# Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs

Authors: Youness El Ansari; Ali El Myr; Lahcen Omari
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

We investigate the various conditions that control the extinction and stability of a nonlinear mathematical spread model with stochastic perturbations. This model describes the spread of viruses into an infected computer network which is powered by a system of antivirus software. The system is analyzed by using the stability theory of stochastic differential equations and the computer simulations. First, we study the global stability of the virus-free equilibrium state and the virus-epidemic equilibrium state. Furthermore, we use the Itô formula and some other theoretical theorems of stochastic differential equation to discuss the extinction and the stationary distribution of our system. The analysis gives a sufficient condition for the infection to be extinct (i.e., the number of viruses tends exponentially to zero). The ergodicity of the solution and the stationary distribution can be obtained if the basic reproduction numberRp is bigger than 1, and the intensities of stochastic fluctuations are small enough. Numerical simulations are carried out to illustrate the theoretical results.


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## Body

## 1. Introduction

A computer virus is a small malicious program that spreads from host to host and has a great ability to replicate itself into or over data files during the execution of programs. On a computer network, the interconnectivity of workstations makes the spread of viruses easier. In fact, an infected computer can spread viruses (malware) through the other connected nodes, which causes severe damage like reformatting the hard drive, unexplained data loss, slowing computer performance, corrupting databases, and so forth. Consequently, mathematics specialists are becoming more aware of the importance of protecting systems. To clean and protect the network, an antivirus is a computer program designed to identify and erase malicious software in the system.Similar to the biological virus systems, many mathematical models have been proposed to describe quantitatively the spread of infections into a computer network [1–9]. Following a deterministic approach, Mishra and Jha [1] have formulated and analyzed the effect of quarantine on recovered nodes. Yuan and Chen [2] discussed an e-SEIR network virus-epidemic model using the theory of stability in differential equations. Shukla et al. [3] modeled and studied the effect of antivirus software on infected computer network; they proved that, under certain conditions, the used antivirus can successfully clean the system.Other authors have followed the stochastic approach to describe the spread of viruses in a computer network [4, 5]. Zhang et al. [4] proposed and investigated a stochastic computer virus spread model. By constructing a suitable Lyapunov function, they established the necessary conditions for the virus-free equilibrium and viral equilibrium to be stabilized.To study the effect of antivirus software on its ability to clean an infected computer network, Shukla et al. [3] have proposed the following deterministic system: (1)dX=A-βXY-dX-kX+πYVadt,dY=βXY-dY-αY-πYVadt,dZ=kX-dZdt,dVa=μY-μ0Va-Va0dt; here X(t) denotes the number of susceptible nodes at time t≥0, Y(t) the number of infected nodes, Z(t) the number of protected nodes, and Va(t) the number of antivirus programs used to clean the network, which is considered to be proportional with the number of infected nodes. The total number of nodes in the network is N(t)=X(t)+Y(t)+Z(t). The significance of each parameter in the model is as follows: A is the inflow rate of susceptible nodes, β is the transition rate from X to Y resulting from the contact between susceptible and infected nodes, d is the rate at which nodes are crashed by another cause other than being attacked by viruses, k is the constant rate at which the nodes are being protected by another antivirus software, α is the rate at which infected nodes become incapable of infecting susceptible nodes, π is the rate at which infected nodes are recovered and become susceptible again, μ is the growth rate of antivirus software used to clean the network, μ0 is the rate by which it fails to work efficiently, and Va0 is the number of antivirus programs permanently present in the system to protect the existing software; all these parameters are assumed to be positive.In real life, the number of contacts of a susceptible node per unit time cannot always increase linearly withY, especially when the number of infectious nodes is large [6], so for more realism, we first suppose that the saturated incidence rate is nonlinear and takes the form βXY/f(Y). Model (1) becomes (2)dX=A-βXYfY-dX-kX+πYVadt,dY=βXYfY-dY-αY-πYVadt,dZ=kX-dZdt,dVa=μY-μ0Va-Va0dt, where f is a positive function satisfying f(0)=1 and f′(Y)≥0. Such function f enables the introduction of several effects, like the psychological effects (e.g., (Y/f(Y)=Y/(1+γY)) see [7]). Conforming to the biological epidemic models, this effect is manifested when the number of infected nodes is very large; the infection force may decrease as the number of infections Y increases. In relation to our model, in the presence of a very large number of infectious nodes, the users of computers in the network may tend to reduce the number of contacts of their computers.In the real world, the parameters of a compartmental model are always subject of random variability that affects the dynamic of the population. There exist several types of noise that may represent the environmental random variability; many studies showed that the white noise is an appropriate representation of environmental random variability in terrestrial systems (see Steele 1985 [10]; Vasseur and Yodzis 2004 [11]). For this reason, we introduce a Gaussian white noise disturbance into this model by considering the case where the parameters d and β are subject to random fluctuations. Hence, by using the technique of parameter perturbation, which represents the commonly used procedure in constructing SDE models (e.g., Zhang et al. [4], Ji et al. [12]), we replace d by d+σ1dB1 and β by β+σ2dB2, where B1(t) and B2(t) are standard one-dimensional independent Brownian motions and σ1 and σ2 their intensities. The other parameters are the same as in system (1). We obtain the following stochastic system:(3)dX=A-βXYfY-dX-kX+πYVadt-σ1XdB1-σ2XYfYdB2,dY=βXYfY-dY-αY-πYVadt-σ1YdB1+σ2XYfYdB2,dZ=kX-dZdt-σ1ZdB1,dVa=μY-μ0Va-Va0dt.The added value of this paper is summarized in the fact that presents a deterministic and stochastic study of a global model that takes into consideration the effect of antivirus programs in the spread of malware in a computer network. Also, it contains a global nonlinear saturated incidence rate which reflects more realism.The remainder of this paper is as follows: in Section2, we present some preliminary results related to the following study. We investigate in Section 3 the stability of the two equilibrium states of the deterministic system (2). In Section 4, we show that system (3) admits a global and unique positive solution, starting from the initial values (X(0),Y(0),Z(0),Va(0)). Next in Section 5, we give a sufficient condition for the system to be extinct. In Section 6, sufficient condition for the existence of a unique stationary distribution is obtained. We conclude in Section 7 by presenting some numerical simulations to illustrate our results.


## 2. Preliminaries

Throughout the rest of this paper, we let(Ω,F,P) be a complete probability space with a filtration (Ft)t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). Bi(t),(i=1,2) are defined on this complete probability space, we also denote (4)R+4=χ1,χ2,χ3,χ4χi>0,i=1,2,3,4.In general, consider thed dimensional stochastic differential equation (5)dχt=fχt,tdt+gχt,tdBt,fort≥0, with initial value χ(0)=x0∈Rd. B(t) denotes an n-dimensional standard Brownian motion defined on the complete probability space (Ω,F,{Ft}t≥0,P). We denote by (C2,1(Rd×[t0,∞],R+) the family of all nonnegative functions V(χ,t) defined on Rd×[t0,∞] such that they are continuously twice differentiable in χ and once in t. The differential operator L is defined in [13].LetX(t) be a homogeneous Markov process in Ed (Ed denotes the d-dimensional Euclidean space) and be described by the following stochastic differential equation: (6)dXt=bXdt+∑r=1kgrXdBrt.The diffusion matrix is defined as follows:(7)Aχ=aijχ,aij=∑r=1kgriχgrjχ.The following lemma is used to prove the theorem related to the stationary distribution for SDE (3) (see [14], Theorem  3.13, p. 1164, Remark  3.2, p. 1160, Theorem  4.3, p. 1168, and Theorem  4.4, p. 1169).Lemma 1.
System (6) is positive recurrent if there is a bounded open subset U of Rn with a regular boundary, and the following holds:(B1)
There exist somel=1,2,…,n and a positive constant ν such that (8)allx≥ν,foranyx∈U.(B2)
There exists a nonnegative functionV:Uc→R such that V is twice continuously differentiable and that for some θ>0(9)LV≤-θ,foranyx∈Uc.
Moreover, the positive recurrent processX(t) has a unique stationary distribution π(·) with density in Rn such that for any Borel set B∈Rn(10)limt→∞⁡Pt,x,B=πB,PχlimT→∞1T∫0TfXtdt=∫Rnfχtπdχ=1,for all χ∈Rn, where f:Rn→R is a function integrable with respect to the measure π.


## 3. Equilibrium Analysis and Global Stability of Equilibria

The basic reproduction number is the number of secondary infectious cases produced by an infectious individual during his or her effective infectious period when introduced in a population of susceptibles [15]. For model (2) the basic reproduction number is (11)Rp=βAd+kd+α+πVa0,where Rp acts as a sharp threshold between extinction and invasion of the disease.Proposition 2.
System (2) admits the virus-free equilibrium state E0(A/(d+k),0,kA/d(d+k),Va0), which exists for all parameters values, and a unique virus-epidemic equilibrium state E∗(X∗,Y∗,Z∗,Va∗) which exists if Rp>1.Proof.
System (2) has the virus-free equilibrium state E0(A/d+k,0,kA/d(d+k),Va0); this equilibrium is obtained by giving Y=0 (i.e., absence of infection). The positive virus-epidemic equilibrium state is the solution of the following system (equivalent to system (2), except Y≠0): (12)βX-fYd+α+πVa=0,X=A-d+αYd+k,Z=kdA-d+αYd+kdt,Va=μμ0Y+Va0.
SubstitutingX and Va by their expression in the first equation of system (12), we get the following equation of Y:(13)ΛY=βAd+k-βd+αd+kY-fYd+α+πμμ0Y+Va0=0,since f′≥0, then Λ is decreasing. Moreover, we have(14)ΛY<βAd+k-βd+αd+kY.
Thus(15)limY→+∞ΛY=-∞, and also, we have f(0)=1; then (16)Λ0=βAd+k-d+α+πVa0=d+α+πVa0Rp-1,
whereΛ is decreasing; then (15) and (16) imply that Λ has a unique positive zero if and only if Rp>1.The following theorem gives sufficient conditions for the global stability of equilibria of model (2).Theorem 3.
IfRp≤1, then the virus-free equilibrium state E0 is globally asymptotically stable. While if Rp>1, then the virus-epidemic equilibrium state E∗ is globally asymptotically stable.Proof.
Using the Lyapunov asymptotic theorem [16] to prove the global asymptotic stability of E0. Let us consider the function (17)V1=ω12Z-Z02+ω22X+Y-X02+ω32Va-Va02+ω4∫0Yfsds+ω52X+Y+Z-X0-Z02, where E0=(X0,Y0,Z0,Va0) is the virus-free equilibrium state determined in the precedent theorem. ω1, ω2, ω3, ω4, and ω5 are positive constants that will be determined later. We have(18)V1E0=0,V1X,Y,Z,Va>0∀X,Y,Z,Va≠E0,V1x⟶∞asx⟶∞,x∈R4.
The time derivative ofV1 along solutions of system (2) is (19)V˙1=ω1Z-Z0kX-dZ+ω2X+Y-X0A-d+kX-d+αY+ω3Va-Va0μY-μ0Va-Va0+ω4fYβXYfY-dY-αY-πYVa+ω5X+Y+Z-X0-Z0A-dX+Y+Z-αY.
AtE0, we have (20)A-dX0-kX0=0,kX0-dZ0=0.
Then(21)V˙1=ω1Z-Z0kX-X0-dZ-Z0+ω2X+Y-X0-d+kX-X0-d+αY+ω3Va-Va0μY-μ0Va-Va0+ω4βXY-ω4fYd+αY+πYVa+ω5X-X0+Z-Z0+Y-dX-X0-dZ-Z0-d+αY=-ω5dZ-Z02-ω3μ0Va-Va02-ω2d+k+ω5dX-X02-ω5d+αY2+ω1k-2ω5dZ-Z0X-X0+ω4β-ω22d+k+α-ω52d+αYX-X0-ω1dZ-Z02+ω52d+αYZ-Z0+ω2d+αY2+ω3μ-ω4πfYYVa-Yω3μVa0+ω4fYd+α-βω4X0.
Let(22)ω1=1d,ω5=ω1k2d,ω2=ω52d+α221d+α,ω4=ω22d+k+α+ω52d+αβ,ω3=ω4πμ.
Then(23)V˙1=-ω5dZ-Z02-ω3μ0Va-Va02-ω2d+k+ω5dX-X02-ω5d+αY2-Z-Z0+2ω52d+α2YZ-Z0+ω52d+α2Y2+ω4π1-fYYVa-ω4YπVa0+fYd+α-βX0.
We havef(Y(t))≥1 for all t≥0; then (24)V˙1≤-ω5dZ-Z02-ω3μ0Va-Va02-ω2d+k+ω5dX-X02-ω5d+αY2-Z-Z0+ω52d+α2Y2-ω4YπVa0+d+α-βAd+k.
Thus(25)V˙1≤-ω5dZ-Z02-ω3μ0Va-Va02-ω2d+k+ω5dX-X02-ω5d+αY2-ω4d+α+πVa0Y1-Rp.
Thus, ifRp≤1 then V˙1<0 for all (X,Y,Z,Va)≠E0. Hence by Lyapunov asymptotic theorem [16], E0 is globally asymptotically stable.
For the stability of the virus-epidemic equilibrium stateE∗, we refer the reader to Remark 13 in Section 6.Remark 4.
Theorem3 showed that the basic reproductive number Rp acts as a sharp threshold determining when the disease (i.e., spread of computer viruses) becomes endemic for model (2). Furthermore, the disease is permanent with simple dynamics whenever Rp>1.


## 4. Existence and Uniqueness of the Global Nonnegative Solution

The solution(X(t),Y(t),Z(t),Va(t)) of (3) presents the sizes of susceptible nodes, infected nodes, protected nodes, and antivirus programs used to clean the network at time t, respectively. Then, they should all be nonnegative. For this reason, and to study the dynamical behavior of system (3), the first concern is whether the solution is of global and positive existence.Theorem 5.
Let(X(0),Y(0),Z(0),Va(0))∈R+4; then there is a unique positive solution to SDE (3) on t≥0; this solution remains in R+4 with probability 1.Proof.
Since the coefficients of system (3) are locally Lipschitz continuous, for any given initial value (X(0),Y(0),Z(0),Va(0))∈R+4 then, there is a unique local solution (X(t),Y(t),Z(t),Va(t)) on t∈[0,τe), where τe is the explosion time [13]. We show next that τe=+∞ almost surely. For this purpose, we define the stopping time(26)τ∞=inf⁡t∈0,τe:Xt≤0orYt≤0orZt≤0orVat≤0.
Throughout this paper we setinf⁡∅=∞, where ∅ denotes the empty set. One can see that τ∞≤τe, if we prove that τ∞=∞ a.s.; then, τe=∞ a.s.; thus (X(t),Y(t),Z(t),Va(t))∈R+4 a.s. If this statement is false, then there is a pair of constants T>0 and ϵ∈(0,1) such that P(τ∞≤T)≥ϵ. We define a C2-function V for (X(t),Y(t),Z(t),Va(t))∈R+4 by V(X,Y,Z,Va)=ln⁡XYZVa. Setting Ω=(τ∞≤T) and using Itô’s formula we get, for all ω∈Ω and t∈[0,τ∞), (27)dVX,Y,Z,Va=LVX,Y,Z,Vadt-3σ1dB1+σ2X-YfYdB2,where (28)LVX,Y,Z,Va=1XA-βXYfY-dX-kX+πYVa-σ122-σ22Y22f2Y+1YβXYfY-dY-αY-πYVa-σ122-σ22X22f2Y+1ZkX-dZ-σ122+1VaμY-μ0Va-Va0≥-3d-k-α-μ0-3σ122-βYfY-σ222Y2+X2f2Y-πVa.
We know thatf is a positive function that verifies f(0)=1 and f′(Y)≥0 for all Y>0; then (29)LVX,Y,Z,Va≥-3d-k-α-μ0-3σ122-βY-σ222Y2+X2-πVa.
Therefore, we get(30)dVX,Y,Z,Va≥hX,Y,Vadt-3σ1dB1+σ2X-YfYdB2,where (31)hX,Y,Va=-3d-k-α-μ0-3σ122-βY-σ222Y2+X2-πVa.
Thereafter, we get(32)VX,Y,Z,Va≥VX0,Y0,Z0,Va0+∫0thXs,Ys,Vasds-3σ1B1t+∫0tσ2Xs-YsfYsdB2s.
Note that, forω∈Ω, there is at least one of X(τ∞,ω), Y(τ∞,ω), Z(τ∞,ω) and Va(τ∞,ω) that equal 0. Thereby(33)limt→τ∞⁡VXt,Yt,Zt,Vat=-∞.
Lettingt→∞, in (32), leads to the contradiction that (34)-∞≥VX0,Y0,Z0,Va0+∫0τ∞hXs,Ys,Vasds-3σ1B1τ∞+∫0τ∞σ2X-YfYdB2s>-∞.
Soτ∞=∞ a.s. This completes the proof of Theorem 5.


## 5. Extinction

One of the main interests of epidemiology is how to control the disease dynamics so that it will be eliminated in a long term. In this section, we shall establish sufficient conditions for the extinction of viruses in the stochastic model of computer network alimented by a system of antivirus software (3).Lemma 6.
Let(X(t),Y(t),Z(t),Va(t)) be the solution of system (3). Then with any initial value (X(0),Y(0),Z(0),Va(0))∈R+4, we have (35)limt→∞Xt+Ytt=0a.s.
Moreover(36)limt→∞Xtt=0a.s.,limt→∞Ytt=0a.s.Proof.
Letv(t)=X(t)+Y(t) and Φ(v)=(1+v)p, where p>0 will be chosen later. Using Itô’s formula, we get(37)dΦvt=LΦvdt+p1+vp-1-σ1XdB1-σ1YdB1,where (38)LΦv=p1+vp-1A-dv-kX-αY+pp-121+vp-2σ12v2=p1+vp-21+vA-dv-kX-αY+p-12σ12v2≤p1+vp-21+vA-dv+p-12σ12v2=p1+vp-2-d-p-12σ12v2+A-dv+A. Let p>0 such that d-(p-1/2)σ12≔λ>0; then (39)LΦv≤p1+vp-2-λv2+A-dv+A, so (40)dΦv≤p1+vp-2-λv2+A-dv+Adt-pσ1v1+vp-1dB1, and let 0<k<pλ; we have (41)dektΦvt=LektΦvdt-pσ1ektv1+vp-1dB1, and thus (42)EektΦvt=Φv0+E∫0tLeksΦvsds, where (43)LektΦvt=kektΦvt+ektLΦvt≤pekt1+vp-2kp1+v2-λv2+A-dv+A=pekt1+vp-2-λ-kpv2+A-d+2kpv+A+kp≤pektC; here(44)C=supv∈R+⁡1+vp-2-λ-kpv2+A-d+2kpv+A+kp;thus from (42)(45)Eekt1+vp≤1+v0p+pCkekt;therefore(46)lim⁡supt→∞⁡E1+vp≤pCk≕C0,a.s.;in addition to this result, v is continuous, which implies that there exists a constant M>0, such that (47)E1+vtp≤M,t≥0, with (47), we can proceed as in [8] to complete the proof.Lemma 7.
For any initial value(X(0),Y(0),Z(0),Va(0))∈R+4, the solution (X(t),Y(t),Z(t),Va(t)) of system (3) verifies (48)limt→∞∫0tXsdB1st=0,a.s.limt→∞∫0tYsdB1st=0,a.s.limt→∞∫0tXs/fYsdB2st=0a.s.Proof.
LetM1=∫0tX(s)/f(Y(s))dB2(s) and p>2. By using the Burkholder-Davis-Gundy inequality [13] and (47), we have (49)Esup0≤s≤tM1sp≤CpE∫0tX2sf2Ysdsp/2≤Cptp/2Esup0≤s≤tXpsfpYs≤MpCptp/2.
Letϵ be an arbitrary positive constant; then according to Chebyshev’s inequality we have (50)Pω:supkδ≤t≤k+1δM1tp>kδ1+ϵ+p/2≤MpCpk+1δp/2kδ1+ϵ+p/2≤2p/2MpCpkδ1+ϵ, an application of Doob’s martingale inequality, and the Borel-Cantelli lemma [13] gives, for almost all ω∈Ω and for all except finitely many k, (51)supkδ≤t≤k+1δM1tp≤kδ1+ϵ+p/2, and then, for almost ω∈Ω, there exists an integer k0(ω) such that, for all k≥k0, (51) holds. Thus, if k≥k0 and kδ≤t≤(k+1)δ, we get (52)ln⁡M1tpln⁡t≤1+ϵ+p/2ln⁡kδln⁡kδ≤1+ϵ+p2,a.s.
Consequently(53)lim⁡supt→+∞ln⁡M1tln⁡t≤1+ϵ+p/2p,a.s.and by letting ϵ→0, we obtain (54)lim⁡supt→+∞ln⁡M1tln⁡t≤1p+12,a.s.; therefore, for arbitrary small constant δ, such that 0<δ<1/2-1/p, there exists a constant K(ω) and a set Ωδ, such that P(Ωδ)≥1-δ and for all t≥K(ω), ω∈Ωδ(55)ln⁡M1t≤12+1p+δln⁡t; thus (56)lim⁡supt→+∞M1tt≤lim⁡supt→+∞t1/2+1/p+δt=0,a.s,
together withliminft→+∞M1t/t≥0,a.s., yields (57)limt→+∞M1tt=0,a.s.
So(58)limt→+∞M1tt=limt→+∞∫0tXs/fYsdB2st=0,a.s.
Takingf=1 and using the same arguments, we can show that (59)limt→∞∫0tXsdB1st=0,a.s.limt→∞∫0tYsdB1st=0,a.s.
This completes the proof of Lemma7.Theorem 8.
Let(X(t),Y(t),Z(t),Va(t)) be the solution of system (3) with any positive initial value (X(0),Y(0),Z(0),Va(0)); then (60)limt→∞ln⁡Yt≤-d+α+σ122-β22σ22,a.s.
Assume thatβ2<2σ22(d+α+σ12/2); then Y(t) tends to zero exponentially almost surely (i.e., the disease dies out with probability one).Proof.
By application of Itô’s formula to system (3) we get (61)dln⁡Yt=βXfY-d-α-πVa-σ122-σ22X22f2Ydt-σ1dB1+σ2XfYdB2, and then (62)ln⁡Y≤ln⁡Y0+∫0tβXsfYs-σ22X2s2f2Ys-πVasds-d+α+σ122t-σ1B1t+σ2M1t, where(63)M1t=∫0tXsfYsdB2s;then (64)ln⁡Yt≤ln⁡Y0-σ222∫0tXsfYs-βσ222ds+β22σ22t-d+α+σ122t-σ1B1t+σ2M1t≤ln⁡Y0-d+α+σ122-β22σ22t-σ1B1t+σ2M1t; therefore, by dividing both sides of (64) by t, we obtain (65)ln⁡Yt≤ln⁡Y0t-d+α+σ122-β22σ22-σ1B1tt+σ2M1tt.
Using the strong law of large numbers [13], we obtain limt→∞(σ1B1t/t)=0,a.s., and by (Lemma 7), we have (66)limt→∞σ2M1tt=0,a.s., taking the limit superior of both sides of (65) and the condition β2<2σ22(d+α+σ12/2) leads to (67)lim⁡supt→∞⁡ln⁡Yt≤-d+α+σ122-β22σ22<0,a.s.; thus limt→∞Y(t)=0,  a.s. This completes the proof of Theorem 8.Remark 9.
Theorem8 shows that the number of infected nodes Y(t) goes to extinction almost surely, where β2<2σ22(d+α+σ12/2). Namely, large white noise stochastic disturbances conduct to control the infection of nodes.


## 6. Stationary Distribution and Positive Recurrence

The following theorem investigates the stability in a stochastic sense; it gives a sufficient condition for the existence of an asymptotically invariant distribution for the solution of model (3). The proof is based on Lemma 1 and uses the Lyapunov function method. However, there exist some other methods to investigate the stationary distribution, like the method based on the Markov semigroups theory (see Rudnicki and Pichór [17], Lin et al. [18]). Furthermore, some other authors have given the explicit expression of the density of the stationary distribution through solving the Fokker Planck equation of their proposed model (see Buonocore et al. [19], Xu [20]).Theorem 10.
Consider the stochastic system (3) with positive initial value; suppose that Rp>1 and (68)0<γ<min⁡γ1X∗2,γ2Y∗2,γ3Z∗2,γ4Va∗2, where (69)γ1=a2d+k-2σ12+a5d-4σ12-a4σ22Y∗,γ2=a5d+α-4σ12-2a2σ12,γ3=a5d-4σ12-a1σ12,γ4=a3μ0,γ=2a2σ12X∗2+Y∗2+a1σ12Z∗2+a4σ22Y∗X∗2+a4σ12Y∗2+4a5σ12X∗2+Y∗2+Z∗2,(70)a1=1d,a5=k2d2,a2=2d+α24d+αa52,a4=a52d+α+a2α+2d+kfY∗β,a3=πμa4.
Then the solution(X,Y,Z,Va) is positive recurrent and admits a unique ergodic stationary distribution.Remark 11.
One can see that condition (68) can be verified for sufficiently small environmental noises σ1 and σ2. In fact, we have (71)limσ1,σ2→0γ1=a2d+k+a5d>0,limσ1,σ2→0γ2=a5d+α>0,limσ1,σ2→0γ3=a5d>0, while (72)limσ1,σ2→0γ=0.Proof.
We consider the Lyapunov functionV defined for (X,Y,Z,Va)∈R+4 by (73)VX,Y,Z,Va=a1V1Z+a2V2X,Y+a3V3Va+a4V4Y+a5V5X,Y,Z, where V1,V2,V3,V4, and V5 are positive functions defined for (X,Y,Z,Va)∈R+4 by (74)V1Z=12Z-Z∗2,V2X,Y=12X+Y-X∗-Y∗2,V3Va=12Va-Va∗2,V4Y=Y-Y∗-Y∗ln⁡YY∗,V5X,Y,Z=12X+Y+Z-X∗-Y∗-Z∗2, and a1, a2, a3, a4, and a5 are well defined in (70). By applying the differential operator L to V1, we obtain(75)LV1Z=Z-Z∗kX-dZ+σ122Z2.At the equilibrium state E∗(X∗,Y∗,Z∗,Va∗), we get kX∗-dZ∗=0; then(76)LV1Z=Z-Z∗kX-X∗-dZ-Z∗+σ122Z-Z∗+Z∗2.
An application of the inequality(x+y)2≤2(x2+y2) for all (x,y)∈R2 yields (77)LV1Z≤-d-σ12Z-Z∗2+kX-X∗Z-Z∗+σ12Z∗2.
For the functionV2, we have(78)LV2X,Y=X+Y-X∗-Y∗A-dX-kX-dY-αY+σ122X+Y2;at the equilibrium state E∗ we have A-dX∗-kX∗-dY∗-αY∗=0; hence (79)LV2X,Y≤X+Y-X∗-Y∗-d+kX-X∗-d+αY-Y∗+σ12X2+Y2≤-d+kX-X∗2-d+αY-Y∗2-2d+k+αX-X∗Y-Y∗+σ12X-X∗+X∗2+σ12Y-Y∗+Y∗2.
Therefore(80)LV2X,Y≤-d+k-2σ12X-X∗2-d+α-2σ12Y-Y∗2-2d+k+αX-X∗Y-Y∗+2σ12X∗2+2σ12Y∗2.
Now, we calculateLV3(81)LV3Va=Va-Va∗μY-μ0Va-Va0.
At the equilibrium stateE∗, we have μY∗-μ0(Va∗-Va0)=0; then (82)LV3Va=Va-Va∗μY-Y∗-μ0Va-Va∗;thus (83)LV3Va=-μ0Va-Va∗2+μVa-Va∗Y-Y∗.
ForLV4, we have (84)LV4Y=1-Y∗YβXYfY-dY-αY-πYVa+Y∗2σ12+σ22X2f2Y≤Y-Y∗βXfY-d-α-πVa+Y∗2σ12+σ22X2f2Y; we have at the equilibrium state E∗, βX∗/f(Y∗)-d-α-πVa∗=0; hence(85)LV4Y≤Y-Y∗βXfY-βX∗fY∗-πVa-Va∗+Y∗2σ12+σ22X2f2Y;the function f is chosen to verify that f is nondecreasing and f(Y)≥1; thus (86)LV4Y≤Y-Y∗βX1fY-1fY∗+βfY∗X-X∗-πVa-Va∗+Y∗2σ12+σ22X2f2Y≤Y-Y∗βfY∗X-X∗-πVa-Va∗+Y∗σ22X-X∗2+σ22Y∗X∗2+σ12Y∗2.
Thus(87)LV4Y≤Y∗σ22X-X∗2+βfY∗X-X∗Y-Y∗-πVa-Va∗Y-Y∗+σ22Y∗X∗2+σ12Y∗2.
Next, we calculateLV5(88)LV5X,Y,Z=X-X∗+Y-Y∗+Z-Z∗A-dX-dY-αY-dZ+σ122X+Y+Z2.
At the equilibrium stateE∗ we have A-dX∗-dY∗-αY∗-dZ∗=0. Using the inequality (x+y+z)2≤4(x2+y2+z2) for all (x,y,z)∈R3 we get (89)LV5X,Y,Z≤X-X∗+Y-Y∗+Z-Z∗-dX-X∗-d+αY-Y∗-dZ-Z∗+2σ12X2+Y2+Z2≤-dX-X∗2-d+αY-Y∗2-dZ-Z∗2-2d+αX-X∗Y-Y∗-2dX-X∗Z-Z∗-2d+αY-Y∗Z-Z∗+4σ12X-X∗2+Y-Y∗2+Z-Z∗2+4σ12X∗2+Y∗2+Z∗2.
So(90)LV5X,Y,Z≤-d-4σ12X-X∗2-d+α-4σ12Y-Y∗2-d-4σ12Z-Z∗2-2d+αX-X∗Y-Y∗-2dX-X∗Z-Z∗-2d+αY-Y∗Z-Z∗+4σ12X∗2+Y∗2+Z∗2.
Combining (77), (80), (83), (87), and (90) and multiplying by coefficients a1,a2,a3,a4, and a5 of (73) determined in (70), we get (91)LVX,Y,Z,Va≤-γ1X-X∗2-γ2Y-Y∗2-γ3Z-Z∗2-γ4V-Va∗2-γ5X-X∗Y-Y∗-γ6X-X∗Z-Z∗-γ7V-Va∗Y-Y∗-a1dZ-Z∗2+a2d+αY-Y∗2+a52d+αY-Y∗Z-Z∗+γ, where (92)γ5=a52d+α+a2α+2d+k-a4βfY∗,γ6=2a5d-ka1,γ7=-μa3+a4π, by substituting a1, a2, a3, a4, and a5 into γ5, γ6, and γ7, We can see that γ5=γ6=γ7=0; thus (93)LVX,Y,Z,Va≤-γ1X-X∗2-γ2Y-Y∗2-γ3Z-Z∗2-γ4V-Va∗2-Z-Z∗+a52d+α2Y-Y∗2+γ.
As a result(94)LVX,Y,Z,Va≤-γ1X-X∗2-γ2Y-Y∗2-γ3Z-Z∗2-γ4V-Va∗2+γ, as well, by letting condition (68) hold, and considering a positive constant γ′ such that(95)0<γ<γ′<min⁡γ1X∗2,γ2Y∗2,γ3Z∗2,γ4Va∗2,we have (96)LVX,Y,Z,Va≤-γ1X-X∗2-γ2Y-Y∗2-γ3Z-Z∗2-γ4V-Va∗2+γ′-γ′-γ; the ellipsoid (97)-γ1X-X∗2-γ2Y-Y∗2-γ3Z-Z∗2-γ4V-Va∗2+γ′, lies entirely in R+4. We can take as U any neighborhood of the ellipsoid such that U¯⊂R+4, where U¯ is closer to U. Thus, LV(X,Y,Z,Va)≤-(γ′-γ) for any (X,Y,Z,Va)∈R+4∖U; this implies that condition (B2) of Lemma 1 is satisfied. Condition (B1) of the same lemma remains to be verified. The diffusion matrix associated with system (3) is given by(98)MX,Y,Z=σ12X2+σ22X2Y2f2Yσ12XY-σ22X2Y2f2Yσ12XZ0σ12XY-σ22X2Y2f2Yσ12Y2+σ22X2Y2f2Yσ12YZ0σ12XZσ12YZσ12Z200000.
SinceU is a bounded open subset that lies entirely in R+4, then for all (X,Y,Z,Va)∈U(99)a33X,Y,Z,Va=σ12Z2≥minX,Y,Z,Va∈U¯⁡σ12Z2>ν,where ν is a positive constant; condition (B1) is met. Consequently we have shown conditions (B1) and (B2) of Lemma 1.Remark 12.
Theorem10 treats the long time behavior of (3). In the case Rp>1, and for sufficiently small environment noises, after some initial transients the solutions X(t), Y(t), V(t), and Va(t) fluctuate around the deterministic steady-state values X∗, Y∗, Z∗, and Va∗, respectively.Remark 13.
By takingσ1=σ2=0 and the same function (73) with the same parameters defined in (70), if Rp>1, one can easily follow the same proof below (B2) to show that (100)V˙X,Y,Z,Va≤-γ1X-X∗2-γ2Y-Y∗2-γ3Z-Z∗2-γ4V-Va∗2.
ThenV˙<0 for all (X,Y,Z,Va)≠E∗. Moreover, V(E∗)=0 and V(X,Y,Z,Va)>0 for all (X,Y,Z,Va)≠E∗ and V(x)→∞ as x→∞. Then, by Lyapunov asymptotic stability theorem [16], the virus-epidemic equilibrium state E∗ is globally asymptotically stable.


## 7. Discussions and Numerical Simulations

In this section, we present some numerical simulations to support our theoretical results. Throughout the following numerical simulations, we choosef(Y)=1+2Y2.We use Milstein’s higher-order method [21, 22] to derive the corresponding discretization equations of model (3).Example 14.
We choose the parameters in systems (2) and (3) as follows: A=10, β=0.45, d=0.5, k=0.3, π=0.4, α=0.4, μ=0.35, μ0=0.4, Va0=10, σ1=0.7, and σ2=0.8.Using the above parameters we haveR
p
=
1.15
>
1, which implies from Theorem 3 that model (2) stabilizes at the virus-epidemic equilibrium state E∗. Figure 1(b) illustrates this case.Figure 1
Computer simulation of the pathX(t),Y(t),Z(t),Va(t) for models (a) (3) and (b) (2), for parameters of Example 14, and (X(0),Y(0),Z(0),Va(0))=(2,4,3,5).
(a)
The stochastic system
(b)
The deterministic systemβ
2
=
0.2
<
2
σ
2
2
(
d
+
α
+
σ
1
2
/
2
)
=
1.47, (limt→∞((ln⁡Y)/t)≤-0.99 a.s.). Theorem 8 shows that the disease dies out under large white noise disturbances σ1 and σ2. The numerical simulation in Figure 1(a) supports this result; Y(t) represented by the green color tends to zero almost surely.Example 15.
We choose the parameters in systems (2) and (3) as follows: A=1, β=3, d=0.5, k=0.3, π=0.4, α=0.4, μ=0.35, μ0=0.4, Va0=0.5, σ1=0.08, and σ2=0.08.The parameters in Example15 are chosen to verify the conditions of Theorem 10. Using these parameters, we have Rp=3.41>1 and the intensities σ1 and σ2 are small enough. Thus, there exists a unique ergodic stationary distribution for system (3). The following simulations illustrate this result.The red lines represent the densities of each stochastic variableX(t),Y(t),Z(t), and Va(t), respectively, while the blue ones represent the densities of each deterministic variable. Based on these lines and their corresponding histograms, we can observe the existence of a stationary distribution. The result implies that small intensities of the Brownian motions will strengthen the stability of the system.


## 8. Conclusion

In the real world, problems are not deterministic, including stochastic effects into a model that may give more accuracy to investigate and study phenomena. In this paper, we have studied the stability of a deterministic model powered by a system of antivirus programs and with general nonlinear incidence rate. We have next considered a stochastic version of this system with noises introduced in the rate at which nodes are crashed due to reasons other than the attacks of virusesd and the transition rate β. We first proved the existence and positivity of a unique global solution of this model. Then, with this system of antivirus programs, we investigate the extinction of viruses, we have proved that, under large noise intensities σ1 and σ2, the number of infected nodes Y(t) tends to zero exponentially almost surely, and thus the disease dies out with probability one; Figure 1(a) illustrates this case. We have also proved that system (3) has the ergodic property as Rp>1 and the fluctuations are very small. Numerical simulations in Figure 2 are carried out to illustrate this theoretical result.Figure 2
The solution of stochastic model (3) and its histograms with parameters in Example 15 and initial values (X(0),Y(0),Z(0),Va(0))=(5,1,0.8,0.45).Also, it is interesting to investigate the stochastic persistence in mean. Furthermore, we may study a stochastic version of (3) including Markovian switching into all parameters [23]. These studies are in progress.



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*Source: 3540278-2017-10-18.xml*