A Stochastic Dynamic Model of Computer Viruses

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 264874, 16 pages doi:10.1155/2012/264874

Research Article A Stochastic Dynamic Model of Computer Viruses Chunming Zhang, Yun Zhao, Yingjiang Wu, and Shuwen Deng School of Information Engineering, Guangdong Medical College, Dongguan 523808, China Correspondence should be addressed to Yun Zhao, [email protected] Received 8 June 2012; Accepted 16 July 2012 Academic Editor: Bimal Kumar Mishra Copyright q 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. 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 μN.

2

Discrete Dynamics in Nature and Society ρER μN

αrE/N

S μ

α

E μ

γ

I μ

R I μ

ρSR

Figure 1

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: αr ˙ St μN − EtSt − ρSR St − μSt, N αr ˙ Et EtSt − α ρER μ Et, N ˙ αEt − γ μ It, It

1.1

˙ Rt ρSR St ρER Et − γIt − μRt. Notably, the first three equations in 1.1 do not depend on the fourth equation, since ˙ Et ˙ It ˙ Rt ˙ St 1. Therefore, the forth equation can be omitted and the model 1.1 can be rewritten as αrEtSt ˙ St μN − − ρSR St − μSt, N αrEtSt ˙ Et − α ρER μ Et, N ˙ αEt − γ μ It. It

1.2

In 13, authors have proved the virus-free equilibrium EQvf μ/ρSR μN, 0, 0 is globally asymptotically stable if R0 αrμ/αrμρSR μ ≤ 1, and the viral equilibrium EQve is globally asymptotically stable if R0 > 1, where EQve

ρER μ N ρER μ N αρER μ μN μN α N, , . − − αr αr γ μ αρER μ αr αρER μ 1.3

Discrete Dynamics in Nature and Society

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 μ → μ σ1 B˙ 1 t, μ → μ σ2 B˙ 2 t, and μ → μ σ3 B˙ 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 αrEtSt ˙ St μN − − ρSR St − μSt − σ1 StB˙ 1 t, N αrEtSt ˙ − α ρER μ Et − σ2 EtB˙ 1 t, Et N ˙ αEt − γ μ It − σ3 ItB˙ 1 t. It

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 dxt fxt, tdt gxt, tdBt,

on t ≥ t0 ,

1.5

with the initial value xt0 x0 ∈ Rn . Bt denotes n-dimensional standard Brownian motion defined on the above probability space. Define the differential operator L associated with 1.4 by L

n

∂2 1 ∂ g T x, tgx, t . ∂xk 2 k,j1 ∂xk ∂xj

1.6

If L acts on a function V , then

1 LV x, t Vt x, t Vx x, tfx, t trace g T x, tVxx x, tgx, t , 2

1.7

where Vt ∂V/∂t , Vx ∂V/∂x1 , . . . , ∂V/∂xn , Vxx ∂2 V/∂xk ∂xk n∗n . By Ito’s formula, if xt ∈ Rn , then for 1.4, assume that f0, t 0, g0, t 0 for all t ≥ t0 . So xt ≡ 0 is a solution of 1.4, called the trivial solution or equilibrium position.

4


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 St, Et, It on t ≥ 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 St, Et, It 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,

1 1 1 , k or Et ∈ / , k or It ∈ / ,k , τk inf t ∈ 0, τe : St ∈ / k 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∞ limk → ∞ τk , whence τ∞ ≤ τe a. s. If we can show that τ∞ ∞ a. s., then τe ∞ and St, Et, It 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 ε ∈ 0, 1 such that P τ∞ ≤ T > ε.

2.2

Hence, there is an integer k1 ≥ k0 such that P {τ∞ ≤ T } > ε

∀k > k1 .

2.3

Define a C2 -function V for XS, E, I ∈ R3 by V X

S S − a − log a

E − 1 − log E I − 1 − log I .

2.4

The nonnegativity of this function can be seen from μ 1 − log μ ≥ 0, for all μ > 0. Using Ito’s formula we get

a a 1 1 1 1 2 dS 2 dS2 1 − dE dI 2 dI2 1 − dE S E I 2S 2E2 2I 2.5 . ˙ ˙ ˙ LV dt − σ1 S − aB1 t σ2 E − 1B2 t σ3 I − 1B3 t ,

dV X

a−


5

where

aσ 2 a αrEtSt − ρSR St − μSt 1 LV 1 − μN − S N 2

σ2 αrEtSt 1 − α ρSR μ Et 1 1− E N 2

σ2 1 αEt − γ μ It 3 1− I 2 aσ12 σ22 σ32 μN aρSR μa α ρSR μ γ μ 2 2 2 α a αr αra E − ρSR S − μS − μN − ρSR E − μE − S − γI − μI − E N S N I

≤ μN aρSR μa α ρSR μ γ μ

2.6

aσ12 σ22 σ32 αra E − ρER E − μE. 2 2 2 N

By choosing a ρER μN/αr, then

LV ≤ μN aρSR μa α ρSR μ γ μ

aσ12 σ22 σ32 . ˙ M. 2 2 2

2.7

Therefore, τm ∧T 0

dV X ≤

τm ∧T

˙ Mdt −

τm ∧T

0

σ1 S − adB1 t σ2 E − 1dB2 t σ3 I − 1dB3 t,

0

EV Xτm ∧ T ≤ V X0 E

τ ∧T m

˙ ˙ Mdt ≤ V X0 MT.

0

2.8 Setting Ωm {τm ≤ T } for m ≥ m1 , then by 2.3, we know that P Ωm ≥ ε. Note that for every ω ∈ Ωm , there is at least one of SΩm , ω, EΩm , ω, and IΩm , ω that equals either m or 1/m. Then V Xτm ≥ m − 1 − log m ∧

m 1 1 − 1 log m ∧ m − a − a log ∧ − a a log am , m a m 2.9

where 1Ωm ω is the indicator function of Ωm . Let m → ∞ lead to the contradiction that ˙ ∞. So τ∞ ∞ is necessary. The proof of Theorem 2.1 is completed. ∞ > V X0 MT

6


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 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 ρSR μ > σ12 , 3α2 2ρSR 2μ > σ22 , 2γ 2μ − α > σ32 and R0 ≤ 1, then the solution Xt of system 1.3 with initial value X0 ∈ R3 has the property

1 lim sup E x→∞ t

t 0

1 1 α ρSR μ − σ22 σ 2 s 1 b ρSR μ − σ12 μ2 s 2 2

μ α 1 N , γ μ − − σ32 w2 s ds ≤ 1 − bσ12 2 2 ρSR μ

3.1

where b is positive constants, defined as in the proof. Proof. For simplicity, let ut St − μN/ρSR μ, vt Et, wt It, system 1.3 can be written as

μ μ αrvt ˙ N − ρSR μ ut − σ1 ut N Bt, ut N ρSR μ ρSR μ μ αrvt ˙ vt ˙ ut N − α ρSR μ vt − σ2 vtBt, N ρSR μ wt ˙ αvt − γ μ wt − σ3 wtB˙ 3 t.

ut ˙ −

3.2

Let

V x

1 1 1 1 u v2 bu2 bv w2 2 2 2 2

V1 bV2 bV3 V4 ,

then b is positive constants to be determined later. By Ito’s formula, we compute . dV1 LV1 dt − ut vt σ1 ut

μ ˙ ˙ N Bt σ2 vtB2 t , ρSR μ

3.3


7

2 1 μ N LV1 ut vt − ρSR μ ut − α ρSR μ vt σ12 ut 2 ρSR μ 1 σ22 v2 t 2 ≤ ut vt − ρSR μ ut − α ρSR μ vt σ12 u2 t σ12

μ N ρSR μ

2

1 σ22 v2 t 2

1 2 2 2 2 − ρSR μ − σ1 u t α ρSR μ − σ2 v t α 2ρSR 2μ utvt 2 −σ12

μ N ρSR μ

2 ,

. dV2 LV2 dt − σ1 ut ut

μ ˙ Bt, ρSR μ

μ αrvt LV2 ut − N − ρSR μ ut ut N ρSR μ

2 μ 1 N σ12 ut 2 ρSR μ μ αrvt N − ρSR μ ut σ12 u2 t ut ≤ ut − N ρSR μ

−

σ12

μ N ρSR μ

ρSR μ − σ12 u2 t

σ12

≤−

2

μ N ρSR μ

ρSR μ −

σ12

αrμ αr utvt vtu2 t ρSR μ N

2

2 μ αrμ 2 , utvt σ1 N u t ρSR μ ρSR μ

2

μ αrvt ˙ ut N dt − α ρSR μ vtdt − σ2 vtBt N ρSR μ

αrμ αr ˙ vtut − α ρSR μ vt dt − σ2 vtBt N ρSR μ . ˙ LV3 − σ2 vtBt,

dV3

8

Discrete Dynamics in Nature and Society 1 ˙ dV4 wt αvt − γ μ wt σ32 w2 t dt − σ3 w2 tBt 2 1 ˙ αvtwt − γ μ w2 t σ32 w2 t dt − σ3 w2 tBt 2 2 α 2 1 2 2 2 ˙ v t w t − γ μ w t σ3 w t dt − σ3 w2 tBt ≤ 2 2

α 1 2 α 2 2 ˙ − γ − μ σ3 w t v t dt − σ3 w2 tBt 2 2 2 . ˙ LV4 dt − σ3 w2 tBt, LV LV1 bLV2 bLV3 LV4

1 − ρSR μ − σ12 u2 t α ρSR μ − σ22 v2 t 2

2 μ 2 N α 2ρSR 2μ utvt − σ1 ρSR μ

2 μ αrμ 2 2 2 utvt σ1 N − b ρSR μ − σ1 u t ρSR μ ρSR μ

αrμ αr vtut − α ρSR μ vt b N ρSR μ

1 2 α α 2 2 − γ − μ σ3 w t v t . 2 2 2 3.4

Choosing b Nα 2ρSR 2μρSR μ/αrρSR μ − Nμ, then we get

1 1 α 1 LV −1 b ρSR μ − σ12 u2 t − α ρSR μ − σ22 v2 t − γ μ − − σ32 w2 t 2 2 2 2

2 αrμ μ vt 1 − bσ12 , − b α ρSR μ − ρSR μ ρSR μ

1 2 2 1 α 1 2 2 2 α ρSR μ − σ2 v t − γ μ − − σ3 w2 t dV ≤ −1 b ρSR μ − σ1 u t − 2 2 2 2

2

μ μ 1 − bσ12 N − ut vt σ1 ut N B˙ 1 t σ2 vtB˙ 2 t ρSR μ ρSR μ

μ ˙ − σ2 vtB˙ 2 t − σ3 w2 tB˙ 3 t. N Bt − σ1 ut ut ρSR μ 3.5


9

Integrating this from 0 to t and taking the expectation, we have EV t − V 0 t ≤ −E

0

1 2 2 1 2 2 α ρSR μ − σ2 v s 1 b ρSR μ − σ1 u s 2 2

2

μ α 1 2 2 2 ds. N γ μ − − σ3 w s − 1 − bσ1 2 2 ρSR μ

3.6

Hence, 1 lim sup E x→∞ t

t 0

1 1 α ρSR μ − σ22 v2 s 1 b ρSR μ − σ12 u2 s 2 2

2

μ α 1 2 2 2 γ μ − − σ3 w s ds ≤ 1 − bσ1 N . 2 2 ρSR μ

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 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. Assumption B: there exists a bounded domain U ⊂ El 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 Ax is bounded away from zero. B.2 If x ∈ El /U, the mean time τ at which a path issuing from x reaches the set U is finite, and supx∈K Ex τ < ∞ for every compact subset K ⊂ El . If B holds, then the Markov process Xt has a stationary distribution μ•. Let f• be a function integrable with respect to the measure μ. Then Px

1 lim T →∞T

T 0

fXtdt

fxμdx El

1,

∀x ∈ El .

4.1

10


Lemma 4.2 see 14, 15. Let Xt be a regular temporally homogeneous Markov process in El . If Xt is recurrent relative to some bounded domain U, then it is recurrent relative to any nonempty domain in El . Theorem 4.3. If σ 21 < ρSR μ1αr/S∗ NS∗ /S∗ −1, σ22 < α/2ρSR μσ32 < γ μ−α/2, and R0 > 1, then, for any initial value X0 ∈ 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

αrE∗ S∗ − ρSR S∗ μS∗ , N

αrE∗ S∗ α ρSR μ E∗ , N

αE∗ γ μ I ∗ .

4.2

Define

S E E V x a S − S∗ − S∗ log ∗ E − E∗ − E∗ log ∗ E − E∗ − E∗ log ∗ S E E 1 1 1 S − S∗ E − E∗ 2 cS − S∗ 2 I − I ∗ 2 2 2 2

4.3

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 dV1

αrES S∗ μN − − ρSR S − μS dt − σ1 SB˙ 1 t 1− S N

∗ αrES E − α ρSR μ E dt − σ2 EB˙ 2 t 1− E N

1 1 S∗ σ12 dt E∗ σ22 dt 2 2

S∗ E∗ LV1 dt − 1 − σ1 StB˙ 1 t − 1 − σ2 EtB˙ 2 t, S E where

αrES − ρSR S − μS dt N

αrES E∗ − α ρSR μ E dt 1− E N

LV1

1−

S∗ S

μN −

1 1 S∗ σ12 dt E∗ σ22 dt 2 2

S∗ αr ∗ ∗ 1− E S − ESdt ρSR μ S∗ − Sdt S N

4.4


E∗ αrES 1− − α ρSR μ E dt E N

11

1 1 S∗ σ12 dt E∗ σ22 dt 2 2 αr ∗ ∗ S − S∗ 2 ρSR μ E S E∗ α ρSR μ S N αr αr ∗ ∗ αr E∗ S∗2 S∗ − α − ρSR − μ E − E S − N N N S

−

1 1 S∗ σ12 E∗ σ22 2 2 1 1 S − S∗ 2 ρSR μ S∗ σ12 E∗ σ22 , S 2 2

∗ E 1 αrES dV2 1 − − α ρSR μ E dt − σ2 EB˙ 2 t E∗ σ22 dt E N 2

E∗ LV2 dt − 1 − σ2 EtB˙ 2 t. E ≤−

4.5 Let B αr/NE∗ S∗ α ρSR μE∗ and α − 1 − log α > 0, for all α

E∗ 1 αrES − α ρSR μ E dt E∗ σ22 dt 1− E N 2

E∗ E 1 ES 1− B ∗ ∗ − B ∗ E∗ σ22 E E S E 2

ES E S 1 B ∗ ∗ − B ∗ B ∗ B E∗ σ22 E S E S 2

ES E S 1 ≤ B ∗ ∗ − ∗ − 1 log ∗ 1 E∗ σ22 E S E S 2

ES E S 1 ≤ B ∗ ∗ − ∗ − 2 1 E∗ σ22 ∗ E S E S 2

∗

E S S S − 2 1 E∗ σ 2 B − 1 − 1 B 2 E∗ S∗ S S∗ 2

LV2

αr αr ∗ S − S∗ 2 1 ∗ 2 E E σ2 , E − E∗ S − S∗ N N S 2 dV3 S − S∗ E − E∗ μN − ρSR μ S − α ρSR μ E dt

− S − S E − E σ1 SB˙ 1 t σ2 EB˙ 2 t ∗

∗

σ12 2 σ22 2 S E dt, 2 2

12

Discrete Dynamics in Nature and Society LV3 S − S∗ E − E∗ − ρSR μ S − S∗ − α ρSR μ E − E∗ σ12 2 σ22 2 S E 2 2 ≤ − ρSR μ S − S∗ 2 − α ρSR μ E − E∗ 2 − α ρER ρSR 2μ S − S∗ E − E∗

σ12 S − S∗ S∗2 σ22 E − E∗ E∗2

≤ σ12 − ρSR − μ S − S∗ 2 σ22 − α − ρER − μ E − E∗ 2 σ12 S∗2 σ22 E∗2 , αr ˙ 1 σ 2 S2 , ES − ρSR μ S − SS − S∗ σ1 Bt dV4 S − S∗ μN − N 2 1 1 αr ES − ρSR μ S σ12 S2 LV4 S − S∗ μN − N 2 αr

1 S − S∗ − ES − E∗ S∗ − ρSR μ S − S∗ σ12 S2 N 2 αr ∗ αr 1 αr S S − S∗ E − E∗ − ρSR μ S − S∗ 2 σ12 S2 S − S∗ 2 E − N N N 2 αr αr ∗ ≤ − S S − S∗ E − E∗ − ρSR μ − σ12 S − S∗ 2 σ12 S∗2 , N N −

σ2 dV5 I − I ∗ αE − γ μ I 3 I 2 2 LV5 dt − σ3 II − I ∗ B˙ 3 , σ2 LV5 I − I ∗ αE − γ μ I 3 I 2 2 σ2 I − I ∗ αE − E∗ − γ μ I − I ∗ 3 I 2 2 ∗ ∗ 2 ∗ 2 ≤ αE − E I − I − γ μ − σ3 I − I σ32 I ∗2 α α ≤ E − E∗ 2 − γ μ − σ32 − I − I ∗ 2 σ32 I ∗2 . 2 2 4.6 Choosing a αr/ρSR μNE∗ , then 1 ∗ 2 1 ∗ 2 S − S∗ 2 aV1 V2 a − ρSR μ S σ1 E σ2 S 2 2 αr ∗ S − S∗ 2 1 ∗ 2 αr ∗ ∗ E E σ2 E − E S − S N N S 2

αr 1 1 E − E∗ S − S∗ aS∗ σ12 E∗ σ22 . N 2 2

4.7


13

Choosing c 1/S∗ , then aV1 V2 V3 V4 V5 αr 1 1 E − E∗ S − S∗ aS∗ σ12 a 1E∗ σ22 N 2 2

σ12 − ρSR − μ S − S∗ 2 σ22 − α − ρSR − μ E − E∗ 2 σ12 S∗2 σ22 E∗2

≤

αr

αr c − S∗ S − S∗ E − E∗ − ρSR μ − σ12 S − S∗ 2 σ12 S∗2 N N

α α E − E∗ 2 − γ μ − σ32 − I − I ∗ 2 σ32 I ∗2 2 2 1 2 ∗ 1 aσ S σ12 S∗ σ12 S∗2 a 1σ22 E∗ σ22 E∗2 σ32 I ∗2 2 1 2

αr σ12 − ρSR − μ − c ρSR μ − σ12 S − S∗ 2 N α α σ22 − − ρSR − μ E − E∗ 2 − γ μ − σ32 − I − I ∗ 2 2 2

1 αr . 2 ρSR μ − σ1 1 ∗ S − S∗ 2 δ− 1 ∗ SN S α α − − ρSR μ − σ22 E − E∗ 2 − γ μ − σ32 − I − I ∗ 2 . 2 2

4.8

Then the ellipsoid δ−

1

α αr 1 2 ρ μ − σ 1 S − S∗ 2 − − ρSR μ − σ22 E − E∗ 2 0 SR 1 ∗ ∗ SN S 2 4.9

lies entirely in R3 . We can take U to be a neighborhood of the ellipsoid with U ⊂ R3 , so, for x ∈ U/R3 , LV ≤ −K K is a positive constant, which implies that condition B.2 in Lemma 4.1 is satisfied. Hence, the solution Xt is recurrent in the domain U, which, together with Lemma 4.2, implies that Xt is recurrent in any bounded domain D ⊂ R3 . Besides, for all D, there is an M min σ12 S2 , σ22 E2 , σ32 I 2 ∈ D > 0,

4.10

such that 3i,j1 aij ξi ξj σ12 S2 ξ12 σ22 E2 ξ22 σ32 I 2 ξ32 ≥ M ξ2 for all X ∈ D, ξ ∈ R3 which 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

14

Discrete Dynamics in Nature and Society ×104 3.8

0.12

3.7

0.1

3.6 3.5

0.08

3.4

0.06

3.3 3.2

0.04

3.1

0.02

3 2.9

0

0.5

1

1.5

0

2 ×105

0

0.5

a

1

1.5

2 ×105

b

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.5

1

1.5

2 ×105

c

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

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 2a, 2b, and 2c, we have displayed, respectively, the susceptible, infected and recovered computer of system 1.4 with initial conditions: S0 3, E0 0.1 and I0 0.1. 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 3a, 3b, and 3c, we have displayed, respectively, the susceptible and infected population of system 1.4 with initial conditions: S0 15000, E0 2000 and I0 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


15

×104 1.6

5000

1.5

4500

1.4 1.3

4000

1.2

3500

1.1 1

3000

0.9

2500

0.8 0.7

0

0.5

1

1.5

2 ×105

2000

0

0.5

a

1

1.5

2 ×105

b

3600 3400 3200 3000 2800 2600 2400 2200 2000 1800

0

0.5

1

1.5

2 ×105

c

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

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.

References 1 J. O. Kephart and S. R. White, “Directed-graph epidemiological models of computer viruses,” in Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343–359, May 1991. 2 J. O. Kephart, S. R. White, and D. M. Chess, “Computers and epidemiology,” IEEE Spectrum, vol. 30, no. 5, pp. 20–26, 1993.

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