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.


Introduction
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 [1] proposed the first epidemiological model of computer viruses.From then on, much effort has been done in developing virus spreading models [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].On the other hand, it was found [16][17][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  ≤ 3 for most real-world networks.This finding has greatly stimulated the interest in understanding the impact of network topology on virus spreading [16][17][18][19][20][21][22][23][24][25][26][27][28][29].
Recently, Mishra and Jha [2] 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.According to the above assumptions, the following model is derived (see Figure 1): (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 networkbased 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 .
The organization of this paper is as follows.Section 2 determines the equilibria of system (3) and the basic reproduction number  0 .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.

Basic Reproduction Number and Equilibria
The basic reproduction number  0 , 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 ⟨ 2 ⟩ stands for the second origin moment of the node degree, ⟨ 2 ⟩ := ∑   2 ().Then, we have the following theorem.
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.

Stability of the Virus-Free Equilbrium
It is clear that  0 = ( 4 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 0, 0, . . ., 0) is the virus-free equilibrium of system (3).In this section, we will prove that virus-free equilibrium is globally asymptotically stable when  0 < 1.

Proof. The characteristic equation with respect to 𝑃
We obtain This equation has negative roots − and − with multiplicity  and negative roots −, −, and − with multiplicity  − 1.Now let Suppose  0 < 1.Then, ( + ) − (⟨ 2 ⟩/⟨⟩) > 0 and it follows from the Hurwitz criterion that all roots of the characteristic equation have negative real parts, implying that  0 is locally asymptotically stable.Now, assume  0 > 1.Then, ( + ) − (⟨ 2 ⟩/⟨⟩) < 0 and the characteristic equation has exactly one positive root, implying that  0 is a saddle point.
Lemma 4 (see [16]).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 .
Proof.Ω consists of the following 5Δ sets: as their respective outer normal vectors.For 1 ≤  ≤ , we have Thus, the claimed result follows from Lemma 4.
We are ready to prove.

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:  = 0.04,  = 0.8,  = 0.8,  = 0.5,  = 0.2,  = 0.4, which runs on a scale-free network with    = 1000 and  = 2.4.In this case, we have  0 = 0.8825 < 1.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:  = 0.2,  = 0.8,  = 0.8,  = 0.5,  = 0.2,  = 0.4, which runs on a scale-free network with  = 1000 and  = 2.4.In this case, we have  0 = 4.4124 > 1.The time plots of the four relative densities are plotted in Figure 4, from which it can be seen that the virus would persist.

Conclusions
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  0 of the model has been calculated.The global asymptotic stability of the virus-free equilibrium has been shown when  0 is below one, and the global attractivity of the viral equilibrium has been proved if  0 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.

Figure 3 :
Figure 3: Global stability of infection-free solution.

Figure 5
Figure 5 demonstrates how () evolves with time.It can be seen that smaller exponent  favors virus spreading.