Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 842614, 5 pages http://dx.doi.org/10.1155/2013/842614

Research Article

Modeling Computer Virus and Its Dynamics

Mei Peng,1 Xing He,2 Junjian Huang,3 and Tao Dong4

1 College of Mathematical and Computer Science, Yangtze Normal University, Chongqing 400084, China

2 College of Computer Science, Chongqing University, Chongqing 400030, China

3 School of Computer Science, Chongqing University of Education, Chongqing 400067, China

4 College of Software and Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Correspondence should be addressed to Mei Peng; pmgs@qq.com

Received 27 March 2013; Accepted 9 June 2013 Academic Editor: Tingwen Huang

Copyright © 2013 Mei Peng 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.

Based on that the computer will be infected by infected computer and exposed computer, and some of the computers which are in suscepitible status and exposed status can get immunity by antivirus ability, a novel coumputer virus model is established. The dynamic behaviors of this model are investigated. First, the basic reproduction number R0, which is a threshold of the computer virus spreading in internet, is determined. Second, this model has a virus-free equilibrium P0, which means that the infected part of the computer disappears, and the virus dies out, and P° is a globally asymptotically stable equilibrium if R0 < 1. Third, if R0 >1 then this model has only one viral equilibrium P', which means that the computer persists at a constant endemic level, and P* is also globally asymptotically stable. Finally, some numerical examples are given to demonstrate the analytical results.

1. Introduction

Computer virus is a malicious mobile code which including virus, Trojan horses, worm, and logic bomb. It is a program that can copy itself and attack other computers. And they are residing by erasing data, damaging files, or modifying the normal operation. Due to the high similarity between computer virus and biological virus [1], various computer virus propagation models are proposed [2-4]. This dynamical modeling of the spread process of computer virus is an effective approach to the understanding of the behavior of computer viruses because on this basis, some effective measures can be posed to prevent infection.

The computer virus has a latent period, during which individuals are exposed to a computer virus but are not yet infectious. An infected computer which is in latency, called exposed computer, will not infect other computers immediately; however, it still can be infected. Based on these characteristics, delay is used in some models of computer virus to describe that although the exposed computer does not infect other computers, it still has infectivity [5, 6]. Yang et al. [7, 8] proposed an SLB and SLBS models; in these models, the authors considered that the computer virus has

latency, and the computer also has infectivity in the period of latency. However, they do not show the length of latency and take into account the impact of artificial immunization ways such as installing antivirus software. And the newly entered in the internet from the susceptible status to exposed status, the contact rate is the same as that of susceptible status entering into infected status. In this paper, a novel model of computer virus, known as SEIR model, is put forward to describe the susceptible computer which can be infected by the other infected or exposed computer and come into the exposed status. In the SEIR model, based on artificial immunity, we consider the bilinear incidence rate for the latent and infection status. Assume that the computers which newly entered the internet are susceptible, the computers correspond with exposed computers, and their adequate contact rate is denoted by , and computers also correspond with infected computers, and their adequate contact rate is denoted by p2. So, the fraction of the computer which newly entered the internet will enter the class R by anti-virus software; the fraction of computers contact with exposed and infected computer will stay latent before becoming infectious and enter the class E. It is shown that the dynamic behavior of the proposed model is determined by a threshold R0, and this

model has a virus-free equilibrium P0, and P0 is a globally asymptotically stable equilibrium if R0 < 1; if R0 > 1 this model has only one viral equilibrium P*, and it is globally asymptotically stable.

This paper is organized as follows. Section 2 formulates a novel computer virus mode. Section 3 proves the global stability of the virus-free equilibrium. Section 4 discusses the stability of the viral equilibrium. In Section 5, numerical simulations are given to present the effectiveness of the theoretic results. Finally, Section 6 summarizes this work.

2. Model Formulation

At any time, a computer is classified as internal and external depending on weather it is connected to internet or not. At that time, all of the internet computers are further categorized into four classes: (1) susceptible computers, that is, uninfected computers and new computers which connected to network; (2) exposed computers, that is, infected but not yet broken-out; (3) infectious computers; (4) recovered computers, that is, virus-free computer having immunity. Let S(t), E(t), I(t), R(t) denote their corresponding numbers at time t, without ambiguity; S(t), E(t), I(t), R(t) will be abbreviated as S, E, I, R, respectively. The model is formulated as the following system of differential equations:

S' = (1- p)N- p1SI - p2SE -pS- pS, E' = p1SI + p2SE -kE- aE- pE,

i' = aE - ri - pi, R' = pS + kE + ri, N(t) = S(t)+E(t)+i(t) + R(t).

We may see that the first three equations in (1) are independent of the fourth equation, and therefore, the fourth equation can be omitted without loss of generality. Hence, system (1) can be rewritten as

S' = A- ß1SI - ß2SE - aS, E' = ß1SI + ß2SE - bE, i' = aE - ci.

Therefore,

a = p + p, b = k + a + p, c = r + p, (1 - p)N = A,

where N denotes the rate at which external computers are connected to the network; p denotes the recovery rate of susceptible computer due to the anti-virus ability of network; k denotes the recovery rate of exposed computer due to the anti-virus ability of network; p1 denotes the rate at which, when having a connection to one infected computer, one susceptible computer can become exposed but has not broken-out; p2 denotes the rate of which, when

having connection to one exposed computer, one susceptible computer can become exposed; a denotes the rate of the exposed computers cannot be cured by anti-virus software and broken-out; r denotes the recovery rate of infected computers that are cured; p denotes the rate at which one computer is removed from the network. All the parameters are nonnegative.

Moreover, all feasible solutions of the system (3) are bounded and enter the region D, where

D = {(S, E, i) e R+ | S > 0:

>0, E>0, i>0, S + E + i<-

Referring to [9], we define the basic reproduction number of the infection as

A(ß1a + ß2c) a b c

For system (3), there always exists the virus-free equilibrium which is P (A/a, 0, 0); if R0 > 1, then there also exists a viral equilibrium P*(S*, E*, I*). Therefore,

A(Ro -1)

i* = Aa(Ro -1) bcRn .

3. The Virus-Free Equilibrium and Its Stability

Theorem 1. P0 is locally asymptotically stable if R0 < 1. Whereas P0 is unstable if R0 > 1.

Proof. The characteristic equation of (3) at P0 is given by

~ßiS\

0 \-(p2S-b) piS \ 0 a X + c J

which equals to

(X + a) [X2 - (p2S0-b-c)X-bc(R0 - 1)] = 0. Then, (9) has negative real part characteristic roots: Xi = -a,

(p2S -b-c)±

ß2S -b - c < 0.

When R0 < 1, there are no positive real roots of (9) and thus P0 is a local asymptotically stable equilibrium. While R0 > 1,(9) has positive real roots, which means P0 is unstable. The proof is completed. □

Theorem 2. P0 is globally asymptotically stable with respect to D if R0 < 1.

Proof. Let L = ((fac + p2a)/bc)E + ¡32I/c. Obviously

= {ßlc + ß2a)^l + ß2 ^

(ßlC + ß2a) be

(ßlC + ß2<x) be

(ß1SI + ß2SE -bE)+'-+ (cxE - cl) (ßlI + ß2 E)S-(ßiC:ß2a)bE

+ ~2T - ß2l

= (ßlC + ß2«) bc

= (ßlE + ß2l) = (ßlE + ß2l)

(ßiI + ß2E)S-ßiE-ß2l

(ßlC + ß2 bc

1 A R0abc

be a A

= (fiiE + ^2l)(R0 -1)<0

The proofis completed.

4. The Viral Equilibrium and Its Stability

Theorem 3. P* is locally asymptotically stable if R0 > 1. Proof. The Jacobin matrix of system (3) about P* is given by

-ß2S* -ßiS*\

a(R0 -1) ß2S* -b ßiS \ 0 a -c J

which equals to

f (X) = a0X3 + alX2 + a2X + a3 = 0, (15)

ao = 1,

al = aR0 - (ß2S* - b-c)>0, a2 = abR0 + acR0 - aß2S* > abR0 + acR0 - a (b + c) = a (b + c) (R0 - 1) > 0,

ß2S* <b + c, a3 = abc (R0 - 1) > 0.

A l = al > 0,

al 1 a3 a2

al 1 0 as a2 a~i 0 0a3

= aia2 - a3 > 0,

= a3 (aia2 - a3) > 0.

According to the Hurwitz criterion, all roots of (15) have negative real pats. Thus, the claimed result follows. The proof is completed. □

The following result can be proved in the same way (see [9]).

Theorem 4. P* is uniquely globally asymptotically stable if

R0 >1.

Proof. The Jacobin matrix of system (3) about P* is given by

- a R0

-ß2S* -ßiS*\

a(R0 -1) ß2S* -b ßiS \ 0 a -c )

The second compound matrix J^2 of the Jacobin matrix can be calculated as follows (see [10,11]):

j[2] =

/-aR0 + ß2S* -b ßiS* ßiS* \ -aR0 - c -ß2S*

'(R0-1) ß2S-b-cJ

(14) Set P as the following diagonal matrix:

P(X) = (1,J,J

Denote that

-1 j ( E' I' E' I'

PfP 1 = diag (0,---,---

f E IE I

Therefore, the matrix B = P^P 1 +PJ^P 1 can be written in the following block form:

ß = ( ß11 B12

ß21 ß22

ß11 = -aR0 + ß2S* - b, ßl2 = ißiS* (1,1),

E t ß2i = ~(a,0)T,

ß22 =

-ß2S*

iE' Ï

1 e-1-( 0 + c

y a(Ro-1) E-tJ + (ßiS*-b-c) )

Pl (ß11) = ß2S* -aR0

Pi (ß22) = max -

E' i' „

----aR0 - c + aR0 - a,

Ei 0 0

E - Y + ß2S * -b - c - ß2S*

----c - a.

The vector norm || • || in R3 = R( 21 is choosen as \\(u, v, w)|| = max [lu\, |v + .

The Lozinskii measure p(B) with respect to || • || is as follows (see [12]):

p(ß) < sup{01,02},

01 = (ß11) + \ß12\ = ß2S* -aR0 -b+ iß1S

02 = P1 (ß22) + \ß21 \ = -r - -r -a-c+ -a.

From (3), we find that

Eß1S* = E-ß2S* + b,

Ï E — = — a - c, I I

E' „

01 = e- a^

70 60 50 ^ 40

■if 30 20 10 0

0 2 4 6 8 10 12 14 16 18 20 Time (t)

-S(t) E(t) -I(t)

Figure 1: Dynamical behavior of system (3). Time series of susceptible, exposed, and infectious computers S(t), E(t), I(t) with R0 > 1.

Relations (28)-(30) imply that

, N E'

p (ß) < e - a.

H *<\Î (¥

1 {t(E' -a) dr=- ln ^-a. (32) t h\E J t E (0)

If R0 > 1, then the virus-free equilibrium is unstable by Theorem 1. Moreover, the behavior of the local dynamic near D0 as described in Theorem 1 implies that the system (3) is uniformly persistent in D; that is, there exists a constant c1 > 0 and T > 0, such that t > T implies that

lim inf S (t) > c1,

lim inf E (t) > c1,

t ^ OT

lim inf I (t) > c1,

t ^ OT

lim inf [1-S(t)-E(t)-I(t)] > cv

For all (S(0),E(0),I(0) e D) (see [13,14]),

q= lim sup sup- I p(B)dr x£K t J0

(29) The proofis complete.

5. Numerical Examples

< — <0. 2

(34) □

For the system (3), Theorem 2 implies that the virus dies out if R0 < 1, and Theorem 4 implies that the virus persists if R0 >1. Now, we present two numerical examples. (30) Let p = 0.5, p = 0.02, k = 0.4, a = 0.6, r = 0.6, N =

100, p = 0.7, p2 = 0.8, then R0 = 13.8 > 1 and pS* <b + c; Figure 1 shows the solution of system (3) when R0 > 1. We

0 50 100 150 200 250 300 350 400 450 500 Time (t)

-S(t) E(t) -I(t)

Figure 2: Dynamical behavior of system (3). Time series of susceptible, exposed, and infectious computers S(t), E(t), I(t) with

can see that the viral equilibrium P* of system (3) is globally asymptotically stable.

Let p = 0.7, p = 0.001, k = 0.02, a = 0.09, r = 0.04, N = 10, fa = 0.002, fa = 0.003, then R0 = 0.1808 < 1 and faS° < b + c; Figure 2 shows the solution of system (3) when R0 <1. We can see that the virus-free equilibrium P of the system (3) is globally asymptotically stable.

6. Conclusion

We assume that the virus process has a latent period and in these times the infected computers have infectivity also. A compartmental SEIR model for transmission of virus in computer network is formulated. In this paper, the dynamics of this model have been fully studied.

The results show that we should try our best to make R0 less than 1. The most effective way is to increase the parameters p, k, r and decrease fa, fa, a and so on. Maybe in such way, the computer virus can be well predicted and thus controlled.

[3] 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.

[4] 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.

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

[6] Q. Zhu, X. Yang, and J. Ren, "Modeling and analysis of the spread of computer virus," Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 5117-5124, 2012.

[7] 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.

[8] L. X. Yang, X. Yang, L. Wen, and J. Liu, "A novel computer virus propagation model and its dynamics," International Journal of Computer Mathematics, vol. 89, no. 17, pp. 2307-2314, 2012.

[9] P. van den Driessche and J. Watmough, "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission," Mathematical Biosciences, vol. 180, pp. 29-48, 2002.

[10] M. Fiedler, "Additive compound matrices and an inequality for eigenvalues of symmetric stochastic matrices," Czechoslovak Mathematical Journal, vol. 24(99), pp. 392-402,1974.

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

[12] G. Butler, H. I. Freedman, and P. Waltman, "Uniformly persistent systems," Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 425-430, 1986.

[13] 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. 583600, 1994.

[14] P. Waltman, "A brief survey of persistence in dynamical systems," in Delay Differential Equations and Dynamical Systems (Claremont, CA, 1990), S. Busenberg and M. Martelli, Eds., vol. 1475, pp. 31-40, Springer, Berlin, Germany, 1991.

Ro < 1.

Acknowledgment

The work described in this paper was supported by the Science and Technology Project of Chongqing Education Committee under Grant KJ130519.

References

[1] C. Sun and Y.-H. Hsieh, "Global analysis of an SEIR model with varying population size and vaccination," Applied Mathematical Modelling, vol. 34, no. 10, pp. 2685-2697, 2010.

[2] L.-P. Song, Z. Jin, and G.-Q. Sun, "Modeling and analyzing of botnet interactions," Physica A, vol. 390, no. 2, pp. 347-358,2011.

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