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

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 threshold R0. 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.


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][18][19][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 R 0 : the virus-free equilibrium is globally asymptotically stable if R 0 ≤ 1, whereas the viral equilibrium is globally asymptotically stable if R 0 > 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: Sections 2 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 R 0 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.

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 rate μ.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 rate βS L B , where β is positive constant.
A4 Latent computers break out at nonnegative constant rate ε.
A5 Breaking-out computers are cured at nonnegative constant rate γ.
A6 Recovered computers become susceptibly virus-free again at nonnegative constant rate α.
According to the above assumptions, the transfer diagram is depicted in Figure 1.
Therefor, the corresponding computer virus model is of the form:

2.1
Because S L B R 1, system 2.1 simplifies to the following three-dimensional subsystem: Hence, we consider only solutions with initial conditions inside the region Ω, in which the usual existence, uniqueness of solutions, and continuation results hold.
Clearly, the system 2.2 always has the virus-free equilibrium: As one of the most useful threshold parameters mathematically characterizing the spread of virus, the basic reproduction number, R 0 , 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 where Let F be the Jacobian matrix of F at E 0 , and V the Jacobian matrix of V at E 0 , respectively, then we get The spectral radius R 0 of the matrix K FV −1 is exactly the basic reproduction number of the model, that is, R 0 .Hence Further, system 2.2 also has an interior equilibrium called viral equilibrium given by where and In the following sections, we will study the dynamical behavior of the system 2.2 .

Global Stability
In this section, we study the global stabilities of virus-free and viral equilibria, respectively.First, we have the following.
, then E 0 is unstable and there exists a unique vrial equilibrium E * .Furthermore, all solutions starting in Ω and sufficiently close to E 0 move away from Proof.Consider a Lyapunov function: Clearly, V is positively definite.By calculation, we get It follows from LaSalle invariance principle 34 that E 0 is globally asymptotically stable with respect to Ω if R 0 ≤ 1.This proves claim a .We linearize the system 2.2 at E 0 , giving the Jacobian matrix: The corresponding eigenvalues of J E 0 are
Secondly, we study the global stability of viral equilibrium, E * .So we have the following.

Discrete Dynamics in Nature and Society
Proof.The Jacobian matrix of the linearized system of system 2.2 evaluated at E * is The characteristic equation of J E * is p λ λ 3 a 2 λ 2 a 1 λ a 0 0, where

3.6
We have
It is obvious that M { 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 R 0 > 1.Furthermore, the stable set of M, M S {P ∈ Ω : ω P ⊆ M}, is equal to M and on the boundary of Ω.Then, we can derive the following proposition.
System 2.2 is said to be uniformly persistent in Ω, or rather there exists constant c such that 3.9 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, x 0 ⊂ K for sufficiently large t, where x t, x 0 denotes the solution of 2.2 such that x 0, x 0 x 0 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.
Proof.From the Appendix A, we know that system 2.2 satisfies the following assumptions.
H 1 There exists a compact absorbing set K ∈ 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 and its second additive compound Jacobian matrix J 2 39, 40 is

3.11
We consider the matrix-valued function P X P S, L, B as and 0 < a 2 < 1.Here, c is the uniform persistence constant in 3.8 .Then,  We thus obtain

3.22
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 Substituting 3.21 into 3.19 and 3.22 into 3.20 and using 3.23 and our choice of a 1 , we obtain, for t > T,

3.24
Therefore, μ B ≤ L/L − μ for t > T by 3.18 and 3.24 .Along each solution x t, x 0 to 2.2 such that x 0 ∈ K and for t > T, we thus have   For R 0 > 1, by Theorem 3.4, the viral equilibrium of system 2.2 is globally asymptotically stable.Figures 3 a -3

Discussions
As was indicated in the previous section, it is critical to take various actions to control the system parameters so that R 0 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 of R 0 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
It can be seen that, among these four parameters, R 0 , 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 R 0 , an increase in ε or γ, or μ will bring about a decrease in R 0 , 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 parameter β 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 R 0 .
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 R 0 .
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 R 0 .
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.

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 E 0 and the viral equilibrium E * , as well as the basic reproduction ratio R 0 are obtained.2 The dynamical behavior is determined completely by the value of R 0 : R 0 ≤ 1 implies the global stability of E 0 , whereas R 0 > 1 implies the global stability of E * .3 By conducting a sensitive analysis of R 0 with respect to various model parameters and on the condition that R 0 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 when R 0 > 1 utilizes a general approach established in 43 , and relies on the construction of a new Lyapunov function for the second compound system.

A. The General Mathematical Framework of Geometric Approach to Global Stability
The presentation here follows that in 22 .Let x → f x ∈ R n be a C 1 function for x in an open set D ⊂ R n .Consider the differential equation: Denote by x t, x 0 the solution to A.1 such that x 0, x 0 x 0 .We make the following two assumptions.
H1 There exists a compact absorbing set K ⊂ D. H2 Equation A.1 has a unique equilibrium x in D.
The equilibrium x is said to be globally stable in D if it is locally stable and all trajectories in D converge to x.The following global-stability problem is formulated in 22 .
Global-stability problem.Under the assumptions H1 and H2 , find conditions on the vector field f such that the local stability of x implies its global stability in D.
The assumptions H1 and H2 are satisfied if x is globally stable in D. For n ≥ 2, a Bendixson criterion is a condition satisfied by f which precludes the existence of nonconstant periodic solutions of A.1 .A Bendixson criterion is said to be robust under C 1 local perturbations of f at x 1 ∈ D if, for sufficiently small > 0 and neighborhood U of x 1 , it is also satisfied by Such g will be called local ε-perturbations of f at x 1 .It is easy to see that the classical Bendixson's condition f x < 0 for n 2 is robust under C 1 local perturbations of f at each x 1 ∈ R 2 .Bendixson criteria for higher dimensional systems that are C 1 robust are discussed in 21, 22, 44 .
A point x 0 ∈ D is wandering for A.1 if there exists a neighborhood U of x 0 and T > 0 such that U ∩ x t, U is empty for all t > T. Thus, for example, all equilibria and limit points are nonwandering.The following is a version of the local C 1 closing lemma of Pugh 45,46 as stated in 43 .
Lemma A.1.Let f ∈ C 1 D → R n .Suppose that x 0 is a nonwandering point of A.1 and that f x 0 / 0. Also assume that the positive semiorbit of x 0 has compact closure.Then, for each neighborhood U of x 0 and ε > 0, there exists a C 1 local ε-perturbation g of f at x 0 such that 1 sup p f − g ⊂ U and 2 the perturbed system x g x has a nonconstant periodic solution whose trajectory passes through x 0 .
The following general global-stability principle is established in 43 .
Theorem A.2. Suppose that assumptions (H1) and (H2) hold.Assume that A.1 satisfies a Bendixson criterion that is robust under C 1 local perturbations of f at all nonequilibrium nonwandering points for A.1 .Then, x is globally stable in D provided it is stable.
The main idea of the proof in 43 for Theorem A.2 is as follows.Suppose that system A.1 satisfies a Bendixson criterion.Then it does not have any nonconstant periodic solutions.Moreover, the robustness assumption on the Bendixson criterion implies that all nearby differential equations have no nonconstant periodic solutions.Thus, by Lemma A.1, all nonwandering points of A.1 in D must be equilibria.In particular, each omega limit point in D must be an equilibrium.Therefore, ω x 0 {x} for all x 0 ∈ D since x is the only equilibrium in D.

B. The Second Additive Compound Matrix
Let A be a linear operator on R n and also denote its matrix representation with respect to the standard basis of R N .Let ∧ 2 R n denote the exterior product of R n .A induces canonically a linear operator A 2 on ∧ 2 R n : for μ 1 , μ 2 ∈ R n , define For detailed discussions of compound matrices and their properties, we refer the reader to 39, 40 .A comprehensive survey on compound matrices and their relations to differential equations is given in 40 .

Figure 1 :
Figure 1: State transition diagram for the SLBRS model.

Figure 2 :
Figure 2: a Evolutions of S, L, B, and R in the case of R 0 0.2908 and b evolutions of S, L, B, and R in the case of R 0 0.9944.

Figure 3 :
Figure 3: a Evolutions of S, L, B, and R in the case of R 0 1.1607, b evolutions of S, L, B, and R in the case of R 0 4.2273, c evolutions of S, L, B, and R in the case of R 0 9.9440 and d evolutions of S, L, B and R in the case of R 0 109.0909.
3.14and the matrix B P f P −1 PJ 2 P −1 in A.4 can be written in block form:

Table 1 :
Typical R 0 and corresponding parameter values for system 2.2 .
d demonstrate how S t , L t , B t and R t evolve with the elapse of time in the case of R 0 1.1607, 4.2273, 9.9440 and 109.0909 in Table 1, respectively.
B.1 and extend the definition over ∧ 2 R n by linearity.The matrix representation of A 2 with respect to the canonical basis in ∧ 2 R n is called the second additive compound matrix of A. This is an n 2 × n 2 matrix and satisfies the property A B 2 A 2 B 2 .In the special case, when n 2, we have A In general, each entry of A 2 is a linear expression of those of A. For instance, when n 3, the second additive compound matrix of A a ij is