# Stability and Hopf Bifurcation of a Delayed Epidemic Model of Computer Virus with Impact of Antivirus Software

Authors: Zizhen Zhang; Ranjit Kumar Upadhyay; Dianjie Bi; Ruibin Wei
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

In this paper, we investigate an SLBRS computer virus model with time delay and impact of antivirus software. The proposed model considers the entering rates of all computers since every computer can enter or leave the Internet easily. It has been observed that there is a stability switch and the system becomes unstable due to the effect of the time delay. Conditions under which the system remains locally stable and Hopf bifurcation occurs are found. Sufficient conditions for global stability of endemic equilibrium are derived by constructing a Lyapunov function. Formulae for the direction, stability, and period of the bifurcating periodic solutions are conducted with the aid of the normal form theory and center manifold theorem. Numerical simulations are carried out to analyze the effect of some of the parameters in the system on the dynamic behavior of the system.


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## Body

## 1. Introduction

Computer viruses are programs created to carry out activities in a computer without consent of its owner. They not only disrupt the normal functionalities of computer system and damage data files in the computer, but also cause heavy economic losses and tremendous social impacts [1–3]. In recent years, mathematical modeling enjoys popularity in both analyzing and controlling computer viruses based on the similarity between computer viruses and biological viruses. A few works proposing SIR models have appeared in the literatures [4–6]. In [4], Amador studied a stochastic SIRA epidemic model for computer viruses and analyzed the quasistationary distribution, the extinction time, and the number of infections in order to understand the spreading mechanism of computer viruses. In [5], Ozturk and Gulsu proposed an approximate solution to a modified SIR computer virus model by using the shifted Chebyshev collocation method. In [6], Khanh studied the stability and approximate iterative solutions of an SIR computer virus model with antidotal component.Considering the latent period of computer viruses, some models with latency are proposed by some scholars [1, 7–10]. In [7], Yang investigated global stability of a VEISV network worm attack model by using the Li-Muldowney geometric approach. In [8], Keshri et al. proposed a reduced SEIR scale-free network model and studied its stability. In [9], Hosseini et al. formulated a discrete-time SEIRS model of computer virus propagation in scale-free networks and analyzed the local and global stability of the model. In [1], Guillen et al. proposed an improved SEIRS worm model with considering accurate positions for dysfunctional hosts and their replacements in state transition. In [10], Ren and Xu investigated an SEIR-KS computer virus propagation model based on the kill signals. They studied the local and global stability of the model by applying Routh-Hurwitz criterion and Lyapunov functional method. There are also some other models considering the latency of computer viruses with quarantine [11–14] and vaccination [15–19].However, as stated in [20], those above models with the exposed compartment neglect the fact that a computer can infect other computers through file copying or file downloading. Therefore, to overcome the above-mentioned defect, computer virus models with infectivity in latent period have received much attention in recent years [21–26]. Unfortunately, most of these models still have some defects. On the one hand, they ignore the effect of time delay. As is known, there are some time delays of one type or another in the transmission process of computer viruses due to latent period, temporary immunity period, or other reasons. On the other hand, only the susceptible computers are regarded as the entering computers, but every computer can enter or leave the Internet easily in reality. Finally, they neglect the effect of antivirus software, especially the effect of antivirus software on the susceptible computers. Based on the discussion above, we investigated a delayed SLBRS computer virus model with impact of antivirus software based on the following model proposed in [27]: (1)dStdt=b1+γ2Lt+γ2Bt+ηRt-γ1St-μ0St-βLt+BtSt,dLtdt=b2+βLt+BtSt-γ1Lt-γ2Lt-μ0Lt-αLt,dBtdt=b3+αLt-γ1Bt-γ2Bt-μ0Bt,dRtdt=b4+γ1Lt+γ1Bt+γ1St-μ0Rt-ηRt,where S(t), L(t), B(t), and R(t) denote the numbers of susceptible, latent, breaking, and recovered computers at time t, respectively. More parameters are listed in Table 1 as follows.Table 1
Parameters and their meanings in this paper.
Parameter Description b 1 Entering rate of susceptible computers b 2 Entering rate of latent computers b 3 Entering rate of breaking computers b 4 Entering rate of recovered computers α Rate of latent computers break β Infection rate of susceptible computers γ 1 Recovery rate of all computers μ 0 Leaving rate of all computers η Rate of recovered computers lose immunity γ 2 Rate of latent and breaking computers reinstall the operating systemConsidering the temporary immune period of the recovered computers, we incorporate the time delay due to the temporary immunity period into system (1) and obtain the following delayed model:(2)dStdt=b1+γ2Lt+γ2Bt+ηRt-τ-γ1St-μ0St-βLt+BtSt,dLtdt=b2+βLt+BtSt-γ1Lt-γ2Lt-μ0Lt-αLt,dBtdt=b3+αLt-γ1Bt-γ2Bt-μ0Bt,dRtdt=b4+γ1Lt+γ1Bt+γ1St-μ0Rt-ηRt-τ,where τ is the time delay due to the temporary immunity period.The remainder of the paper is structured as follows. In Section2, conditions for local stability of the endemic equilibrium and the existence of Hopf bifurcation are performed. Section 3 deals with global stability of the endemic equilibrium. Section 4 is devoted to establishing the formulae to determine the direction, stability, and period of the bifurcating periodic solutions. Some numerical simulations are presented to illustrate the theoretical results in Section 5. We end the paper with a brief conclusion in Section 6.


## 2. Local Stability and Existence of Hopf Bifurcation

By a direct computation, it can be concluded that system (2) has the endemic equilibrium E∗(S∗,L∗,B∗,R∗) where(3)S∗=γ1+γ2+μ0+αL∗-b2βL∗+b3+αL∗/γ1+γ2+μ0,B∗=b3+αL∗γ1+γ2+μ0,R∗=b4+b1+b2+b3+b4/μ0μ0+η+γ1,where L∗ is the positive root of (4)(4)a~2L2+a~1L+a~0,where(5)a~0=a2a5+a4,a~1=a1a5+a2a6-a3,a~2=a1a6,and(6)a1=β+αβγ1+γ2+μ0,a2=b3βγ1+γ2+μ0,a3=γ1+μ0γ1+γ2+μ0+α,a4=b2γ1+μ0,a5=b1+b2+b3γ2γ1+γ2+μ0+ηR∗,a6=-γ1+μ0γ1+γ2+μ0+αγ1+γ2+μ0.The Jacobian matrix of system (2) evaluated at E∗ is(7)JE∗=p11p12p13q14e-λτp21p22000p32p330p41p42p43p44+q44e-λτ,where(8)p11=-γ1+μ0+βL∗+B∗,p12=γ2-βS∗,p13=γ2-βS∗,p21=βL∗+B∗,p22=βS∗-γ1+γ2+μ0+α,p32=α,p33=-γ1+γ2+μ0,p41=γ1,p42=γ1,p43=γ1,p44=-μ0,q14=η,q44=-η.The corresponding characteristic equations is(9)λ4+P3λ3+P2λ2+P1λ+P0+Q3λ3+Q2λ2+Q1λ+Q0e-λτ=0,with(10)P0=p33p44p11p22-p12p21+p13p32p44q21,P1=p12q21p33+p44-p13p32q21-p11p33p22+p44-p22p44p11+p33,P2=p11p33+p22p44+p11+p33p22+p44-p12q21,P3=-p11+p22+p33+p44,Q0=q21q44p13p32-p12p33+q21q14p33p42-p32p43-p22p33p41q14,Q1=p12q21q44-q44p11p22+p11p33+p22p33+p41q14p22+p33-p21p42q14,Q2=q44p11+p22+p33-p41q14,Q3=-q44.For τ=0, (9) becomes(11)λ4+P3+Q3λ3+P2+Q2λ2+P1+Q1λ+P0+Q0=0.Clearly,P3+Q3=3γ1+4μ0+2γ2+α+η+β(L∗+B∗)>0. Hence, it follows from the Hurwitz criterion that all the roots of (11) have negative real parts, if H1: P0+Q0>0, (P2+Q2)(P3+Q3)>P1+Q1 and (P1+Q1)(P2+Q2)(P3+Q3)>(P0+Q0)(P3+Q3)2+(P1+Q1)2 holds.Forτ>0, let λ=iω(ω>0) be the root of (9). Then,(12)Q1ω-Q3ω3sin⁡τω+Q0-Q2ω2cos⁡τω=P2ω2-ω4-P0,Q1ω-Q3ω3cos⁡τω-Q0-Q2ω2sin⁡τω=P3ω3-P1ω.Thus,(13)ω8+P~3ω6+P~2ω4+P~1ω2+P~0=0,with(14)P~0=P02-Q02,P~1=P12-2P0P2-Q12+2Q0Q2,P~2=P22+2P0-2P1P3+2Q1Q3-Q22,P~3=P32-2P2-Q32.Let ω2=ν, then (13) becomes(15)ν4+P~3ν3+P~2ν2+P~1ν+P~0=0.Suppose that(H2) (15) has a positive root ν0. Then, (13) has a positive root ω0=ν0 such that (9) has a pair of purely imaginary roots ±iω0. For ω0,(16)τ0=1ω0×arccos⁡F1ω0F2ω0,where(17)F1ω0=Q2-P3Q3ω06+P3Q1+P1Q3-P2Q2-Q0ω04+P2Q0+P0Q2-P1Q1ω02-P0Q0,F2ω0=Q32ω06+Q22-2Q1Q3ω04+Q12-2Q0Q2ω02+Q02.Differentiating on both sides of (9) with respect to τ, one can obtain(18)dλdτ-1=-4λ3+3P3λ2+2P2λ+P1λλ4+P3λ3+P2λ2+P1λ+P0+3Q3λ2+2Q2λ+Q1λQ3λ3+Q2λ2+Q1λ+Q0-τλFurther, we have(19)Re⁡dλdττ=τ0-1=f′ν0F2ω02,where f(ν)=ν4+P~3ν3+P~2ν2+P~1ν+P~0 and ν0=ω02.Therefore, if(H3): f′(ν0)≠0, then Re⁡[dλ/dτ]τ=τ0≠0. Thus, we have the following results based the Hopf bifurcation theorem in [28].Theorem 1.
For system (2), if (H1)-(H3) hold, then E∗(S∗,L∗,B∗,R∗) is locally asymptotically stable when τ∈[0,τ0); system (2) undergoes a Hopf bifurcation at E∗(S∗,L∗,B∗,R∗) when τ=τ0 and a family of periodic solutions bifurcate from E∗(S∗,L∗,B∗,R∗). τ0 is defined as in (16).


## 3. Global Stability Analysis

Theorem 2.
Ifmin⁡l1,l2,l3,l4>0, with(20)l1=1M1S∗b1+γ2L∗+γ2B∗+ηR∗-βM3m2+1-γ1m41+ηM4τ1m4+1m1,l2=β+1M2L∗b2+βB∗S∗-γ2m1-αm3-γ1m41+ηM4τ1m4+1m1,l3=β+1M3B∗b3+αL∗-γ2m1-βS∗m2-γ1m41+ηM4τ1m4+1m1,l4=η1M4-1m1+1m41+ητ+ηM4τm4+1R∗1M4-ητ1m1+1m4×b4+S∗+L∗+B∗γ1,where m1<S(t)<M1, m2<L(t)<M2, m3<B(t)<M3, and m4<R(t)<M4 for t>0, then the endemic equilibrium E∗ is globally asymptotically stable.Proof.
Let(21)St=S∗ewt,Lt=L∗ext,Bt=B∗eyt,Rt=R∗ezt.
ThenE∗(S∗,L∗,B∗,R∗) becomes the trivial equilibrium w(t)=x(t)=y(t)=z(t)=0 for all t>0, and system (2) can be reduced to the following form:(22)dwdt=-1Sb1+γ2L∗+γ2B∗+ηR∗ewt-1+L∗γ2S-βext-1+B∗γ2S-βeyt-1+ηR∗Sezt-τ-1,(23)dxdt=-1Lb2+βB∗S∗ext-1+βS∗BL+1ewt-1+βB∗S∗Leyt-1,(24)dydt=αL∗Bext-1-1Bb3+αL∗eyt-1,(25)dzdt=γ1S∗Rewt-1+γ1L∗Rext-1+γ1B∗Reyt-1-ηR∗Rezt-τ-1-1Rb4+γ1S∗+L∗+B∗-ηR∗ezt-1.Now, we have(26)ezt-τ=ezt-∫t-τtezsdzdsds.Now, (22) can be rewritten as follows by using above relation,(27)dwdt=-1Sb1+γ2L∗+γ2B∗+ηR∗ewt-1+L∗γ2S-βext-1+B∗γ2S-βeyt-1+ηR∗Sezt-1-ηR∗S∫t-τtezsdzdsds,=-1Sb1+γ2L∗+γ2B∗+ηR∗ewt-1+L∗γ2S-βext-1+B∗γ2S-βeyt-1+ηR∗Sezt-1-ηR∗S∫t-τtezsγ1S∗Rews-1+γ1L∗Rexs-1+γ1B∗Reys-1-1Rb4+γ1S∗+L∗+B∗-ηR∗ezs-1-ηR∗Rezs-τ-1ds,Let V1(t)=wt. It follows from the above equation(28)D+V1≤-1M1b1+γ2L∗+γ2B∗+ηR∗ewt-1+L∗γ2m1-βext-1+B∗γ2m1-βeyt-1+ηR∗m1ezt-1+ηR∗m1∫t-τtezsγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+1M4b4+S∗+L∗+B∗γ1ezs-1+1m4ηR∗ezs-1+ηR∗M4ezs-τ-1ds,We find that there exists a t1>0, such that R∗ez(t)<M4 for all t>t1 and for t>t1+τ, we have(29)D+V1≤-1M1b1+γ2L∗+γ2B∗+ηR∗ewt-1+L∗γ2m1-βext-1+B∗γ2m1-βeyt-1+ηR∗m1ezt-1+ηM4m1∫t-τtγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+1M4b4+S∗+L∗+B∗γ1ezs-1+1m4ηR∗ezs-1+ηR∗M4ezs-τ-1dsAgain due to form of (29) we consider the following functional:(30)V11t=V1t+ηM4m1∫t-τt∫vtγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+1M4b4+S∗+L∗+B∗γ1ezs-1+1m4ηR∗ezs-1+ηR∗M4ezs-τ-1dsdv+η2R∗τm1∫t-τtezs-1ds,whose derivative along the solution of system (2) is given by(31)D+V11t≤D+V1t+ηM4τm1γ1S∗m4ewt-1+γ1L∗m4ext-1+γ1B∗m4eyt-1+1M4b4+S∗+L∗+B∗γ1ezt-1+ηR∗m4ezt-1+η2R∗τm1ezt-1-ηM4m1∫t-τtγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+1M4b4+S∗+L∗+B∗γ1ezs-1+ηR∗m4ezs-1+ηR∗M4ezs-τ-1ds≤-1M1b1+γ2L∗+γ2B∗+ηR∗+ηM4γ1S∗τm1m4ewt-1+L∗γ2m1-β+ηM4τγ1m1m4ext-1+B∗γ2m1-β+ηM4τγ1m1m4eyt-1+ηR∗m11+ητ+ηM4τm4+ητm1b4+S∗+L∗+B∗γ1ezt-1Again let V2(t)=xt and V3(t)=|y(t)|. Now calculate the derivative of V2(t) and V3(t) with the solution of (2), it follows from, respectively, (23) and (24)(32)D+V2≤βS∗M3m2+1ewt-1-1M2b2+βB∗S∗ext-1+βB∗S∗m2eyt-1,(33)D+V3≤αL∗m3ext-1-1M3b3+αL∗eyt-1,Now, (25) can be rewritten as follows by using (26),(34)dzdt=γ1S∗Rewt-1+γ1L∗Rext-1+γ1B∗Reyt-1-ηR∗Rezt-1-1Rb4+γ1S∗+L∗+B∗-ηR∗ezt-1+ηR∗R∫t-τtezsdzdsds.=γ1S∗Rewt-1+γ1L∗Rext-1+γ1B∗Reyt-1-ηR∗Rezt-1-1Rb4+γ1S∗+L∗+B∗-ηR∗ezt-1+ηR∗R∫t-τtezsγ1S∗Rews-1+γ1L∗Rexs-1+γ1B∗Reys-1-ηR∗Rezs-τ-1-1Rb4+γ1S∗+L∗+B∗-ηR∗ezs-1ds.Again let V4(t)=zt. It follows from the above equation(35)D+V4≤γ1S∗m4ewt-1+γ1L∗m4ext-1+γ1B∗m4eyt-1-ηR∗M4ezt-1+-b4+γ1S∗+L∗+B∗M4+ηR∗m4ezt-1+ηR∗m4∫t-τtezsγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+ηR∗M4ezs-τ-1+b4+γ1S∗+L∗+B∗M4+ηR∗m4ezs-1ds.We find that there exists a t1>0, such that R∗ez(t)<M4 for all t>t1 and for t>t1+τ, we have(36)D+V4≤γ1S∗m4ewt-1+γ1L∗m4ext-1+γ1B∗m4eyt-1-ηR∗M4ezt-1+-b4+γ1S∗+L∗+B∗M4+ηR∗m4ezt-1+ηM4m4∫t-τtγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+ηR∗M4ezs-τ-1+b4+γ1S∗+L∗+B∗M4+ηR∗m4ezs-1ds.Again due to the above form of (36) we consider the following functional:(37)V44t=V4t+ηM4m4∫t-τt∫vtγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+ηR∗M4ezs-τ-1+b4+γ1S∗+L∗+B∗M4+ηR∗m4ezs-1dsdv+η2R∗τm4∫t-τtezs-1ds,whose right derivative along the solution of the system (2) is given by(38)D+V44t≤D+V4t+ηM4τm4γ1S∗m4ewt-1+γ1L∗m4ext-1+γ1B∗m4eyt-1+b4+γ1S∗+L∗+B∗M4+ηR∗m4ezt-1+η2R∗τm4ezt-1-ηM4m4∫t-τtγ1S∗m4ews-1+γ1L∗m4exs-1+γ1B∗m4eys-1+ηR∗M4ezs-τ-1+b4+γ1S∗+L∗+B∗M4+ηR∗m4ezs-1ds,≤γ1S∗m41+ηM4τm4ewt-1+γ1L∗m41+ηM4τm4ext-1+γ1B∗m41+ηM4τm4eyt-1+ηR∗m41+ητ-m4M4+ηM4τm4+ητm4-1M4b4+γ1S∗+L∗+B∗ezt-1Let us define a Lyapunov functional V(t) as (39)Vt=V11t+V2t+V3+V44t>wt+xt+yt+zt.Computing the upper right derivative of V(t) along the solution of system (2) and by using (31)-(33) and (38), we obtain(40)
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,where l1, l2, l3, and l4 are defined above in (20).
Since the model system (2) is positive invariant, therefore, for all t>t1∗, we have (41)S∗ewt=St>S_,L∗ext=Lt>L_,B∗eyt=Bt>B_,R∗ezt=Rt>R_.Using the mean value theorem, we have(42)S∗ewt-1=S∗eθ1twt>m1wt,L∗ext-1=L∗eθ2txt>m2xt,B∗eyt-1=B∗eθ3tyt>m3yt,R∗ezt-1=R∗eθ4tzt>m4zt,where S∗eθ1(t) lies between S∗ and S(t), L∗eθ2(t) lies between L∗ and L(t), B∗eθ3(t) lies between B∗ and B(t), and R∗eθ4(t) lies between R∗ and R(t). Therefore,(43)D+Vt≤-l1S_wt-l2L_xt-l3B_yt-l4R_zt≤-lwt+xt+yt+zt,wherel=min⁡l1S_,l2L_,l3B_,l4R_.
Note thatV(t)>wt+xt+yt+zt. Hence, from theory of global stability and (43), we conclude that the zero solution of the reduced system (22)-(25) is globally asymptotically stable. Therefore, the endemic equilibrium E∗ of model system (2) is globally asymptotically stable.


## 4. Direction and Stability of Hopf Bifurcation

Letτ=τ0+ζ(ζ∈R), u1=S(τt), u2=L(τt), u3=B(τt), and u4=R(τt). System (2) becomes(44)u˙t=Lζut+Fζ,ut,where u(t)=(u1,u2,u3,u4)T∈C=C([-1,0],R4) and Lζ: C→R4 and F: R×C→R4 are defined as follows:(45)Lζϕ=τ0+ζPmaxϕ0+Qmaxϕ-1,and(46)Fζ,ϕ=τ0+ζ-βϕ10ϕ20+ϕ30βϕ10ϕ20+ϕ3000with(47)Pmax=p11p12p130p21p22000p32p330p41p42p43p44,Qmax=000q1400000000000q44,Thus, there exists η(θ,ζ) such that(48)Lζϕ=∫-10dηθ,ζϕθ,forϕ∈C.In fact,(49)ηθ,ζ=τ0+ζPmaxδθ+Qmaxδθ+1,where δ(θ) is the Dirac delta function.Forϕ∈C([-1,0],R4), define(50)Aζϕ=dϕθdθ,-1≤θ<0,∫-10dηθ,ζϕθ,θ=0,and(51)Rζϕ=0,-1≤θ<0,Fζ,ϕ,θ=0.Then system (44) is equivalent to(52)u˙t=Aζut+Rζut.For φ∈C1([0,1],(R4)∗), define(53)A∗φ=-dφsds,0<s≤1,∫-10dηTs,0φ-s,s=0,and the bilinear inner form for A and A∗(54)φs,ϕθ=φ¯0ϕ0-∫θ=-10∫ξ=0θφ¯ξ-θdηθϕξdξ,where η(θ)=η(θ,0).Letv(θ)=(1,v2,v3,v4)Teiτ0ω0θ be the eigenvector of A(0) corresponding to +iτ0ω0 and v∗(s)=D(1,v2∗,v3∗,v4∗)Teiτ0ω0s be the eigenvector of A∗(0) corresponding to -iτ0ω0, respectively. Based on the definition of A(0) and A∗, we obtain (55)v2=p21iω0-p22,v3=p32v2iω0-p33,v4=iω0-p11-p12v2-p13v3q14e-iτ0ω0,v2∗=iω0+p11+p41v4p21,v3∗=-p13+p43v4iω0+p33,v4∗=-q14eτ0ω0iω0+p44+q44eτ0ω0.From (54), the expression of Q can be obtain as follows:(56)D¯=1+v2v¯2∗+v3v¯3∗+v4v¯4∗+τ0e-τ0ω0v4q14+q44v¯4∗-1,such that 〈v∗,v〉=1 and 〈v∗,v¯〉=0.Next, we can obtain the expressions ofg20, g11, g02, and g21 by the algorithms in [28] and the computation process in [29–31]:(57)g20=2τ0D¯βv¯2∗-1v2+v3,g11=τ0D¯βv¯2∗-1Re⁡v2+Re⁡v3,g02=2τ0D¯βv¯2∗-1v2¯+v3¯,g21=2βτ0D¯v¯2∗-1W1110v2+12W2010v¯2+W1120+12W2020+W1110v3+12W2010v¯3+W1130+12W2030,with(58)W20θ=ig20v0τ0ω0eiτ0ω0θ+ig¯02v¯03τ0ω0e-iτ0ω0θ+E1e2iτ0ω0θ,W11θ=-ig11v0τ0ω0eiτ0ω0θ+ig¯11v¯0τ0ω0e-iτ0ω0θ+E2.E 1 and E2 can be obtained by the following two equations:(59)E1=22iω0-p11-p12-p13-q14e-2iτ0ω0-p212iω0-p22000-p322iω0-p330-p41-p42-p432iω0-p44-q44e-2iτ0ω0-1×-βv2+v3βv2+v300,E2=-p11p12p13q14p21p22000p32p330p41p42p43p44+q44-1×-βRe⁡v2+Re⁡v3βRe⁡v2+Re⁡v300.Then, one can obtain(60)C10=i2τ0ω0g11g20-2g112-g0223+g212μ2=-Re⁡C10Re⁡λ′τ0,β2=2Re⁡C10,T2=-Im⁡C10+μ2Im⁡λ′τ0τ0ω0,In conclusion, we have the following results.Theorem 3.
For system (2), if μ2>0 (μ2<0), then the Hopf bifurcation is supercritical (subcritical); if β2<0 (β2>0), then the bifurcating periodic solutions are stable (unstable); if T2>0 (T2<0), then the period of the bifurcating periodic solutions increases (decrease).


## 5. Numerical Simulations

In this section, we develop some numerical simulations in order to support the obtained results in our paper. A set of parameters of system (2) are chosen as follows: b1=1, b2=1, b3=2, b4=1, α=0.35, β=0.03, γ1=0.1, γ2=0.3, η=0.5, and μ0=0.01. Then, we obtain the following specific case of system (2):(61)dStdt=1+0.3Lt+0.3Bt+0.5Rt-τ-0.11St-0.03Lt+BtSt,dLtdt=1+0.03Lt+BtSt-0.76Lt,dBtdt=2+0.35Lt-0.41Bt,dRtdt=1+0.1Lt+0.1Bt+0.1St-0.01Rt-0.5Rt-τ,Then, (4) becomes the following form:(62)-0.0113L2+2.4034L+6.7325=0,from which we can obtain the unique positive root L∗=215.4556. Further, we can verify that system (61) has a unique endemic equilibrium E∗(13.4193,215.4556,188.8036,83.6066) and all the conditions given in Theorem 1 are satisfied.By means of Matlab software, we getω0=2.0684, τ0=3.4685, and λ′(τ0)=0.0081+1.0307i. Thus, we can obtain C1(0)=-0.0560+0.0092i, μ2=6.9136, β2=-0.1120, and T2=-0.9945. It follows that μ2>0, β2<0, and T2>0. Fix τ=3.2575<τ0, then we can see that the solution of system (61) would tend to the endemic equilibrium E∗(13.4193,215.4556,188.8036,83.6066). In other words, E∗(13.4193,215.4556,188.8036,83.6066) is locally asymptotically stable, which can be illustrated by Figures 1–3. However, when τ passes through the critical value τ0, E∗(13.4193,215.4556, 188.8036,83.6066) loses its stability, and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from E∗(13.4193,215.4556, 188.8036,83.6066). This property can be shown as in Figures 4–6. Since μ2>0, β2<0, and T2<0, we can conclude that the Hopf bifurcation occurring at τ0=3.4685 is supercritical and the bifurcating periodic solutions are stable and decrease. Next, we are interested to study the effect of some other parameters on the dynamics of system (62).Figure 1
Time plots ofS, L, B, and R with τ=3.2575<τ0=3.4685.Figure 2
Dynamic behavior of system (61): projection on S-L-B with τ=3.2575<τ0=3.4685.Figure 3
Dynamic behavior of system (61): projection on L-B-R with τ=3.2575<τ0=3.4685.Figure 4
Time plots ofS, L, B, and R with τ=3.6755>τ0=3.4685.Figure 5
Dynamic behavior of system (61): projection on S-L-B with τ=3.6755>τ0=3.4685.Figure 6
Dynamic behavior of system (61): projection on L-B-R with τ=3.6755>τ0=3.4685.(i) Effect of the recovered rate (γ1): in Figures 7(a)–7(d), we can see that the numbers of susceptible and recovered computers increase; nevertheless, the numbers of latent and breaking computers decrease, when the number of γ1 increases. And the system changes its behavior from limit cycle to stable focus as we increase the value of γ1, from 0.1 to 0.3, which can be shown as in Figure 8.Figure 7
Time plots ofS, L, B, and R for different γ1 at τ=3.2575. Rest of the parameters are taken as given in the text.
(a) (b) (c) (d)Figure 8
Dynamic behavior of system (61): projection on L-B-R with τ=3.6755>τ0=3.4685 for different γ1. Rest of the parameters are taken as given in the text.(ii) Effect of the rate of latent and breaking computers reinstall the operating system (γ2): in the same manner, we can see from Figures 9(a)–9(d) that the numbers of susceptible and latent computers increase and the number of breaking computers decreases, when the number of γ2 increases. But it does not affect the number of recovered computers, which can be also seen from the expression of R∗ in Section 2. Also, we observe that γ2 does not affect the dynamics of the system; it remains at limit cycle when we choose τ=3.6755. This property can be illustrated by Figure 10.Figure 9
Time plots ofS, L, B, and R for different γ2 at τ=3.2575. Rest of the parameters are taken as given in the text.
(a) (b) (c) (d)Figure 10
Dynamic behavior of system (61): projection on L-B-R with τ=3.6755>τ0=3.4685 for different γ2. Rest of the parameters are taken as given in the text.(iii) Effect of the entering rates (b1, b2, b3, b4): as is shown in Figures 11–14, the numbers of all computers increase when the numbers of b1 and b4 increase. However, the number of susceptible computers decreases and the numbers of latent, breaking, and recovered computers increase, when the numbers of b2 and b3 increase. In addition, we find that the entering rates does not affect the dynamics of the systems.Figure 11
Time plots ofS, L, B, and R for different b1 at τ=3.2575. Rest of the parameters are taken as given in the text.
(a) (b) (c) (d)Figure 12
Time plots ofS, L, B, and R for different b2 at τ=3.2575. Rest of the parameters are taken as given in the text.
(a) (b) (c) (d)Figure 13
Time plots ofS, L, B, and R for different b3 at τ=3.2575. Rest of the parameters are taken as given in the text.
(a) (b) (c) (d)Figure 14
Time plots ofS, L, B, and R for different b3 at τ=3.2575. Rest of the parameters are taken as given in the text.
(a) (b) (c) (d)


## 6. Conclusions

In this paper, a delayed SLBRS computer virus model is presented by incorporating the time delay due to the temporary immunity period of the recovered computers based on the model proposed in [27]. Compared with the model in [27], we mainly consider the effect of the time delay on its dynamic behavior. Compared with other computer virus models, we assume that every computer can enter the Internet, which is consistent with the reality. Further, we also consider the effect of antivirus software on the susceptible computers in the presented model. Thus, the computer virus model proposed in our paper is more general.It has been shown that the endemic equilibriumE∗(S∗,L∗,B∗,R∗) is locally asymptotically stable when τ∈[0,τ0) under some certain conditions. In this case, the propagation of the computer virus in system (2) can be controlled easily. Once the value of the time delay passes through τ0, E∗(S∗,L∗,B∗,R∗) loses its stability and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from E∗(S∗,L∗,B∗,R∗). In this case, the numbers of the four classes computers in system (2) will oscillate in the vicinity of E∗(S∗,L∗,B∗,R∗). Namely, the propagation of the computer virus will be out of control. Therefore, the results obtained in the present paper can help us to gain insight into the spreading process of computer viruses. Also, sufficient conditions for global stability of the endemic equilibrium are derived by constructing a suitable Lyapunov function. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and center manifold theorem. Numerical simulations are presented to verify the analytical predictions. In addition, it has been observed in our simulations that the recovered rate γ1 can change the dynamics of the system from limit cycle to stable focus as its value increases. Thus, it is strongly recommended that users of computers connected to Internet should periodically run antivirus software of the newest version. From the point of this view, we can conclude that the results of the proposed model in our paper can be used to evaluate the effectiveness of antivirus software. In addition, the numbers of latent and breaking computers decrease, when the reinstalling of the operating system rate increases. Thus, it can be concluded that users should reinstall operating system if necessary. Finally, the numbers of latent and breaking computer will also increase, when the values of entering rates of all computers b1, b2, b3, and b4 increase. Therefore, the manager of a network should control the number of computers connected to the network properly.Of course, when we pursue a low level of infections, we should also consider the cost of the measures we carry out. In addition, it should be pointed out that the model investigated in the literature [27] and our present paper assumes that the latent computers and the breaking computers have the same infection rate β. In the near future, we will investigate the optimal control problem of the following general system (63) so as to achieve a low level of infections at a low cost by using the method introduced in [32]:(63)dStdt=b1+γ2Lt+γ2Bt+ηRt-τ-γ1St-μ0St-β1StLt-β2StBt,dLtdt=b2+β1StLt+β2StBt-γ1Lt-γ2Lt-μ0Lt-αLt,dBtdt=b3+αLt-γ1Bt-γ2Bt-μ0Bt,dRtdt=b4+γ1Lt+γ1Bt+γ1St-μ0Rt-ηRt-τ,where β1 and β2 are the infection rate of the latent computers and the breaking computers, respectively.



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*Source: 8239823-2018-11-04.xml*