Abstract
A four-compartment computer virus propagation model with two delays and graded infection rate is investigated in this paper. The critical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the corresponding characteristic equation. In succession, direction and stability of the Hopf bifurcation when the two delays are not equal are determined by using normal form theory and center manifold theorem. Finally, some numerical simulations are also carried out to justify the obtained theoretical results.
1. Introduction
In recent years, with the fast development and popularization of computer technologies and network, Internet has offered numerous functionalities and facilities to the world. Meanwhile, Internet has also become a powerful mechanism for propagating computer viruses. Computer viruses are computer programs which have serious effects on individual and corporate computer systems in the network, such as modifying data and formatting disks [1, 2].
In order to analyze the propagation laws of computer viruses in the network, many epidemiological models have been borrowed to depict the spread of computer viruses because of the high similarity between the computer viruses and the biological viruses [3–5]. In [6–11], Mishra et al. proposed SIRS computer virus models in different forms. Yuan and Chen presented the SEIR computer virus propagation model in [12] and they studied the stability of the model. Based on the the work in [12], Dong et al. proposed the SEIR computer virus model with time delay in [13] and they investigated the Hopf bifurcation of the model. There are also some other different computer virus models which have been proposed by other scholars in recent years and one can refer to [14–18]. However, all the computer virus models above which incorporate the latent status of the viruses assume that the latent computers have no infection ability. This is not consistent with the reality, because an infected computer which is in latency can also infect other computers through file copying or file downloading. Based on this fact, Yang et al. established a computer virus propagation model with graded infection rate in [19]:where , , , and are the percentages of susceptible computers, latent computers, active computers, and recovered computers on the Internet, at time , respectively. is the rate at which external computers are connected to the Internet and it is also the rate at which internal computers are disconnected from the Internet; is the infected rate of the susceptible computers by the latent computers; is the infected rate of the susceptible computers by the active computers; is the rate at which the recovered computers become susceptibly virus-free again; is the rate at which the latent computers break out; and is the rate at which the active computers are cured by the antivirus software.
As pointed out in [9], one of the typical features of computer viruses is their latent characteristic. Therefore, they need a period to become active computers for the latent ones. Likewise, the antivirus software needs a period to clean the viruses in the active computers. Based on this and motivated by the work about the dynamical system with delay in [20–24], we incorporate two delays into system (1) and obtain the following delayed computer virus model:where is the latent period of the computer viruses and is the period that the antivirus software needs to clean the viruses in the active computers.
The rest of this paper is organized as follows. In Section 2, we present the existence of the viral equilibrium and conditions for the local stability of the viral equilibrium and existence of the Hopf bifurcation are derived. Direction and stability of the Hopf bifurcation are studied in Section 3 and some numerical simulations are performed in Section 4 to justify the obtained theoretical findings by taking some relevant values of the parameters in system (2) and using the Matlab software package. Finally, we end this paper with concluding remarks in Section 5.
2. Existence of Local Hopf Bifurcation
By a simple computation, we know that if and , then system (2) has a unique viral equilibrium , where
Let , , , . Dropping the bars, system (2) becomeswhereThe linear system of system (4) isThe corresponding characteristic equation iswhere
Case 1 (). For , (7) becomeswhereThus, according to the Routh-Hurwithz theorem, we know that if conditions , , and hold, then viral equilibrium of system (2) without delay is locally asymptotically stable.
Case 2 (, ). For and , we can get the following from (7):whereWe assume that is a root of (11). Then,which implies thatwithLet ; then (14) becomes
Discussion about distribution of roots for (16) is similar to that in [25]. Therefore, we directly assume that (16) has at least one positive equilibrium .
If holds, we know that (14) has at least one positive root such that (11) has a pair of purely imaginary roots . For , where Differentiating both sides of (11) with respect to , one can obtain Thus, where and . Therefore, if condition holds, then . Based on the discussion above and according to the Hopf bifurcation theorem in [26], we obtain the following.
Theorem 1. If conditions hold, then(i)viral equilibrium of system (2) is locally asymptotically stable for ;(ii)system (2) undergoes a Hopf bifurcation at viral equilibrium when and a family of periodic solutions bifurcate from .
Case 3 (). For and , (7) becomeswhereLet be a root of (21). Then,It follows thatwithLet ; then, we have
Similar to Case 2, we make the following assumption. (26) has at least one positive root . If condition holds, then there exists such that (21) has a pair of purely imaginary roots . For , where In addition, we have Further, where and . Therefore, if condition holds, then . Based on the discussion above and according to the Hopf bifurcation theorem in [26], we obtain the following.
Theorem 2. If conditions hold, then(i)viral equilibrium of system (2) is locally asymptotically stable for ;(ii)system (2) undergoes a Hopf bifurcation at viral equilibrium when and a family of periodic solutions bifurcate from .
Case 4 (). For , we havewith Multiplying by , (31) becomes the following:Let be a root of (37); it is easy to getwhereIt leads to Thus, we can get the following equation with respect to :
Next, we make the following assumption. (37) has at least one positive root . Then, for , we have Taking the derivative of with respect to , we obtain where Then, we can get that where
We assume that . Clearly, if condition holds, then we can conclude that . Therefore, according to the Hopf bifurcation theorem in [26], we obtain the following.
Theorem 3. If conditions hold, then(i)viral equilibrium of system (2) is locally asymptotically stable for ;(ii)system (2) undergoes a Hopf bifurcation at viral equilibrium when and a family of periodic solutions bifurcate from .
Case 5 (, ). Let be the root of (7). Then,whereThus, one can get the following equation with respect to :
Similar to Case 4, we assume that (45) has at least one positive root . Then, for , we have Differentiating (7) with respect to , we get withThen, we obtain where
We assume that . Thus, we know that , if condition holds. Therefore, according to the Hopf bifurcation theorem in [26], we obtain the following.
Theorem 4. If conditions hold and , then(i)viral equilibrium of system (2) is locally asymptotically stable for ;(ii)system (2) undergoes a Hopf bifurcation at viral equilibrium when and a family of periodic solutions bifurcate from .
3. Properties of the Hopf Bifurcation
In this section, we shall investigate direction of the Hopf bifurcation and stability of the bifurcating periodic solution of system (2) when and by using the center manifold theorem and the normal form theory which has been developed by Hassard et al. [26].
Let , , , , , . Then, is the Hopf bifurcation value of system (2) and system (2) can be rewritten aswhere and : and are given, respectively, by with Using Riesz representation theorem, there exists matrix such that In fact, choose For , define
Then, system (51) can be rewritten in the following form:For , , and bilinear formare defined, where , , and are adjoint operators.
Based on the discussion above, one can conclude that are common eigenvalues of and . The eigenvectors of and can be calculated corresponding to and , respectively. Let be the eigenvector of corresponding to and be the eigenvector of corresponding to . By some complex computations, we obtainwith From (59), we get such that . In what follows, we can obtain the coefficients by using the method introduced in [26]: with where with Thus, we can compute the following coefficients:In conclusion, we have the following results in this section.
Theorem 5. For system (2),(i)the direction of the Hopf bifurcation is determined by : if , the Hopf bifurcation is supercritical; if , the Hopf bifurcation is subcritical;(ii)the stability of the bifurcating periodic solutions is determined by : if , the bifurcating periodic solutions are stable; if , the bifurcating periodic solutions are unstable;(iii)the period of the bifurcating periodic solution is determined by : if , the period of the bifurcating periodic solutions increases; if , the period of the bifurcating periodic solutions decreases.
4. Numerical Simulations
In this section, we present some numerical simulation results of system (2) to illustrate our theoretical results. We choose a set of parameters as follows: , , , , , and . Then, we get the following system:
It is easy to verify that and , which ensures the fact that system (68) has a unique viral equilibrium .
For , by direct computation by Matlab 7.0, we can get , , and , which means that viral equilibrium is locally asymptotically stable.
For , we obtain that (16) has a unique positive root and (14) has a unique positive root . Further, we get the critical value of delay and . Thus, we can see that the conditions in Theorem 1 hold true. It follows that viral equilibrium is locally asymptotically stable for and system (68) undergoes a Hopf bifurcation at viral equilibrium when . This property can be depicted in Figures 1–4. Similarly, we can also obtain , for and , for , respectively. The corresponding phase plots are depicted in Figures 5–8 and Figures 9–12, respectively.
For and , we obtain , . The simulation results can be seen in Figures 13–16. In addition, we obtain , by some complex computations. Further, we have , , . Therefore, we can conclude that the Hopf bifurcation is supercritical; the bifurcating periodic solutions are stable; and the period of the bifurcating periodic solutions increases.
5. Conclusions
In the present paper, an improved model for propagation of computer virus propagation model in the network is introduced and studied by incorporating the delay due to the latent period of the computer viruses and the delay due to the period that the antivirus software needs to clean the viruses in the active computers into the model proposed in [19]. We mainly investigate effect of the two delays on the model.
By choosing different combination of the two delays as a bifurcation parameter, it has been found that both the two delays can change the stability of the viral equilibrium of the model under some conditions. When the value of the delay is below corresponding critical value, the model is locally asymptotically stable which indicates that the law of propagation of the computer viruses in system (2) can be predicted. However, when the value of the delay is above the corresponding critical value, a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the viral equilibrium, which suggests that the percentages of susceptible, latent, active, and recovered computers in system (2) will fluctuate periodically in a range. This is not helpful in predicting the law of propagation of the computer viruses. Therefore, we should control the occurrence of the Hopf bifurcation by using some bifurcation control strategies and we leave this as our near future work. Furthermore, the properties of the Hopf bifurcation when and have been investigated in detail. Finally, some numerical simulations are also included to support the theoretical results obtained in the paper.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
The research was supported by Natural Science Foundation of Anhui Province (no. 1608085QF151 and no. 1608085QF145).