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.
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:where denotes the number of the recovery computers at time , is the rate at which one computer is removed from the network, and 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 of the recovery computers depends on the length of the recovery time. To this end, we assume that the removal rate in (1) should be replaced with the following piecewise function:where , is a positive constant, and 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 aswhere denotes the density of the recovery computers with respect to the acquired immunity age at time . Obviously, the loss rate of the acquired immunity of the recovery computers obeys a non-Markovian process.
Let and be the number of the susceptible computers and infected computers at time , respectively, be the rate at which external computers are connected to the network, be the vaccination proportion of computers which are connected to the network, be the transmission rate, be the rate at which the infected computers recover due to the antivirus treatment, be the self-recovery rate of the infected computers, and 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 conditionand the initial condition
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 and discuss the existence of all the feasible equilibria. In section 4, we study the global stability of the virus-free equilibrium when and local stability of the computer virus equilibrium when and . In section 5, we study the Hopf bifurcations occurring from the computer virus equilibrium with the increase in . 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 in Lemma A. 1.
Theorem 1. If , then the solution of system (4) is non-negative for all and the setwhich is positively invariant with respect to system (4); moreover, it is ultimately bounded for large enough.
Proof. The second equation of system (4) implies thatwhich means that, for any initial value , remains non-negative for .
For any , we claim that remains non-negative for any . 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 such that for , , and . Then, by using of the third and fourth equation of system (4), we can getIt is clear that since , , and . Plugging such a value into the first equation of (4) leads towhich contradicts with . Hence, the claim holds and remains non-negative for any if .
Based on the above analysis, we directly integrate the third equation in system (4) along the characteristic line yields thatThe non-negativity of and together with ensures that remains non-negative for all .
In the following, we proceed with the ultimate boundedness of the solutions of system (4). Let , which represents the total number of the recovery individuals at time . Then, adding those equations in system (4) yieldsIt is reasonable to assume that according to the biological significance. Noting that , we obtainTherefore,It is not hard to see that the setwhich 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 .
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:
The virus-free equilibrium means that there is no virus-infected computers in the entire network; therefore, in order to get the virus-free equilibrium of system (4), we assume . Then, equation (16) can be rewritten as the following equations:
The second and third equations in (17) yields
Taking into the first equation in (17), we obtainwith
Then, we know that system (4) always has the virus-free equilibrium , where
The computer virus equilibrium means that there are always virus-infected computers in the entire network; therefore, in order to get the computer virus equilibrium of system (4), we assume . Then, equation (16) can be rewritten as the following equations:
The second equation in (22) states that . Furthermore,
Substituting and into the first equation in (22), we can getwith
Obviously, inequalities guarantee . That is, when , system (4) has the computer virus equilibrium , where
Therefore, the following theorem gives the existence of the equilibria of system (4).
Theorem 2. System (4) has always the virus-free equilibrium ; In addition, when , system (4) also has a unique computer virus equilibrium .
In fact, 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, is the infection rate by an infected computer, is the average infection period of the infected computers, and 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 equilibrium by the fluctuation lemma and the local stability of the computer virus equilibrium by analyzing the linearizing system (4).
4.1. The Stability of the Virus-Free Equilibrium
Theorem 3. If , then the virus-free equilibrium of system (4) is locally asymptotically stable while if , is unstable.
Proof. Linearizing system (4) at the virus-free equilibrium and considering the exponential forms of that linear system, we obtain the characteristic equation of system (4) at as follows:It is clear that is the root of the characteristic (27). The characteristic root when and when .
In addition, the remaining roots of (27) satisfy the following equation:If , then is a continuous real function strictly increasing and satisfies thatIt implies that has no positive real root. Suppose that is an arbitrary complex root of and it satisfies , then we havewhich implies that the real part and the imaginary part . Separating the real and imaginary part of , we obtainIt is obvious that since and . Based on the above analysis, we know cannot be the root of . That is, has no complex root with positive real part.
Summarizing the above analysis, we can see that the virus-free equilibrium is locally asymptotically stable when and the virus-free equilibrium is unstable when .
In the following, we discuss the global stability of by employing the fluctuation lemma. Letand the fluctuation lemma is given as follows.
Lemma 1 (fluctuation lemma [16]). Let be a bounded and continuously differentiable function. Then, there exist sequences and such that , , , , , and as .
Lemma 2 (see [17]). Suppose is a bounded function. Then,where .
Theorem 4. If , then the virus-free equilibrium of system (4) is globally asymptotically stable for any .
Proof. In order to establish the globally asymptotical stability of the virus-free equilibrium , according to Theorem 3, it is sufficient to prove that is attractive in . Let be a solution of system (4) with . Integrating the third equation of system (4) with the boundary condition yieldsWith the assistance of the fluctuation lemma, it is easy to getFurthermore, it follows from the second equations of system (4) thatIt is clear that under the condition .
According to Lemma 1, we can select a sequence such that , , and as . Therefore,Employing Lemma 2, one admits thatLet . Equation (37) indicates thatNote that , then . And then, we immediately admit thatTherefore, . It follows from (34) thatConsequently, if , then for . This completes the proof.
4.2. The Stability of the Computer Virus Equilibrium
In this subsection, we will discuss the stability of the computer virus equilibrium of system (4) when . Then, similar to Theorem 3, linearizing system (4) around the computer virus equilibrium 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 as follows:where
It is clear that , , and , .
In the case where , the following result holds.
Theorem 5. If , then the computer virus equilibrium of system (4) is locally asymptotically stable for .
Proof. When , by the direct computation, the coefficients of (42) becomeTherefore, the Routh–Hurwitz criterion ensures that all the roots of have negative real parts. Namely, is locally asymptotically stable for and .
In fact, in the case , the characteristic roots have continuous dependence on which implies that Theorem 4 is still valid for sufficiently small. However, some roots of (34) may cross the imaginary axis to the right part as increases. Therefore, we will further insight into the stability of when and in the next section.
5. Stability and Hopf Bifurcation When and
When , we rewrite the characteristic equation as a transcendental equation as follows:where
It is clear that and are both analytic function with respect to and differentiable with respect to . Following section 2 in [18], we need to justify the following hypotheses:(i)(ii)(iii), (iv)(v)Each positive roots of is continuous and differentiable in whenever it exists
Through a tedious manipulation, we derivewhich implies that conditions (i), (ii), and (iii) are satisfied.
Noting thatwhich admits
Obviously, condition (iv) readily holds, and the implicit function theorem ensures that condition (v) is also satisfied.
Let , , be one purely imaginary root of . Then, we calculate that
Setting , then (49) can be rewrittten aswhere , , and . It is easy to verify .
Let . If , then has no real roots. While if , then has two real roots, which are and , respectively. The following lemma gives the results on the positive root of the equation .
Lemma 3. (i)If , then has no positive root(ii)If and , then has no positive root(iii)If , , and , then has no positive root(iv)If , , and , then has the positive rootsIf (51) does not have any positive root, then the stability of will not change as increases. Therefore, we have the following result.
Theorem 6. Suppose that and .(i)If , then the computer virus equilibrium of system (4) is locally asymptotically stable(ii)If and , then the computer virus equilibrium of system (4) is locally asymptotically stable(iii)If , , and , then the computer virus equilibrium of system (4) 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 may change once passes through some specific values. Let be the root of . Namely, is the unique positive real root of . For the sake of convenience, let us define a set by
That is, for , there exists such that .
Let be a solution of the following equations:
Then, we conclude that . Hence, is a purely imaginary root of (45) if and only if is a zero of for some , which is defined by
Theorem 2.2 in [18] implies that the following lemma is true.
Lemma 4. (see [18]). Assume that is a positive real root of for , and at some ,Then, a pair of simple conjugate pure imaginary roots and of the characteristic (45) exists at which crosses the imaginary axis from left to right if and crosses the imaginary axis from right to left if , whereThe relationship between and leads towhich implies that the transversality condition holds and a Hopf bifurcation occurs at when . According to the Hopf bifurcation theorem for functional differential equations [19], we have the following result.
Theorem 7. Suppose that and . If , , and , then the computer virus equilibrium of system (4) is locally asymptotically stable for , and system (4) undergoes a Hopf bifurcation at the computer virus equilibrium when .
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, , , , , , , and . Then, we first illustrate the impact of on the number of the susceptible computers , the infected computers , and the recovery computers . Taking , then when and when . Figure 2 displays that the number of the susceptible computers and the infected computers decreases as the delay increases and the number of the recovery computers increases as the delay increases.
(a)
(b)
Next, we show that the stability of the computer virus equilibrium and the Hopf bifurcation happens around the computer virus equilibrium under different conditions. Choosing , , and , we can obtain and . Figure 3 exhibits the solution of system (4) with different initial values which will tend to as tends to infinity.
(a)
(b)
If we take , , and , then , , and . Figure 4(a) shows that the computer virus equilibrium is a stable focus. While if we take , , and , then , , , and . Accordingly, Figure 4(b) shows that the computer virus equilibrium is a stable focus again.
(a)
(b)
Finally, setting , , and , we have , , , and . Figure 5 displays that system (4) exists the periodic solutions.
(a)
(b)
In addition, choosing the same parameter values as these of Figure 5 and using as the bifurcation parameter, we can display the complete dynamic behavior of system (4) as the delay increases in Figure 6. It illustrates that system (4) exists the periodic solutions for . When , we obtain , namely, system (4) has a unique computer virus equilibrium , which is stable. When , we obtain , namely, system (4) has a unique virus-free equilibrium , which is stable.
(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 and proved that is the threshold that determines whether the epidemic persists or not by studying the stability of both virus-free equilibrium and the computer virus equilibrium . The virus-free equilibrium is globally asymptotically stable if and is unstable if . 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 when 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 equilibrium when and . In addition, the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions from have not been resolved in this paper. We will continue to discuss these aspects in the future.
Appendix
In this section, we show Theorem 1 of [20] as the following Lemma A. 1. Some definitions and notations are first listed as follows.
Let be a Banach space with norm and be the maximal age. We set where . Let and . Let be a diagonal matrix of functions which may not be bounded on ; that is, we may have for some or all and if is finite. Consider the systemwhere , , and are generally nonlinear. Let denote the positive cone in . Let , be a bounded invariant set for system (A.1).
Lemma 5. Assume the following:(1), , and are Lipschitz continuous; that is, for all and all (or , ), and all , we have(2) where , for all , , , and and for all , , , and or all .(3) where , for all , , and and for all , , and or all .(4) for all , , and .(5), , and are bounded. Then, for every or in , system (A.1) has a unique non-negative solution.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors also would like to thank Professor Jianquan Li for his suggestions, which also helped us to improve this paper. This work was supported by National Natural Science Foundation of China grants 12071268 and 11971281, by Natural Science Foundation of Jiangsu Province of China grant BK20200749, by Nanjing University of Posts and Telecommunications Science Foundation grant NY220093, by Natural Science Foundation of Shaanxi Provincial Department of Education in China grant 18JK0092, and by Youth Innovation Team on Computationally Efficient Numerical Methods Based on New Energy Problems in Shannxi Province.