Abstract
This paper is devoted to the study of an SIRS computer virus propagation model with two delays and multistate antivirus measures. We demonstrate that the system loses its stability and a Hopf bifurcation occurs when the delay passes through the corresponding critical value by choosing the possible combination of the two delays as the bifurcation parameter. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by means of the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate the obtained results.
1. Introduction
With the rapid development of computer technologies and network applications, the threat of computer viruses to the world would become increasingly serious. It is of vital importance to understand how computer viruses spread over computer network and to control the computer viruses’ propagation in computer networks. To this end, many mathematical models have been studied to illustrate the dynamical behavior of computer viruses spreading since Murray [1] suggested that computer viruses share some traits of biological viruses. In [2, 3], Kephart and White used the SIS model to describe the propagation of computer viruses. In [4], Zou et al. investigated how the spread of red worms is affected by the worm characteristics based on the SIR model. In [5, 6], Yuan et al. proposed the SEIR computer virus model and studied the dynamics of the model, respectively. In [7], Mishra and Pandey formulated an SEIRS model for the transmission of worms in computer network through vertical transmission. In addition, there are also some researchers who proposed the computer virus models with vaccination and quarantine strategy [8–10].
In fact, many computer viruses have different kinds of delays when the viruses spread, such as latent period delay [11, 12], immunity period delay [12–15], and the delay due to the period that the anti-virus software needs to clean the viruses [6]. In [12], Feng et al. proposed the following computer virus propagation model with dual delays and multistate antivirus measures based on the classical SIR epidemic model in [16]: where , , and represent the numbers of susceptible, infected, and recovered hosts in computer networks at time , respectively. is the number of the hosts which are attached to the computer networks and is the proportion of the new hosts which are susceptible. is the death rate of the hosts. , , , and are the state transition rates between the classes , , and . is the latent period of the computer viruses and is the temporary immune period of the recovered hosts. For the convenience of analysis, Feng et al. [12] let ; then, system (1) becomes the following form:
By regarding the time delay as the bifurcation parameter, Feng et al. [12] studied the existence and properties of Hopf bifurcation of system (2). As is known, it needs some time to clean the viruses in the infected hosts for the antivirus software. Therefore, it is reasonable to take into account the time delay due to the period that the antivirus software uses to clean the viruses in the infected hosts in system (2). To this end, we consider the following system with two delays: where is the time delay due to the latent period of the computer viruses and the temporary immune period of the recovered hosts. is the time delay due to the period that the antivirus software uses to clean the viruses in the infected hosts.
The remaining materials of this paper are organized in this fashion: local stability and existence of local Hopf bifurcation are discussed in Section 2. Properties of the Hopf bifurcation such as the direction and stability are investigated in Section 3. Some numerical simulations are carried out to verify the theoretical results in Section 4 and, finally, this work is summarized in Section 5.
2. Local Stability and Existence of Local Hopf Bifurcation
By direct computation, it can be concluded that if , then system (3) has a unique positive equilibrium , where The characteristic equation of system (3) at is from which one can obtain where with
From the expressions of , , , , , and , one can obtain . Therefore, (6) can be transformed into the following form:
Case 1 (). When , (9) is equivalent to
where
It is easy to get that . Therefore, according to the Routh-Hurwitz criterion, we can conclude that if , then the positive equilibrium of system (3) is locally asymptotically stable when .
Case 2 (, ). When and , (9) becomes the following:
where
Multiplying on both sides of (12), it is easy to get
Let be the root of (14). Then,
Then, one can obtain
where
Since , we have
where
Let ; then, (18) becomes
In order to give the main results in the present paper, we make the following assumption.Equation (20) has at least one positive real root.
If the condition holds, then there exists a positive root of (20) which can make (14) have a pair of purely imaginary roots . For , the corresponding critical value of delay is
Differentiating (14) with respect to , we get
Thus,
where
It is obvious that if the condition holds, then . According to the Hopf bifurcation theorem in [17], the following results hold.
Theorem 1. If the conditions - hold, the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at the positive equilibrium when .
Case 3 (, ). When and , (9) becomes where Let be the root of (25). Then, which follows that with Let ; then, (28) becomes Let Discussion about the roots of (30) is similar to that in [18].
Lemma 2. (i) If , then (30) has at least one positive root.
(ii) If and , then (30) has no positive root.
(iii) If and , then (30) has positive root if and only if and .
In what follows, we suppose that the coefficients in (30) satisfy the following condition:(a) or (b) , , , and .
If the condition holds, we know that there exists a positive root of (30) such that (25) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is Differentiating two sides of (25) with respect to , we have Thus, where and .
Obviously, if the condition holds, then . According to the Hopf bifurcation theorem in [17], the following results hold.
Theorem 3. If the conditions - hold, the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at when .
Case 4 (, , ). We consider (9) with in its stable interval and choose as a bifurcation parameter. Multiplying by , (9) becomes
Let be the root of (35). Then,
where
Then, we can obtain
where
Then, we can get a function with respect to :
Next, we suppose that : (40) has at least one positive real root.
If the condition holds, then there exists a such that (35) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is
Taking the derivative with respect to in (35), we get
where
Thus,
where
Thus, if the condition holds, then , which implies that the transversality condition is satisfied. According to the Hopf bifurcation theorem in [17], we can conclude the discussions above as follows.
Theorem 4. If the conditions - hold and , the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at when .
Case 5 (, and ). We consider (9) with in its stable interval and is considered as a bifurcation parameter.
Let be the root of (9). Then,
where
Then, we have
where
Similar to Case 4, we suppose that : (48) has at least one positive real root. If the condition holds, then there exists a such that (9) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is
Differentiating (9) with respect to , we have
where
Define
If the condition holds, then . Therefore, according to the Hopf bifurcation theorem in [17], we can conclude the discussions above as follows.
Theorem 5. If the conditions - hold and , the positive equilibrium of system (3) is locally asymptotically stable for and system (3) undergoes a Hopf bifurcation at when .
3. Direction and Stability of the Hopf Bifurcation
In this section, we determine the properties of the Hopf bifurcation of system (3) with respect to for . Throughout this section, we assume that , where .
Let , ; then, is the Hopf bifurcation value of system (3). Rescale the time delay . Let , let , and let ; then, system (3) can be transformed into an FDE in : where and and are given, respectively, by with
By the Riesz representation theorem, there exists a function of bounded variation for such that In fact, we can choose For , we define Then, system (54) is equivalent to where for .
For , define and the bilinear form where .
Let and be the eigenvectors of and corresponding to and , respectively. By a direction computation, we get From (62), we obtain Then, one can see that and .
Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [17] and using a computation process similar to that in [19, 20]: with where and can be calculated by the following two equations: with Then, we can get the following coefficients:
By the discussion above, we have the following results about the properties of the Hopf bifurcation.
Theorem 6. For system (3), the direction of the Hopf bifurcation is determined by the sign of : if , the Hopf bifurcation is supercritical (subcritical); the stability of bifurcating periodic solutions is determined by the sign of : if , the bifurcating periodic solutions are stable (unstable); the period of the bifurcating periodic solutions is determined by the sign of : if , the period of the bifurcating periodic solutions increases (decreases).
4. Numerical Simulations and Discussion
In this section, in order to support our theoretical results, we will show the interesting dynamical behaviors of system (3) by a special case of system (3). Let , , let , , , , and and we consider the following system: from which one can get and the unique positive equilibrium . By computing, we obtain , , and . Obviously, .
For , . Equation (18) has a unique positive root and one can obtain from (21). Further, the characteristic equation (14) has a pair of purely imaginary roots . The computer simulations in Figures 1 and 2 show that is asymptotically stable when and when passes through the critical value , loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from . Similarly, we obtain and . The corresponding trajectories graphs and phase graphs are shown in Figures 3 and 4.
Let and choose as a bifurcation parameter. Then, we have and . The computer simulations in Figures 5 and 6 show that is asymptotically stable when and loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from , . Similarly, by some complex computations, we have and when and choose as a bifurcation parameter. The corresponding trajectories graphs and phase graphs are shown in Figures 7 and 8. Furthermore, we can compute and obtain and . It follows from (69) that , , and . According to Theorem 6, we can conclude that the Hopf bifurcation of system (70) is supercritical, the bifurcating periodic solutions are stable, and the period of the periodic solutions decreases.
In addition, it can be seen from the expression of the positive equilibrium of system (3) that the more the hosts are attached to the computer networks, the more the hosts in networks will be infected. Therefore, the managers of the real networks should properly control the number of the new hosts attached to networks. According to the numerical simulations, we also find that the onset of the Hopf bifurcation can be delayed by the values of the parameters and in system (3), which can be controlled by the managers of the real networks. Therefore, the managers of the real networks should properly control the number of the hosts attached to the networks and properly strengthen the immunization of the new hosts in order to control the onset of the Hopf bifurcation, so as to make the propagation of computer viruses predicted and controlled easily.
5. Conclusion
In this paper, an SIRS computer virus propagation model with two delays and multistate antivirus measures is investigated. By choosing the possible combination of the two delays as the bifurcation parameter and analyzing the distribution of the roots of the associated characteristic equation, sufficient conditions for the local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained. Furthermore, the properties of the Hopf bifurcation are determined by using the method in [17].
Compared to the model considered in [12], we consider not only the delay due to the latent period of computer virus and the delay due to the temporary immune period of the recovered hosts, but also the delay due to the period that the antivirus software uses to clean the viruses in the infected hosts. All the possible delays are incorporated into the model and the model considered in this paper is more general. Our analysis shows that the new delay we incorporate into the model can also change the stability of the positive equilibrium of the model and numerical simulations show that our results obtained in the present paper improve some of the existing results on this system that are obtained in [12].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the anonymous referees and the editor for their valuable comments and suggestions on the paper. This work was supported by the National Natural Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2014A005).