# A New Model for Capturing the Spread of Computer Viruses on Complex-Networks

Authors: Chunming Zhang; Tianliang Feng; Yun Zhao; Guifeng Jiang
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

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.


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## Body

## 1. 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 [1–15]. 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 k other nodes follows a power-law of the form P(k)~k-τ, with the remarkable feature that τ≤3 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 [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. LetS, E, I, Q, and R denote the numbers of susceptible, exposed, infected, quarantine, and recovered computers, respectively.(H3)
New computers are attached to the Internet at rateA.(H4)
Computers are disconnected from the Internet naturally at a constant rated and removed with probability α due to the attack of malicious objects.(H5)
S computers become E with constant rate ρ; R computers become S with constant rate η; E computers become I with constant rate μ; I computers become Q with constant rate δ; I computers become R with constant rate γ; Q computers become R with constant rate ε.According to the above assumptions, the following model is derived (see Figure1):
(1)S′(t)=A-ρSI-dS+ηR,E′(t)=ρSI-(d+μ)E,I′(t)=μE-(d+α+γ+δ)I,Q′(t)=δI-(α+ε+d)Q,R′(t)=γI+εQ-(d+η)R.Figure 1
Original model.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,P(k)~k-τ, where P(k) stands for the probability that a node chosen randomly from the Internet is of degree k.(H7)
The total number of nodes does not change or, equivalently,A=0, d=0, and α=0.(H8)
Sk(t): the relative density of k-degree S-nodes; Ek(t): the relative density of k-degree E-nodes; Ik(t): the relative density of k-degree I-nodes; Qk(t): the relative density of k-degree Q-nodes; Rk(t): the relative density of k-degree R-nodes; Sk(t)+Ek(t)+Ik(t)+Qk(t)+Rk(t)=1.(H9)
The probability that a link has anI-node as one endpoint does not depend on the degree of the other endpoint of the link and, hence, is only a function of I(t):=(I1(t),I2(t),…,In(t)). Let Θ(I(t)) denote the probability, Θ(I(t))=(1/〈k〉)∑kkP(k)Ik, where, 〈k〉:=∑kkP(k).By applying the mean-field technique to the above assumptions, we get a new epidemic model of computer virus, which is formulated as (see Figure2)
(2)Sk′(t)=-kρΘ(t)Sk(t)+ηRk(t),Ek′(t)=kρΘ(t)Sk(t)-μEk(t),Ik′(t)=μEk(t)-(γ+δ)Ik(t),Qk′(t)=δIk(t)-εQk(t),Rk′(t)=γIk(t)+εQk(t)-ηRk(t),k=1,…,n,Figure 2
Our model.with initial conditionsSk(0), Ek(0), Ik(0), Qk(0), and Rk(0)≥0, 1≤k≤n.Note that, for everyk, we have Sk(t)+Ek(t)+Ik(t)+Qk(t)+Rk(t)=1; thus, the first set of equations in system (2) can be removed, yielding the following system,
(3)Ek′(t)=kρΘ(t)(1-Ek(t)-Ik(t)-Qk(t)-Rk(t))-μEk(t),Ik′(t)=μEk(t)-(γ+δ)Ik(t),Qk′(t)=δIk(t)-εQk(t),Rk′(t)=γIk(t)+εQk(t)-ηRk(t),k=1,…,n,with initial conditionsEk(0), Ik(0), Qk(0), Rk(0)≥0 and Ek(0)+Ik(0)+Qk(0)+Rk(0)≤1.The organization of this paper is as follows. Section2 determines the equilibria of system (3) and the basic reproduction number R0. 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 numberR0, which can be explained as the average number of secondary infections produced by a single infected node during its infection time, is calculated as
(4)R0=〈k2〉〈k〉ργ+δ,where〈k2〉 stands for the second origin moment of the node degree, 〈k2〉:=∑kk2P(k). Then, we have the following theorem.Theorem 1.
Consider system (3). The following assertions hold.
(
1
) There always exists a virus-free equilibrium P0=(0,0,…,0︷4n)T.
(
2
) There is no viral equilibrium if R0≤1.
(
3
) There exists a unique viral equilibrium
(5)P*=(E*,I*,Q*,R*)P*=(E1*,…,En*,I1*,…,In*,Q1*,…,Qn*,R1*,…,Rn*)T,
ifR0≤1, where
(6)Ek*=γ+δμkx1+(k-1)xΔ,Ik*=kx1+(k-1)xΔ,Qk*=δεkx1+(k-1)xΔ,Rk*=γ+δηkx1+(k-1)xΔ,Δ=γ+δμ+δε+γ+δη+1,
x is the unique positive root of the equation
(7)f(x)=ρ∑k[k2P(k)1+(k-1)xΔ][1-Δx]-γ-δ=0.Proof.
After imposing the stationarity condition, we have(8)kρΘ(t)(1-Ek(t)-Ik(t)-Qk(t)-Rk(t))-μEk(t)=0,μEk(t)-(γ+δ)Ik(t)=0,δIk(t)-εQk(t)=0,γIk(t)+εQk(t)-ηRk(t)=0.
It is easily verified thatP0=(0,0,…,0︷4n)T is always a root of this system. Solving the system, we get
(9)Ek*=γ+δμkx1+(k-1)xΔ,Ik*=kx1+(k-1)xΔ,Qk*=δεkx1+(k-1)xΔ,Rk*=γ+δηkx1+(k-1)xΔ,
wherex is the unique positive root of the equation
(10)f(x)=ρ∑k[k2P(k)1+(k-1)xΔ][1-Δx]-γ-δ=0.
IfR0≤1, we have E*=I*=Q*=R*=0, implying that f(x)=0 and, thus, (10) has no positive roots. Hence, assertion (3) holds. Now, assume R0>1. The observations that (a) f(0)>0, (b) f′(x)<0 for x≥0, and (c) f(+∞)>0 imply that (10) has a unique positive root. Hence, assertion (8) also holds.Remark 2.
It can be seen from Theorem1 that E1*<E2*<⋯<En*,I1*<I2*<⋯<In*,Q1*<Q2*<⋯<Qn* and R1*<R2*<⋯<Rn*. This shows that, when in the steady state P*, 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 thatP0=(0,0,…,0︷4n) is the virus-free equilibrium of system (3). In this section, we will prove that virus-free equilibrium is globally asymptotically stable when R0<1.For convenience, let(11)Ω={x=(x1,x2,…,x4n)∣xi≥∀1≤i≤4n,xi+xi+n+xi+2n+xi+3n≤1∀1≤i≤n}.Letx(t)=(E(t),I(t),Q(t),R(t))T and rewrite system (3) in matrix-vector notation as
(12)x(t)=Ax(t)+H(x(t)),with initial conditionx(0)∈Ω, where(13)A=[A110A21A22](4n×4n),A11=[-μ0⋯0p(1)ρ〈k〉2p(2)ρ〈k〉⋯np(n)ρ〈k〉0-μ⋯02p(1)ρ〈k〉4p(2)ρ〈k〉⋯2np(n)ρ〈k〉⋮⋮⋮⋮⋮⋮00⋯-μnp(1)ρ〈k〉2np(2)ρ〈k〉⋯n2p(n)ρ〈k〉μ0⋯0-(γ+δ)0⋯00μ⋯00-(γ+δ)⋯0⋮⋮⋮⋮⋮⋮00⋯μ00⋯-(γ+δ)](2n×2n),A21=[00⋯0δ0⋯000⋯00δ⋯0⋮⋮⋮⋮⋮⋮00⋯000⋯δ00⋯0γ0⋯000⋯00γ⋯0⋮⋮⋮⋮⋮⋮00⋯000⋯γ](2n×2n),A22=[-ε0⋯000⋯00-ε⋯000⋯0⋮⋮⋮⋮⋮⋮00⋯-ε00⋯0ε0⋯0-η0⋯00ε⋯00-η⋯0⋮⋮⋮⋮⋮⋮00⋯ε00⋯-η](2n×2n),H(x(t))=-ρΘ(E1(t)+I1(t)+Q1(t)+R1(t),…,n(En(t)+In(t)+Qn(t)+Rn(t)),0,…,0︷3n).Theorem 3.
Consider system (12); P0=(0,0,…,0︷4n) is locally asymptotically stable if R0<1, whereas P0 is a saddle point if R0<1.Proof.
The characteristic equation with respect toP0 is
(14)det(λE4n-A)=det(λE2n-A110-A21λE2n-A22)det(λE4n-A)=det(λE2n-A11)(λE2n-A22)=0.
We obtain(15)det(λE4n-A)=(λ+η)n(λ+ε)n(λ+μ)n-1×(λ+δ+γ)n-1[-λ-μρ〈k2〉〈k〉μ-λ-γ-δ]=(λ+η)n(λ+ε)n(λ+μ)n-1(λ+δ+γ)n-1×((λ+μ)(λ+γ+δ)-μρ〈k2〉〈k〉)=(λ+η)n(λ+ε)n(λ+μ)n-1(λ+δ+γ)n-1×(λ2+(γ+δ+μ)λ+μ(γ+δ)-μρ〈k2〉〈k〉)=0.
This equation has negative roots-η and -ε with multiplicity n and negative roots -μ, -δ, and -γ with multiplicity n-1. Now let
(16)g(λ)=λ2+(γ+δ+μ)λ+μ(γ+δ)-μρ〈k2〉〈k〉=0.
SupposeR0<1. Then, (γ+δ)-ρ(〈k2〉/〈k〉)>0 and it follows from the Hurwitz criterion that all roots of the characteristic equation have negative real parts, implying that P0 is locally asymptotically stable. Now, assume R0>1. Then, (γ+δ)-ρ(〈k2〉/〈k〉)<0 and the characteristic equation has exactly one positive root, implying that P0 is a saddle point.Lemma 4 (see [16]).
Consider a systemdx/dt=f(x) defined at least in a compact set C. Then, C is invariant if, for every point y on ∂C, the vector f(y) is tangent to or pointing into C.Lemma 5.
The setΩ is positively invariant for system (12). That is, x(0)∈Ω implies x(t)∈Ω for all t>0.Proof.
∂
Ω consists of the following 5Δ sets:
(17)Si={x∈Ω∣xi=0},Ti={x∈Ω∣xi+n=0},Ui={x∈Ω∣xi+2n=0},Vi={x∈Ω∣xi+3n=0},Wi={x∈Ω∣xi+xi+n+xi+2n+xi+3n=1},
which have(18)φi=(0,…,0,-1i,0,…,0),ςi=(0,…,0,-1i+n,0,…,0),ξi=(0,…,0,-1i+2n,0,…,0),ψi=(0,…,0,-1i+3n,0,…,0),ζi=(0,…,0,1i,0,…,0,1i+n,0,…,0,1i+2n,0,…,0,1i+3n,0,…,0)
as their respective outer normal vectors. For1≤i≤n, we have
(19)(dxdt|x∈Si·φi)=-iρ∑kkp(k)〈k〉(1-xi+n-xi+2n-xi+3n)≤0,(dxdt|x∈Ti·ςi)=-μxi≤0,(dxdt|x∈Ui·ξi)=-δxi+n≤0,(dxdt|x∈Vi·ψi)=-(γxi+n+εxi+2n)≤0,(dxdt|x∈Wi·ζi)=-ηxi+3n≤0.
Thus, the claimed result follows from Lemma4.Lemma 6 (see [16]).
Consider ann-dimensional autonomous system
(20)dx(t)dt=Ax(t)+H(x(t)),x∈D,
whereA is an irreducible n×n matrix, D is a region containing the origin, H(x)∈C1(D), and limx→0∥H(x)∥/∥x∥=0. Assume there exist, a positively invariant compact convex set C⊂D containing the origin, a positive number r, and a real eigenvector ω of AT, such that(C1)
(x,ω)≥r∥x∥ for all x∈C,(C2)
(H(x),ω)≤0 for all x∈C,(C3)
the origin forms the largest positively invariant set included inN={x∈C∣(H(x),ω)=0}.
Then, one has that(1)
s(AT)<0 implies that the origin is globally asymptotically stable in C, and(2)
s(AT)>0 implies that there exists m>0 such that, for each x0∈C-{0}, the solution ϕ(t,x0) to system (12) satisfies limt→∞inf∥ϕ(t,0)∥≥m.We are ready to prove.Theorem 7.
Consider system (12). Then, P0 is globally asymptotically stable in Ω, if R0<1.Proof.
LetC=Ω and look at (12). As matrix AT is irreducible and all of its nondiagonal entries are nonnegative, it follows from [13] that AT has a positive eigenvector ω=(ω1,…,ω4n) corresponding to its eigenvalue s(AT). Let ω0=miniωi>0. Then, for all x∈Ω, we have
(21)(x,ω)≥ω0∑i=14nxi≥ω0(∑i=14nxi2)1/2=ω0∥x∥,(H(x),ω)=-ρΘ∑i=1niωi(xi+xi+n+xi+2n+xi+3n)≤0.
Moreover,(H(x),ω)=0 implies that x=0. 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.
IfR0>1, then the infection solution of (12) P*=(P1*,P2*,…,P4n*︷4n) is globally attractive in Ω-{0}.Proof.
Theorem3 ensures the existence of the viral equilibrium. We need to prove that if R0>1, there is a unique constant equilibrium P* in Ω-{0}. Let x*=P*, Assume that x=x*>0 and y=y*>0 are two constant solutions of (12) in Ω-{0}. If x*≠y*, then there exists i0, i0=1,2,…,4n, such that xi0*≠yi0*, where xi0* is the i0th component of the vector x*. Without loss of generality, assume xi0*>yi0*, and xi0*/yi0*>xi*/yi* for all i=1,…,4n. Since x* and y* are constant solutions of (12), we substitute them into (12). And if 1≤i0≤n, we obtain,
(22)kρΘ(x*)(1-xi0*-xi0+n*-xi0+2n*-xi0+3n*)-μxi0*=kρΘ(y*)(1-yi0*-yi0+n*-yi0+2n*-yi0+3n*)-μyi0*=0,
whereΘ(x*)=(1/〈k〉)∑kkP(k)xi*.
After equivalent deformation, it follows that(23)kρΘ(x*)(1-xi0*-xi0+n*-xi0+2n*-xi0+3n*)yi0*xi0*-μyi0*=kρΘ(y*)(1-yi0*-yi0+n*-yi0+2n*-yi0+3n*)-μyi0*=0.
Butxi0*/yi0*>xi*/yi* for all i and
(24)(1-xi0*-xi0+n*-xi0+2n*-xi0+3n*)yi0*xi0*<(1-yi0*-yi0+n*-yi0+2n*-yi0+3n*).
Thus, from the above inequality, we get(25)kρΘ(x*)(1-xi0*-xi0+n*-xi0+2n*-xi0+3n*)yi0*xi0*<kρΘ(y*)(1-yi0*-yi0+n*-yi0+2n*-yi0+3n*).
This is a contradiction. Similarly, we can also get contradictions whenn+1≤i0≤2n, 2n+1≤i0≤3n, and 3n+1≤i0≤4n. Therefore, there exists a unique constant solution P*=(P1*,P2*,…,P4n*) of (3) in Ω-{0}. Now, we shall prove that x* is globally attractive in Ω-{0}. To find the asymptotic behavior of the solutions of (12) in Ω, we define the following functions, F:Ω→R and f:Ω→R for P∈Ω, where F(x)=maxi(xi/xi*), f(x)=mini(xi/xi*), F(x) and f(x) are continuous and the right-hand derivative exists along solutions of (12). Let x=x(t) be a solution of (12), we may assume that F(x(t))=xi0(t)/xi0*, 1≤i0≤4n, t∈[t0,t0+ε] for a given t0 and a sufficiently small ε>0. Then,
(26)F′(x(t0))=xi0′(t0)xi0*,t∈[t0,t0+ε],
whereF′ is defined as
(27)F′=limh→0+supF(x(t+h))-F(x(t))h.
If1≤i0≤n, from (12) we have
(28)xi0*xi0′(t0)xi0(t0)=xi0*xi0(t0)kρΘ(xi0+n(t0))×(1-xi0(t0)-xi0+n(t0)-xi0+2n(t0)-xi0+3n(t0))-μxi0*.
And we obtain(29)xi0*xi0′(t0)xi0(t0)=μxi0*xi0(t0)xi0-n(t0)-(γ+δ)xi0*,whenn+1≤i0≤n;xi0*xi0′(t0)xi0(t0)=δxi0*xi0(t0)xi0-n(t0)-εxi0*,when2n+1≤i0≤3n;xi0*xi0′(t0)xi0(t0)=γxi0*xi0(t0)xi0-2n(t0)+εxi0*xi0(t0)xi0-n(t0)-ηxi0*,when3n+1≤i0≤4n.
According to the definition ofF(x(t)), we have
(30)xi0(t0)xi0*≥xi(t0)xi*,i=1,2,…,4n.
Then ifF(x(t0))>1, we obtain
(31)xi0*xi0′(t0)xi0(t0)≤kρΘ(xi0+n*)(1-xi0*-xi0+n*-xi0+2n*-xi0+3n*)xi0*xi0′(t0)xi0(t0)-μxi0*=0,xi0*xi0′(t0)xi0(t0)≤μxi0-n*-(γ+δ)xi0*=0,xi0*xi0′(t0)xi0(t0)≤δxi0-n*-εxi0*=0,
or(32)xi0*xi0′(t0)xi0(t0)≤γxi0-2n*+εxi0-n*-ηxi0*=0.
Sincexi0*>0 and xi0(t0)>0, we conclude that xi0′(t0)>0. Therefore, if F(x(t0))>1, F′(x(t0))<0.
Similarly, we can testify thatF(x(t0))=1 imples F′(x(t0))≤0, and f(x(t0))<1 implies f′(x(t0))>0. If f(x(t0))1, then f′(x(t0))≥0. Denote
(33)U(x)=max{F(x)-1,0},V(x)=max{1-f(x),0}.
BothU(x) and V(x) are continuous and non-negative for x∈Ω. Notice that U′(x(t))≤0, V′(x(t))≤0. Let
(34)HU={x∈Ω∣U′(x)=0},HV={x∈Ω∣V′(x)=0},
then we have(35)HU={x∣0≤xi≤xi*},HV={x∣xi*≤xi≤1}∪{0}.
According to the LaSalle invariant set principle, any solution of (12) starting in Ω will approach HU∩HV. And HU∩HV={x*}∪{0}. But if x(t)≠0, by Lemma 6 we know that limt→∞inf∥x(t)∥≥m>0. Then we conclude that any solution x(t) of (12), such that x(0)∈Ω-{0}, satisfies limt→∞x(t)=x*, so x=P* is globally attractive in Ω-{0}.Conjecture 9.
Consider system (12) and suppose R0>1. Then the infection equilibrium, P*, is globally asymptotically stable in Ω-{0}.


## 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: ρ=0.04, μ=0.8, γ=0.8, ε=0.5, δ=0.2, η=0.4, which runs on a scale-free network with n=1000 and τ=2.4. In this case, we have R0=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.Figure 3
Global stability of infection-free solution.To demonstrate the global attractivity of the viral equilibrium of system (3), we take the following set of parameter values: ρ=0.2, μ=0.8, γ=0.8, ε=0.5, δ=0.2, η=0.4, which runs on a scale-free network with n=1000 and τ=2.4. In this case, we have R0=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.Figure 4
Global attractivity of infection solution.Consider system (12) with ρ=0.2, μ=0.8, γ=0.8, ε=0.5, δ=0.2, η=0.4 and n=1000. For τ∈{2.0,2.2,2.4,2.6}, Figure 5 demonstrates how I(t) evolves with time. It can be seen that smaller exponent τ favors virus spreading.Figure 5
Evolution ofI(t) for different τ values.


## 6. 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 numberR0 of the model has been calculated. The global asymptotic stability of the virus-free equilibrium has been shown when R0 is below one, and the global attractivity of the viral equilibrium has been proved if R0 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.



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