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A New Model for Capturing the Spread of Computer Viruses on Complex-Networks
Based on complex network, this paper proposes a novel computer virus propagation model which is motivated by the traditional SEIRQ model. A systematic analysis of this new model shows that the virus-free equilibrium is globally asymptotically stable when its basic reproduction is less than one, and the viral equilibrium is globally attractive when the basic reproduction is greater than one. Some numerical simulations are finally given to illustrate the main results, implying that these results are applicable to depict the dynamics of virus propagation.
Computer viruses, including the narrowly defined viruses and network worms, are loosely defined as malicious codes that can replicate themselves and spread among computers. Usually, computer viruses attack computer systems directly, while worms mainly attack computers by searching for system or software vulnerabilities. With the rapid popularization of the Internet and mobile wireless networks, network viruses have posed a major threat to our work and life. To thwart the fast spread of computer viruses, it is critical to have a comprehensive understanding of the way that computer viruses propagate. Kephart and White  proposed the first epidemiological model of computer viruses. From then on, much effort has been done in developing virus spreading models [1–15]. On the other hand, it was found [16–18] that the Internet topology follows the “scale-free” (SF) networks; that is, the probability that a given node is connected to other nodes follows a power-law of the form , with the remarkable feature that for most real-world networks. This finding has greatly stimulated the interest in understanding the impact of network topology on virus spreading [16–29].
Recently, Mishra and Jha  investigated a so-called SEIQRS model on a homogeneous network by making the following assumptions.(H1)The population has a homogeneous degree distribution.(H2)The total population of computers is divided into five groups: susceptible, exposed, infected, quarantine and recovered computers. Let , , , , and denote the numbers of susceptible, exposed, infected, quarantine, and recovered computers, respectively.(H3)New computers are attached to the Internet at rate .(H4)Computers are disconnected from the Internet naturally at a constant rate and removed with probability due to the attack of malicious objects.(H5) computers become with constant rate ; computers become with constant rate ; computers become with constant rate ; computers become with constant rate ; computers become with constant rate ; computers become with constant rate .
According to the above assumptions, the following model is derived (see Figure 1):
In view of the fact that the Internet topology is scale-free rather than exponential in its degree distribution [17, 18, 23], this paper addresses the dynamics of a scale-free network-based SEIQRS model.
For convenience, computers on the Internet are called as nodes in the sequel. For our purpose, the following additional assumptions are imposed on the previous SEIQRS model.(H6)The node degrees of the network asymptotically follow a power-law distribution, , where stands for the probability that a node chosen randomly from the Internet is of degree .(H7)The total number of nodes does not change or, equivalently, , , and .(H8): the relative density of -degree -nodes; : the relative density of -degree -nodes; : the relative density of -degree -nodes; : the relative density of -degree -nodes; : the relative density of -degree -nodes; .(H9)The probability that a link has an -node as one endpoint does not depend on the degree of the other endpoint of the link and, hence, is only a function of . Let denote the probability, , where, .
By applying the mean-field technique to the above assumptions, we get a new epidemic model of computer virus, which is formulated as (see Figure 2)
with initial conditions , , , , and , .
Note that, for every , we have ; thus, the first set of equations in system (2) can be removed, yielding the following system,
with initial conditions , , , and .
The organization of this paper is as follows. Section 2 determines the equilibria of system (3) and the basic reproduction number . Sections 3 and 4 address the global stability of the virus-free equilibrium and the global attractivity of the viral equilibrium, respectively. Numerical examples are provided in Section 5 to support our theoretical results. In the final section, a brief conclusion is given and some future research topics are also pointed out.
2. Basic Reproduction Number and Equilibria
The basic reproduction number , which can be explained as the average number of secondary infections produced by a single infected node during its infection time, is calculated as
where stands for the second origin moment of the node degree, . Then, we have the following theorem.
Theorem 1. Consider system (3). The following assertions hold.
There always exists a virus-free equilibrium .
There is no viral equilibrium if .
There exists a unique viral equilibrium
if , where
is the unique positive root of the equation
Proof. After imposing the stationarity condition, we have
It is easily verified that is always a root of this system. Solving the system, we get
where is the unique positive root of the equation
If , we have , implying that and, thus, (10) has no positive roots. Hence, assertion (3) holds. Now, assume . The observations that (a) , (b) for , and (c) imply that (10) has a unique positive root. Hence, assertion (8) also holds.
Remark 2. It can be seen from Theorem 1 that and . This shows that, when in the steady state , the infection density for a higher node degree is higher than that for a lower node degree.
3. Stability of the Virus-Free Equilbrium
It is clear that is the virus-free equilibrium of system (3). In this section, we will prove that virus-free equilibrium is globally asymptotically stable when .
For convenience, let
Let and rewrite system (3) in matrix-vector notation as
with initial condition , where
Theorem 3. Consider system (12); is locally asymptotically stable if , whereas is a saddle point if .
Proof. The characteristic equation with respect to is
This equation has negative roots and with multiplicity and negative roots , , and with multiplicity . Now let
Suppose . Then, and it follows from the Hurwitz criterion that all roots of the characteristic equation have negative real parts, implying that is locally asymptotically stable. Now, assume . Then, and the characteristic equation has exactly one positive root, implying that is a saddle point.
Lemma 4 (see ). Consider a system defined at least in a compact set . Then, is invariant if, for every point y on , the vector is tangent to or pointing into .
Lemma 5. The set is positively invariant for system (12). That is, implies for all .
Proof. consists of the following sets:
as their respective outer normal vectors. For , we have
Thus, the claimed result follows from Lemma 4.
Lemma 6 (see ). Consider an -dimensional autonomous system
where is an irreducible matrix, is a region containing the origin, , and . Assume there exist, a positively invariant compact convex set containing the origin, a positive number , and a real eigenvector of , such that(C1) for all ,(C2) for all ,(C3)the origin forms the largest positively invariant set included in .
Then, one has that(1) implies that the origin is globally asymptotically stable in , and(2) implies that there exists such that, for each , the solution to system (12) satisfies .
We are ready to prove.
Theorem 7. Consider system (12). Then, is globally asymptotically stable in , if .
Proof. Let and look at (12). As matrix is irreducible and all of its nondiagonal entries are nonnegative, it follows from  that has a positive eigenvector corresponding to its eigenvalue . Let . Then, for all , we have
Moreover, implies that . Hence, the claimed result follows from assertion (2) of Lemma 6.
4. Global Attractivity of the Viral Equilibrium
We will ascertain the global attractivity of the viral equilibrium.
Theorem 8. If , then the infection solution of (12) is globally attractive in .
Proof. Theorem 3 ensures the existence of the viral equilibrium. We need to prove that if , there is a unique constant equilibrium in . Let , Assume that and are two constant solutions of (12) in . If , then there exists , , such that , where is the th component of the vector . Without loss of generality, assume , and for all . Since and are constant solutions of (12), we substitute them into (12). And if , we obtain,
After equivalent deformation, it follows that
But for all and
Thus, from the above inequality, we get
This is a contradiction. Similarly, we can also get contradictions when , , and . Therefore, there exists a unique constant solution of (3) in . Now, we shall prove that is globally attractive in . To find the asymptotic behavior of the solutions of (12) in , we define the following functions, and for , where , , and are continuous and the right-hand derivative exists along solutions of (12). Let be a solution of (12), we may assume that , , for a given and a sufficiently small . Then,
where is defined as
If , from (12) we have
And we obtain
According to the definition of , we have
Then if , we obtain
Since and , we conclude that . Therefore, if , .
Similarly, we can testify that imples , and implies . If , then . Denote
Both and are continuous and non-negative for . Notice that , . Let
then we have
According to the LaSalle invariant set principle, any solution of (12) starting in will approach . And . But if , by Lemma 6 we know that . Then we conclude that any solution of (12), such that , satisfies , so is globally attractive in .
Conjecture 9. Consider system (12) and suppose . Then the infection equilibrium, , is globally asymptotically stable in .
5. Numerical Examples
In this section, some numerical simulations are given to support our results. To demonstrate the global stability of the infection-free solution of system (3), we take the following set of parameter values: , , , , , , which runs on a scale-free network with and . In this case, we have . The time plots of the four relative densities are plotted in Figure 3, from which it can be seen that the virus would die out.
To demonstrate the global attractivity of the viral equilibrium of system (3), we take the following set of parameter values: , , , , , , which runs on a scale-free network with and . In this case, we have . The time plots of the four relative densities are plotted in Figure 4, from which it can be seen that the virus would persist.
To clearly understand how the Internet topology affects the spread of computer viruses, a new model capturing the epidemics of computer viruses on scale-free networks has been proposed. The basic reproduction number of the model has been calculated. The global asymptotic stability of the virus-free equilibrium has been shown when is below one, and the global attractivity of the viral equilibrium has been proved if is above one. Our future work will focus on establishing impulsive models on complex networks and studying the effect of impulsive immunization on computer virus propagation.
The work is supported by the National Natural Science Foundation of China under Grant no. 61304117, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant no. 13KJB520008, the doctorate teacher support project of JiangSu Normal University under Grant no. 12XLR021.
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