# An SLBRS Model with Vertical Transmission of Computer Virus over the Internet

Authors: Maobin Yang; Zhufan Zhang; Qiang Li; Gang Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

By incorporating an additional recovery compartment in the SLBS model, a new model, known as the SLBRS model, is proposed in this paper. The qualitative properties of this model are investigated. The result shows that the dynamic behavior of the model is determined by a thresholdℛ0. Specially, virus-free equilibrium is globally asymptotically stable if ℛ0≤1, whereas the viral equilibrium is globally asymptotically stable if ℛ0>1. Next, the sensitivity analysis of ℛ0 to four system parameters is also analyzed. On this basis, a collection of strategies are advised for eradicating viruses spreading across the Internet effectively.


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## Body

## 1. Introduction

Malicious computer virus programs form a great threat to information society by acquiring information and hard drives illegally, damaging motherboard, congesting network traffic, making devices out of control, and the like [1–5]. The popularization of the Internet further strengthens this threat; computer viruses can propagate through downloading files, opening email attachments, or even instant messaging [3, 4]. Although a considerable amount of work has been done to protect against the propagation of virus, the human effort to struggle against virus is still in its infancy [1–8].Because of the development and unpredictability of computer virus, the creation of antivirus software, which aims to analyze the structures of viral codes, always lags behind the creation of virus programs. As a result, antivirus techniques are incompetent to effectively forecast the trend of evolution of virus [3, 4, 7, 8]. For the purpose of understanding the long-term behavior of viruses and posing effective strategies of controlling their spread across the Internet, and in view of that the propagation of computer virus across the Internet resembles that of biological virus across a population, it is appropriate to macroscopically establish and study computer virus propagation models by properly modifying their biological counterparts. Indeed, some classical epidemiological models were simply borrowed to establish computer virus models so as to characterize the way virus propagates [9–16], which share a common assumption that an infected computer in which the virus resides is in latency cannot infect other computers. Very recently, Yang et al. [17–20] proposed a new model, the SLBS model, by taking into account the fact that a computer immediately possesses infection ability once it is infected. This model, however, does not consider that those recovered computers can gain temporary immunity.In this paper, we propose a new computer virus model, the SLBRS model, which incorporates an additional recovery compartment. In SLBRS model, all the computers connected to the Internet are partitioned into four compartments: uninfected computers having no immunity (S computers), infected computers that are latent (L computers), infected computers that are breaking out (B computers), and uninfected computers having temporary immunity (R computers) from which virus is removed after its outbreak. Because the incidence rate of SLBRS model is f(S;L,B), not conventional f(S,I) [11, 12], it is a challenge to research this new model. Specially, the global stability of the viral equilibrium for SLBRS model is mostly difficult to resolve by means of constructive Lyapunov function.To establish our results, we apply a so-called geometric approach to global stability to obtain sufficient conditions for global stability of viral equilibrium, namely, equilibrium with all positive components. It is a generalization of the Poincare-Bendixson criterion for systems of ordinary differential equations, which first appeared in Smith [21] and was further developed in Li and Muldowney [22, 23] and Li [24, 25], the majority of applications refer to epidemic models, as SIR, SEIR, SEIS, SEIRS models (see, e.g., [26–31]), as well as other population dynamics context [32, 33]. It is proved that the dynamical behavior of the SLBRS model is fully determined by a threshold ℛ0: the virus-free equilibrium is globally asymptotically stable if ℛ0≤1, whereas the viral equilibrium is globally asymptotically stable if ℛ0>1. The theoretical results obtained imply some practical means of eradicating computer viruses distributed over the Internet.The subsequent materials are organized in this pattern: Sections2 formulates the SLBRS model and gives its basic reproduction number as well as virus-free and viral equilibria, respectively. Sections 3 examine the global stabilities of the virus-free and viral equilibria, respectively. By analyzing the dependence of ℛ0 on the system parameters, Section 4 poses a set of effective strategies for controlling the spread of computer virus across the Internet. Finally, Section 5 makes a brief summary of this work.


## 2. SLBRS Model

In this paper, all computers connected to the Internet are partitioned into four compartments: uninfected computers having no immunity (S computers), infected computers that are latent (L computers), infected computers that are breaking out (B computers), and uninfected computers having temporary immunity (R computers). Let S(t), L(t), B(t), and R(t) denote their corresponding percentages of all computers at time t, respectively. The basic assumptions in concern with our model are presented below. (A1)
All newly connected computers are all virus free.(A2)
External computers are connected to the Internet at positive constant rateμ. Also, internal   computers are disconnected from the Internet at the same rate μ.(A3)
Each virus-free computer gets contact with an infected computer at a bilinear incidence rateβS(L+B), where β is positive constant.(A4)
Latent computers break out at nonnegative constant rateε.(A5)
Breaking-out computers are cured at nonnegative constant rateγ.(A6)
Recovered computers become susceptibly virus-free again at nonnegative constant rateα. According to the above assumptions, the transfer diagram is depicted in Figure 1.Figure 1
State transition diagram for the SLBRS model.Therefor, the corresponding computer virus model is of the form:(2.1)S˙=μ-βS(L+B)+αR-μS,L˙=βS(L+B)-εL-μL,B˙=εL-γB-μB,R˙=γB-αR-μR.BecauseS+L+B+R=1, system (2.1) simplifies to the following three-dimensional subsystem:
(2.2)S˙=μ(1-S)-βS(L+B)+α(1-L-B-S),L˙=βS(L+B)-(ε+μ)L,B˙=εL-(γ+μ)B,
on the closed, positively invariant set Ω={(S,L,B):S≥0,L≥0,B≥0,S+L+B≤1}, with initial conditions S(0)≥0, L(0)≥0, B(0)≥0.Hence, we consider only solutions with initial conditions inside the regionΩ, in which the usual existence, uniqueness of solutions, and continuation results hold.Clearly, the system (2.2) always has the virus-free equilibrium:
(2.3)E0=(S0,L0,B0)=(1,0,0).As one of the most useful threshold parameters mathematically characterizing the spread of virus, the basic reproduction number,ℛ0, is defined as the expected number of secondary cases produced by a single (typical) infection in a population of susceptible computers [24]. Let X=(L,B), it follows from system (2.1) that
(2.4)dXdt=ℱ-𝒱,
where
(2.5)ℱ=[βS(L+B)0],𝒱=[(μ+ε)L-εL+(μ+γ)B].
Let F be the Jacobian matrix of ℱ at E0, and V the Jacobian matrix of 𝒱 at E0, respectively, then we get
(2.6)F=[ββ00],V=[(μ+ε)0-εμ+γ].
The spectral radius ℛ0 of the matrix K=FV-1 is exactly the basic reproduction number of the model, that is, ℛ0. Hence
(2.7)ℛ0=βμ+γ+ε(μ+ε)(μ+γ).Further, system (2.2) also has an interior equilibrium called viral equilibrium given by
(2.8)E*=(S*,L*,B*),
where
(2.9)S*=(μ+γ)(μ+ε)β(μ+γ+ε)=1ℛ0,L*=K*(μ+γ)(μ+α),B*=K*ε(μ+α),
and K*=((μ+ε)(μ+γ)(ℛ0-1))/(β(μ+γ+ε)[γ(ε+α+μ)+(μ+ε)(μ+γ)]).  E*=(S*,L*,B*)∈Ω′=Ω-{E0}. It is clear that the viral equilibrium exists if and only if ℛ0>1.In the following sections, we will study the dynamical behavior of the system (2.2).


## 3. Global Stability

In this section, we study the global stabilities of virus-free and viral equilibria, respectively. First, we have the following.Theorem 3.1.
(a) Ifℛ0≤1, then E0 is the only equilibrium and it is globally stable in Ω. (b) If ℛ0>1, then E0 is unstable and there exists a unique vrial equilibrium E*. Furthermore, all solutions starting in Ω and sufficiently close to E0 move away from E0 if ℛ0>1.Proof.
Consider a Lyapunov function:(3.1)V(L,B)=12[L2+(μ+ε)(μ+γ)ε(μ+γ+ε)B2].
Clearly, V is positively definite. By calculation, we get
(3.2)dvdt|(2)=βSL2+βSLB-(μ+ε)L2+(μ+ε)(μ+γ)ε(μ+γ+ε)[εL-(μ+α)]B=[βS-(μ+ε)(μ+γ)μ+γ+ε](L+B)L-(μ+ε)εμ+γ+ε(L-μ+γεB)2.
Because ℛ0≤1, that is, β(μ+γ+ε)<(μ+ε)(μ+γ). It can be seen that dV/dt≤0 holds for (S,L,B)∈Ω. Furthermore, dV/dt=0 if and only if (L,B)=(0,0). Besides, V(L,B)→∞ as L→∞ or B→∞. It follows from LaSalle invariance principle [34] that E0 is globally asymptotically stable with respect to Ω if ℛ0≤1. This proves claim (a).
We linearize the system (2.2) at E0, giving the Jacobian matrix:
(3.3)JE0=[β-μ-εβ0ε-μ-γ00γ-μ-α].
The corresponding eigenvalues of JE0 are
(3.4)λ1=-μ-α,λ2=12(-γ-ε-2μ+β+β2+2β(ε+γ)+(ε-γ)2),λ3=12(-γ-ε-2μ+β-β2+2β(ε+γ)+(ε-γ)2).
Obviously λ1<0 and λ2>0 if ℛ0>1 which implies β(μ+γ+ε)>μ2+μγ+με+εγ. By the stability theory [35], E0 is unstable and there exists a unique vrial equilibrium E* if ℛ0>1. This proves claim (b).Secondly, we study the global stability of viral equilibrium,E*. So we have the following.Lemma 3.2.
The viral equilibriumE* is locally asymptotically stable with respect to Ω' if ℛ0>1.Proof.
The Jacobian matrix of the linearized system of system (2.2) evaluated at E* is
(3.5)JE*=[-β(L*+B*-S*)-μ-ε-β(L*+B*-S*)-β(L*+B*)ε-μ-γ00γ-μ-α].
The characteristic equation of JE* is p(λ)=λ3+a2λ2+a1λ+a0=0, where
(3.6)a2=β(L*+B*+S*B*L*)+εL*B*+γB*R*,a1=εγL*+βε(L*+B*)2B*+βγB*L*2+B*2L*+S*B*2R*L*,a1=βγε(R*+L*+B*)(L*+B*)R*.
We have
(3.7)a2a1-a0=β2ε[(L*+B*)3R*+S*(L*+B*)2L*]+βεγ[L*2+B*L*+S*B*R*+L*2B*]+β2γ[(L*2+B*L*+S*B*)(B*L*2+B*2+B*2L*+S*B*2)B*L*2]+βε2L*2(L*+2B*)B*2+βγ2B*(B*L*2+B*2L*+S*B*2)R*L*+εγ2L*R*+βεμB*+βεγS*+ε2γ.
Clearly, a2>0, a1>0, a2>0, and (a2a1-a0)>0 if ℛ0>1. By Routh-Hurwitz criterion, E* is locally asymptotically stable.It is obvious thatM={(S,0,0):0≤S≤1} is the maximum invariant set on the boundary of Ω. By Theorem 3.1, all orbits started from the interior of Ω will not get away from M if ℛ0>1. Furthermore, the stable set of M, MS={P∈Ω:ω(P)⊆M}, is equal to M and on the boundary of Ω. Then, we can derive the following proposition.Proposition 3.3.
Ifℛ0>1, the system (2.2) is uniformly persistent.System (2.2) is said to be uniformly persistent in Ω, or rather there exists constant c such that
(3.8)liminft→∞S(t)>c,liminft→∞L(t)>c,liminft→∞B(t)>c,(3.9)liminft→∞[1-S(t)-L(t)-B(t)]>c.
provided (S(0),L(0),B(0))∈Ω′ [9, 36–38]. Here, c is independent of initial conditions.The uniform persistence of (2.2) in the bounded set Ω is equivalent to the existence of a compact K∈Ω′ that is absorbing for (2.2), namely, each compact set K∈Ω′ satisfies x(t,x0)⊂K for sufficiently large t, where x(t,x0) denotes the solution of (2.2) such that x(0,x0)=x0 [38]. We state our main result in the following theorem. Appendix A outlines a general mathematical framework for providing global stability, which will be used in the following to prove the Theorem 3.4.Theorem 3.4.
E
* is globally stable in Ω′ if ℛ0>1.Proof.
From the AppendixA, we know that system (2.2) satisfies the following assumptions.  (H1)
There exists a compact absorbing setK∈Ω′.(H2)
System (2.2) has a unique equilibrium E*∈Ω′. Let X=(S,L,B) and F(X) denote the vector field of system (2.2), the Jacobian matrix J=∂F/∂X associated with a general solution x(t) of system (2.2) is
(3.10)J=[-β(L+B)-(μ+α)-βS-α-βS-αβ(L+B)βS-(μ+ε)βS0ε-(μ+γ)],
and its second additive compound Jacobian matrix J[2] [39, 40] is
(3.11)J[2]=[-β(L+B-S)-(2μ+ε+α)βSβS+αε-β(L+B)-(2μ+ε+γ)-βS-α0β(L+B)βS-(2μ+ε+γ)].
We consider the matrix-valued function P(X)=P(S,L,B) as
(3.12)P(S,L,B)=[a1000(1-a2)LB00a2LBLB],
where
(3.13)1<a1<1+α(1-c)+2βcα+β,
and 0<a2<1. Here, c is the uniform persistence constant in (3.8). Then,
(3.14)PfP-1=[0000L˙L-B˙B000L˙L-B˙B],
and the matrix B=PfP-1+PJ[2]P-1 in (A.4) can be written in block form:
(3.15)B=[B11B12B21B22],
where
(3.16)B11=[-β(L+B-S)-(2μ+ε+α)],B12=[a1βSB-a1a2(βS+α)BL(1-a2)a1(βS+α)BL],B21=[1-a2a1LBεa2a1LBε],B22=[ℑ-β(L+B)-(2μ+γ+α)+a2(βS+α)(a2-1)(βS+α)β(L+B)-a2(βS+α)+a21-a2εℑ-a2(βS+α)+βS-(2μ+γ+ε)],
where ℑ donates L˙/L-B˙/B.
Letz=(u,v,w) denote the vectors in ℝ3, we select a norm in ℝ3 as
(3.17)|(u,v,w)|=max{|u|,|v|+|w|},
and let μ denote the Lozinskii measure with respect to this norm.
As described in [41], we have the estimate:
(3.18)μ(B)≤sup{g1,g2}=sup{μ1(B11)+|B12|,μ1(B22)+|B21|}.
Here, |B21|, |B12| are matrix norms with respect to the L1 vector norm, and μ1 denotes the Lozinskii measure with respect to the L1 norm.
We thus obtain(3.19)g1=μ1(B11)+|B12|=-β(L+B-S)-(2μ+ε+α)+max{a1βSBL-αa1a21-a2BL,a1(βS+α)BL},(3.20)g2=|B21|+μ1(B22)=L˙L-B˙B+εa1BL-(2μ+γ+α)+max{a2ε1-a2,2(1-a2)(βS+α)-ε}.
Rewriting (2.2), we have
(3.21)L˙L=βS+βSBL-(μ+ε),(3.22)B˙B=εLB-(μ+γ).
The uniform persistence constant c in (3.8) can be adjusted so that there exists T>0 independent of x(0)∈K the compact absorbing set, such that
(3.23)S(t)≥c,L(t)≥c,B(t)≥cfort>T.
Substituting (3.21) into (3.19) and (3.22) into (3.20) and using (3.23) and our choice of a1, we obtain, for t>T,
(3.24)g1=μ1(B11)+|B12|=L˙L-μ-α-β(L+B)-βSBL+a1(βS+α)BL≤L˙L-μ,g2≤L˙L-μ-α+a2ε1-a2≤L˙L-μ.
Therefore, μ(B)≤L˙/L-μ for t>T by (3.18) and (3.24). Along each solution x(t,x0) to (2.2) such that x0∈K and for t>T, we thus have
(3.25)1t∫0t‍μ(B)ds≤1t∫0T‍μ(B)ds+1tlogL(t)L(T)-μ,
which implies that
(3.26)q-2=limsupt→∞supx0∈Ω1t∫0t‍μ(B(x,x0))ds≤-μ2<0,
from (3.18), proving Theorem 3.4.Consider system (2.2) with parameter values shown in Table 1.Table 1
Typicalℛ0  and corresponding parameter values for system (2.2).
ℛ
0
β
ε
γ
α
μ
S
(
0
)
L
(
0
)
B
(
0
)
0.2908
0.015
0.002
0.2
0.1
0.05
0.2
0.4
0.3
0.9944
0.05
0.001
0.02
0.02
0.05
0.2
0.4
0.3
1.1607
0.025
0.03
0.025
0.02
0.01
0.5
0.2
0.2
4.2273
0.045
0.001
0.02
0.02
0.01
0.5
0.2
0.2
9.9440
0.15
0.05
0.02
0.01
0.001
0.5
0.2
0.2
109.0909
0.2
0.002
0.005
0.01
0.0005
0.5
0.2
0.2Forℛ0≤1, by Theorem 3.1, the virus-free equilibrium of system (2.2) is globally asymptotically stable. Figures 2(a)-2(b) demonstrate how S(t), L(t), B(t), and R(t) evolve with the elapse of time in the case of ℛ0=0.2908, and 0.9944 in Table 1, respectively.(a) Evolutions ofS, L, B, and R in the case of ℛ0=0.2908 and (b) evolutions of S, L, B, and R in the case of ℛ0=0.9944.
(a)
(b)Forℛ0>1, by Theorem 3.4, the viral equilibrium of system (2.2) is globally asymptotically stable. Figures 3(a)–3(d) demonstrate how S(t), L(t), B(t) and R(t) evolve with the elapse of time in the case of ℛ0=1.1607, 4.2273, 9.9440 and 109.0909 in Table 1, respectively.(a) Evolutions ofS, L, B, and R in the case of ℛ0=1.1607, (b) evolutions of S, L, B, and R in the case of ℛ0=4.2273, (c) evolutions of S, L, B, and R in the case of ℛ0=9.9440 and (d) evolutions of S, L, B and R in the case of ℛ0=109.0909.
(a)
(b)
(c)
(d)


## 4. Discussions

As was indicated in the previous section, it is critical to take various actions to control the system parameters so thatℛ0 is remarkably below one. This section is intended to propose some effective measures for achieving this goal.For our purpose, it is instructive to examine the sensitivities ofℛ0 to four system parameters: β, ε, γ, and μ, respectively. Following Arriola and Hyman [42], the normalized forward sensitivity indices with respect to β, ε, γ, and μ are calculated, respectively, as follows:
(4.1)∂ℛ0/ℛ0∂β/β=βℛ0μ+γ+ε(μ+γ)(μ+ε)=1>0,∂ℛ0/ℛ0∂ε/ε=εℛ0∂ℛ0∂ε=-εγ(μ+γ+ε)(μ+ε)<0,∂ℛ0/ℛ0∂γ/γ=γℛ0∂ℛ0∂γ=-εγ(μ+γ+ε)(μ+γ)<0,∂ℛ0/ℛ0∂μ/μ=μℛ0∂ℛ0∂μ=-μ[γ(μ+γ)+(μ+γ+ε)(μ+γ)](μ+ε)(μ+γ(μ+γ+ε))<0.
It can be seen that, among these four parameters, ℛ0, in proportion with β, is the most sensitive to the change in β. As opposed to this, the other three parameters ε, γ, and μ have an inversely proportional relationship with ℛ0, an increase in ε or γ, or μ will bring about a decrease in ℛ0, with a proportionally smaller size of decrease.Below, let us explain how these properties of model (2.2) can be utilized to control the spread of computer virus.(1)
Filtering and blocking suspicious messages with firewall located at the gateway of a domain, the parameterβ can be kept low and, hence, the chance that a virus-free computer within the domain is infected by a viral computer outside the domain can be significantly decreased, yielding a lower threshold value ℛ0.(2)
Timely updating and running antivirus software of the newest version on computers, the breaking out rate of latent computers,ε, and the cure rate of breaking out computers, γ, can be remarkably enhanced, leading to a low ℛ0.(3)
Timely disconnecting computers from the Internet when the connections are unnecessary, the disconnecting rate of computers,μ, can be made high, bringing about an ideal ℛ0.In practice, all of these measures are strongly recommended to achieve a threshold value well below unity, so that viruses within the Internet approach extinction.


## 5. Conclusion

In nearly all previous computer virus propagation models with latent compartment, to our knowledge, latent computers are assumed not to infect other computers, which does not accord with the real situations. To overcome this defect, the SLBRS model proposed in this paper assumes that all latent computers have infectivity. The dynamics of this model has been fully studied. The results concerning this model include the following.(1) two equilibria, the virus-free equilibrium E0 and the viral equilibrium E*, as well as the basic reproduction ratio ℛ0 are obtained. (2) The dynamical behavior is determined completely by the value of ℛ0: ℛ0≤1 implies the global stability of E0, whereas ℛ0>1 implies the global stability of E*. (3) By conducting a sensitive analysis of ℛ0 with respect to various model parameters and on the condition that ℛ0≪1, a series of measures of strategies is proposed for controlling the spread of virus through the Internet effectively.Our proof of the global stability of viral equilibrium whenℛ0>1 utilizes a general approach established in [43], and relies on the construction of a new Lyapunov function for the second compound system.



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