Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 925648,17 pages doi:10.1155/2012/925648

Research Article

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

Maobin Yang, Zhufan Zhang, Qiang Li, and Gang Zhang

School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Correspondence should be addressed to Maobin Yang, mbyang0808@gmail.com Received 5 July 2012; Revised 20 August 2012; Accepted 23 August 2012 Academic Editor: Yanbing Liu

Copyright © 2012 Maobin Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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. [1720] 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: 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 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 rate ¡. Also, internal computers are disconnected from the Internet at the same rate ¡.

P f\PS(L +B) eL ^^ yb UK

©———»GH1^ i ^ ^ I

pS pL pB pK

Figure 1: State transition diagram for the SLBRS model.

(A3) Each virus-free computer gets contact with an infected computer at a bilinear incidence rate ¡5S(L + B), where ¡5 is positive constant.

(A4) Latent computers break out at nonnegative constant rate e.

(A5) Breaking-out computers are cured at nonnegative constant rate j.

(A6) Recovered computers become susceptibly virus-free again at nonnegative constant rate a.

According to the above assumptions, the transfer diagram is depicted in Figure 1. Therefor, the corresponding computer virus model is of the form:

S = u - ¡5S(L + B) + aR - ¡S, L = 5S(L + B) - eL - uL,

B = eL - jB - uB, R = jB - aR - uR.

Because S + L + B + R = 1, system (2.1) simplifies to the following three-dimensional subsystem:

S = ¡(1 - S) - ¡5S(L + B)+ a(1 - L - B - S),

L = ¡5S(L + B) - (e + u) L, (2.2)

B = eL - (j + u)B,

on the closed, positively invariant set Q = {(SrL,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 region Q, in which the usual existence, uniqueness of solutions, and continuation results hold. Clearly, the system (2.2) always has the virus-free equilibrium:

E0 = (S0,L0 ,B0) = (1,0,0). (2.3)

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

= F-V,

ßS(L + B) 0

(ß + e) L -eL + (ß + y) B

Let F be the Jacobian matrix of F at E0, and V the Jacobian matrix of V at E0, respectively, then we get

ß ß 0 0

(ß + e) 0 -e ß + J

The spectral radius R0 of the matrix K = FV 1 is exactly the basic reproduction number of the model, that is, 0. Hence

R0 = ß

(ß + e)(ß + r)

Further, system (2.2) also has an interior equilibrium called viral equilibrium given by

E„ = (S„, L„, B„),

(ß + y)(ß + e) = ß(ß + y + e) R0'

L* = K (ß+y) (ß+a, B» = K*e(ß + a),

and K* = ((p + e)(n + j)(R0 - 1))/(p(p + j + e)[j(e + a + ¡¿) + (p + e)(p + j)])- E* = (S*,L*,B*) e Q' = Q — {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) If R0 < 1, then E0 is the only equilibrium and it is globally stable in Q. (b) If R0 > 1, then E0 is unstable and there exists a unique vrial equilibrium E,. Furthermore, all solutions starting in Q and sufficiently close to E0 move away from E0 if R0 > 1.

Proof. Consider a Lyapunov function:

V (L,B) = 2

L2 + (h + e)(h + Y) B2

e(h + y + e)

Clearly, V is positively definite. By calculation, we get

dv ~dt

= PSL2 + pSLB - h + e)L2 + (h + + Y)

e(h + y + e)

[eL - (h + a)]B

(h + e)(h + y)

h + Y + e

(L + B)L-ihl±(L - h^B h + Y + e

Because R0 < 1, that is, ¡(h + Y + e) < (h + e)(h + Y). It can be seen that dV/dt < 0 holds for (S,L,B) e Q. Furthermore, dV/dt = 0 if and only if (L,B) = (0,0). Besides, V(L,B) ^ to as L ^ to or B ^ to. It follows from LaSalle invariance principle [34] that E0 is globally asymptotically stable with respect to Q if R0 < 1. This proves claim (a). We linearize the system (2.2) at E0, giving the Jacobian matrix:

'p - h - e ¡ 0 e -h - Y 0

Y -h -

The corresponding eigenvalues of JE0 are

li = -h - a,

12 = ^-Y - e - 2h + p + y/¡2 + 2p(e + y) + (e - y)'

13 = -Y - e - 2h + p p2 + 2p(e + y) + (e - y):

Obviously 11 < 0 and 12 > 0 if R0 > 1 which implies p(h + y + e) > h2 + hY + he + eY. 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 equilibrium E, is locally asymptotically stable with respect to Q' if R0 > 1.

Proof. The Jacobian matrix of the linearized system of system (2.2) evaluated at E, is

-p(L, + B, - S,) - ¡i - e -p(L, + B, - S,) -p(L, + B,)

-p - a

The characteristic equation of JE, is p(1) - I3 + a212 + a11 + a0 - 0, where

V S, B,\

a2 - plL, + B, + )

eYL, + pe(L, + B,)

B, Y r, B L2 + B2L + S B2

a1 - pYe

(R, + L, + B,)(L, + B,)

We have

a2ai - ao - p2e

(L, + B, )3 S,(L, + B,)2

L2 + B,L, + S, B, L2 RB

(l2 + b,l, + s,b,) (b,l2 + b2 + b,2l, + s,b2)

„ 2 L2(L, + 2B,) ^ 2 B, (B,Ll + B2l, + S,B2)

Pe2--+ py —

+ eY2 r + pe^B, + peYS* + e2Y-R

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 that M - {(S, 0,0) : 0 < S < 1} is the maximum invariant set on the boundary of By Theorem 3.1, all orbits started from the interior of ^ will not get away from M if R0 > 1. Furthermore, the stable set of M, MS - {P e Q : w(P) c M}, is equal to M and on the boundary of Q. Then, we can derive the following proposition.

Proposition 3.3. If R0 > 1, the system (2.2) is uniformly persistent.

System (2.2) is said to be uniformly persistent in Q, or rather there exists constant c such that

lim inf S(t) > c, lim inf L(t) > c, lim inf B(t) > c,

t^^ t^^ t^^

lim inf [1 - S(t) - L(t) - B(t)] > c.

provided (S(0),L(0), B(0)) e Q' [9, 36-38]. Here, c is independent of initial conditions.

The uniform persistence of (2.2) in the bounded set O is equivalent to the existence of a compact K e O' that is absorbing for (2.2), namely, each compact set K e O' satisfies x(t,x0) c K for sufficiently large t, where x(t,xo) denotes the solution of (2.2) such that x(0,x0) = xo [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 O' if R0 > 1.

Proof. From the Appendix A, we know that system (2.2) satisfies the following assumptions.

(H1) There exists a compact absorbing set K e O'.

(H2) System (2.2) has a unique equilibrium E, e O'.

Let X = (S,L,B) and F(X) denote the vector field of system (2.2), the Jacobian matrix J = dF/dX associated with a general solution x(t) of system (2.2) is

-p(L + B) - (p + a) -pS - a -pS - a p(L + B) pS - (p + e) pS 0 e -(P + i).

(3.10)

and its second additive compound Jacobian matrix J[2] [39, 40] is

J [2] =

-p(L + B - S) - (2p + e + a)

pS + a -pS - a

-p(L + B)

-(2P + e + r) p(L + B) pS - (2p + e + y)

(3.11)

We consider the matrix-valued function P(X) = P(S,L,B) as

P (S,L,B) =

ai 0 0'

0 (1 - a2) B 0

0 a2 L B

(3.12)

1 < a1 < 1 +

a(1 - c) + 2pc

and 0 < a2 < 1. Here, c is the uniform persistence constant in (3.8). Then,

(3.13)

PfP-1 =

0 0 0 L B

0 0 L " B.

(3.14)

and the matrix B = PfP-1 + PJ[2]P-1 in (A.4) can be written in block form:

B11 B12 B21 B22

(3.15)

Bn = [~p(L + B - S) - (2p + e + a)},

a16SB - a1a2(IS + a)B XB

-LCT-OT)- a^IS + a)L

1 - a2 L

a1 B a2 L

■ a1 B

% -l(L + B) - (2p + y + a) + a2(pS + a)

(a2 - 1)(IS + a)

I(L + B) - a2(fS + a) + —— e % - a2(|S + a) + IS - (2p + y + e) 1 - a2

(3.16)

where % donates L/L - B/B.

Let z = (u, v, w) denote the vectors in R3, we select a norm in R3 as

\(u,v,w)\ = max{|u|, |v| + |w|},

(3.17)

and let p denote the Lozinskii measure with respect to this norm. As described in [41], we have the estimate:

p(B) < sup jgig = sup jp1(Bn) + №121^1^22) + |B21Q.

(3.18)

Here, |B211, |B12| are matrix norms with respect to the L1 vector norm, and denotes the Lozinskii measure with respect to the L1 norm. We thus obtain

g1 = p.1(Bn) + | B121

= - I(L + B - S) - (2p + e + a)

f B a1a2 B, s B

+ max^ a1lb— - a--—, a1 (IS + a) —

L 1 - a2 L L

g2 = | B211 + (B22)

L B e B . = L - B + * L - ^ + Y + ^

a2e 1 - a2

.2(1 - a2)(IS + a) - e

(3.19)

(3.20)

Discrete Dynamics in Nature and Society Rewriting (2.2), we have

L = pS + psB - (p + £), (3.21)

B=4 - (p+y )• (3.22)

The uniform persistence constant c in (3.8) can be adjusted so that there exists T > 0 independent of x(0) e K the compact absorbing set, such that

S(t) > c, L(t) > c, B(t) > c for t>T. (3.23)

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,

gi = pi(Bn) + |Bi21

= — -p - a - p(L + B) -pSB + ai(pS + a)b

p, (3.24)

g2 ^ l - p - a + ta~a2

< — - p.

Therefore, p(B) < L/L - p for t>T by (3.18) and (3.24). Along each solution x(t,xo) to (2.2) such that x0 e K and for t > T, we thus have

T{ p(B)ds < T p(B)ds + T log —T - p, (3.25)

which implies that

- 1 f p q2 = limsupsup- p(B(x,x0))ds <- < 0,

t^^ x0eQ t J0 2

(3.26)

from (3.18), proving Theorem 3.4. □

Consider system (2.2) with parameter values shown in Table 1. For R0 < 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.

Table 1: Typical R0 and corresponding parameter values for system (2.2).

R0 P e Y a P 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 0.2

1 ^-1-'-'- 1

0.9 0.9

R 0.8 / R 0.8

P ra 0.7 e ti 0.7

CQ 0.6 CQ 0.6

i-i i4

CO 0.5 C/i 0.5

"o 0.4 ts 0.4

e 0.3 e 0.3

> 0.2 > 0.2

0.1 IV 0.1

100 200 300 400 Time t

200 400 600 800 Time t

Figure 2: (a) Evolutions of S, 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.

For R0 > 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.

4. Discussions

As was indicated in the previous section, it is critical to take various actions to control the system parameters so that R0 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 R0 to four system parameters: ¡5, e, j, and p, respectively. Following Arriola and Hyman [42], the normalized forward sensitivity indices with respect to ¡5, e, j, and p are calculated, respectively, as follows:

dR0/R0 ¡5 p + Y + e

dp/p R0 (p + Y)(p + e)

= 1 > 0,

400 600 Time t

800 1000

400 600 Time t

800 1000

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

400 600 Time t

400 600 Time t

Figure 3: (a) Evolutions of S, 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 R° = 109.0909.

dRVR0 = g dR0 de/e = R0 de

ÔR°/R° y ÔR°

(p + y + e)(p + e) -ej

dj/j R0 dj (p + y + ^(p + y) dR0/R0 p ÔR° -p[j(p + y) + (P + Y + e)(p + j)]

dp/p R0 dp

(p + ^ (p + j(p + Y + e))

It can be seen that, among these four parameters, R0, in proportion with ¡5, is the most sensitive to the change in ¡5. As opposed to this, the other three parameters e, y, and p have

an inversely proportional relationship with R0, an increase in e or y, or p 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 parameter ¡5 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, e, and the cure rate of breaking out computers, y, 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, p, 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 when R0 > 1 utilizes a general approach established in [43], and relies on the construction of a new Lyapunov function for the second compound system.

Appendices

A. The General Mathematical Framework of Geometric Approach to Global Stability

The presentation here follows that in [22]. Let x ^ f (x) e R" be a C1 function for x in an open set D c R". Consider the differential equation:

x' = f (x).

Denote by x(t,x0) the solution to (A.1) such that x(0,x0) = x0. We make the following two assumptions.

(H1) There exists a compact absorbing set K c 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 C1 local perturbations of f at xi e D if, for sufficiently small e > 0 and neighborhood U of xi, it is also satisfied by g e C1(D ^ Rn) such that the support (f - g) c U and |f - g|C1 < e, where

Such g will be called local e-perturbations of f at x1. It is easy to see that the classical Bendixson's condition f (x) < 0 for n = 2 is robust under C1 local perturbations of f at each x1 e R2. Bendixson criteria for higher dimensional systems that are C1 robust are discussed in [21, 22, 44].

A point x0 e D is wandering for (A.1) if there exists a neighborhood U of x0 and T > 0 such that U n 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 C1 closing lemma of Pugh [45, 46] as stated in [43] .

Lemma A.1. Let f e C1(D ^ Rn). Suppose that x0 is a nonwandering point of (A.1) and that f (x0) / 0. Also assume that the positive semiorbit of x0 has compact closure. Then, for each neighborhood U of x0 and e> 0, there exists a C1 local e-perturbation g of f at x0 such that

(1) sup p(f - g) c U and

(2) the perturbed system x' = g(x) has a nonconstant periodic solution whose trajectory passes through x0.

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 C1 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, w(x0) = {x} for all x0 e D since x is the only equilibrium in D.

A method of deriving a Bendixson criterion in R" is developed in [35]. The idea is to show that the second compound equation,

z'(t) = f (x(i,xc))z(i), (A.3)

with respect to a solution x(t,x0) c D to (A.1), is uniformly asymptotically stable, and the exponential decay rate of all solutions to (A.3) is uniform for xo in each compact subset of D. Here, df[2]/dx is the second additive compound matrix of the Jacobian matrix df/dx; see Appendix B. It is an (") x (") matrix, and thus (A.3) is a linear system of dimension ("). If D is simply connected, the above-mentioned stability property of (A.3) implies the exponential decay of the surface area of any compact two-dimensional surface in D, which in turn precludes the existence of any invariant simple closed rectifiable curve in D, including periodic orbits. The required uniform asymptotic stability of the linear system (A.3) can be proved by constructing a suitable Lyapunov function.

Let x ^ p(x) be an (") x (") matrix-valued function that is C1 for x e D. Assume that P(x) exists and is continuous for x e K, the compact absorbing set.

B = PfP-1 + Pdf[2] P-\ (A.4)

where the matrix Pf is obtained by replacing each entry pij of P by its derivative in the direction of f, pijf. Let Z be a suitable vector norm for z e R", N = ("), and let p(B) be the Lozinskii measure of B with respect to the induced matrix norm | -1 in R", defined by

p(B) = lim 11 + hB1- 1 ■ (A.5)

' h ^ o+ h

Define a quantity q2 as

- 1 (t

q2 = limsupsup- p(B(x(s,x0)))ds. (A.6)

t^>X> xoeQ t Jo

It is shown in [22] that if D is simply connected and q2, the function |P(x)z| is a Lyapunov function for (A.3), and hence (A.1) has no orbit that gives rise to an invariant simple closed rectifiable curve, such as periodic orbits, homoclinic orbits, and heteroclinic cycles. Hence, q2 < 0 is a Bendixson criterion for (A.1) in D. Moreover, it is robust under C1 local perturbations of f near any nonequilibrium point that is nonwandering. In particular, the following global-stability result is proved in Theorem 3.5 of [22].

Theorem A.3. Assume that D is simply connected and that the assumptions (H1), (H2) hold. Then, the unique equilibrium x of (A.1) is globally stable in D if q2 < 0.

We remark that, under the assumptions of Theorem A.3, the condition q2 < 0 also implies the local stability of x, since, assuming the contrary, x is both the alpha and the omega limit point of a homoclinic orbit that is ruled out by the condition q2 < 0.

B. The Second Additive Compound Matrix

Let A be a linear operator on R" and also denote its matrix representation with respect to the standard basis of RN . Let A2R" denote the exterior product of R". A induces canonically a linear operator A[2] on A2R": for p1,p2 e R", define

A[2] (pi A p2) := A(pi) A p2 + A A(p2),

and extend the definition over A2R" by linearity. The matrix representation of A[2] with respect to the canonical basis in A2R" is called the second additive compound matrix of A. This is an (") x (2) matrix and satisfies the property (A + B)[2] = A[2] + B[2]. In the special case, when n = 2, we have Aj,22 = tr 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 = (aij) is

an + a22

a32 . -a3i

a23 aii + a33

a22 + a33

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].

Acknowledgment

This work is partially supported by the National Natural Science Foundation of China (Grant no. 61071195) and Chongqing Municipal Natural Science Foundation (Grants no. CSTC2009BA2090, CSTCjjA40014 and CSTC, 2010bb2417).

References

[1] K. Ashton, "That 'Internet of Things' Thing," In: RFID Journal, Abgerufen, 2009, Abgerufen, 2011.

[2] T. Chen and J. -M. Robert, "The evolution of viruses and worms," in Statistical Methods in Computer Security, 2004.

[3] F. Cohen, A Short Course on Computer Viruses, John Wiley & Sons, New York, NY, USA, 2nd edition, 1994.

[4] F. Cohen, "Computer virus: theory and experiments," Computers & Security, vol. 6, no. 1, pp. 22-35, 1987.

[5] "An architectural approach towards the future internet of things," in Architecting the Internet of Things, U. Dieter, H. Mark, and F. Michahelles, Eds., p. 8, Springer, Berlin, Germany, 2011.

[6] A. Solomon, "A brief history of PC viruses," Computer Fraud & Security Bulletin, vol. 1993, no. 12, pp. 9-19, 1993.

[7] P. Szor, The Art of Computer Virus Research and Defense, Addison-Wesley Professional, 2005.

[8] J. von Neumann and A. Burks, Theory of Self-Reproducing Automata, U. of Illinois Press, Urbana, Ill, USA, 1966.

[9] X. Han and Q. Tan, "Dynamical behavior of computer virus on Internet," Applied Mathematics and Computation, vol. 217, no. 6, pp. 2520-2526, 2010.

[10] B. K. Mishra and N. Jha, "Fixed period of temporary immunity after run of anti-malicious software on computer nodes," Applied Mathematics and Computation, vol. 190, pp. 1207-1212, 2007.

B. K. Mishra and N. Jha, "SEIQRS model for the transmission of malicious objects in computer network," Applied Mathematical Modelling, vol. 34, no. 3, pp. 710-715, 2010.

B. K. Mishra and S. K. Pandey, "Dynamic model of worms with vertical transmission in computer network," Applied Mathematics and Computation, vol. 217, no. 21, pp. 8438-8446, 2011. B. K. Mishra and S. K. Pandey, "Fuzzy epidemic model for the transmission of worms in computer network," Nonlinear Analysis, vol. 11, pp. 4335-4341, 2010.

J. R. C. Piqueira and V. O. Araujo, "A modified epidemiological model for computer viruses," Applied Mathematics and Computation, vol. 213, no. 2, pp. 355-360, 2009.

J. Ren, X. Yang, L.-X. Yang, Y. Xu, and F. Yang, "A delayed computer virus propagation model and its dynamics," Chaos, Solitons & Fractals, vol. 45, no. 1, pp. 74-79, 2012.

J. Ren, X. Yang, Q. Zhu, L.-X. Yang, and C. Zhang, "A novel computer virus model and its dynamics," Nonlinear Analysis, vol. 13, no. 1, pp. 376-384, 2012.

L.-X. Yang, X. Yang, L. Wen, and J. Liu, "A novel computer virus propagation model and its dynamics," International Journal of Computer Mathematics. In press.

L.-X. Yang, X. Yang, L. Wen, and J. Liu, "Propagation behavior of virus codes in the situation that infected computers are connected to the Internet with positive probability," Discrete Dynamics in Nature and Society, vol. 2012, Article ID 693695, 13 pages, 2012.

L.-X. Yang, X. Yang, Q. Zhu, and L. Wen, "A computer virus model with graded cure rates," Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 414-422, 2013.

X. Yang and L.-X. Yang, "Towards the epidemiological modeling of computer viruses," Discrete Dynamics in Nature and Society, In press.

R. A. Smith, "Some applications of Hausdorff dimension inequalities for ordinary differential equations," Proceedings of the Royal Society of Edinburgh A, vol. 104, no. 3-4, pp. 235-259,1986. M. Y. Li and J. S. Muldowney, "A geometric approach to global-stability problems," SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070-1083,1996.

M. Y. Li and J. S. Muldowney, "On R. A. Smith's autonomous convergence theorem," The Rocky Mountain Journal of Mathematics, vol. 25, no. 1, pp. 365-379,1995.

M. Y. Li, "Dulac criteria for autonomous systems having an invariant affine manifold," Journal of Mathematical Analysis and Applications, vol. 199, no. 2, pp. 374-390,1996.

M. Y. Li, "Periodic solutions to a class of autonomous systems," in Differential Equations and Applications to Biology and to Industry, M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme, Eds., pp. 299-308, World Scientific, Singapore, 1996.

J. Arino, C. C. McCluskey, and P. van den Driessche, "Global results for an epidemic model with vaccination that exhibits backward bifurcation," SIAM Journal on Applied Mathematics, vol. 64, no. 1, pp. 260-276, 2003.

B. Buonomo, A. d'Onofrio, and D. Lacitignola, "Global stability of an SIR epidemic model with information dependent vaccination," Mathematical Biosciences, vol. 216, no. 1, pp. 9-16, 2008. B. Buonomo and D. Lacitignola, "On the dynamics of an SEIR epidemic model with a convex incidence rate," Ricerche di Matematica, vol. 57, no. 2, pp. 261-281, 2008.

M. Y. Li and J. S. Muldowney, "Global stability for the SEIR model in epidemiology," Mathematical Biosciences, vol. 125, no. 2, pp. 155-164,1995.

M. Y. Li, H. L. Smith, and L. Wang, "Global dynamics an SEIR epidemic model with vertical transmission," SIAM Journal on Applied Mathematics, vol. 62, no. 1, pp. 58-69, 2001. L. Wang and M. Y. Li, "Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells," Mathematical Biosciences, vol. 200, no. 1, pp. 44-57, 2006.

E. Beretta, F. Solimano, and Y. Takeuchi, "Negative criteria for the existence of periodic solutions in a class of delay-differential equations," Nonlinear Analysis, vol. 50, no. 7, pp. 941-966, 2002. B. Buonomo and D. Lacitignola, "General conditions for global stability in a single species population-toxicant model," Nonlinear Analysis, vol. 5, no. 4, pp. 749-762, 2004.

J. P. LaSalle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1976.

J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.

G. J. Butler and P. Waltman, "Persistence in dynamical systems," Proceedings of the American Mathematical Society, vol. 96, pp. 425-430, 1986.

H. I. Freedman, S. G. Ruan, and M. X. Tang, "Uniform persistence and flows near a closed positively invariant set," Journal of Dynamics and Differential Equations, vol. 6, no. 4, pp. 583-600,1994.

P. Waltman, "A brief survey of persistence in dynamical systems," in Delay Differential Equations and Dynamical Systems, vol. 1475, pp. 31-40, Springer, Berlin, Germany, 1991.

[39] M. Fiedler, "Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices," Czechoslovak Mathematical Journal, vol. 24, no. 99, pp. 392-402,1974.

[40] J. S. Muldowney, "Compound matrices and ordinary differential equations," The Rocky Mountain Journal of Mathematics, vol. 20, no. 4, pp. 857-872,1990.

[41] R. H. Martin, Jr., "Logarithmic norms and projections applied to linear differential systems," Journal of Mathematical Analysis and Applications, vol. 45, pp. 432-454,1974.

[42] L. Arriola and J. Hyman, "Forward and adjoint sensitivity analysis with applications in dynamical systems," Lecture Notes in Linear Algebra and Optimization, 2005.

[43] M. W. Hirsch, "Systems of differential equations that are competitive or cooperative. VI. A local C closing lemma for 3-dimensional systems," Ergodic Theory and Dynamical Systems, vol. 11, no. 3, pp. 443-454,1991.

[44] M. Y. LI and J. S. Muldowney, "On R.A. Smith's autonomous convergence theorem," Rocky Mountain Journal of Mathematics, vol. 25, pp. 365-379,1995.

[45] C. C. Pugh, "The closing lemma," American Journal of Mathematics, vol. 89, pp. 956-1009,1967.

[46] C. C. Pugh and C. Robinson, "The C1 closing lemma, including Hamiltonians," Ergodic Theory and Dynamical Systems, vol. 3, no. 2, pp. 261-313,1983.

Copyright of Discrete Dynamics in Nature & Society is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.