A Stochastic Dynamic Model of Computer Viruses

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 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 𝜌_𝑆𝑅.(𝐻₅)𝐸 computers become 𝐼 computers with constant rate 𝛼; 𝐸 computers become 𝑅 computers with constant rate 𝜌_𝑆𝑅; 𝐼 computers become 𝑅 computers with constant rate 𝛾.

According to the above assumptions, the following model (see Figure 1) is derived: 𝑆̇(𝑡)=𝜇𝑁−𝛼𝑟𝑁𝐸(𝑡)𝑆(𝑡)−𝜌^𝑆𝑅𝐸̇(𝑡)=𝛼𝑟𝑆(𝑡)−𝜇𝑆(𝑡),𝑁𝐸(𝑡)𝑆(𝑡)−𝛼+𝜌^𝐸𝑅𝑅̇(𝑡)=𝜌𝐼̇_𝑆𝑅(𝑡)=𝛼𝐸(𝑡)−(𝛾+𝜇)𝐼+(𝑡)𝜇,𝐸(𝑡),𝑆(𝑡)+𝜌_𝐸𝑅𝐸(𝑡)−𝛾𝐼(𝑡)−𝜇𝑅(𝑡).(1.1)

Figure 1 

Notably, the first three equations in (1.1) do not depend on the fourth equation, since 𝑆̇(𝑡)+𝐸̇(𝑡)+𝐼̇(t)+𝑅̇(𝑡)=1. Therefore, the forth equation can be omitted and the model (1.1) can be rewritten as 𝑆̇(𝑡)=𝜇𝑁−𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁−𝜌^𝑆𝑅𝐸̇(𝑡)=𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑆(𝑡)−𝜇𝑆(𝑡),𝑁−𝛼+𝜌^𝐸𝑅𝐼̇(𝑡)=𝛼𝐸(𝑡)−(𝛾+𝜇)𝐼+(𝑡)𝜇.𝐸(𝑡),(1.2)

In [13], authors have proved the virus-free equilibrium EQ_vf=((𝜇/(𝜌_𝑆𝑅+𝜇))𝑁,0,0) is globally asymptotically stable if 𝑅₀=(𝛼𝑟𝜇/(𝛼𝑟𝜇)(𝜌_𝑆𝑅+𝜇))≤1, and the viral equilibrium EQ_ve is globally asymptotically stable if 𝑅₀>1, where EQ_ve=𝛼+𝜌^𝐸𝑅+𝜇𝛼𝑟𝑁,𝛼+𝜌𝜇𝑁_𝐸𝑅+𝜇−𝜌^𝐸𝑅+𝜇𝑁𝛼𝑟,𝛾+𝛼𝜇𝛼+𝜌𝜇𝑁_𝐸𝑅+𝜇−𝜌^𝐸𝑅+𝜇𝑁𝛼𝑟(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 𝜇→𝜇+𝜎₁𝐵̇₁(𝑡),𝜇→𝜇+𝜎₂𝐵̇₂(𝑡), and 𝜇→𝜇+𝜎₃𝐵̇₃(𝑡), where 𝐵̇₁(𝑡),𝐵̇₂(𝑡), and 𝐵̇₃(𝑡) are mutual independent standard Brownian motions with 𝐵₁(0)=0,𝐵₂(0)=0, and 𝐵₃(0)=0, and intensity of white noise 𝜎²¹≥0,𝜎²²≥0 and 𝜎²³≥0, respectively. Then the stochastic system is 𝑆̇(𝑡)=𝜇𝑁−𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑁−𝜌^𝑆𝑅𝐸̇(𝑡)=𝛼𝑟𝐸(𝑡)𝑆(𝑡)𝑆(𝑡)−𝜇𝑆(𝑡)−𝜎₁𝑆(𝑡)𝐵̇₁(𝑡),𝑁−𝛼+𝜌^𝐸𝑅𝐼̇(𝑡)=𝛼𝐸(𝑡)−(𝛾+𝜇)𝐼+(𝑡)𝜇−𝐸𝜎₃(𝑡)−𝜎₂𝐸(𝑡)𝐵̇₁(𝑡),𝐼(𝑡)𝐵̇₁(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 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 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑡)𝑑𝑡+𝑔(𝑥(𝑡),𝑡)𝑑𝐵(𝑡),on𝑡≥𝑡₀,(1.5) with the initial value 𝑥(𝑡₀)=𝑥₀∈𝑅^𝑛. 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=𝜕𝜕𝑥_𝑘+12^𝑛_𝑘,𝑗=1𝑔^𝑇(𝑥,𝑡)𝑔(𝑥,𝑡)𝜕²𝜕𝑥_𝑘𝜕𝑥_𝑗.(1.6) If 𝐿 acts on a function 𝑉, then 𝐿𝑉(𝑥,𝑡)=𝑉_𝑡(𝑥,𝑡)+𝑉_𝑥(𝑥,𝑡)𝑓(𝑥,𝑡)+12trace𝑔^𝑇(𝑥,𝑡)𝑉_𝑥𝑥(𝑥,𝑡)𝑔(𝑥,𝑡),(1.7) where 𝑉_𝑡=𝜕𝑉/𝜕_𝑡,𝑉_𝑥=(𝜕𝑉/𝜕𝑥₁,…,𝜕𝑉/𝜕𝑥_𝑛),𝑉_𝑥𝑥=(𝜕²𝑉/𝜕𝑥_𝑘𝜕𝑥_𝑘)_𝑛∗𝑛.

By Ito’s formula, if 𝑥(𝑡)∈𝑅^𝑛, 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 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,𝜏_𝑒), where 𝜏_𝑒 is the explosion time [2, 13]. To show this solution is global, we need to show that 𝜏_𝑒=∞ 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, 𝜏_𝑘=inf𝑡∈0,𝜏_𝑒∶𝑆(𝑡)∉1𝑘,𝑘or𝐸(𝑡)∉𝑘1,𝑘or𝐼(𝑡)∉𝑘1,𝑘,(2.1) where throughout this paper we set inf∅=∞ (as usual ∅ denotes the empty set). Clearly, 𝜏_𝑘 is increasing as 𝑘→∞. Set 𝜏_∞=lim_𝑘→∞𝜏_𝑘, whence 𝜏_∞≤𝜏_𝑒 a. s. If we can show that 𝜏_∞=∞ a. s., then 𝜏_𝑒=∞ 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 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 𝑋(𝑆,𝐸,𝐼)∈𝑅³⁺ by 𝑉(𝑋)=𝑆−𝑎−log𝑆𝑎+(𝐸−1−log𝐸)+(𝐼−1−log𝐼).(2.4) The nonnegativity of this function can be seen from 𝜇+1−log𝜇≥0, for  all 𝜇>0. Using Ito’s formula we get 𝑑𝑉(𝑋)=𝑎−𝑎𝑆𝑑𝑆+2𝑆𝑎²(𝑑𝑆)²+1−1𝐸𝑑𝐸+2𝐸1²(𝑑𝐸)²+1−1𝐼𝑑𝐼+21𝐼²(𝑑𝐼)²≐𝐿𝑉𝑑𝑡−𝜎₁(𝑆−𝑎)𝐵̇₁(𝑡)+𝜎₂(𝐸−1)𝐵̇₂(𝑡)+𝜎₃(𝐼−1)𝐵̇₃(𝑡),(2.5) where 𝐿𝑉=1−𝑎𝑆𝜇𝑁−𝛼𝑟𝐸(𝑁𝑡)𝑆(𝑡)−𝜌^𝑆𝑅𝑆(𝑡)−𝜇𝑆(𝑡)+𝑎𝜎²¹+1−12𝐸𝛼𝑟𝐸(𝑁𝑡)𝑆(𝑡)−𝛼+𝜌^𝑆𝑅+𝜇𝐸(𝑡)+𝜎²¹+1−12=𝜇𝑁+𝐼𝑎𝜌[𝛼_𝑆𝑅𝐸(𝑡)−(𝛾+𝜇)𝐼(𝑡)]+𝜎2²³+𝜇𝑎+𝛼+𝜌_𝑆𝑅+𝜇+𝛾+𝜇+𝑎𝜎²¹+𝛼𝑟𝑎2+𝜎2²²+𝜎2²³𝑁𝐸−𝜌^𝑆𝑅𝑆−𝜇𝑆−𝑎𝑆𝜇𝑁−𝜌^𝑆𝑅𝐸−𝜇𝐸−𝛼𝑟≤𝜇𝑁+𝑎𝜌_𝑆𝑅𝑁𝑆−𝛾𝐼−𝜇𝐼−𝐼𝛼𝐸+𝜇𝑎+𝛼+𝜌_𝑆𝑅+𝜇+𝛾+𝜇+𝑎𝜎²¹2+𝜎2²²+𝜎2²³+𝛼𝑁𝑟𝑎𝐸−𝜌^𝐸𝑅𝐸−𝜇𝐸.(2.6) By choosing 𝑎=(𝜌_𝐸𝑅+𝜇)𝑁/𝛼𝑟, then 𝐿𝑉≤𝜇𝑁+𝑎𝜌_𝑆𝑅+𝜇𝑎+𝛼+𝜌_𝑆𝑅+𝜇+𝛾+𝜇+𝑎𝜎²¹2+𝜎2²²+𝜎2²³≐𝑀̇.(2.7) Therefore, ^{𝜏𝑚0∧𝑇}𝑑𝑉(𝑋)𝐸≤𝑉𝑋^{𝜏𝑚0∧𝑇}𝜏𝑀_𝑚̇𝑑𝑡−^{𝜏𝑚0∧𝑇}𝜎₁(𝑆−𝑎)𝑑𝐵₁(𝑡)+𝜎₂(𝐸−1)𝑑𝐵₂(𝑡)+𝜎₃(𝐼−1)𝑑𝐵₃(𝑡),∧𝑇≤𝑉(𝑋(0))+𝐸^{𝜏𝑚0∧𝑇}𝑀̇𝑑𝑡≤𝑉(𝑋(0))+𝑀̇𝑇.(2.8) Setting Ω_𝑚={𝜏_𝑚≤𝑇} for 𝑚≥𝑚₁, then by (2.3), we know that 𝑃(Ω_𝑚)≥𝜀. Note that for every 𝜔∈Ω_𝑚, there is at least one of 𝑆(Ω_𝑚,𝜔), 𝐸(Ω_𝑚,𝜔), and 𝐼(Ω_𝑚,𝜔) that equals either 𝑚 or 1/𝑚. Then 𝑉𝑋𝜏_𝑚≥(𝑚−1−log𝑚)∧1𝑚−1+log𝑚∧𝑚−𝑎−𝑎log𝑚𝑎∧𝑚1−𝑎+𝑎log𝑎𝑚,(2.9) where 1Ω_𝑚(𝜔) is the indicator function of Ω_𝑚. Let 𝑚→∞ lead to the contradiction that ∞>𝑉(𝑋(0))+𝑀̇𝑇=∞. So 𝜏_∞=∞ is necessary. The proof of Theorem 2.1 is completed.

3. Stability of Virus-Free Equilibrium

It is clear that EQ_vf=(𝜇𝑁/(𝜌_𝑆𝑅+𝜇),0,0) is the virus-free equilibrium of system (1.3), which has been mentioned above, and EQ_vf 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 EQ_vf if the white noise is small.

Theorem 3.1. If 𝜌_𝑆𝑅+𝜇>𝜎²¹,3𝛼²+2𝜌_𝑆𝑅+2𝜇>𝜎²²,2𝛾+2𝜇−𝛼>𝜎²³ and 𝑅₀≤1, then the solution 𝑋(𝑡) of system (1.3) with initial value 𝑋(0)∈𝑅³⁺ has the property lim_𝑥→∞sup1𝑡𝐸^𝑡0(1+𝑏)𝜌^𝑆𝑅+𝜇−𝜎²¹𝜇²(𝑠)+12𝛼+𝜌^𝑆𝑅+𝜇−12𝜎²²𝜎²+𝛾+𝜇−𝛼(𝑠)2−12𝜎²³𝑤²(𝑠)𝑑𝑠≤(1−𝑏)𝜎²¹𝜌_𝑆𝑅𝜇+𝜇𝑁,(3.1) where 𝑏 is positive constants, defined as in the proof.

Proof. For simplicity, let 𝑢(𝑡)=𝑆(𝑡)−𝜇𝑁/(𝜌_𝑆𝑅+𝜇),𝑣(𝑡)=𝐸(𝑡),𝑤(𝑡)=𝐼(𝑡), system (1.3) can be written as 𝑢̇(𝑡)=−𝛼𝑟𝑣(𝑡)𝑁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁−𝜌^𝑆𝑅+𝜇𝑢(𝑡)−𝜎₁𝑣̇(𝑡)=𝛼𝑟𝑣(𝑡)𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁𝐵̇(𝑡),𝑁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁−𝛼+𝜌^𝑆𝑅𝑤̇(𝑡)=𝛼𝑣(𝑡)−(𝛾+𝜇)𝑤(𝑡)−𝜎₃+𝜇𝑣(𝑡)−𝜎₂𝑣(𝑡)𝐵̇(𝑡),𝑤(𝑡)𝐵̇₃(𝑡).(3.2) Let 𝑉(𝑥)=12(𝑢+𝑣)²+12𝑏𝑢²+12𝑏𝑣+12𝑤²=𝑉₁+𝑏𝑉₂+𝑏𝑉₃+𝑉₄,(3.3) then 𝑏 is positive constants to be determined later. By Ito’s formula, we compute 𝑑𝑉₁≐𝐿𝑉₁𝑑𝑡−(𝑢(𝑡)+𝑣(𝑡))𝜎₁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁𝐵̇(𝑡)+𝜎²𝐿𝑉₁𝑣(𝑡)𝐵̇₂(𝑡),=(𝑢(𝑡)+𝑣(𝑡))−𝜌_𝑆𝑅+𝜇𝑢(𝑡)−𝛼+𝜌_𝑆𝑅+𝜇𝑣(𝑡)+12𝜎²¹𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁²+12𝜎²²≤(𝑢(𝑡)+𝑣²𝑣(𝑡()𝑡))−𝜌_𝑆𝑅+𝜇𝑢(𝑡)−𝛼+𝜌_𝑆𝑅+𝜇𝑣(𝑡)+𝜎²¹𝑢²(𝑡)+𝜎²¹𝜌_𝑆𝑅𝜇+𝜇𝑁²+12𝜎²²=−𝜌𝑣²_𝑆𝑅(𝑡)+𝜇−𝜎²¹𝑢²(𝑡)+𝛼+𝜌_𝑆𝑅+𝜇−12𝜎²²𝑣²−𝜎₂₁(𝑡)+𝛼+2𝜌_𝑆𝑅+2𝜇𝑢(𝑡)𝑣(𝑡)𝜌_𝑆𝑅𝜇𝑑+𝑉₂𝜇𝑁²,≐𝐿𝑉₂𝑑𝑡−𝜎₁𝑢(𝑡)𝑢(𝑡)+𝜇𝐿𝑉₂𝜌_𝑆𝑅+𝜇𝐵̇(𝑡),=𝑢(𝑡)−𝛼𝑟𝑣(𝑡)𝑁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁−𝜌^𝑆𝑅+1+𝜇𝑢(𝑡)2𝜎²¹𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁²≤𝑢(𝑡)−𝛼𝑟𝑣(𝑡)𝑁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁−𝜌^𝑆𝑅+𝜎₂₁+𝜇𝑢(𝑡)+𝜎²¹𝑢²(𝑡)𝜌_𝑆𝑅𝜇+𝜇𝑁²=−𝜌_𝑆𝑅+𝜇−𝜎²¹𝑢²(𝑡)+𝛼𝑟𝜇𝜌_𝑆𝑅+𝜇𝑢(𝑡)𝑣(𝑡)+𝛼𝑁𝑟𝑣(𝑡)𝑢²+𝜎₂₁(𝑡)≤−𝜌_𝑆𝑅𝜌_𝑆𝑅𝜇+𝜇𝑁²+𝜇−𝜎²¹𝑢²(𝑡)+𝛼𝑟𝜇𝑑𝑉₃𝜌_𝑆𝑅+𝜇𝑢(𝑡)𝑣(𝑡)+𝜎²¹𝜌_𝑆𝑅𝜇+𝜇𝑁²,=𝛼𝑟𝑣(𝑡)𝑁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁𝑑𝑡−𝛼+𝜌^𝑆𝑅=𝛼𝑟+𝜇𝑣(𝑡)𝑑𝑡−𝜎₂𝑣(𝑡)𝐵̇(𝑡)𝑁𝑣(𝑡)𝑢(𝑡)+𝜌𝛼_𝑆𝑅𝑟+𝜇𝜇−𝛼+𝜌^𝑆𝑅≐𝐿𝑉₃+𝜇𝑣(𝑡)𝑑𝑡−𝜎₂𝑣(𝑡)𝐵̇(𝑡)𝑑𝑉₄−𝜎₂𝑣(𝑡)𝐵̇(𝑡),=𝑤(𝑡)(𝛼𝑣(𝑡)−(𝛾+𝜇)𝑤(𝑡))+12𝜎²³=𝛼𝑣(𝑡)𝑤(𝑡)−(𝛾+𝜇)𝑤₂𝑤²(𝑡)𝑑𝑡−𝜎₃𝑤²(𝑡)𝐵̇(𝑡)(𝑡)+12𝜎²³≤𝛼𝑤²(𝑡)𝑑𝑡−𝜎₃𝑤²(𝑡)𝐵̇(𝑡)2𝑣²(𝑡)+𝑤²(𝑡)−(𝛾+𝜇)𝑤²(𝑡)+12𝜎²³=𝛼𝑤²(𝑡)𝑑𝑡−𝜎₃𝑤²(𝑡)𝐵̇(𝑡)2−𝛾−𝜇+12𝜎²³𝑤²(𝑡)+𝛼2𝑣²≐𝐿𝑉₄(𝑡)𝑑𝑡−𝜎₃𝑤²(𝑡)𝐵̇(𝑡)𝐿𝑉=𝐿𝑉₁𝑑𝑡−𝜎₃𝑤²(𝑡)𝐵̇(𝑡),+𝑏𝐿𝑉₂+𝑏𝐿𝑉₃+𝐿𝑉₄=−𝜌_𝑆𝑅+𝜇−𝜎²¹𝑢²(𝑡)+𝛼+𝜌_𝑆𝑅+𝜇−12𝜎²²𝑣²+𝛼+2𝜌_𝑆𝑅(𝑡)+2𝜇𝑢(𝑡)𝑣(𝑡)−𝜎²¹−𝑏𝜌_𝑆𝑅𝜌_𝑆𝑅𝜇+𝜇𝑁²+𝜇−𝜎²¹𝑢²(𝑡)+𝛼𝑟𝜇+𝑏𝛼𝑟𝜌_𝑆𝑅+𝜇𝑢(𝑡)𝑣(𝑡)+𝜎²¹𝜌_𝑆𝑅𝜇+𝜇𝑁²𝑁𝑣(𝑡)𝑢(𝑡)+𝜌𝛼_𝑆𝑅𝑟+𝜇𝜇−𝛼+𝜌^𝑆𝑅+𝛼+𝜇𝑣(𝑡)2−𝛾−𝜇+21𝜎²³𝑤²(𝑡)+𝛼2𝑣²(𝑡).(3.4)
Choosing 𝑏=(𝑁(𝛼+2𝜌_𝑆𝑅+2𝜇)(𝜌_𝑆𝑅+𝜇))/(𝛼𝑟(𝜌_𝑆𝑅+𝜇−𝑁𝜇)), then we get 𝐿𝑉=−(1+𝑏)𝜌_𝑆𝑅+𝜇−𝜎²¹𝑢²(𝑡)−12𝛼+𝜌^𝑆𝑅+𝜇−12𝜎²²𝑣²(𝑡)−𝛾+𝜇−𝛼2−21𝜎²³𝑤²−𝑏𝛼+𝜌_𝑆𝑅(𝑡)+𝜇−𝛼𝑟𝜇𝜌_𝑆𝑅+𝜇𝑣(𝑡)+(1−𝑏)𝜎²¹𝜌_𝑆𝑅𝜇+𝜇²𝑑𝑉≤−(1+𝑏)𝜌_𝑆𝑅,+𝜇−𝜎²¹𝑢²(𝑡)−12𝛼+𝜌^𝑆𝑅+𝜇−12𝜎²²𝑣²(𝑡)−𝛾+𝜇−𝛼2−12𝜎²³𝑤²+(1−𝑏)𝜎₂₁(𝑡)𝜌_𝑆𝑅𝜇+𝜇𝑁²−(𝑢(𝑡)+𝑣(𝑡))𝜎₁𝑢(𝑡)+𝜌_𝑆𝑅𝜇+𝜇𝑁𝐵̇¹−𝜎₁(𝑡)+𝜎₂𝑣(𝑡)𝐵̇₂(𝑡)𝑢(𝑡)𝑢(𝑡)+𝜇𝜌_𝑆𝑅+𝜇𝑁𝐵̇(𝑡)−𝜎²𝑣(𝑡)𝐵̇₂(𝑡)−𝜎₃𝑤²(𝑡)𝐵̇₃(𝑡).(3.5) Integrating this from 0 to 𝑡 and taking the expectation, we have 𝐸[𝑉≤(𝑡−)]𝐸−𝑉_𝑡0(0)(1+𝑏)𝜌^𝑆𝑅+𝜇−𝜎²¹𝑢²(𝑠)+12𝛼+𝜌^𝑆𝑅+𝜇−12𝜎²²𝑣²+𝛾+𝜇−𝛼(𝑠)2−12𝜎²³𝑤²(𝑠)−(1−𝑏)𝜎²¹𝜌_𝑆𝑅𝜇+𝜇𝑁²𝑑𝑠.(3.6) Hence, lim_𝑥→∞sup1𝑡𝐸^𝑡0(1+𝑏)𝜌^𝑆𝑅+𝜇−𝜎²¹𝑢²(𝑠)+12𝛼+𝜌^𝑆𝑅+𝜇−12𝜎²²𝑣²+𝛾+𝜇−𝛼(𝑠)2−12𝜎²³𝑤²(𝑠)𝑑𝑠≤(1−𝑏)𝜎²¹𝜌_𝑆𝑅𝜇+𝜇𝑁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 𝐵̇₁(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⊂𝐸_𝑙 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 𝑥∈𝐸_𝑙/𝑈, the mean time 𝜏 at which a path issuing from 𝑥 reaches the set 𝑈 is finite, and sup_𝑥∈𝐾𝐸_𝑥𝜏<∞ for every compact subset 𝐾⊂𝐸_𝑙. 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 P_𝑥lim_𝑇→∞1𝑇^𝑇0𝑓(𝑋(𝑡))𝑑𝑡=_𝐸𝑙𝑓(𝑥)𝜇𝑑(𝑥)=1,∀𝑥∈𝐸_𝑙.(4.1)

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

Theorem 4.3. If𝜎²¹<(𝜌_𝑆𝑅+𝜇)(1+(𝛼𝑟/𝑆^∗𝑁))(𝑆^∗/(𝑆^∗−1)),𝜎²²<(𝛼/2)+𝜌_𝑆𝑅+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, Q_ve=(𝑆^∗,𝐸^∗,𝐼^∗) is the viral equilibrium of system.

Proof. When 𝑅₀>1, there is an viral equilibrium EQ_ve of system (1.3). Then 𝜇𝑁=𝛼𝑟𝐸^∗𝑆^∗𝑁−𝜌^𝑆𝑅𝑆^∗+𝜇𝑆^∗,𝛼𝑟𝐸^∗𝑆^∗𝑁=𝛼+𝜌^𝑆𝑅+𝜇𝐸^∗,𝛼𝐸^∗=(𝛾+𝜇)𝐼^∗.(4.2) Define 𝑉(𝑥)=𝑎𝑆−𝑆^∗−𝑆^∗log𝑆𝑆^∗+𝐸−𝐸^∗−𝐸^∗log𝐸𝐸^∗+𝐸−𝐸^∗−𝐸^∗log𝐸+1𝐸^∗2𝑆−𝑆^∗+𝐸−𝐸^∗²+12𝑐𝑆−𝑆^∗²+12𝐼−𝐼^∗²=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=1−𝑆^∗𝑆𝜇𝑁−𝛼𝑟𝑁𝐸𝑆−𝜌^𝑆𝑅+1−𝐸_∗𝑆−𝜇𝑆𝑑𝑡−𝜎₁𝑆𝐵̇₁(𝑡)𝐸𝛼𝑟𝑁𝐸𝑆−𝛼+𝜌^𝑆𝑅+1+𝜇𝐸𝑑𝑡−𝜎₂𝐸𝐵̇₂(𝑡)2𝑆^∗𝜎²¹𝑑𝑡+12𝐸^∗=𝐿𝑉₁𝜎²²𝑑𝑡𝑑𝑡−1−𝑆^∗𝑆𝜎¹𝑆(𝑡)𝐵̇₁(𝑡)−1−𝐸^∗𝐸𝜎²𝐸(𝑡)𝐵̇₂(𝑡),(4.4) where 𝐿𝑉₁=1−𝑆^∗𝑆𝜇𝑁−𝛼𝑟𝑁𝐸𝑆−𝜌^𝑆𝑅+1−𝐸_∗𝑆−𝜇𝑆𝑑𝑡𝐸𝛼𝑟𝑁𝐸𝑆−𝛼+𝜌^𝑆𝑅+1+𝜇𝐸𝑑𝑡2𝑆^∗𝜎²¹𝑑𝑡+12𝐸^∗=1−𝑆_∗𝜎²²𝑑𝑡𝑆𝛼𝑁𝑟𝐸^∗+1−𝐸_∗𝑆^∗−𝐸𝑆𝑑𝑡+𝜌_𝑆𝑅+𝜇𝑆^∗−𝑆𝑑𝑡𝐸𝛼𝑟𝑁𝐸𝑆−𝛼+𝜌^𝑆𝑅+1+𝜇𝐸𝑑𝑡2𝑆^∗𝜎²¹𝑑𝑡+12𝐸^∗=−𝑆−𝑆_∗𝜎²²𝑑𝑡²𝑆𝜌^𝑆𝑅+𝜇+𝛼𝑟𝑁𝐸^∗+𝛼𝑟𝑆^∗+𝐸^∗𝛼+𝜌_𝑆𝑅+𝜇𝑁𝑆^∗−𝛼−𝜌_𝑆𝑅−𝜇𝐸−𝛼𝑟𝑁𝐸^∗𝑆^∗−𝛼𝑟𝑁𝐸^∗𝑆^∗2+1𝑆2𝑆^∗𝜎²¹+12𝐸^∗𝜎²²≤−𝑆−𝑆^∗²𝑆𝜌^𝑆𝑅+𝜇12𝑆^∗𝜎²¹+12𝐸^∗𝑑𝑉₂𝜎²²,=1−𝐸^∗𝐸𝛼𝑟𝑁𝐸𝑆−𝛼+𝜌^𝑆𝑅+𝜇𝐸𝑑𝑡−𝜎₂𝐸𝐵̇₂(𝑡)+12𝐸^∗=𝐿𝑉₂𝜎²²𝑑𝑡𝑑𝑡−1−𝐸^∗𝐸𝜎²𝐸(𝑡)𝐵̇₂(𝑡).(4.5) Let 𝐵=(𝛼𝑟/𝑁)𝐸^∗𝑆^∗=(𝛼+𝜌_𝑆𝑅+𝜇)𝐸^∗ and 𝛼−1−log𝛼>0, for all 𝛼𝐿𝑉₂=1−𝐸^∗𝐸𝛼𝑟𝑁𝐸𝑆−𝛼+𝜌^𝑆𝑅+𝜇𝐸𝑑𝑡+12𝐸^∗=1−𝐸_∗𝜎²²𝑑𝑡𝐸𝐵𝐸𝐸^∗𝑆𝑆^∗−𝐵𝐸𝐸^∗+12𝐸^∗𝜎²²=𝐵𝐸𝑆𝐸^∗𝑆^∗−𝐵𝐸𝐸^∗+𝐵𝑆𝑆^∗+𝐵+12𝐸^∗𝜎²²≤𝐵𝐸𝑆𝐸^∗𝑆^∗−𝐸𝐸^∗−1+log𝑆𝑆^∗+1+12𝐸^∗𝜎²²≤𝐵𝐸𝑆𝐸^∗𝑆^∗−𝐸𝐸^∗+𝑆𝑆^∗−2+1+12𝐸^∗𝜎²²=𝐵𝐸𝐸^∗−1𝑆𝑆^∗−1+𝐵𝑆^∗𝑆+𝑆𝑆^∗−2+12𝐸^∗𝜎²²=𝛼𝑟𝑁𝐸−𝐸^∗𝑆−𝑆^∗+𝛼𝑁𝑟𝐸^∗𝑆−𝑆^∗²𝑆+21𝐸^∗𝑑𝑉₃𝜎²²,=−𝑆𝑆−−𝑆^∗𝑆+_∗𝐸−𝐸^∗𝜇𝑁−𝜌_𝑆𝑅+𝜇𝑆−𝛼+𝜌_𝑆𝑅+𝜇𝐸𝑑𝑡+𝐸−𝐸^∗𝜎₁𝑆𝐵̇₁(𝑡)+𝜎₂𝐸𝐵̇₂(𝑡)+𝜎²¹2𝑆²+𝜎²²𝐿𝑉₃2𝐸²𝑑𝑡,=+𝑆𝜎−𝑆^∗+𝐸−𝐸^∗−𝜌_𝑆𝑅+𝜇𝑆−𝑆^∗−𝛼+𝜌_𝑆𝑅+𝜇𝐸−𝐸^∗₂₁2𝑆²+𝜎²²2𝐸²≤−𝜌_𝑆𝑅+𝜇𝑆−𝑆^∗²−𝛼+𝜌_𝑆𝑅+𝜇𝐸−𝐸^∗²−+𝜎𝛼₂₁+𝜌_𝐸𝑅+𝜌_𝑆𝑅+2𝜇𝑆−𝑆^∗𝐸−𝐸^∗≤𝜎₂₁𝑆−𝑆^∗+𝑆^∗2+𝜎²²𝐸−𝐸^∗+𝐸^∗2𝑑𝑉₄−𝜌_𝑆𝑅−𝜇𝑆−𝑆^∗²+𝜎²²−𝛼−𝜌_𝐸𝑅−𝜇𝐸−𝐸^∗²+𝜎²¹𝑆^∗2+𝜎²²𝐸^∗2,=𝑆−𝑆^∗𝜇𝑁−𝛼𝑟𝑁𝐸𝑆−𝜌^𝑆𝑅+𝜇𝑆−𝑆𝑆−𝑆^∗𝜎₁𝐵̇(𝑡)+12𝜎²¹𝐿𝑉₄𝑆²,=𝑆−𝑆^∗𝜇𝑁−𝛼𝑟𝑁𝐸𝑆−𝜌^𝑆𝑅+𝜇𝑆+12𝜎²¹𝑆²=𝑆−𝑆^∗−𝛼𝑟𝑁𝐸𝑆−𝐸^∗𝑆^∗−𝜌_𝑆𝑅+𝜇𝑆−𝑆^∗+12𝜎²¹𝑆²=−𝛼𝑟𝑁𝑆^∗𝑆−𝑆^∗𝐸−𝐸^∗−𝛼𝑁𝑟𝑆−𝑆^∗²𝐸−𝛼𝑟𝑁𝜌^𝑆𝑅+𝜇𝑆−𝑆^∗²+12𝜎²¹𝑆²≤−𝛼𝑟𝑁𝑆^∗𝑆−𝑆^∗𝐸−𝐸^∗−𝛼𝑁𝑟𝜌^𝑆𝑅𝑑𝑉₅+𝜇−𝜎²¹𝑆−𝑆^∗²+𝜎²¹𝑆^∗2,=𝐼−𝐼^∗[𝛼𝐸−(𝛾+𝜇)𝐼]+𝜎²³2𝐼²𝐿𝑉₅=𝐿𝑉₅𝑑𝑡−𝜎₃𝐼𝐼−𝐼^∗𝐵̇₃,=𝐼−𝐼^∗[𝛼𝐸−(𝛾+𝜇)𝐼]+𝜎²³2𝐼²=𝐼−𝐼^∗𝛼𝐸−𝐸^∗−(𝛾+𝜇)𝐼−𝐼^∗+𝜎²³2𝐼²≤𝛼𝐸−𝐸^∗𝐼−𝐼^∗−𝛾+𝜇−𝜎²³𝐼−𝐼^∗²+𝜎²³𝐼^∗2≤𝛼2𝐸−𝐸^∗²−𝛾+𝜇−𝜎²³−𝛼2𝐼−𝐼^∗²+𝜎²³𝐼^∗2.(4.6)
Choosing 𝑎=(𝛼𝑟/(𝜌_𝑆𝑅+𝜇)𝑁)𝐸^∗, then 𝑎𝑉₁+𝑉₂=𝑎−𝑆−𝑆^∗²𝑆𝜌^𝑆𝑅+𝜇+12𝑆^∗𝜎²¹+12𝐸^∗𝜎²²+𝛼𝑟𝑁𝐸−𝐸^∗𝑆−𝑆^∗+𝛼𝑁𝑟𝐸^∗𝑆−𝑆^∗²𝑆+12𝐸^∗𝜎²²=𝛼𝑟𝑁𝐸−𝐸^∗𝑆−𝑆^∗+12𝑎𝑆^∗𝜎²¹+12𝐸^∗𝜎²².(4.7)
Choosing 𝑐=1/𝑆^∗, then 𝑎𝑉₁+𝑉₂+𝑉₃+𝑉₄+𝑉₅≤𝛼𝑟𝑁𝐸−𝐸^∗𝑆−𝑆^∗+12𝑎𝑆^∗𝜎²¹+12(𝑎+1)𝐸^∗𝜎²²++𝑐𝜎−²¹𝛼−𝑟𝜌_𝑆𝑅−𝜇𝑆−𝑆^∗²+𝜎²²−𝛼−𝜌_𝑆𝑅−𝜇𝐸−𝐸^∗²+𝜎²¹𝑆^∗2+𝜎²²𝐸^∗2𝑁𝑆^∗𝑆−𝑆^∗𝐸−𝐸^∗−𝛼𝑁𝑟𝜌^S𝑅+𝛼+𝜇−𝜎²¹𝑆−𝑆^∗²+𝜎²¹𝑆^∗22𝐸−𝐸^∗²−𝛾+𝜇−𝜎²³−𝛼2𝐼−𝐼^∗²=1+𝜎²³𝐼^∗22𝑎𝜎²¹𝑆^∗+𝜎²¹𝑆^∗+𝜎²¹𝑆^∗2+12(𝑎+1)𝜎²²𝐸^∗+𝜎²²𝐸^∗2+𝜎²³𝐼^∗2+𝜎²¹−𝜌_𝑆𝑅−𝜇−𝑐𝛼𝑟𝑁𝜌^𝑆𝑅+𝜇−𝜎²¹𝑆−𝑆^∗²+𝜎²²−𝛼2−𝜌^𝑆𝑅−𝜇𝐸−𝐸^∗²−𝛾+𝜇−𝜎²³−𝛼2𝐼−𝐼^∗²≐𝛿−1+𝛼𝑟𝑆^∗𝑁𝜌^𝑆𝑅+𝜇−𝜎²¹1+1𝑆^∗𝑆−𝑆^∗²−−𝛼2+𝜌^𝑆𝑅+𝜇−𝜎²²𝐸−𝐸^∗²−𝛾+𝜇−𝜎²³−𝛼2𝐼−𝐼^∗².(4.8)
Then the ellipsoid 𝛿−1+𝛼𝑟𝑆^∗𝑁𝜌^𝑆𝑅+𝜇−𝜎²¹1+1𝑆^∗𝑆−𝑆^∗²−−𝛼2+𝜌^𝑆𝑅+𝜇−𝜎²²𝐸−𝐸^∗²=0(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 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⊂𝑅³⁺. Besides, for  all 𝐷, there is an 𝑀=min𝜎²¹𝑆²,𝜎²²𝐸²,𝜎²³𝐼²∈𝐷>0,(4.10) such that ∑^3𝑖,𝑗=1𝑎_𝑖𝑗𝜉_𝑖𝜉_𝑗=𝜎²¹𝑆²𝜉²¹+𝜎²²𝐸²𝜉²²+𝜎²³𝐼²𝜉²³≥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, 𝜌_𝑆𝑅=1/2500, 𝜌_𝐸𝑅=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.

(a)

(b)

(c)

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, 𝜌_𝑆𝑅=1/2500, 𝜌_𝐸𝑅=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.

(a)

(b)

(c)

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).