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

. 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.


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. e 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. erefore, 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][6][7], SEIR models [8], SEIS models [9,10], and so on [11][12][13][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]. e 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. at 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: where R(t) denotes the number of the recovery computers at time t, μ is the rate at which one computer is removed from the network, and c 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 c of the recovery computers depends on the length of the recovery time. To this end, we assume that the removal rate c in (1) should be replaced with the following piecewise function: where c(a) ∈ L ∞ + ((0, +∞), R), c * is a positive constant, and τ is the shortest time for a recovery computer to maintain its acquired immunity. erefore, the outflow of the recovery computers in (1) can be rewritten by a partial differential equation as where R(t, a) denotes the density of the recovery computers with respect to the acquired immunity age a at time t.
Obviously, the loss rate c(a) of the acquired immunity of the recovery computers obeys a non-Markovian process. Let S(t) and I(t) 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, r 2 be the self-recovery rate of the infected computers, and α 1 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.
Following the transmission mechanism and schematic diagram, we propose the computer virus model with age structure and delay in the following: with the boundary condition and the initial condition is 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 R 0 (τ) and discuss the existence of all the feasible equilibria. In section 4, we study the global stability of the virus-free equilibrium E 0 when R 0 (τ) < 1 and local stability of the computer virus equilibrium E * when R 0 (τ) > 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.

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. e 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.
which is positively invariant with respect to system (4); moreover, it is ultimately bounded for t large enough.
Proof. e second equation of system (4) implies that which means that, for any initial value I 0 ∈ R + , I(t) remains non-negative for t ≥ 0. For any S 0 ∈ R + , we claim that S(t) 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 t S such that S(t) ≥ 0 for t ∈ [0, t S ), S(t S ) � 0, and S ′ (t S ) < 0. en, by using of the third and fourth equation of system (4), we can get It is clear that R(t S , a) ≥ 0 since S(t S − a) > 0, I(t S − a) ≥ 0, and R 0 (·) ∈ L 1 + (0, +∞). Plugging such a value into the first equation of (4) leads to which contradicts with S ′ (t S ) < 0. Hence, the claim holds and S(t) remains non-negative for any t ∈ R + if S 0 ≥ 0. Based on the above analysis, we directly integrate the third equation in system (4) along the characteristic line yields that e non-negativity of S and I together with R 0 (·) ∈ L 1 + (0, +∞) ensures that R(t, ·) 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 R(t, a)da, which represents the total number of the recovery individuals at time t.
en, adding those equations in system (4) yields It is reasonable to assume that lim a⟶+∞ R(t, a) � 0 according to the biological significance. Noting that It is not hard to see that the set 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. □ ......

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: e virus-free equilibrium E 0 means that there is no virus-infected computers in the entire network; therefore, in order to get the virus-free equilibrium E 0 of system (4), we assume I � 0. en, equation (16) can be rewritten as the following equations: e second and third equations in (17) yields Taking R(a) � (pb + α 1 S)e − a 0 (μ+c(θ))dθ into the first equation in (17), we obtain with en, we know that system (4) always has the virus-free e computer virus equilibrium E * 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. en, equation (16) can be rewritten as the following equations: μ+c(θ)dθ into the first equation in (22), we can get Obviously, inequalities R 0 (τ) > 1 guarantee I > 0. at is, when R 0 (τ) > 1, system (4) has the computer virus equilibrium E * � (S * , I * , R * (a)), where erefore, the following theorem gives the existence of the equilibria of system (4). (4) also has a unique computer virus equilibrium E * � (S * , I * , R * (a)).

Theorem 2. System (4) has always the virus-free equilibrium
In fact, R 0 (τ) 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 + r 2 + μ) is the average infection period of the infected computers, and (1 − p)b + pb(c * e − μτ /μ + c * )/ μ + α 1 − α 1 (c * e − μτ /μ + c * ) denotes the total number of susceptible individuals.

The Stability of the Equilibria
In this section, we study the global stability of the virus-free equilibrium E 0 by the fluctuation lemma and the local stability of the computer virus equilibrium E * by analyzing the linearizing system (4).

e Stability of the Virus-Free Equilibrium
Proof. Linearizing system (4) at the virus-free equilibrium E 0 and considering the exponential forms of that linear system, we obtain the characteristic equation of system (4) at E 0 as follows: In addition, the remaining roots of (27) satisfy the following equation: If λ ∈ R, then Λ(λ) is a continuous real function strictly increasing and satisfies that It implies that Λ(λ) � 0 has no positive real root. Suppose that λ � u + iv is an arbitrary complex root of Λ(λ) � 0 and it satisfies u > 0, then we have which implies that the real part Re(Λ(u + iv)) � 0 and the imaginary part Im(Λ(u + iv)) � 0. Separating the real and imaginary part of Λ(u + iv), we obtain It is obvious that Re(Λ(u + iv)) > 0 since Based on the above analysis, we know λ � u + iv(u > 0) cannot be the root of Λ(λ) � 0.
at is, Λ(λ) � 0 has no complex root with positive real part.
Summarizing the above analysis, we can see that the virus-free equilibrium E 0 is locally asymptotically stable when R 0 (τ) < 1 and the virus-free equilibrium E 0 is unstable when R 0 (τ) > 1.
In the following, we discuss the global stability of E 0 by employing the fluctuation lemma. Let and the fluctuation lemma is given as follows.
Proof. In order to establish the globally asymptotical stability of the virus-free equilibrium E 0 , according to eorem 3, it is sufficient to prove that E 0 is attractive in Γ. Let (S(t), I(t), R(t, a)) be a solution of system (4) with ϕ ∈ Γ.
Integrating the third equation of system (4) with the boundary condition yields With the assistance of the fluctuation lemma, it is easy to get Furthermore, it follows from the second equations of system (4) that It is clear that I ∞ ⟶ 0 under the condition R 0 (τ) < 1. According to Lemma 1, we can select a sequence t n such that t n ⟶ ∞, S(t n ) ⟶ S ∞ , and S ′ (t n ) ⟶ 0 as n ⟶ ∞. erefore, (37) Employing Lemma 2, one admits that Let n ⟶ ∞. Equation (37) indicates that Note that lim n⟶∞ I(t n ) � 0, then I ∞ ≤ I ∞ ⟶ 0. And then, we immediately admit that erefore, lim t⟶∞ S(t) � (1 − p)b + pbM(τ)/μ + α 1 − α 1 M(τ). It follows from (34) that 6 Discrete Dynamics in Nature and Society Consequently, if R 0 (τ) < 1, then S(t), I(t), R(t, ·) ⟶ E 0 for ϕ ∈ Γ. is completes the proof. In this subsection, we will discuss the stability of the computer virus equilibrium E * of system (4) when R 0 (τ) > 1. en, similar to eorem 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: where m 2 (τ) � μ + βI * + μ + α 1 + c * , It is clear that m i (τ) > 0, i � 0, 1, 2, and n i (τ) < 0, i � 0, 1. In the case where τ � 0, the following result holds.
In fact, in the case R 0 (τ) > 1, the characteristic roots have continuous dependence on τ which implies that eorem 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. erefore, we will further insight into the stability of E * when R 0 (τ) > 1 and τ > 0 in the next section.

(46)
It is clear that P(λ, τ) and Q(λ, 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: F(ω, τ) � 0 is continuous and differentiable in τ whenever it exists rough a tedious manipulation, we derive which implies that conditions (i), (ii), and (iii) are satisfied. Noting that which admits Obviously, condition (iv) readily holds, and the implicit function theorem ensures that condition (v) is also satisfied.

is locally asymptotically stable
In 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 Q(Θ) � 0. Namely, ω(τ * ) � �� � Θ * is the unique positive real root of F(ω, τ) � 0. For the sake of convenience, let us define a set by 8 Discrete Dynamics in Nature and Society at is, for τ ∈ Σ, there exists ω � ω(τ) > 0 such that F(ω, τ) � 0.
According to the Hopf bifurcation theorem for functional differential equations [19], we have the following result.

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, c * � 0.9, r 2 � 0.05, and R(t) � +∞ 0 R(t, a)da. en, we first illustrate the impact of τ on the number of the susceptible computers S(t), the infected computers I(t), and the recovery computers R(t). Taking τ � 1, 3, 5, then R 0 (τ) < 1 when β � 0.02 and R 0 (τ) > 1 when β � 0.08. Figure 2 displays that the number of the susceptible computers S(t) and the infected computers I(t) decreases as the delay τ increases and the number of the recovery computers R(t) increases as the delay τ increases.
Next, we show that the stability of the computer virus equilibrium E * and the Hopf bifurcation happens around the computer virus equilibrium E * under different conditions. Choosing β � 0.08, τ � 2, and α 1 � 0.5, we can obtain R 0 (2) � 2.1978 > 1 and q 2 2 (τ) − 3q 1 (τ) � − 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.

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. e 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. e 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 R 0 (τ) and proved that R 0 (τ) � 1 is the threshold that determines whether the epidemic persists or not by studying the stability of both virus-free equilibrium E 0 and the computer virus equilibrium E * . e virus-free equilibrium E 0 is globally asymptotically stable if R 0 (τ) < 1 and is unstable if R 0 (τ) > 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.  Discrete Dynamics in Nature and Society e 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. e limitation of this paper is that there is no discussion about the global stability of the computer virus equilibrium E * when R 0 (τ) > 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.