A Stochastic Dynamic Model of Computer Viruses

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.


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.
H 1 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.
H 2 New computers are attached to the computer network with rate μN.

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.

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 R 0 ≤ 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 R 0 > 1.Some numerical simulations are performed in Section 5.In Section 6, a brief conclusion is given. .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 ≥ t 0 .So x t ≡ 0 is a solution of 1.4 , called the trivial solution or equilibrium position.

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 S 0 , E 0 , I 0 ∈ 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 S 0 , E 0 , I 0 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 k 0 > 0 be sufficiently large so that every component of x 0 lies within the interval 1/k 0 , k 0 .For each integer k ≥ k 0 , define the stopping time, where throughout this paper we set inf ∅ ∞ as usual ∅ denotes the empty set .Clearly, 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 Hence, there is an integer k 1 ≥ k 0 such that The nonnegativity of this function can be seen from μ 1 − log μ ≥ 0, for all μ > 0. Using Ito's formula we get where

2.8
Setting Ω m {τ m ≤ T } for m ≥ m 1 , 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 where

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 R 0 ≤ 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.
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 then b is positive constants to be determined later.By Ito's formula, we compute

3.5
Integrating this from 0 to t and taking the expectation, we have

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 σ 2 1 , 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.

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. < γ μ− α/2 , and R 0 > 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, Q ve S * , E * , I * is the viral equilibrium of system.
Proof.When R 0 > 1, there is an viral equilibrium EQ ve of system 1.3 .Then where a, c, are positive constants to be determined later.Then V is positive definite.By Ito's formula, we compute where Choosing a αr/ ρ SR μ N E * , then

4.7
Discrete Dynamics in Nature and Society

4.8
Then the ellipsoid 4.9 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 3 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.

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.
Computers are disconnected to the computer network with constant rate μ.H 4 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 .
H 5 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 Figure1is derived: Ṡ t μN − αr N E t S t − ρ SR S t − μS t , Ė t αr N E t S t − α ρ ER μ E t , İ t αE t − γ μ I t , Ṙ t ρ SR S t ρ ER E t − γI t − μR t .
Lemma 4.1 see 14, 15 .Assumption B: there exists a bounded domain U ⊂ E l with regular boundary Γ, having the following properties.In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A x is bounded away from zero.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∈K E x τ < ∞ 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 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 .