Research Article Analysis of a Model for Computer Virus ...

Research ArticleAnalysis of a Model for Computer Virus Transmission

Peng Qin1,2

1 Institute of Signal Capturing and Processing Technology, North University of China, Taiyuan 030051, China2School of Computer Science and Control Engineering, North University of China, Taiyuan 030051, China

Correspondence should be addressed to Peng Qin; [email protected]

Received 28 May 2015; Revised 23 August 2015; Accepted 25 August 2015

Academic Editor: Ivanka Stamova

Copyright © 2015 Peng Qin. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Computer viruses remain a significant threat to computer networks. In this paper, the incorporation of new computers to thenetwork and the removing of old computers from the network are considered. Meanwhile, the computers are equipped withantivirus software on the computer network. The computer virus model is established. Through the analysis of the model, disease-free and endemic equilibriumpoints are calculated.The stability conditions of the equilibria are derived. To illustrate our theoreticalanalysis, some numerical simulations are also included. The results provide a theoretical basis to control the spread of computervirus.

1. Introduction

With the rapid development of computer, communication,and network technology, network information system hasbecome an important way of the development of countriesand industries, amongst others. Information security hasbecome one of the most important and challenging issuesfaced in the age of information sharing. The computer virusis one common information security threat.

Computer virus is not only destructive, but also highlycontagious. Once the virus is copied or generates variety,its speed is difficult to be controlled. Infectivity is thebasic characteristic of the virus. In biology, the virus candiffuse from one organism to another. Cohen, Kephart andWhite pointed out that the spread mechanisms of computerviruses and biological viruses have many similarities [1, 2].Under appropriate conditions, biological viruses canmultiplyquickly, and the infected organisms exhibit symptoms oreven die. Similarly, the computer virus can also spread touninfected computers from the infected computer throughmany kinds of ways. In some cases, the infected computersdo not work and even get paralysed. Unlike biologicalviruses, the computer virus is a software program designed toreplicate itself and spread to other machines. Computer virusenters the computer and gets executed; it will search for otherprograms or storagemedia in line with the conditions of their

infection and target and then insert the code itself, achievingthe purpose of self-reproduction. As long as a computer isinfected (if not promptly treated), the viruswill spread rapidlyon this computer, in which a large number of files (usually anexecutable file) will be infected. The infected file has becomea new source of infection and then will exchange data withother machines or over the network exposure; the virus willcontinue to be contagious. A computer virus can enter yourcomputer in any number of ways, such as via mobile harddisk, via an email attachment, during file downloads from theInternet, or even upon a visit to a contaminated web site. Bythe time you find the virus that infected the computer, themobile hard disk which is often used on this computer hasbeen infected with the virus. Other computers connected tothe machine network might be infected with the virus. Thenetwork has no permanent immunity to the computer virus.Therefore, there is always computer virus.

Kephart et al. [2, 3] study the spread of computer virususing the biological virus model; they mainly focus on theinfluence of network topology on the spread of computervirus. In the homogeneousmixing nodes and only susceptiblenode input cases, Mishra et al. [4, 5] establish the mathe-matical model of Internet spread of computer virus. Theyanalyze the propagation law of virus using threshold theoryof infectious disease dynamics and predict the developmenttrend of computer virus. Piqueira et al. [6] study the effects

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 720696, 10 pageshttp://dx.doi.org/10.1155/2015/720696

2 Mathematical Problems in Engineering

of equipping antivirus software on the virus in computernetwork. They prove the stability of the disease-free equilib-rium and endemic equilibrium without incorporating newcomputers and removing old computers. For more studies onvirus in computer network, see [7–13] and so on.

In this paper, the new computers are incorporated to thenetwork, and the old computers are removed from the com-puter network. Meanwhile, the computers are equipped withantivirus software on the computer network. The remainderof this paper presents the model and results. In Section 2,the mathematical model on computer virus is described,and in Section 3 we obtain the equilibriums. In Section 4,the analysis of this model is derived. Numerical simulationssupporting the theoretical analysis are given in Section 5.Thepaper ends with a conclusion and discussion in Section 6.

2. The Dynamic Model

The model proposed here is based on the compartmentalSAIR model [6–8], including an antidotal population com-partment (𝐴) representing nodes of the network equippedwith fully effective antivirus programs, a susceptible compart-ment (𝑆), an infective compartment (𝐼), and a temporarilyimmune compartment (𝑅). Connections between the com-partments represent operational parameters of the networkand their control can be used as a strategy to maintainthe reliability of the whole system, even in the presence ofinfections (see Figure 1).

The total population 𝑁 is divided into four groups. 𝑆(𝑡)denote noninfected computers subjected to possible infec-tion. 𝐼(𝑡) denote infected computers. 𝑅(𝑡) denote removedcomputers due to infection or not. 𝐴(𝑡) denote noninfectedcomputers equipped with antivirus.

The SAIR model for computer viruses propagation wasproposed and can be described by

𝑑𝑆 (𝑡)

𝑑𝑡= 𝐶 − 𝛼

𝑆𝐴𝑆𝐴 − 𝛽𝑆𝐼 − 𝜇𝑆 + 𝜎𝑅,

𝑑𝐼 (𝑡)

𝑑𝑡= 𝛽𝑆𝐼 − 𝛼

𝐼𝐴𝐼𝐴 − 𝛿𝐼 − 𝜇𝐼,

𝑑𝑅 (𝑡)

𝑑𝑡= 𝛿𝐼 − 𝜎𝑅 − 𝜇𝑅,

𝑑𝐴 (𝑡)

𝑑𝑡= 𝛼𝑆𝐴𝑆𝐴 + 𝛼

𝐼𝐴𝐼𝐴 − 𝜇𝐴,

(1)

where 𝐶 is influx rate, representing the incorporation of newcomputers to the network; 𝛽 is infection rate of susceptiblecomputers; 𝜇 is proportion coefficient for the mortality rate,not due to the virus; 𝛿 is removal rate of infected computers;𝜎 is recovering rate of removed computers, with an operatorintervention; 𝛼𝑆𝐴 is conversion of susceptible computersinto antidotal ones, occurring when susceptible computersestablish effective communication with antidotal ones andthe antidotal one installs the antivirus in the susceptible ones;𝛼𝐼𝐴

represents infected computers that can be fixed by usingantivirus programs being converted into antidotal ones.

CS

𝜇S

𝛼SASA

𝛽SI

𝛼IAIA

𝛿I

𝜇I

A

I R

𝛿R

𝜇R

Figure 1: Transfer diagram of the model.

Let𝑁 = 𝑆 + 𝐼 + 𝑅 + 𝐴; then 𝑑𝑁(𝑡)/𝑑𝑡 = 𝐶 − 𝜇𝑁. Let

Ω

= {(𝑆, 𝐼, 𝑅, 𝐴) : 𝑆, 𝐼, 𝑅, 𝐴 ≤ 0, 𝑆 + 𝐼 + 𝑅 + 𝐴 ≤𝐶

𝜇} .

(2)

Then, it is clear that Ω is a positive invariant set. Hence, wewill focus our attention only on the regionΩ.

But [6, 7] think the influx rate is considered to be 𝐶 = 0,𝜇 = 0. In fact, every day new computers can be incorporatedto the network or old computers can be removed from thenetwork. Therefore, they can not be ignored. In this paper,we analyse completely dynamical behavior of the spreadof the virus in computer network with incorporating newcomputers and removing old computers.

3. The Disease-Free and Endemic Equilibrium

When 𝐼 = 0, if 𝐴 = 0, the disease-free equilibrium of system(1) is 𝑃

1= (𝑆1, 𝐼1, 𝑅1, 𝐴1) = (𝐶/𝜇, 0, 0, 0). If 𝐴 ̸= 0, we obtain

the threshold 𝑅01= 𝐶𝛼

𝑆𝐴/𝜇2. When 𝑅

01> 1, the disease-

free equilibrium of system (1) is 𝑃2= (𝑆2, 𝐼2, 𝑅2, 𝐴2) =

(𝜇/𝛼𝑆𝐴, 0, 0, 𝐶/𝜇 − 𝜇/𝛼𝑆𝐴).In the following, we compute the positive equilibrium;

namely, 𝐼 ̸= 0.According to the third equation of system (1), we have

𝑅 =𝛿

𝜎 + 𝜇𝐼. (3)

When𝐴 = 0, using the second equation of system (1), we have

𝑆 =𝛿 + 𝜇

𝛽. (4)

According to the first equation of system (1), we obtain

𝐼 =(𝛽𝐶 − 𝜇 (𝛿 + 𝜇)) (𝜎 + 𝜇)

𝛽𝜇 (𝜎 + 𝛿 + 𝜇). (5)

Mathematical Problems in Engineering 3

Therefore, when the threshold 𝑅02= 𝛽𝐶/𝜇(𝛿 + 𝜇) > 1, we

have the positive equilibrium

𝑃3= (𝑆3, 𝐼3, 𝑅3, 𝐴3)

= (𝛿 + 𝜇

𝛽,(𝑅02 − 1) (𝛿 + 𝜇) (𝜎 + 𝜇)

𝛽 (𝜎 + 𝛿 + 𝜇),𝛿

𝜎 + 𝜇𝐼3, 0) .

(6)

When 𝐴 ̸= 0, taking advantage of the second equation ofsystem (1), we have

𝐴 =𝛽𝑆 − (𝛿 + 𝜇)

𝛼𝐼𝐴

. (7)

According to the fourth equation of system (1), we have

𝐼 =𝜇 − 𝛼𝑆𝐴𝑆

𝛼𝐼𝐴

. (8)

Substituting (7) and (8) into the first equation of system (1),we have

𝐶 − 𝛼𝑆𝐴𝑆𝛽𝑆 − (𝛿 + 𝜇)

𝛼𝐼𝐴

− 𝛽𝑆𝜇 − 𝛼𝑆𝐴𝑆

𝛼𝐼𝐴

− 𝜇𝑆

+𝜎𝛿

𝜎 + 𝜇

𝜇 − 𝛼𝑆𝐴𝑆

𝛼𝐼𝐴

= 0.

(9)

Namely,

[𝛽𝜇 + 𝛼𝐼𝐴𝜇 +𝜎𝛿𝛼𝑆𝐴

𝜎 + 𝜇− (𝛿 + 𝜇) 𝛼𝑆𝐴] 𝑆

= 𝐶𝛼𝐼𝐴 +𝜎𝛿𝜇

𝜎 + 𝜇.

(10)

Hence,

𝑆 =𝐶𝛼𝐼𝐴+ 𝜎𝛿𝜇/ (𝜎 + 𝜇)

𝛽𝜇 + 𝛼𝐼𝐴𝜇 + 𝜎𝛿𝛼

𝑆𝐴/ (𝜎 + 𝜇) − (𝛿 + 𝜇) 𝛼

𝑆𝐴

. (11)

In order to make 𝑆 > 0, it must satisfy the followingconditions:

𝛽𝜇 + 𝛼𝐼𝐴𝜇 +𝜎𝛿𝛼𝑆𝐴

𝜎 + 𝜇− (𝛿 + 𝜇) 𝛼𝑆𝐴 > 0. (12)

Namely,

𝑅03=𝛽𝜇 + 𝛼

𝐼𝐴𝜇 + 𝜎𝛿𝛼

𝑆𝐴/ (𝜎 + 𝜇)

(𝛿 + 𝜇) 𝛼𝑆𝐴

> 1. (13)

Therefore, if the threshold 𝑅03> 1, we obtain

𝑆4=𝐶𝛼𝐼𝐴+ 𝜎𝛿𝜇/ (𝜎 + 𝜇)

(𝛿 + 𝜇) 𝛼𝑆𝐴(𝑅03− 1),

𝐼4=𝜇 − 𝛼𝑆𝐴𝑆4

𝛼𝐼𝐴

,

𝐴4=𝛽𝑆4− 𝛿 − 𝜇

𝛼𝐼𝐴

.

(14)

In order to make 𝐼4, 𝐴4> 0, it must satisfy

𝑅03> 1 +𝐶𝛼𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇

𝜇 (𝜎 + 𝜇) (𝛿 + 𝜇),

𝑅03< 1 +

𝛽𝜇

(𝛿 + 𝜇) 𝛼𝑆𝐴

𝐶𝛼𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇

𝜇 (𝜎 + 𝜇) (𝛿 + 𝜇).

(15)

When 1 + (𝐶𝛼𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇) < 𝑅

03<

1 + (𝛽𝜇/(𝛿 + 𝜇)𝛼𝑆𝐴)((𝐶𝛼

𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇))

(namely, the threshold 𝑅04= 𝛽𝜇/(𝛿+𝜇)𝛼

𝑆𝐴> 1), we have the

positive equilibrium

𝑃4 = (𝑆4, 𝐼4, 𝑅4, 𝐴4)

= (𝐶𝛼𝐼𝐴+ 𝜎𝛿𝜇/ (𝜎 + 𝜇)

(𝛿 + 𝜇) 𝛼𝑆𝐴(𝑅03− 1),𝜇 − 𝛼𝑆𝐴𝑆4

𝛼𝐼𝐴

,𝛿

𝜎 + 𝜇

⋅ 𝐼4,𝛽𝑆4 − 𝛿 − 𝜇

𝛼𝐼𝐴

) .

(16)

4. Stability of Equilibrium

4.1. Stability of the Disease-Free Equilibrium. The Jacobinmatrix of system (1) at the disease-free equilibrium 𝑃1 is

𝐽 (𝑃1)

=

((((

(

−𝜇 −𝛽𝐶

𝜇𝜎 −

𝐶𝛼𝑆𝐴

𝜇

0𝛽𝐶

𝜇− 𝛿 − 𝜇 0 0

0 𝛿 − (𝜎 + 𝜇) 0

0 0 0𝐶𝛼𝑆𝐴

𝜇− 𝜇

))))

)

.

(17)

The characteristic equation of 𝐽(𝑃1) is given by

(𝜆 + 𝜇) (𝜆 + 𝑙1) (𝜆 + 𝜎 + 𝜇) (𝜆 + 𝑙2) = 0, (18)

where 𝑙1= −𝛽𝐶/𝜇 + 𝛿 + 𝜇 and 𝑙

2= −𝐶𝛼

𝑆𝐴/𝜇 + 𝜇. Therefore,

we have 𝑙1> 0 if and only if𝑅

02= 𝛽𝐶/𝜇(𝛿+𝜇) < 1, and 𝑙

2> 0

if and only if 𝑅01 = 𝐶𝛼𝑆𝐴/𝜇

2< 1. It follows from the Routh-

Hurwitz criterion that the eigenvalues have negative real partsif and only if 𝑅

01< 1 and 𝑅

02< 1. Hence, the disease-free

equilibrium 𝑃1of model (1) is locally asymptotically stable if

𝑅01< 1 and 𝑅

02< 1 and unstable if 𝑅

01> 1 or 𝑅

02> 1.

If𝑅01> 1, the disease-free equilibrium𝑃

1is unstable, and

system (1) exhibits the other disease-free equilibrium 𝑃2.


If𝑅02> 1, the disease-free equilibrium𝑃

2is unstable, and

system (1) exhibits the positive equilibrium 𝑃3.

The Jacobin matrix of system (1) at the disease-freeequilibrium 𝑃

2is

𝐽 (𝑃2) = (

−𝜇 − 𝛼𝑆𝐴𝐴2

−𝛽𝑆2

𝜎 −𝛼𝑆𝐴𝑆2

0 𝛽𝑆2 − 𝛼𝐼𝐴𝐴2 − 𝛿 − 𝜇 0 0

0 𝛿 − (𝜎 + 𝜇) 0

𝛼𝑆𝐴𝐴2

𝛼𝐼𝐴𝐴2

0 𝛼𝑆𝐴𝑆2− 𝜇

). (19)

Since 𝛼𝑆𝐴𝑆2− 𝜇 = 0, the characteristic equation of 𝐽(𝑃

2) is

given by

[𝜆 + (𝜎 + 𝜇)] [𝜆 − (𝛽𝑆2− 𝛼𝐼𝐴𝐴2− 𝛿 − 𝜇)]

⋅ [𝜆2+ (𝜇 + 𝛼

𝑆𝐴𝑆2) 𝜆 + 𝜇𝛼

𝑆𝐴𝑆2] = 0.

(20)

By calculating, we obtain that the characteristic equationhas the four eigenvalues: −(𝜎 + 𝜇), 𝛽𝑆2 − 𝛼𝐼𝐴𝐴2 − 𝛿 −𝜇,−𝜇, and−𝛼𝑆𝐴𝐴2. When 𝑅01 > 1, −𝛼𝑆𝐴𝐴2 < 0. We have𝛽𝑆2 − 𝛼𝐼𝐴𝐴2 − 𝛿 − 𝜇 < 0 if and only if 𝑅05 = (𝛽 +𝛼𝐼𝐴)𝜇2/(𝐶𝛼𝑆𝐴𝛼𝐼𝐴+ 𝛼𝑆𝐴(𝛿 + 𝜇)) < 1. Hence, the disease-free

equilibrium 𝑃2of model (1) is locally asymptotically stable if

𝑅01> 1 and𝑅

05< 1.Thedisease-free equilibrium𝑃

2ofmodel

(1) is unstable if 𝑅01> 1 and 𝑅

05> 1.

Theorem 1. (1) If 𝑅01< 1 and 𝑅

02< 1, the disease-free

equilibrium 𝑃1of system (1) is locally asymptotically stable. If

𝑅01> 1 or 𝑅

02> 1, the disease-free equilibrium 𝑃

1of system

(1) is unstable.(2) If 𝑅

01> 1, the disease-free equilibrium 𝑃

1is unstable,

and system (1) exhibits the other disease-free equilibrium 𝑃2.

(3) If 𝑅02 > 1, the disease-free equilibrium 𝑃1 is unstable,and system (1) exhibits the positive equilibrium 𝑃3.

(4) If 𝑅01 > 1 and 𝑅05 < 1, the disease-free equilibrium𝑃2 is locally asymptotically stable. If 𝑅05 > 1, the disease-freeequilibrium 𝑃2 is unstable.

4.2. Stability of the Endemic Equilibrium. In the following, westudy the stability of the endemic equilibriums.

The Jacobin matrix of system (1) at the endemic equilib-rium 𝑃3 is

𝐽 (𝑃3)

= (

−𝛽𝐼3− 𝜇 −𝛽𝑆

3𝜎 −𝛼

𝑆𝐴𝑆3

𝛽𝐼3 𝛽𝑆3 − 𝛿 − 𝜇 0 −𝛼𝐼𝐴𝐼3

0 𝛿 − (𝜎 + 𝜇) 0

0 0 0 𝛼𝑆𝐴𝑆3+ 𝛼𝐼𝐴𝐼3− 𝜇

).

(21)

Since 𝛽𝑆3 − 𝛿 − 𝜇 = 0, the characteristic equation of 𝐽(𝑃3) is

given by

(𝜆 − (𝛼𝑆𝐴𝑆3+ 𝛼𝐼𝐴𝐼3− 𝜇)) 𝐹 (𝜆) = 0, (22)

where

𝐹 (𝜆) = 𝜆3+ 𝑎1𝜆2+ 𝑎2𝜆 + 𝑎3,

𝑎1= 𝛽𝐼3+ 𝜇 + 𝜎 + 𝜇 > 0,

𝑎2 = (𝛽𝐼3 + 𝜇) (𝜎 + 𝜇) + 𝛽2𝑆3𝐼3 > 0,

𝑎3= 𝛽2𝑆3𝐼3(𝜎 + 𝜇) − 𝛽𝜎𝛿𝐼

3

= 𝛽𝐼3[(𝛿 + 𝜇) (𝜎 + 𝜇) − 𝛿𝜎] > 0.

(23)

It is easy to calculate 𝑎1𝑎2− 𝑎3> 0. Hence, the eigenvalues of

𝐹(𝜆) = 0 have negative real parts.Therefore, if the eigenvalues of the characteristic equation

in𝑃3want to have negative real parts, only 𝛼

𝑆𝐴𝑆3+𝛼𝐼𝐴𝐼3−𝜇 <

0. Namely,

𝑅02

< 1

+𝛼𝑆𝐴 (𝜎 + 𝛿 + 𝜇) (𝛿 + 𝜇) (𝛽𝜇/𝛼𝑆𝐴 (𝛿 + 𝜇) − 1)

𝛼𝐼𝐴 (𝛿 + 𝜇) (𝜎 + 𝜇)

= 𝑅07.

(24)

According to Routh-Hurwitz criterion, the epidemicequilibrium𝑃

3is locally asymptotically stable if 1 < 𝑅

02< 𝑅07

and unstable if 𝑅02> 𝑅07.

System (1) has the following limiting system:

𝑑𝐼 (𝑡)

𝑑𝑡= 𝛽(𝐶

𝜇− 𝐼 − 𝑅 − 𝐴) 𝐼 − 𝛼𝐼𝐴𝐴𝐼 − 𝛿𝐼 − 𝜇𝐼,

𝑑𝑅 (𝑡)

𝑑𝑡= 𝛿𝐼 − 𝜎𝑅 − 𝜇𝑅,

𝑑𝐴 (𝑡)

𝑑𝑡= 𝛼𝑆𝐴(𝐶

𝜇− 𝐼 − 𝑅 − 𝐴)𝐴 + 𝛼

𝐼𝐴𝐴𝐼 − 𝜇𝐴.

(25)

In the following, we prove the stability of the endemicequilibrium 𝑃

4by the limit system. The Jacobin matrix of

system (25) at the endemic equilibrium 𝑃4 is


𝐽 (𝑃4) = (

𝛽𝑆4− 𝛽𝐼4− 𝛼𝐼𝐴𝐴4− 𝜇 − 𝛿 −𝛽𝐼

4−𝛽𝐼4− 𝛼𝐼𝐴𝐼4

𝛿 − (𝜎 + 𝜇) 0

−𝛼𝑆𝐴𝐴4+ 𝛼𝐼𝐴𝐴4

−𝛼𝑆𝐴𝐴4𝛼𝑆𝐴𝑆4− 𝛼𝑆𝐴𝐴4+ 𝛼𝐼𝐴𝐼4− 𝜇

) . (26)

Because 𝛽𝑆4 − 𝛼𝐼𝐴𝐴4 − 𝜇 − 𝛿 = 0, 𝛼𝑆𝐴𝑆4 + 𝛼𝐼𝐴𝐼4 − 𝜇 = 0, theabove matrix becomes

𝐽 (𝑃4)

= (

−𝛽𝐼4

−𝛽𝐼4−𝛽𝐼4− 𝛼𝐼𝐴𝐼4

𝛿 − (𝜎 + 𝜇) 0

−𝛼𝑆𝐴𝐴4+ 𝛼𝐼𝐴𝐴4−𝛼𝑆𝐴𝐴4−𝛼𝑆𝐴𝐴4

).

(27)

The characteristic equation of 𝐽(𝑃4) is given by

𝜆3+ 𝑏1𝜆2+ 𝑏2𝜆 + 𝑏3= 0, (28)

where

𝑏1= 𝛽𝐼4+ 𝜎 + 𝜇 + 𝛼

𝑆𝐴𝐴4> 0,

𝑏2= 𝛼𝑆𝐴𝛽𝐼4𝐴4+ 𝛼𝑆𝐴𝐴4(𝜎 + 𝜇) + 𝛽𝐼

4(𝜎 + 𝜇)

− (𝛼𝑆𝐴𝐴4 − 𝛼𝐼𝐴𝐴4) (𝛽𝐼4 + 𝛼𝐼𝐴𝐼4) + 𝛿𝛽𝐼4,

𝑏3= 𝛼𝑆𝐴𝛽𝐼4𝐴4(𝜎 + 𝜇) − 𝛿𝛼

𝑆𝐴𝐴4(𝛽𝐼4+ 𝛼𝐼𝐴𝐼4)

− (𝜎 + 𝜇) (𝛼𝑆𝐴𝐴4 − 𝛼𝐼𝐴𝐴4) (𝛽𝐼4 + 𝛼𝐼𝐴𝐼4)

+ 𝛿𝛼𝑆𝐴𝛽𝐼4𝐴4.

(29)

Because of 𝛼𝐼𝐴𝐼4= 𝜇 − 𝛼

𝑆𝐴𝑆4, we have

𝑏2= 𝛼𝑆𝐴𝛽𝐼4𝐴4+ 𝛼𝑆𝐴𝐴4(𝜎 + 𝜇) + 𝛽𝐼

4(𝜎 + 𝜇)

− (𝛼𝑆𝐴𝐴4 − 𝛼𝐼𝐴𝐴4) (𝛽𝐼4 + 𝛼𝐼𝐴𝐼4) + 𝛿𝛽𝐼4

= 𝛼𝑆𝐴𝐴4(𝜎 + 𝜇) + 𝛽𝐼

4(𝜎 + 𝜇) − 𝛼

𝑆𝐴𝐴4𝛼𝐼𝐴𝐼4

+ 𝛼𝐼𝐴𝐴4(𝛽𝐼4+ 𝛼𝐼𝐴𝐼4) + 𝛿𝛽𝐼

4

= 𝛼𝑆𝐴𝐴4(𝜎 + 𝜇) + 𝛽𝐼

4(𝜎 + 𝜇)

− 𝛼𝑆𝐴𝐴4(𝜇 − 𝛼

𝑆𝐴𝑆4) + 𝛼𝐼𝐴𝐴4(𝛽𝐼4+ 𝛼𝐼𝐴𝐼4)

+ 𝛿𝛽𝐼4

= 𝛼𝑆𝐴𝐴4𝜎 + 𝛽𝐼

4(𝜎 + 𝜇) + 𝛼

𝑆𝐴𝐴4𝛼𝑆𝐴𝑆4

+ 𝛼𝐼𝐴𝐴4(𝛽𝐼4+ 𝛼𝐼𝐴𝐼4) + 𝛿𝛽𝐼

4> 0.

𝑏3= 𝛼𝑆𝐴𝛽𝐼4𝐴4(𝜎 + 𝜇) − 𝛿𝛼

𝑆𝐴𝐴4(𝛽𝐼4+ 𝛼𝐼𝐴𝐼4)

− (𝜎 + 𝜇) (𝛼𝑆𝐴𝐴4− 𝛼𝐼𝐴𝐴4) (𝛽𝐼4+ 𝛼𝐼𝐴𝐼4)

+ 𝛿𝛼𝑆𝐴𝛽𝐼4𝐴4

= 𝛼𝐼𝐴𝐴4𝐼4(𝜎 + 𝜇) (𝛽 + 𝛼

𝐼𝐴) − 𝛿𝛼

𝑆𝐴𝐴4𝛼𝐼𝐴𝐼4

− (𝜎 + 𝜇) 𝛼𝑆𝐴𝛼𝐼𝐴𝐴4𝐼4.

(30)

It is easy to calculate 𝑏1𝑏2−𝑏3 > 0.Therefore, in order tomake𝑏3 > 0, only there is 𝑅06 = (𝜎+𝜇)(𝛽+𝛼𝐼𝐴)/𝛼𝑆𝐴(𝜎+𝛿+𝜇) > 1.Hence, if the threshold𝑅06 = (𝜎+𝜇)(𝛽+𝛼𝐼𝐴)/𝛼𝑆𝐴(𝜎+𝛿+𝜇) >1, the endemic equilibrium 𝑃4 is locally asymptotically stable.

Theorem 2. (1) If 1 < 𝑅02< 𝑅07, the positive equilibrium

𝑃3is locally asymptotically stable. If 𝑅

02> 𝑅07, the positive

equilibrium 𝑃3is unstable.

(2) If 1 + (𝐶𝛼𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇) < 𝑅

03<

1 + (𝛽𝜇/(𝛿 + 𝜇)𝛼𝑆𝐴)((𝐶𝛼

𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇))

and 𝑅06= (𝜎 + 𝜇)(𝛽 + 𝛼

𝐼𝐴)/𝛼𝑆𝐴(𝜎 + 𝛿 + 𝜇) > 1, the positive

equilibrium 𝑃4 is locally asymptotically stable. If 𝑅06 = (𝜎 +𝜇)(𝛽 + 𝛼𝐼𝐴)/𝛼𝑆𝐴(𝜎 + 𝛿 + 𝜇) < 1, the positive equilibrium 𝑃4 isunstable.

5. Numerical Simulation

In this section, we will perform a series of numerical simula-tions to verify the mathematical analysis. In Figure 2, when𝑅01< 1 and 𝑅

02< 1, the disease-free equilibrium 𝑃

1is

asymptotically stable, where the initial values are 𝑆 = 50,𝐼 = 20, 𝑅 = 1, and 𝐴 = 1. The parameters are 𝐶 = 1,𝛼𝑆𝐴= 0.00045, 𝛽 = 0.05, 𝜇 = 0.05, 𝜎 = 0.8, 𝛼

𝐼𝐴= 0.0025, and

𝛿 = 0.96.If 𝑅01> 1 and 𝑅

05< 1, Figure 3 signifies that the disease-

free equilibrium 𝑃2of system (1) is asymptotically stable. The

initial values are 𝑆 = 50, 𝐼 = 20,𝑅 = 1, and𝐴 = 1. Choose theparameters as follows:𝐶 = 6, 𝛼

𝑆𝐴= 0.0045, 𝛽 = 0.1, 𝜇 = 0.05,

𝜎 = 0.8, 𝛼𝐼𝐴= 0.0025, and 𝛿 = 0.8.

If 1 < 𝑅02< 𝑅07, the positive equilibrium 𝑃

3of system (1)

is asymptotically stable (see Figure 4). The initial values are𝑆 = 50, 𝐼 = 20, 𝑅 = 1, and 𝐴 = 1. Choose the parameters asfollows: 𝐶 = 1, 𝛼𝑆𝐴 = 0.00045, 𝛽 = 0.2, 𝜇 = 0.05, 𝜎 = 0.8,𝛼𝐼𝐴 = 0.0025, and 𝛿 = 0.96.

If 1 + (𝐶𝛼𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇) < 𝑅03 < 1 +(𝛽𝜇/(𝛿+𝜇)𝛼𝑆𝐴)((𝐶𝛼𝐼𝐴(𝜎+𝜇)+𝜎𝛿𝜇)/𝜇(𝜎+𝜇)(𝛿+𝜇)) and𝑅06 =(𝜎 + 𝜇)(𝛽 + 𝛼𝐼𝐴)/𝛼𝑆𝐴(𝜎 + 𝛿 + 𝜇) > 1, the positive equilibrium𝑃4 of system (1) is asymptotically stable (see Figure 5). Theinitial values are 𝑆 = 50, 𝐼 = 20, 𝑅 = 1, and 𝐴 = 1. Choosethe parameters as follows: 𝐶 = 10, 𝛼

𝑆𝐴= 0.00045, 𝛽 = 0.2,

𝜇 = 0.05, 𝜎 = 0.8, 𝛼𝐼𝐴= 0.0025, and 𝛿 = 17.

6. Conclusion

This paper mainly considers the incorporation of new com-puters to the network, the removal of old computers fromthe network, the computer equipped with antivirus software,and so forth. They affect the spread of the virus. The modelfor computer virus transmission is established. Through theanalysis of the model, two disease-free and two positiveequilibriums are obtained. The stability conditions of theequilibriums are derived.


20 40 60 80 1000Time t

S(t)

15

20

25

30

35

40

45

50S(t)

(a)

0

5

10

15

20

25

I(t)

20 40 60 80 1000Time t

I(t)

(b)

0

5

10

15

20

25

R(t)

20 40 60 80 1000Time t

R(t)

(c)

0

0.2

0.4

0.6

0.8

1

1.2

1.4A(t)

20 40 60 80 1000Time t

A(t)

(d)

0

5

10

15

20

25

30

35

40

45

50

20 40 60 80 1000Time t

S(t)

I(t)

R(t)

A(t)

(e)

1020

3040

50

S(t)

0

5

10

15

20

25

R(t)

3020

I(t)10

0

(f)

Figure 2: 𝑅01< 1 and 𝑅

02< 1. The disease-free equilibrium 𝑃

1of system (1) is asymptotically stable.


5

10

15

20

25

30

35

40

45

50S(t)

20 40 60 80 100 1200Time t

S(t)

(a)

0

5

10

15

20

25

30

35

40

45

I(t)

20 40 60 80 100 1200Time t

I(t)

(b)

0

5

10

15

20

25

30

35

40

45

R(t)

20 40 60 80 100 1200Time t

R(t)

(c)

0

20

40

60

80

100

120A(t)

20 40 60 80 100 1200Time t

A(t)

(d)

R(t)

A(t)

0

20

40

60

80

100

120

20 40 60 80 100 1200Time t

S(t)

I(t)

(e)

010

2030

4050

020

4060

S(t)

I(t)

0

10

20

30

40

50

R(t)

(f)

Figure 3: 𝑅01> 1 and 𝑅

05< 1. The disease-free equilibrium 𝑃



5

10

15

20

25

30

35

40

45

50S(t)

20 40 60 80 1000Time t

S(t)

(a)

0

5

10

15

20

25

30

35

40

I(t)

20 40 60 80 1000Time t

I(t)

(b)

0

5

10

15

20

25

30

R(t)

20 40 60 80 1000Time t

R(t)

(c)

0

0.2

0.4

0.6

0.8

1

1.2

1.4A(t)

20 40 60 80 1000Time t

A(t)

(d)

0

5

10

15

20

25

30

35

40

45

50

20 40 60 80 1000Time t

S(t)

I(t)

R(t)

A(t)

(e)

010

2030

4050

0

10

2030

40

S(t)

I(t)

0

5

10

15

20

25

30

R(t)

(f)

Figure 4: 1 < 𝑅02< 𝑅07. The positive equilibrium 𝑃



40

50

60

70

80

90

100

110S(t)

20 40 60 80 1000

Time t

S(t)

(a)

0

1

2

3

4

5

6

7

8

9

10

I(t)

20 40 60 80 1000

Time t

I(t)

(b)

0

20

40

60

80

100

120

R(t)

20 40 60 80 1000

Time t

R(t)

(c)

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

A(t)

20 40 60 80 1000

Time t

A(t)

(d)

0

20

40

60

80

100

120

20 40 60 80 1000

Time t

S(t)

I(t)

R(t)

A(t)

(e)

4060

80100

120

0

5

10

S(t)

I(t)

0

20

40

60

80

100

120

R(t)

(f)

Figure 5: 1 + (𝐶𝛼𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇) < 𝑅

03< 1 + (𝛽𝜇/(𝛿 + 𝜇)𝛼

𝑆𝐴)((𝐶𝛼

𝐼𝐴(𝜎 + 𝜇) + 𝜎𝛿𝜇)/𝜇(𝜎 + 𝜇)(𝛿 + 𝜇)), and 𝑅

06=

(𝜎 + 𝜇)(𝛽 + 𝛼𝐼𝐴)/𝛼𝑆𝐴(𝜎 + 𝛿 + 𝜇) > 1. The positive equilibrium 𝑃



Through the qualitative analysis of computer virus prop-agation model, mastering the virus prevention and controltechnology is very necessary. In the meantime, computerusers have been advised to update their security settings. Wecan strengthen the knowledge of the computer virus spread(e.g., enhance the user’s information security awareness) ina timely manner to install antivirus software or fix bugs, tominimize the impact on network computer virus.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is partially supported by the National Natural Sci-ence Foundation of China (11301491) and the Youth ScienceFund of Shanxi Province (2011021001-2).

References

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[2] J. O. Kephart and S. R. White, “Directed-graph epidemiologicalmodels of computer viruses,” in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacy,pp. 343–358, May 1991.

[3] J. O. Kephart and S. R. White, “Measuring and modeling com-puter virus prevalence,” in Proceedings of the IEEE ComputerSociety Symposium onResearch in Security and Privacy, pp. 2–15,IEEE, Oakland, Calif, USA, May 1993.

[4] B. K. Mishra and N. Jha, “Fixed period of temporary immu-nity after run of anti-malicious software on computer nodes,”AppliedMathematics and Computation, vol. 190, no. 2, pp. 1207–1212, 2007.

[5] B. K.Mishra andD. K. Saini, “SEIRS epidemicmodel with delayfor transmission of malicious objects in computer network,”AppliedMathematics and Computation, vol. 188, no. 2, pp. 1476–1482, 2007.

[6] J. R. C. Piqueira, B. F.Navarro, andH.A.M. Luiz, “Epidemiolog-ical models applied to viruses in computer networks,” Journal ofComputer Science, vol. 1, no. 1, pp. 31–34, 2005.

[7] J. R. Piqueira and V. O. Araujo, “A modified epidemiologicalmodel for computer viruses,”Applied Mathematics and Compu-tation, vol. 213, no. 2, pp. 355–360, 2009.

[8] J. R. C. Piqueira, A. A. de Vasconcelos, C. E. C. J. Gabriel, and V.O. Araujo, “Dynamic models for computer viruses,” Computersand Security, vol. 27, no. 7-8, pp. 355–359, 2008.

[9] F. W. Wang, Y. K. Zhang, C. G. Wang, J. F. Ma, and S.Moon, “Stability analysis of a SEIQV epidemic model for rapidspreading worms,” Computers and Security, vol. 29, no. 4, pp.410–418, 2010.

[10] B. K.Mishra andN. Jha, “SEIQRSmodel for the transmission ofmalicious objects in computer network,” Applied MathematicalModelling, vol. 34, no. 3, pp. 710–715, 2010.

[11] B. K. Mishra and S. K. Pandey, “Fuzzy epidemic model forthe transmission of worms in computer network,” NonlinearAnalysis: Real World Applications, vol. 11, no. 5, pp. 4335–4341,2010.

[12] B. K. Mishra and G. M. Ansari, “Differential epidemic model ofvirus and worms in computer network,” International Journal ofNetwork Security, vol. 14, no. 3, pp. 149–155, 2012.

[13] Z. Zhang and H. Yang, “Hopf bifurcation of an SIQR computervirus model with time delay,” Discrete Dynamics in Nature andSociety, vol. 2015, Article ID 101874, 8 pages, 2015.

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