)+αrNE*(S-S*)2S+12E*σ22]=αrN(E-E*)(S-S*)+12aS*σ12+12E*σ22.
Choosingc=1/S*, then
(4.8)aV1+V2+V3+V4+V5≤αrN(E-E*)(S-S*)+12aS*σ12+12(a+1)E*σ22aV1+[(σ12-ρSR-μ)(S-S*)2+(σ22-α-ρSR-μ)(E-E*)2+σ12S*2+σ22E*2]aV1+c[-αrNS*(S-S*)(E-E*)-(αrN(ρSR+μ)-σ12)(S-S*)2+σ12S*2]aV1+[α2(E-E*)2-(γ+μ-σ32-α2)(I-I*)2+σ32I*2]=12aσ12S*+σ12S*+σ12S*2+12(a+1)σ22E*+σ22E*2+σ32I*2aV1+[(σ12-ρSR-μ)-c(αrN(ρSR+μ)-σ12)](S-S*)2aV1+(σ22-α2-ρSR-μ)(E-E*)2-(γ+μ-σ32-α2)(I-I*)2≐δ-[(1+αrS*N)(ρSR+μ)-σ12(1+1S*)](S-S*)2aV1-(-α2+ρSR+μ-σ22)(E-E*)2-(γ+μ-σ32-α2)(I-I*)2.
Then the ellipsoid(4.9)δ-[(1+αrS*N)(ρSR+μ)-σ12(1+1S*)](S-S*)2-(-α2+ρSR+μ-σ22)(E-E*)2=0
lies entirely in R+3. We can take U to be a neighborhood of the ellipsoid with U⊂R+3, so, for x∈U/R+3, LV≤-K (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 D⊂R+3. Besides, for  all D, there is an
(4.10)M=min⁡{σ12S2,σ22E2,σ32I2∈D}>0,
such that ∑i,j=13aijξiξj=σ12S2ξ12+σ22E2ξ22+σ32I2ξ32≥M∥ξ2∥ for all X∈D,ξ∈R3which 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,σ12=0.0006, σ22=0.001, σ32=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.Deterministic and stochastic trajectories around infection-free solution.
(a)
(b)
(c)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, σ12=0.0006, σ22=0.001, σ32=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.Deterministic and stochastic trajectories around virus endemic equilibrium.
(a)
(b)
(c)


## 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.



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*Source: 264874-2012-08-26.xml*