Abstract
A further generalization of an SEIQRS-V (susceptible-exposed-infectious-quarantined-recovered-susceptible with vaccination) computer virus propagation model is the main topic of the present paper. This paper specifically analyzes effects on the asymptotic dynamics of the computer virus propagation model when two time delays are introduced. Sufficient conditions for the asymptotic stability and existence of the Hopf bifurcation are established by regarding different combination of the two delays as the bifurcation parameter. Moreover, explicit formulas that determine the stability, direction, and period of the bifurcating periodic solutions are obtained with the help of the normal form theory and center manifold theorem. Finally, numerical simulations are employed for supporting the obtained analytical results.
1. Introduction
Computer viruses, including conventional viruses and network worms, can propagate among computers with no human awareness and popularization of Internet has been the major propagation channel of viruses [1, 2]. The past few decades have witnessed the great financial losses caused by computer viruses. Therefore, it is of considerable importance to investigate the laws describing propagation of computer viruses in order to provide some help with preventing computer viruses. For that purpose and in view of the fact that propagation of computer viruses among computers resembles that of biological viruses among a population, many dynamical models describing propagation of computer viruses across the Internet have been established by the scholars at home and abroad, such as conventional models [3–8], stochastic models [9–12], and delayed models [13–18]. There are also some other computer virus models [19–21] combined with network theory to investigate the impact of the network topology, the patch forwarding, and the network eigenvalue on the viral prevalence.
As is known, vaccination is regarded as one of the most effective measures of preventing computer viruses and the awareness that there exist many infected computers would enhance the probability that the user of a susceptible computer will make his computer vaccinated [22, 23]. However, the mentioned models above neglect the influence of vaccination strategy on the propagation of computer viruses. Recently, considering the importance of vaccination, Kumar et al. [24] proposed the following SEIQRS-V computer virus propagation model:where , , , , , and denote the numbers of the uninfected computers, the exposed computers, the infected computers, the quarantined computers, recovered computers, and vaccinated computers at time , respectively. is the birth rate of new computers in the network; is the death rate of the computers due to the reason other than the attack of viruses; is the death rate of computers due to the attack of viruses; is the contact rate of the uninfected computers; , , , , , , and are the transition rates between the states in system (1).
Obviously, system (1) neglects the delays in the procedure of viruses’ propagation and it is investigated under the assumption that the transition between the states is instantaneous. This is not reasonable with reality. For example, it needs a period to clean the viruses in the infected and quarantined computers for antivirus software and there is usually a temporary immunity period for the recovered and the vaccinated computers because of the effect of the antivirus software. In addition, a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this, we introduce two delays into system (1) and get the following delayed system: where is the time delay due to the period that antivirus software uses to clean the viruses in the infected and quarantined computers and is the time delay due to the temporary immunity period of the recovered and the vaccinated computers.
To the best of our knowledge, until now, there is no good analysis on system (2). Therefore, it is meaningful to analyze the proposed system with two delays.
The rest of this paper is organized as follows. In the next section, we analyze the threshold of Hopf bifurcation of system (2) by regarding different combination of the two delays as the bifurcation parameter. In Section 3, by means of the normal form theory and center manifold theorem, direction and stability of the Hopf bifurcation for and are investigated. Simulation results of system (2) are shown in Section 4. Finally, we finish the paper with conclusions in Section 5.
2. Analysis of Hopf Bifurcation
By direct computation, we know that if and , then system (2) has a unique viral equilibrium , where The linearized section of system (2) at is as follows: where Then, the characteristic equation for system (4) can be obtained: with
Case 1 (). When , (6) becomes where Clearly, . Thus, if condition (see (10)) holds, then system (2) without delay is locally asymptotically stable:
Case 2 (; ). Equation (6) equals where Multiplying on left and right of (11), one has Assume that is the root of (13): with Thus, one can obtain the expressions of and as follows: Then, we can get Suppose that (see (17)) has at least one positive root.
If condition holds, then there exists such that (13) has a pair of purely imaginary roots . For , Differentiating (13) with respect to , one has where Thus, with Thus, if condition holds, then . Based on the Hopf bifurcation theorem in [25], we have the following results.
Theorem 1. Suppose that conditions , , and hold for system (2). The viral equilibrium is locally asymptotically stable when and a Hopf bifurcation occurs at the viral equilibrium when .
Case 3 (; ). Equation (6) becomeswhereMultiplying on left and right of (23), one has Let be the root of (25):withThen,And the equation following equation regarding can be obtained: Suppose that (see (29)) has at least one positive root.
If condition holds, then there exists such that (25) has a pair of purely imaginary roots . For , Differentiate both sides of (25) with respect to . Then, whereThus,withSimilar to Case 2, we know that if condition holds, then . In conclusion, we have the following results.
Theorem 2. Suppose that conditions , , and hold for system (2). The viral equilibrium is locally asymptotically stable when and a Hopf bifurcation occurs at the viral equilibrium when .
Case 4 (; ). Regarding as the bifurcation parameter when , multiplying by , (6) becomes Let be the root of (35), and for the convenience we still denote as ; then, whereThus,Then, we getSuppose that (see (39)) has at least one positive root.
If holds, then there exists such that (35) has a pair of purely imaginary roots . For ,Differentiating both sides of (25) with respect to ,whereDefine Similar to Case 2, we know that if condition holds, then . Thus, we have the following results.
Theorem 3. Let and suppose that conditions , , and hold for system (2). The viral equilibrium is locally asymptotically stable when and a Hopf bifurcation occurs at the viral equilibrium when .
Case 5 (; ). Regarding as the bifurcation parameter when , multiplying by , (6) becomes Let be the root of (44), and for the convenience we still denote as ; then, whereThus,Then, we get If holds, then there exists such that (35) has a pair of purely imaginary roots . For ,Differentiating (25) with respect to , one can get whereThus,Therefore, we know that if condition holds, then . Then, we have the following results.
Theorem 4. Let and suppose that conditions , , and hold for system (2). The viral equilibrium is locally asymptotically stable when and a Hopf bifurcation occurs at the viral equilibrium when .
3. Properties of the Hopf Bifurcation
In this section, we shall investigate direction and stability of the Hopf bifurcation under the case where and . Set , , , , , , and . For convenience, we assume that throughout this section. Then, system (2) becomes functional differential equations in : with where Based on the Riesz representation theorem, there exists a function such that In fact, we choose For , we define Then, system (53) becomes where for .
Define as follows: and a bilinear form where .
Let be the eigenvector of with and let be the eigenvector of with . Then, according to the definition of and , we obtain In addition, from (61), we have Thus, we can choose such that , .
Then, using the algorithms from Hassard et al. [25] and the similar computation process in [26–29], we obtain with where
Then, we can get the following coefficients: Thus, we have the following results.
Theorem 5. The sign of determines direction of the Hopf bifurcation: if (), then the Hopf bifurcation is supercritical (subcritical); the sign of determines stability of the bifurcating periodic solutions: if (), then the bifurcating periodic solutions are stable (unstable); the sign of determines period of the bifurcating solutions: if (), then the period of the bifurcating periodic solutions increases (decreases).
4. Numerical Simulation
In this section, we present some numerical results of system (2) in order to validate the analytical predictions obtained in Sections 2 and 3. We choose a set of parameters as follows: , , , , , , , , , , and , and consider the following special case of (2): from which we can get the unique viral equilibrium , , 71.7439). It can be easily verified that condition is satisfied when .
By computation, we have and . Then, we get . Thus, we know that conditions and hold. We can conclude that all roots that cross the imaginary axis at cross from left to right as increases by the theory in [22]. According to Theorem 1, , 71.7439) is asymptotically stable when . This property can be illustrated by Figures 1 and 2. In this case, spreading law of the computer viruses can be predicted and the viruses can be controlled and eliminated. However, once the value of passes through the critical value , , 71.7439) loses its stability and a Hopf bifurcation occurs, which can be shown in Figures 3 and 4. The occurrence of a Hopf bifurcation means that the state of computer viruses propagation changes from the viral equilibrium point to a limit cycle. This makes spreading of the computer viruses be out of control.
Similarly, we have the following: and when and ; and when and ; and when and . The corresponding phase plots are shown in Figures 5–8, Figures 9–12, and Figures 13–16, respectively. In addition, for and , we have and by some complex computation. Based on (68), we get , , and . Therefore, the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the bifurcating periodic solutions decreases.
According to the numerical simulation results, we know that the time delay should remain less than the corresponding threshold in order to control and predict the viruses’ propagation by decreasing the period that antivirus software uses to clean the computer viruses and the temporary immunity period of the recovered and the vaccinated computers. To this end, we can adjust the parameters of our proposed model in real-world networks, such as timely updating the antivirus software on computers, properly controlling the number of computers attached to the network, and timely disconnecting computers from the network when the connections are unnecessary. Of course, in the next step, we also need to collect large amount of relevant data and estimate the parameters involved in our proposed model through statistical analysis in real-world networks. Namely, we have to adjust the parameters in the model so as to control viruses’ propagation effectively if it is necessary.
5. Conclusions
It is definitely an interesting work to consider the effect of delays on dynamical systems, because a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this fact, we introduce the time delay due to the period that antivirus software uses to clean the computer viruses in the infectious and quarantined computers () and the time delay due to the temporary immunity period of the recovered and the vaccinated computers () into the SEIQRS-V computer virus propagation model considered in [21]. We obtain some conditions for local stability and Hopf bifurcation occurring by analyzing distribution of roots of the associated characteristic equation.
By computation, there exists a corresponding critical value of the time delay below which system (2) is stable and above which system (2) is unstable. When the system is stable, the characteristics of the propagation of computer viruses can be easily predicted and then the computer viruses can get eliminated. Otherwise, the propagation of the computer viruses is out of control. Therefore, stability of the computer virus propagation system must be guaranteed in practice. In addition, we find that the effect of on system (2) is marked compared with , because the critical value of is much smaller when we only consider it. At last, we have also derived the explicit formula which can determine direction and stability of the Hopf bifurcation under the case where and .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by Natural Science Foundation of Anhui Province (nos. 1608085QF151, 1608085QF145, and 1708085MA17) and Natural Science Foundation of the Higher Education Institutions of Anhui Province (nos. KJ2014A006 and KJ2015A144).