# Dynamics of a Delay-Varying Computer Virus Propagation Model

Authors: Jianguo Ren; Yonghong Xu; Yongchang Zhang; Yongquan Dong; Guosheng Hao
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

By considering the varying latency period of computer virus, we propose a novel model for computer virus propagation in network. Under this model, we give the threshold value determining whether or not the virus finally dies out, and study the local stability of the virus-free and virus equilibrium. It is found that the model may undergo a Hopf bifurcation. Next, we use different methods to prove the global asymptotic stability of the equilibria: the virus-free equilibrium by using the direct Lyapunov method and virus equilibrium by using a geometric approach. Finally, some numerical examples are given to support our conclusions.


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## Body

## 1. Introduction

With the advance of computer software and hardware and communication technologies, the number and sort of computer viruses have increased dramatically, which causes huge losses to the human society. Therefore, establishing reasonable computer-virus-propagation models by considering the characteristics of computer virus and, by model analysis, understanding the spread law of the virus over the network, are a currently hot topic of research.Towards this goal, the classical SIR (susceptible-infected-recovered) model [1, 2], as well as its extensions [3–5], is extended to explore the behavior of computer virus propagation in network. Based on these classical models and by considering the computer virus fixed latent period, Mishra et al. [6, 7] proposed delayed SIRS, SEIR computer virus models with a fixed period of temporary immunity, which accounts for the temporary recovery from the infection of virus. In [8], Tan and Han proposed an SIRS computer virus model with fixed latency and temporal immune periods, studied the effect of time delays on the stability of the equilibria, and gave some conditions for the equilibria to be locally asymptotically stable for all delays.Motivated by the previous work, this paper proposes and studies a computer-virus-propagation model with varying latency period, known as the SIRC model. We obtain the threshold value determining whether the virus dies out completely, study the local asymptotic stabilities of the equilibria of the model and it is found that, model may undergo a Hopf bifurcation. Next, we prove the global asymptotic stability of the virus-free equilibrium by using the direct Lyapunov method, prove the global asymptotic stability of the virus equilibrium by using a geometric approach. By introducing varying time delay, the model may truly reflect the virus propagation and hence, the corresponding results may help understand and prevent the spread of computer virus over a computer network.The remaining materials of this paper are organized this way: Section2 introduces the mathematical model to be discussed; Section 3 studies the local stability of the virus-free and virus equilibrium of model, respectively, examines the stability switch for a virus equilibrium, and shows that our model may admit a Hopf bifurcation; Section 4 uses different methods to prove the global asymptotic stability of the equilibria. In Section 5, some numerical examples are given to support our conclusions. We end the paper with a brief discussion in Section 6.


## 2. Mathematical Model

Consider the classical SIR computer virus model proposed in [1, 2]
(2.1)dSdt=b-βS(t)I(t)-μS(t),dIdt=βS(t)I(t)-(γ+μ)I(t),dRdt=γI(t)-μR(t).
Here it is assumed that all the computers connected to the network in concern are classified into three categories: susceptible, infected, and recovered computers. Let S(t), I(t), and R(t) denote their corresponding numbers at time t. This model involves four positive parameters: b denotes the rate at which external computers are connected to the network, γ denotes the recovery rate of infected computers due to the antivirus ability of the network, μ denotes the rate at which one computer is removed from the network, β denotes the rate at which, when having connection to one infected computer, one susceptible computer can become infected. For some variants of this model, see [1–5, 9, 10].The computer virus has latent and unpredictable characteristics [11]. A sophisticated computer virus program, when entering into the computer system, does not immediately break out. The longer the latency of a computer virus, the wider its spreading scope will be. On one hand, the computer virus program can not be detected without use of the specialized programs. The virus can stay quietly in the disk or CD a few days, even years, and when the time comes, it will break out to reproduce, spread, and continue to harm. On the other hand, there is a trigger mechanism within the computer virus, if the trigger conditions are not met, the computer virus does not do any other damage. Only when the trigger conditions are met, can the virus be activated to do some damages. Without loss of reality, the following assumptions are made:(1)
The virus in susceptible computer has a latency period. Moreover, this latency period is varying, which can be reflected by the following expression:(2.2)∫-∞tS(τ)D(t-τ)dτ,
where D is the delay kernel [12], τ is the distributed delay, S(τ) indicates how S(t) is affected by their previous values.(2)
Only when the virus breaks out can the susceptible computers become the infected ones.We choose a typical class of kernels(2.3)D(t-τ)={σn+1(t-τ)nexp⁡[-σ(t-τ)]}n!,t≥0,n=0,1,2,…,
where σ is a positive constant indicating the average delay of the collected information on the virus infection. In this paper, we simply take the weak kernel
(2.4)D(t-τ)=σe-σ(t-τ),σ>0,
which implies that the effect of previous events decreases exponentially.By incorporating these factors into model (2.1), we get the following model:
(2.5)dSdt=b-β∫-∞tS(τ)D(t-τ)dτI(t)-μS(t),dIdt=β∫-∞tS(τ)D(t-τ)dτI(t)-(μ+γ)I(t),dRdt=γI(t)-μR(t).
We define a new variable
(2.6)C(t)=∫-∞tS(τ)D(t-τ)dτ,
which indicates in susceptible computer, the effect of latent virus on infection. Then model (2.5) becomes
(2.7)dSdt=b-βC(t)I(t)-μS(t),dIdt=βC(t)I(t)-(μ+γ)I(t),dRdt=γI(t)-μR(t),dCdt=1σS(t)-1σC(t).
Because S, I, and C are independent of variable R, this paper focuses on the following model:
(2.8)dSdt=b-βC(t)I(t)-μS(t),dIdt=βC(t)I(t)-(μ+γ)I(t),dCdt=1σS(t)-1σC(t).
Adding the first two equations of model (2.8), we can obtain
(2.9)d(S+I)dt=b-μ(S+I)-γI,
Therefore, S(t) and I(t) are bounded, that is, S+I≤b/μ. From the third equation of model (2.8), we can obtain
(2.10)dCdt=1σS-1σC≤bσμ-1σC.
It is easy to see that C(t) is bounded, that is, C≤b/u. Thus, the set
(2.11)Ω={(S,I,C)∈R+3,S+I≤bμ,C≤bμ}
is the positively invariant set of model (2.8).


## 3. The Equilibria and Local Stability

This section investigates the equilibria of model (2.8) and their stability. For that purpose, let us introduce the basic reproduction number, which is defined as R0=bβ/μ(μ+γ).First, model (2.8) has a virus-free equilibrium E0=(b/μ,0,b/μ). The characteristic equation of the corresponding linearized system with respect to E0 is
(3.1)det⁡(-μ-λ-βbμ00βbμ-(μ+γ)-λ01σ0-1σ-λ)=0.
The three eigenvalues are -μ,-1/σ,andbβ/μ-(γ+μ). Thus, we immediately getTheorem 3.1.
Consider model (2.8).(a)
The virus-free equilibriumE0 is locally asymptotically stable if R0<1.(b)
E0 is unstable if R0>1.Next, whenR0>1, model (2.8) has a positive virus equilibrium E*=(S*,I*,C*), where
(3.2)S*=bμR0,I*=μ(R0-1)β,C*=bμR0.
The characteristic equation of the corresponding linearized system near E* is
(3.3)det⁡(-μ-λ-βC*-βI*0-λβI*1σ0-1σ-λ)=0,
which equals
(3.4)λ3+p0λ2+p1λ+p2=0,
where
(3.5)p0=μ+1σ,p1=(βI*+μ)1σ,p2=1σβ2C*I*.
A simple calculation gives
(3.6)p0p1-p2=(1σ+μ)(βI*+μ)1σ-β2C*I*1σ=1σ{μ(βI*+μ)-β2C*I*+(βI*+μ)1σ}.
If p0p1-p2>0, that is, σ<σ*, E* is locally asymptotically stable, where σ*=R0/(R0-1)(γ+μ)-μR0, and σ*>0 is equivalent to R0>1+μ/γ. From the above analysis, we obtain the following Theorem:Theorem 3.2.
Consider model (2.8). Suppose R0>1+μ/γ.(a)
The virus equilibriumE* is locally asymptotically stable if σ<σ*.(b)
E* is unstable if σ>σ*.Remark 3.3.
From the above analysis, we can see that that there exists a stability switch forE*: E* changes its stability when σ goes across the critical value σ*, which may result in a Hopf bifurcation and, hence, can be exploited to find an effective strategy for preventing the spread of computer virus.
Indeed, whenσ=σ*, (3.4) has two complex conjugate roots, λ1,2=α(T)±iω(T). It is noted that α(σ*)=0, ω(σ*)=p1>0, and
(3.7)Δ=dαdσ|σ*=σ*μ(γ+μ)(R0-1)-σ*μ2R0-2μR02σ*3(μ+1/σ*)2+2σ*2μR0.
Let f(R0)=σ*μ(γ+μ)(R0-1)-σ*μ2R0-2μR0≠0, which leads to Δ≠0. Thus, we obtain.Theorem 3.4.
IfR0>1+μ/γ, model (2.8) undergoes a Hopf bifurcation with respect to the virus equilibrium when σ goes across the value of σ*.


## 4. Global Stability

In this section, we will discuss the global stability of the model.Theorem 4.1.
whenR0<1, the virus-free equilibrium E0 is global stability.Proof.
Define(4.1)V(S,I,C)=12I2.
If R0<1, then
(4.2)V˙(S,I,C)=[βC-(μ+γ)]I2≤[βbμ-(μ+γ)]I2≤[(μ+γ)(R0-1)]I2≤0.
Since all the model parameters are positive, it follows that V˙(S,I,C)<0 for R0<1 with V˙(S,I,C)=0 if and only if I=0 or R0=1. Hence, V is a Lyapunov function on Ω. Thus, I→0 as t→∞. Using I=0 in the first equation of (2.8) shows that S→b/μ as t→∞. Therefore, it follows from the Lasalle’s invariance principle, that every solution of the model, starting from within Ω, approaches E0 as t→∞.
In the following, we use the geometrical approach [13, 14] to discuss the global stability of virus equilibrium E*. First, we give a brief outline of this approach.
Letx→f(x)∈Rn be a C1 function for x in an open set D∈Rn. Consider the following equation:
(4.3)x˙=f(x).
Denote byx(t,x0) the solution with x(t,x0)=x0. Then, the following assumptions are made:(H1)
There exists a compact absorbing setK⊂D.(H2)
Equation (4.3) has a unique equilibrium x0 in D.
Letx→p(x) be an (n2)×(n2) matrix-valued function that is C1 for x∈D. Assume that p-1(x) exists and is continuous for x∈K, the compact absorbing set. A quantity q-2 is defined as
(4.4)q-2=limsup⁡x→∞sup⁡x0∈K1t∫0tμ(B(x(s,x0)))ds,
where
(4.5)B=pfp-1+p∂f[2]∂xp-1.
The matrix pf is obtained by replacing each entry of p by its derivative in the direction of f, and μ(B) is defined by
(4.6)μ(B)=lim⁡h→0+|I+hB|-1h,
which is the Lozinskil measure of B with respect to a vector norm |·| in RN.From the above outline, a theorem can be given as follows:Theorem 4.2 (see [13]).
Assume thatD is simply connected, and that the assumptions (H1) and (H2) hold, if  q-2<0, then the unique equilibrium x0 of (4.3) is globally asymptotically stable.Now, we discuss the global stability of the virus equilibriumE* of model (2.8).Model (2.8) has a unique virus equilibrium E* in Ω, hence it satisfies the assumption (H1). If R0>1, then virus-free equilibrium is not stable, and the solutions of model (2.8) are bounded, which ensure model (2.8) has a compact set in Ω. Therefore, the assumption (H2) is met.The Jacobian matrix of model (2.8) is
(4.7)J=(-μ-βC-βI0βC-μ-γβI-1σ0-1σ),
and its second additive compound matrix is
(4.8)J[2]=(βC-2μ-γβIβI0-μ-1σ-βC-1σ0βC-(μ+γ)-1σ).
Set the function
(4.9)P(X)=P(S,I,C)=diag⁡{SI,SI,SI}.
Then
(4.10)PfP-1=diag⁡{S′S-I′I,S′S-I′I,S′S-I′I},
and the matrix B=PfP-1+PJ[2]P-1 can be written as
(4.11)B=(B11B12B21B22),
where B11=S′/S-I′/I+βC-2μ-r,B12=(βI,βI),B12=(0,1/σ)T, and
(4.12)B22=(S′S-I′I-μ-1σ-βC0S′S-I′I-μ-γ-1σ+βC).
Select the norm in R3 as the following:
(4.13)|u,v,ω|=max⁡{|u|,|v|+|ω|},
where (u,v,ω) denotes the vector in R3, let μ denote the Lozinskii measure with respect to this norm, then
(4.14)μ(B)≤sup⁡{g1,g2}=sup⁡{μ(B11)+|B12|,μ(B22)+|B21|},
where |B12|, |B21| are matrix norms with respect to the L1 vector norm. Thus,
(4.15)μ(B11)=S′S-I′I+βC-2μ-r,|B12|=βI,|B21|=1σ.
Under the condition of C≤γ/2β,
(4.16)μ(B22)=S′S-I′I-μ-1σ.
Therefore
(4.17)g1=S′S-I′I+βC+βI-2μ-r,g2=S′S-I′I-μ.
According to the second equation of model (2.8), we can obtain
(4.18)I′I=βC-μ-r.
Hence,
(4.19)g1=S′S-I′I+βI+βC-2μ-γ≤S′S-(2μ+γ-bβμ),g2=S′S-I′I-μ≤S′S-μ-(μ+γ)(R0-1)≤S′S-μ.
Therefore,
(4.20)μ(B)≤S′S-b-.
For t≥t1, where b-=min{μ,2μ+γ-bβ/μ}.This leads to(4.21)1t∫0tμ(B)ds=1t∫0t1μ(B)ds+1t∫t1tμ(B)ds≤1t∫0t1μ(B)ds+1tlog⁡S(t)S(t1)-b-,
which implies that q-2≤-b-/2<0.From the above discussions, we can obtain the following theorem:Theorem 4.3.
WhenR0>1, if then the unique equilibrium E* is globally asymptotically stable.


## 5. Numerical Simulations

In this section, we make some numerical simulations to understand the obtained theorems. Letb=20,β=0.01,γ=0.4,μ=0.2,andσ=5, then R0≈0.83<1. Hence, the virus-free equilibrium E0(50.00,0) is asymptotically stable (see Figure 1), that is, the virus would extinguish after a period of time. In contrast, let b=20,β=0.02,γ=0.6,andμ=0.1 yield σ*≈5.91. In this case, when σ=4<σ* and σ=7.5>σ*, the virus equilibrium E*(35.00,9.29) would become stable (see Figure 2) and unstable (see Figure 3), respectively.Figure 1
I versus S, where b=20,β=0.01,γ=0.4,μ=0.2,σ=5, and R0≈0.83 satisfy stable condition for the virus-free equilibrium E0(50.00, 0).Figure 2
Distribution of computers versus time whenb=10,β=0.02,γ=0.6,μ=0.1, and σ=4, where R0=2.86>1 and σ*≈5.91>σ, satisfy the stable condition for the virus equilibrium E*(35.00,9.29).Figure 3
Distribution of computers versus time whenb=20,β=0.02,γ=0.6,μ=0.1,andσ=7.5, where R0=2.86>1 and σ*≈5.91<σ satisfies the unstable condition for the virus equilibrium E*.


## 6. Discussions

In this paper, by considering varying latency period of computer virus, we propose a model for computer virus propagation in network. First, we give the threshold valueR0 determining whether the virus extinguishes, and study the local stabilities of the virus-free equilibrium E0 and virus equilibrium E* under this model. It is found that R0 changes the stability of E0 and time delay parameter σ changes the stability of E*, and that the model may undergo a Hopf bifurcation. Next, we use two different methods to prove the global asymptotic stabilities of the equilibria: the virus-free equilibrium by using the direct Lyapunov method and virus equilibrium by using a geometric approach. Finally, some numerical examples are given to support our conclusions.



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*Source: 371792-2012-08-22.xml*