# The Dynamical Analysis of Computer Viruses Model with Age Structure and Delay

Authors: Hui Cao; Si Wang; Dongxue Yan; Hongwu Tan; Hemiao Xu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper deals with the dynamical behaviors for a computer viruses model with age structure, where the loss of the acquired immunity and delay are incorporated. Through some rigorous analyses, an explicit formula for the basic reproduction number of the model is calculated, and some results about stability and instability of equilibria for the model are established. These findings show that the age structure and delay can produce Hopf bifurcation for the computer viruses model. The numerical examples are executed to validate the theoretical results.


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## Body

## 1. Introduction

With the popularization of computers and the rapid development of information network technology, the network has brought great convenience to human work, study, and life. However, the network has brought us a lot of harm as well. The spread of computer viruses in the network is a common phenomenon. Once the computers in the network are infected with the viruses, the normal programs of the computers may not be able to run, the files in the computers may be damaged, the important information in the computer are lost, and so on. Therefore, it is more important issue to better understand the dynamical spread of computer viruses in the network.Since the spread of computer viruses in the network is very similar to the spread of biological viruses in populations [1], in recent years, many authors have constructed the computer viruses transmission models based on the epidemic model framework, including SIS models [2, 3], SIR models [4], SIRS models [5–7], SEIR models [8], SEIS models [9, 10], and so on [11–14]. In particular, for the different types of the computer viruses models, the term vaccination is introduced to describe the process of installing the newest version antivirus software in the uninfected computer [5, 6, 14, 15]. The process can lead to the uninfected computer acquire temporary immunity.In fact, after the virus-infected computers are successfully disinfected, the computer’s antivirus system will inevitably be upgraded to strengthen defenses, which will cause the recovery computers to obtain short-term immune protection. That is, the recovery computers will stay in the recovery class for a while. For the classic SIRS model, the outflow of the recovery computers is often described by an ordinary equation as follows:(1)dRtdt=−μRt−γRt,where Rt denotes the number of the recovery computers at time t, μ is the rate at which one computer is removed from the network, and γ is the removal rate of the recovery computers, which describes the recovery computers leaving the recovery class and entering the susceptible class again since they lose its immunity. However, the diversity of computer virus leads the recovery computers must stay in the recovery class for some time, and then they lose immune protection and become susceptible ones again. It means that the removal rate γ of the recovery computers depends on the length of the recovery time. To this end, we assume that the removal rate γ in (1) should be replaced with the following piecewise function:(2)γa=γ∗,a>τ,0,a≤τ,where γa∈L+∞0,+∞,ℝ, γ∗ is a positive constant, and τ is the shortest time for a recovery computer to maintain its acquired immunity. Therefore, the outflow of the recovery computers in (1) can be rewritten by a partial differential equation as(3)∂Rt,a∂t+∂Rt,a∂a=−μRt,a−γaRt,a,t≥0,a≥0,where Rt,a denotes the density of the recovery computers with respect to the acquired immunity age a at time t. Obviously, the loss rate γa of the acquired immunity of the recovery computers obeys a non-Markovian process.LetSt and It be the number of the susceptible computers and infected computers at time t, respectively, b be the rate at which external computers are connected to the network, p be the vaccination proportion of computers which are connected to the network, β be the transmission rate, α be the rate at which the infected computers recover due to the antivirus treatment, r2 be the self-recovery rate of the infected computers, and α1 be the acquired temporary immunity rate of the susceptible computers. Based on the model in [5, 6], we display the flowchart of infection progression in Figure 1.Figure 1
The flowchart of infection progression for the computer viruses based on the SIRS model.Following the transmission mechanism and schematic diagram, we propose the computer virus model with age structure and delay in the following:(4)dStdt=1−pb−βStIt−α1+μSt+r2It+∫0+∞γaRt,ada,dItdt=βStIt−α+r2+μIt,∂Rt,a∂t+∂Rt,a∂a=−μRt,a−γaRt,a,t≥0,a≥0,with the boundary condition(5)Rt,0=pb+α1St+αIt,t≥0,and the initial condition(6)S0=S0>0,I0=I0≥0,R0,⋅=R0⋅∈L+10,+∞.This paper is organized as follows. Some preliminaries results and the well-posedness of system (4) are presented in section 2. In section 3, we give an explicit expression of basic reproduction number ℛ0τ and discuss the existence of all the feasible equilibria. In section 4, we study the global stability of the virus-free equilibrium E0 when ℛ0τ<1 and local stability of the computer virus equilibrium E∗ when ℛ0τ>1 and τ=0. In section 5, we study the Hopf bifurcations occurring from the computer virus equilibrium E∗ with the increase in τ. In section 6, we present some numerical examples to illustrate our theoretical results and give the conclusions.


## 2. Preliminaries

In this section, we will mainly discuss the non-negativity and ultimately boundedness of the solutions of system (4) with non-negative initial condition. The existence and uniqueness of the solution of system (4) directly follows from Lemma A. 1 in the Appendix since the structure of model (4) satisfies the assumptions 1−5 in Lemma A. 1.Theorem 1.
Ifϕ=S0,I0,R0⋅∈ℝ+2×L+1, then the solution of system (4) is non-negative for all t≥0 and the set(7)Γ=S,I,R∈ℝ+2×L+10,+∞,ℝ:S+I+∫0+∞Rt,ada≤bμ,which is positively invariant with respect to system (4); moreover, it is ultimately bounded for t large enough.Proof.
The second equation of system (4) implies that(8)It=I0e∫0tβSθ−α−r2−μdθ≥0,which means that, for any initial value I0∈ℝ+, It remains non-negative for t≥0.
For anyS0∈ℝ+, we claim that St remains non-negative for any t∈ℝ+. Suppose that the claim does not hold and then it follows from the continuity of the solution of system (4) associated with the initial condition that there exists a tS such that St≥0 for t∈0,tS, StS=0, and S′tS<0. Then, by using of the third and fourth equation of system (4), we can get(9)RtS,a=pb+α1StS−a+αItS−ae−∫0aμ+γθdθ,a≤tS,R0a−tSe−∫a−tSaμ+γθdθ,a>tS.
It is clear thatRtS,a≥0 since StS−a>0, ItS−a≥0, and R0⋅∈L+10,+∞. Plugging such a value into the first equation of (4) leads to(10)dStdt|t=tS=1−pb+r2ItS+∫0+∞γaRtS,ada>0,which contradicts with S′tS<0. Hence, the claim holds and St remains non-negative for any t∈ℝ+ if S0≥0.
Based on the above analysis, we directly integrate the third equation in system (4) along the characteristic line yields that(11)Rt,a=pb+α1St−a+αIt−ae−∫0aμ+γθdθ,a≤t,R0a−te−∫a−taμ+γθdθ,a>t.
The non-negativity ofS and I together with R0⋅∈L+10,+∞ ensures that Rt,⋅ remains non-negative for all t≥0.
In the following, we proceed with the ultimate boundedness of the solutions of system (4). Let R¯t=∫0+∞Rt,ada, which represents the total number of the recovery individuals at time t. Then, adding those equations in system (4) yields(12)dSt+It+R¯tdt=1−pb−μ+α1St+∫0+∞γaRt,ada−α+μIt−0∫0+∞∂Rt,a∂a−μ+γaRt,ada=1−pb−α1St+αIt−μSt+It+R¯t−∫0+∞∂Rt,a∂ada=1−pb−α1St+αIt−μSt+It+R¯t−lima⟶+∞Rt,a+Rt,0.
It is reasonable to assume thatlima⟶+∞Rt,a=0 according to the biological significance. Noting that Rt,0=pb+α1St+αIt, we obtain(13)dSt+It+R¯tdt=b−μSt+It+R¯t.
Therefore,(14)limt⟶+∞St+It+R¯t=bμ.
It is not hard to see that the set(15)Γ=S,I,R∈ℝ+2×L+10,+∞,ℝ:S+I+∫0+∞Rt,ada≤bμ,which is positively invariant with respect to system (4). Consequently, system (4) is ultimately bounded.
For the sake of convenience, we account for the dynamics of system (4) taken the initial values from Γ.


## 3. The Existence of the Equilibria

In this section, we focus on the existence of the virus-free equilibrium and the computer virus equilibrium of system (4). To this end, we need to solve the following equation:(16)1−pb−βSI−α1+μS+r2I+∫0+∞γaRada=0,βSI−α+r2+μI=0,dRada=−μ+γaRa,R0=pb+α1S+αI.The virus-free equilibriumE0 means that there is no virus-infected computers in the entire network; therefore, in order to get the virus-free equilibrium E0 of system (4), we assume I=0. Then, equation (16) can be rewritten as the following equations:(17)1−pb−α1+μ+∫0+∞γaRada=0,dRada=−μ+γaRa,R0=pb+α1S.The second and third equations in (17) yields(18)Ra=pb+α1Se−∫0aμ+γθdθ.TakingRa=pb+α1Se−∫0aμ+γθdθ into the first equation in (17), we obtain(19)S=1−pb+pbMτμ+α1−α1Mτ,with(20)Mτ=∫0+∞γae−∫0aμ+γθdθda=γ∗e−μτμ+γ∗.Then, we know that system (4) always has the virus-free equilibrium E0=S0,0,R0a, where(21)S0=1−pb+pbMτμ+α1−α1Mτ,R0a=pb+α1S0e−∫0aμ+γθdθ.The computer virus equilibriumE∗ means that there are always virus-infected computers in the entire network; therefore, in order to get the computer virus equilibrium E∗ of system (4), we assume I>0. Then, equation (16) can be rewritten as the following equations:(22)1−pb−βSI−α1+μS+r2I+∫0+∞γaRada=0,βS−α+r2+μ=0,dRada=−μ+γaRa,R0=pb+α1S+αI.The second equation in (22) states that S=α+r2+μ/β. Furthermore,(23)Ra=pb+α1α+r2+μβ+αIe−∫0aμ+γθdθ.SubstitutingS=α+r2+μ/β and Ra=pb+α1α+r2+μ/β+αIe−∫0aμ+γθdθ into the first equation in (22), we can get(24)I=α+r2+μα1+μ−α1Mτβμ+α1−αMτℛ0τ−1,with(25)ℛ0τ=β×1α+r2+μ×1−pb+pbγ∗e−μτ/μ+γ∗μ+α1−α1γ∗e−μτ/μ+γ∗.Obviously, inequalitiesℛ0τ>1 guarantee I>0. That is, when ℛ0τ>1, system (4) has the computer virus equilibrium E∗=S∗,I∗,R∗a, where(26)S∗=α+r2+μβ,R∗a=pb+α1α+r2+μβ+αI∗e−∫0aμ+γθdθ,I∗=α+r2+μμ+α1−α1Mτβμ+α−αMτℛ0τ−1.Therefore, the following theorem gives the existence of the equilibria of system (4).Theorem 2.
System (4) has always the virus-free equilibrium E0=S0,0,R0a; In addition, when ℛ0τ>1, system (4) also has a unique computer virus equilibrium E∗=S∗,I∗,R∗a.In fact,ℛ0τ in (25) is the basic reproduction number of system (4); that is, it represents the total number of the newly infected cases by an infectious individual in the entire infection period. Here, β is the infection rate by an infected computer, 1/α+r2+μ is the average infection period of the infected computers, and 1−pb+pbγ∗e−μτ/μ+γ∗/μ+α1−α1γ∗e−μτ/μ+γ∗ denotes the total number of susceptible individuals.


## 4. The Stability of the Equilibria

In this section, we study the global stability of the virus-free equilibriumE0 by the fluctuation lemma and the local stability of the computer virus equilibrium E∗ by analyzing the linearizing system (4).


### 4.1. The Stability of the Virus-Free EquilibriumE0

Theorem 3.
Ifℛ0τ<1, then the virus-free equilibrium E0=S0,0,R0a of system (4) is locally asymptotically stable while if ℛ0τ>1, E0 is unstable.Proof.
Linearizing system (4) at the virus-free equilibrium E0 and considering the exponential forms of that linear system, we obtain the characteristic equation of system (4) at E0 as follows:(27)λ+α+r2+μ−βS0λ+μ+α1−α1∫0+∞γae−∫0aλ+μ+γθdθda=0.
It is clear thatλ1=βS0−α+r2+μ=α+r2+μℛ0τ−1 is the root of the characteristic (27). The characteristic root λ1<0 when ℛ0τ<1 and λ1>0 when ℛ0τ>1.
In addition, the remaining roots of (27) satisfy the following equation:(28)Λλ=λ+μ+α1−α1∫0+∞γae−∫0aλ+μ+γθdθda=0.
Ifλ∈ℝ, then Λλ is a continuous real function strictly increasing and satisfies that(29)Λ0=μ+α1−α1Mτ>0,limλ⟶+∞Λλ=+∞.
It implies thatΛλ=0 has no positive real root. Suppose that λ=u+iv is an arbitrary complex root of Λλ=0 and it satisfies u>0, then we have(30)Λu+iv=u+iv+μ+α1−α1∫0+∞γae−∫0au+iv+μ+γθdθda=0,which implies that the real part ReΛu+iv=0 and the imaginary part ImΛu+iv=0. Separating the real and imaginary part of Λu+iv, we obtain(31)ReΛu+iv=u+μ+α1−α1∫0+∞γae−uacosvae−∫0aμ+γθdθda=u+μ+α11−γ∗eγ∗τ∫τ+∞e−u+μ+γ∗acosvada=u+μ+α11−γ∗u+μ+γ∗e−u+μτcosvτ−v/u+μ+γ∗sinvτ1+v2/u+μ+γ∗2=u+μ+α11−γ∗u+μ+γ∗e−u+μτcosarccos1/1+v2/u+μ+γ∗2+vτ1+v2/u+μ+γ∗2.
It is obvious thatReΛu+iv>0 since 1+v2/u+μ+γ∗2>1 and −1≤cosarccos1/1+v2/u+μ+γ∗2+vτ≤1. Based on the above analysis, we know λ=u+ivu>0 cannot be the root of Λλ=0. That is, Λλ=0 has no complex root with positive real part.
Summarizing the above analysis, we can see that the virus-free equilibriumE0 is locally asymptotically stable when ℛ0τ<1 and the virus-free equilibrium E0 is unstable when ℛ0τ>1.
In the following, we discuss the global stability ofE0 by employing the fluctuation lemma. Let(32)ψ∞=liminft⟶∞ψt,andψ∞=limsupt⟶∞ψt,and the fluctuation lemma is given as follows.Lemma 1 (fluctuation lemma [16]).
Letψ:ℝ+⟶ℝ be a bounded and continuously differentiable function. Then, there exist sequences qn and tn such that qn⟶∞, tn⟶∞, ψqn⟶ψ∞, ψ′qn⟶0, ψtn⟶ψ∞, and ψtn⟶0 as n⟶∞.Lemma 2 (see [17]).
Supposeg:ℝ+⟶ℝ is a bounded function. Then,(33)liminft⟶∞∫0tkθgt−θdθ≥g∞k1,where k1=∫0∞ksds.Theorem 4.
Ifℛ0τ<1, then the virus-free equilibrium E0 of system (4) is globally asymptotically stable for any ϕ∈Γ.Proof.
In order to establish the globally asymptotical stability of the virus-free equilibriumE0, according to Theorem 3, it is sufficient to prove that E0 is attractive in Γ. Let St,It,Rt,a be a solution of system (4) with ϕ∈Γ. Integrating the third equation of system (4) with the boundary condition yields(34)Rt,a=pb+α1St−a+αIt−ae−∫0aμ+γθdθ,a≤t,R0a−te−∫a−taμ+γθdθ,a>t.
With the assistance of the fluctuation lemma, it is easy to get(35)S∞≤1−pb+pbMτμ+α1−α1Mτ.
Furthermore, it follows from the second equations of system (4) that(36)dItdt≤β1−pb+pbMτμ+α1−α1MτIt−α+r2+μIt=α+r2+μℛ0τ−1It.
It is clear thatI∞⟶0 under the condition ℛ0τ<1.
According to Lemma1, we can select a sequence tn such that tn⟶∞, Stn⟶S∞, and S′tn⟶0 as n⟶∞. Therefore,(37)dStndt=1−pb−βStnItn+∫0tnγapb+α1Stn−a+αItn−ae−∫0aμ+γθdθda+∫tn+∞γaR0a−tne−∫a−tnaμ+γθdθda−μ+α1Stn+r2Itn.
Employing Lemma2, one admits that(38)limn⟶∞∫0tnγapb+α1Stn−a+αItn−ae−∫0aμ+γθdθda≥pb+α1S∞+αI∞Mτ,limn⟶∞∫tn+∞γaR0a−tne−∫a−tnaμ+γθdθda=0.
Letn⟶∞. Equation (37) indicates that(39)0≥1−pb−βS∞I∞−μ+α1S∞+r2I∞+pb+α1S∞+αI∞Mτ.
Note thatlimn⟶∞Itn=0, then I∞≤I∞⟶0. And then, we immediately admit that(40)1−pb+pbMτμ+α1−α1Mτ≤S∞≤S∞≤1−pb+pbMτμ+α1−α1Mτ.
Therefore,limt⟶∞St=1−pb+pbMτ/μ+α1−α1Mτ. It follows from (34) that(41)limt⟶∞Rt,a=pb+α11−pb+pbMτμ+α1−α1Mτe−∫0aμ+γθdθ.
Consequently, ifℛ0τ<1, then St,It,Rt,⋅⟶E0 for ϕ∈Γ. This completes the proof.


### 4.2. The Stability of the Computer Virus EquilibriumE∗

In this subsection, we will discuss the stability of the computer virus equilibriumE∗ of system (4) when ℛ0τ>1. Then, similar to Theorem 3, linearizing system (4) around the computer virus equilibrium E∗ and accouting for exponential forms of solution for that linear system and taking (2) into the linear system, we obtain the characteristic equation of system (4) at the computer virus equilibrium E∗ as follows:(42)fλ,τ=λ3+m2τλ2+m1τλ+m0τ+n1τλ+n0τe−λτ=0,where(43)m2τ=μ+βI∗+μ+α1+γ∗,m1τ=βI∗μ+α+γ∗+μ+α1μ+γ∗,m0τ=βI∗αμ+γ∗,n1τ=−α1γ∗e−μτ,n0τ=−βI∗αγ∗e−μτ.It is clear thatmiτ>0, i=0,1,2, and niτ<0, i=0,1.In the case whereτ=0, the following result holds.Theorem 5.
Ifℛ0τ>1, then the computer virus equilibrium E∗ of system (4) is locally asymptotically stable for τ=0.Proof.
Whenτ=0, by the direct computation, the coefficients of (42) become(44)m10+n10=α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1μ+α+γ∗+μ+α1μ+γ∗−α1γ∗>0,m00+n00=α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1αμ>0,m10+n10m20−m00+n00=α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−12μ+α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1+α1+γ∗μ+α+γ∗−αμ+2μ+α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1+α1+γ∗μ+α1μ+γ∗−α1γ∗.
Therefore, the Routh–Hurwitz criterion ensures that all the roots offλ,0 have negative real parts. Namely, E∗ is locally asymptotically stable for ℛ0τ>1 and τ=0.
In fact, in the caseℛ0τ>1, the characteristic roots have continuous dependence on τ which implies that Theorem 4 is still valid for τ>0 sufficiently small. However, some roots of (34) may cross the imaginary axis to the right part as τ increases. Therefore, we will further insight into the stability of E∗ when ℛ0τ>1 and τ>0 in the next section.


## 4.1. The Stability of the Virus-Free EquilibriumE0

Theorem 3.
Ifℛ0τ<1, then the virus-free equilibrium E0=S0,0,R0a of system (4) is locally asymptotically stable while if ℛ0τ>1, E0 is unstable.Proof.
Linearizing system (4) at the virus-free equilibrium E0 and considering the exponential forms of that linear system, we obtain the characteristic equation of system (4) at E0 as follows:(27)λ+α+r2+μ−βS0λ+μ+α1−α1∫0+∞γae−∫0aλ+μ+γθdθda=0.
It is clear thatλ1=βS0−α+r2+μ=α+r2+μℛ0τ−1 is the root of the characteristic (27). The characteristic root λ1<0 when ℛ0τ<1 and λ1>0 when ℛ0τ>1.
In addition, the remaining roots of (27) satisfy the following equation:(28)Λλ=λ+μ+α1−α1∫0+∞γae−∫0aλ+μ+γθdθda=0.
Ifλ∈ℝ, then Λλ is a continuous real function strictly increasing and satisfies that(29)Λ0=μ+α1−α1Mτ>0,limλ⟶+∞Λλ=+∞.
It implies thatΛλ=0 has no positive real root. Suppose that λ=u+iv is an arbitrary complex root of Λλ=0 and it satisfies u>0, then we have(30)Λu+iv=u+iv+μ+α1−α1∫0+∞γae−∫0au+iv+μ+γθdθda=0,which implies that the real part ReΛu+iv=0 and the imaginary part ImΛu+iv=0. Separating the real and imaginary part of Λu+iv, we obtain(31)ReΛu+iv=u+μ+α1−α1∫0+∞γae−uacosvae−∫0aμ+γθdθda=u+μ+α11−γ∗eγ∗τ∫τ+∞e−u+μ+γ∗acosvada=u+μ+α11−γ∗u+μ+γ∗e−u+μτcosvτ−v/u+μ+γ∗sinvτ1+v2/u+μ+γ∗2=u+μ+α11−γ∗u+μ+γ∗e−u+μτcosarccos1/1+v2/u+μ+γ∗2+vτ1+v2/u+μ+γ∗2.
It is obvious thatReΛu+iv>0 since 1+v2/u+μ+γ∗2>1 and −1≤cosarccos1/1+v2/u+μ+γ∗2+vτ≤1. Based on the above analysis, we know λ=u+ivu>0 cannot be the root of Λλ=0. That is, Λλ=0 has no complex root with positive real part.
Summarizing the above analysis, we can see that the virus-free equilibriumE0 is locally asymptotically stable when ℛ0τ<1 and the virus-free equilibrium E0 is unstable when ℛ0τ>1.
In the following, we discuss the global stability ofE0 by employing the fluctuation lemma. Let(32)ψ∞=liminft⟶∞ψt,andψ∞=limsupt⟶∞ψt,and the fluctuation lemma is given as follows.Lemma 1 (fluctuation lemma [16]).
Letψ:ℝ+⟶ℝ be a bounded and continuously differentiable function. Then, there exist sequences qn and tn such that qn⟶∞, tn⟶∞, ψqn⟶ψ∞, ψ′qn⟶0, ψtn⟶ψ∞, and ψtn⟶0 as n⟶∞.Lemma 2 (see [17]).
Supposeg:ℝ+⟶ℝ is a bounded function. Then,(33)liminft⟶∞∫0tkθgt−θdθ≥g∞k1,where k1=∫0∞ksds.Theorem 4.
Ifℛ0τ<1, then the virus-free equilibrium E0 of system (4) is globally asymptotically stable for any ϕ∈Γ.Proof.
In order to establish the globally asymptotical stability of the virus-free equilibriumE0, according to Theorem 3, it is sufficient to prove that E0 is attractive in Γ. Let St,It,Rt,a be a solution of system (4) with ϕ∈Γ. Integrating the third equation of system (4) with the boundary condition yields(34)Rt,a=pb+α1St−a+αIt−ae−∫0aμ+γθdθ,a≤t,R0a−te−∫a−taμ+γθdθ,a>t.
With the assistance of the fluctuation lemma, it is easy to get(35)S∞≤1−pb+pbMτμ+α1−α1Mτ.
Furthermore, it follows from the second equations of system (4) that(36)dItdt≤β1−pb+pbMτμ+α1−α1MτIt−α+r2+μIt=α+r2+μℛ0τ−1It.
It is clear thatI∞⟶0 under the condition ℛ0τ<1.
According to Lemma1, we can select a sequence tn such that tn⟶∞, Stn⟶S∞, and S′tn⟶0 as n⟶∞. Therefore,(37)dStndt=1−pb−βStnItn+∫0tnγapb+α1Stn−a+αItn−ae−∫0aμ+γθdθda+∫tn+∞γaR0a−tne−∫a−tnaμ+γθdθda−μ+α1Stn+r2Itn.
Employing Lemma2, one admits that(38)limn⟶∞∫0tnγapb+α1Stn−a+αItn−ae−∫0aμ+γθdθda≥pb+α1S∞+αI∞Mτ,limn⟶∞∫tn+∞γaR0a−tne−∫a−tnaμ+γθdθda=0.
Letn⟶∞. Equation (37) indicates that(39)0≥1−pb−βS∞I∞−μ+α1S∞+r2I∞+pb+α1S∞+αI∞Mτ.
Note thatlimn⟶∞Itn=0, then I∞≤I∞⟶0. And then, we immediately admit that(40)1−pb+pbMτμ+α1−α1Mτ≤S∞≤S∞≤1−pb+pbMτμ+α1−α1Mτ.
Therefore,limt⟶∞St=1−pb+pbMτ/μ+α1−α1Mτ. It follows from (34) that(41)limt⟶∞Rt,a=pb+α11−pb+pbMτμ+α1−α1Mτe−∫0aμ+γθdθ.
Consequently, ifℛ0τ<1, then St,It,Rt,⋅⟶E0 for ϕ∈Γ. This completes the proof.


## 4.2. The Stability of the Computer Virus EquilibriumE∗

In this subsection, we will discuss the stability of the computer virus equilibriumE∗ of system (4) when ℛ0τ>1. Then, similar to Theorem 3, linearizing system (4) around the computer virus equilibrium E∗ and accouting for exponential forms of solution for that linear system and taking (2) into the linear system, we obtain the characteristic equation of system (4) at the computer virus equilibrium E∗ as follows:(42)fλ,τ=λ3+m2τλ2+m1τλ+m0τ+n1τλ+n0τe−λτ=0,where(43)m2τ=μ+βI∗+μ+α1+γ∗,m1τ=βI∗μ+α+γ∗+μ+α1μ+γ∗,m0τ=βI∗αμ+γ∗,n1τ=−α1γ∗e−μτ,n0τ=−βI∗αγ∗e−μτ.It is clear thatmiτ>0, i=0,1,2, and niτ<0, i=0,1.In the case whereτ=0, the following result holds.Theorem 5.
Ifℛ0τ>1, then the computer virus equilibrium E∗ of system (4) is locally asymptotically stable for τ=0.Proof.
Whenτ=0, by the direct computation, the coefficients of (42) become(44)m10+n10=α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1μ+α+γ∗+μ+α1μ+γ∗−α1γ∗>0,m00+n00=α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1αμ>0,m10+n10m20−m00+n00=α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−12μ+α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1+α1+γ∗μ+α+γ∗−αμ+2μ+α+r2+μμ+α1−α1γ∗/μ+γ∗μ+α−αγ∗/μ+γ∗ℛ00−1+α1+γ∗μ+α1μ+γ∗−α1γ∗.
Therefore, the Routh–Hurwitz criterion ensures that all the roots offλ,0 have negative real parts. Namely, E∗ is locally asymptotically stable for ℛ0τ>1 and τ=0.
In fact, in the caseℛ0τ>1, the characteristic roots have continuous dependence on τ which implies that Theorem 4 is still valid for τ>0 sufficiently small. However, some roots of (34) may cross the imaginary axis to the right part as τ increases. Therefore, we will further insight into the stability of E∗ when ℛ0τ>1 and τ>0 in the next section.


## 5. Stability and Hopf Bifurcation WhenR0τ>1 and τ>0

Whenτ>0, we rewrite the characteristic equation fλ,τ=0 as a transcendental equation as follows:(45)Pλ,τ+Qλ,τe−λτ=0,where(46)Pλ,τ=λ3+m2τλ2+m1τλ+m0τ,Qλ,τ=n1τλ+n0τ.It is clear thatPλ,τ and Qλ,τ are both analytic function with respect to λ and differentiable with respect to τ. Following section 2 in [18], we need to justify the following hypotheses:(i)
P0,τ+Q0,τ≠0(ii)
Piω,τ+Qiω,τ≠0(iii)
limsupQ/P:λ⟶∞,Reλ≥0<1, ∀τ∈R+(iv)
Fω,τ=Piω,τ2−Qiω,τ2(v)
Each positive rootsωτ of Fω,τ=0 is continuous and differentiable in τ whenever it existsThrough a tedious manipulation, we derive(47)P0,τ+Q0,τ=m0τ+n0τ=βI∗αμ+γ∗−γ∗e−μτ>0,Piω,τ+Qiω,τ=−2μ+βI∗+α1+γ∗+βI∗αμ+γ∗−γ∗e−μτ+iωβI∗μ+α+γ∗+μ+α1μ+γ∗−α1γ∗e−μτ−ω2≠0,limλ⟶∞Qλ,τPλ,τ=limλ⟶∞n1τλ+n0τλ3+d2τλ2+d1τλ+d0τ=0,which implies that conditions (i), (ii), and (iii) are satisfied.Noting that(48)Piω,τ2=ω6+m22τ−2m1τω4+m12τ−2m2τm0τω2+m02τ,Qiω,τ2=n12τω2+n02τ,which admits(49)Fω,τ=ω6+m22τ−2m1τω4+m12τ−2m2τm0τ−n12τω2+m02τ−n02τ.Obviously, condition (iv) readily holds, and the implicit function theorem ensures that condition (v) is also satisfied.Letλ=iω, ω>0, be one purely imaginary root of fλ,τ=0. Then, we calculate that(50)n1τωsinωτ+n0τcosωτ=m2τω2−m0τ,n0τsinωτ−n1τωcosωτ=m1τω−ω3.SettingΘ=ω2, then (49) can be rewrittten as(51)QΘ≔Θ3+q2τΘ2+q1τΘ+q0τ,where q2τ=m22τ−2m1τ, q1τ=m12τ−2m2τm0τ−n12τ, and q0τ=m02τ−n02τ. It is easy to verify q0τ>0.LetQ′Θ=3Θ2+2q2τΘ+q1τ. If q22τ−3q1τ<0, then Q′Θ=0 has no real roots. While if q22τ−3q1τ≥0, then Q′Θ=0 has two real roots, which are Θ1=−q2τ−q22τ−3q1τ/3 and Θ2=−q2τ−q22τ−3q1τ/3, respectively. The following lemma gives the results on the positive root of the equation QΘ=0.Lemma 3.
(i)
Ifq22τ<3q1τ, then QΘ=0 has no positive root(ii)
Ifq22τ≥3q1τ and Θ2≤0, then QΘ=0 has no positive root(iii)
Ifq22τ≥3q1τ, Θ2>0, and QΘ2>0, then QΘ=0 has no positive root(iv)
Ifq22τ≥3q1τ, Θ2>0, and QΘ2≤0, then QΘ=0 has the positive roots
If (51) does not have any positive root, then the stability of E∗ will not change as τ increases. Therefore, we have the following result.Theorem 6.
Suppose thatℛ0τ>1 and τ>0.(i)
Ifq22τ<3q1τ, then the computer virus equilibrium E∗ of system (4) is locally asymptotically stable(ii)
Ifq22τ≥3q1τ and Θ2≤0, then the computer virus equilibrium E∗ of system (4) is locally asymptotically stable(iii)
Ifq22τ≥3q1τ, Θ2>0, and QΘ2>0, then the computer virus equilibrium E∗ of system (4) is locally asymptotically stableIn what follows, we assume that (51) has one positive root. It implies that the stability of the computer virus equilibrium E∗ may change once τ passes through some specific values. Let Θ∗ be the root of QΘ=0. Namely, ωτ∗=Θ∗ is the unique positive real root of Fω,τ=0. For the sake of convenience, let us define a set by(52)Σ=τ>0:q22τ≥3q1τ,Θ2>0,QΘ2≤0.That is, forτ∈Σ, there exists ω=ωτ>0 such that Fω,τ=0.Letθτ∈0,2πτ∈Σ be a solution of the following equations:(53)cosθτ=n1τω4+m2τn0τ−m1τn1τω2−m0τn0τn02τ+n12τω2,sinθτ=m2τn1τ−n0τω3+m1τn0τ−m0τn1τωn02τ+n12τω2.Then, we conclude thatωττ=θτ+2nπ. Hence, ω∗=ωτ∗ω∗>0 is a purely imaginary root of (45) if and only if τ∗ is a zero of Cnτ for some n∈ℕ, which is defined by(54)Cnτ=τ−θτ+2nπωτ,τ∈Σ,n∈ℕ.Theorem 2.2 in [18] implies that the following lemma is true.Lemma 4.
(see [18]). Assume that ωτ is a positive real root of Fω,τ=0 for τ∈Σ, and at some τ∗∈Σ,(55)Cnτ∗=0,for somen∈ℕ.
Then, a pair of simple conjugate pure imaginary rootsλ+τ∗=iωτ∗ and λ−τ∗=−iωτ∗ of the characteristic (45) exists at τ=τ∗ which crosses the imaginary axis from left to right if Hτ∗>0 and crosses the imaginary axis from right to left if Hτ∗<0, where(56)Hτ∗=signdReλdτ|λ=iωτ∗=signFω′ωτ∗,τ∗signdCnτdτ|τ=τ∗.
The relationship betweenFω,λ and QΘ leads to(57)Hτ∗=signdReλdτ|λ=iωτ∗=signQ′Θ∗signdCnτdτ|τ=τ∗,which implies that the transversality condition holds and a Hopf bifurcation occurs at τ=τ∗ when Q′Θ∗≠0. According to the Hopf bifurcation theorem for functional differential equations [19], we have the following result.Theorem 7.
Suppose thatℛ0τ>1 and Q′Θ∗≠0. If q22τ≥3q1τ, Θ2>0, and QΘ2≤0, then the computer virus equilibrium E∗ of system (4) is locally asymptotically stable for τ∈0,τ∗, and system (4) undergoes a Hopf bifurcation at the computer virus equilibrium E∗ when τ=τ∗.


## 6. The Numerical Simulations

In the following, we will proceed with Matlab to validate the oscillation behaviors of system (4). Let the maximum acquired immunity age be 100, p=0.2, b=1, μ=0.005, α=2.8, γ∗=0.9, r2=0.05, and Rt=∫0+∞Rt,ada. Then, we first illustrate the impact of τ on the number of the susceptible computers St, the infected computers It, and the recovery computers Rt. Taking τ=1,3,5, then ℛ0τ<1 when β=0.02 and ℛ0τ>1 when β=0.08. Figure 2 displays that the number of the susceptible computers St and the infected computers It decreases as the delay τ increases and the number of the recovery computers Rt increases as the delay τ increases.Figure 2
The impact ofτ on the number of the susceptible computers St, the infected computers It, and the recovery computers Rt: (a) ℛ0τ<1; (b) ℛ0τ>1.
(a)(b)Next, we show that the stability of the computer virus equilibriumE∗ and the Hopf bifurcation happens around the computer virus equilibrium E∗ under different conditions. Choosing β=0.08, τ=2, and α1=0.5, we can obtain ℛ02=2.1978>1 and q22τ−3q1τ=−5.9250<0. Figure 3 exhibits the solution of system (4) with different initial values which will tend to E∗ as t tends to infinity.Figure 3
The stability of the computer virus equilibriumE∗ of system (4) when q22τ<3q1τ and ℛ0τ>1: (a) the infectious computer; (b) the recovered computer.
(a)(b)If we takeβ=0.2, τ=2, and α1=2.5, then ℛ02=1.6036>1, q22τ−3q1τ=12.6772>0, and Θ2=−1.6804<0. Figure 4(a) shows that the computer virus equilibrium E∗ is a stable focus. While if we take β=0.06, τ=2, and α1=0.5, then ℛ02=1.6484>1, q22τ−3q1τ=0.1442>0, Θ2=0.4363>0, and QΘ2=0.0576>0. Accordingly, Figure 4(b) shows that the computer virus equilibrium E∗ is a stable focus again.Figure 4
The stability of the computer virus equilibriumE∗ of system (4) when ℛ0τ>1 and q22τ≥3q1τ: (a) Θ2 = −1.6804; (b) Θ2 = 0.4363 and Q(Θ2) = 0.0576.
(a)(b)Finally, settingβ=0.05, α1=0.1, and τ=2, we have ℛ02=2.6688>1, q22τ−3q1τ=2.7028>0, Θ2=1.2942>0, and QΘ2=−0.5037<0. Figure 5 displays that system (4) exists the periodic solutions.Figure 5
The hopf bifurcation around the computer virus equilibriumE∗ when ℛ0τ>1: (a) the periodic solution; (b) the phase diagram.
(a)(b)In addition, choosing the same parameter values as these of Figure5 and using τ as the bifurcation parameter, we can display the complete dynamic behavior of system (4) as the delay τ increases in Figure 6. It illustrates that system (4) exists the periodic solutions for τ∈2,23. When τ∈0,2, we obtain ℛ0τ>1, namely, system (4) has a unique computer virus equilibrium E∗, which is stable. When τ≥23, we obtain ℛ0τ<1, namely, system (4) has a unique virus-free equilibrium E0, which is stable.Figure 6
The dynamic behaviors of system (4) as the delay τ increases: (a) the virus-infected computers; (b) the recovery computers.
(a)(b)


## 7. Conclusion and Discussion

In this paper, we have proposed and analyzed a computer virus model by using of the classic SIRS model with the age structure and delay. The age structure and delay are combined to describe the phenomenon that the recovery computers stay in the recovery class for a short time and eventually become the susceptible computers again since the loss of immune protection. The purpose of this paper is to explore the impact of the age structure and delay on the transmission of the computer virus.On the dynamic behavior analysis of system (4), we showed that the non-negativity of the solutions of system (4) and the boundedness of system (4) gave the basic reproduction number ℛ0τ and proved that ℛ0τ=1 is the threshold that determines whether the epidemic persists or not by studying the stability of both virus-free equilibrium E0 and the computer virus equilibrium E∗. The virus-free equilibrium E0 is globally asymptotically stable if ℛ0τ<1 and is unstable if ℛ0τ>1. Moreover, the computer virus persists in the later case, in the sense that infected computers survive above a certain number for any initial infection numbers. We also proved the existence of Hopf bifurcation around the computer virus equilibrium E∗ when E∗ is unstable.The existence of Hopf bifurcation of system (4) means that, if the age structure and delay are introduced together into the computer virus SIRS model, the simple threshold dynamic behavior will be destroyed. We speculate that the essential reason for the changes in the dynamic behavior of system (4) is that the acquired immunity age is subject to a non-Markovian process.The limitation of this paper is that there is no discussion about the global stability of the computer virus equilibriumE∗ when ℛ0τ>1 and τ=0. In addition, the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions from E∗ have not been resolved in this paper. We will continue to discuss these aspects in the future.



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