Abstract
Hopf bifurcation for an SEIRS-V model with delays on the transmission of worms in a wireless sensor network is investigated. We focus on existence of the Hopf bifurcation by regarding the diverse delay as a bifurcation parameter. The results show that propagation of worms in the wireless sensor network can be controlled when the delay is suitably small under some certain conditions. Then, we study properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Finally, we give a numerical example to support the theoretical results.
1. Introduction
In recent years, wireless sensor networks have received extensive attention due to their vast potential in many application environments. However, security of wireless sensor networks still remains one of the most critical challenges because sensor nodes are often placed in a hostile or dangerous environment [1]. Many epidemiological models [2–6] have been proposed to study and predict the spread of viruses in wireless networks motivated by the pioneering work of Murray [7] and Kephart and White [8, 9]. In [10], Mishra and Keshri proposed the following SEIRS-V model to describe the propagation of worms in a wireless sensor network:where , , , , and denote the number of susceptible, exposed (infected, but not infectious), infectious, recovered, and vaccinated sensor nodes at time , respectively. , , , , , , , , and are the positive parameters of system (1) and for the specific meanings of them one can refer to [10]. Considering the time delays in system (1), Zhang and Si [11] proposed the following delayed SEIRS-V system:where is the time delay due to the period that antivirus software uses to clean worms in the infected nodes; is the time delay due to the temporary immunity period of the recovered and the vaccinated nodes. For the convenience of analysis, Zhang and Si [11] let and considered the following system:Zhang and Si [11] investigated existence and properties of the Hopf bifurcation of system (3).
It should be pointed out that one of the significant features of worms in networks is its latent characteristic. Therefore, there exists a certain period before the exposed nodes develop themselves into the infectious ones. In addition, as far as we know, there have been some papers that deal with research of Hopf bifurcation of dynamical systems with multiple delays in recent years [12–15]. In [12], Xu et al. studied Hopf bifurcation of a ring of five neurons with delays. In [14], Bianca et al. investigated Hopf bifurcation of an economic growth model with two delays. Considering that there is a latent period of worms in the exposed nodes in system (3), we study the following system with delays:where is the time delay due to the latent period of worms in the exposed nodes and is the time delay due to the period that the antivirus software uses to clean worms in the infected nodes and that due to the temporary immunity period of the recovered and the vaccinated nodes.
The structure of this paper is as follows. In Section 2, we obtain sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation of system (4). In Section 3, we deal with the properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Some numerical simulations are carried out in Section 4 with the aim of verifying the obtained analytic results. Finally, conclusions and future work are summarized.
2. Hopf Bifurcation Analysis
By a direct computation, we know that if , then system (4) has a unique positive equilibrium in whichThe characteristic equation of system (4) at iswherewith
Case 1. When , (6) becomeswhereIt is easy to get that . Thus, we have the following condition: If (11) holds, is locally asymptotically stable:
Case 2. When , , (6) becomeswhereLet () be a root of (12). Then,Thus, we havewhereLet ; then, (15) becomesAccording to the analysis of roots of (17) in [16], we assume that()Equation (17) has at least one positive root.Then, there exists a positive root for (17). Thus, we obtain . Further,whereDifferentiating (12) regarding , we haveThen,whereand .
Thus, we can conclude that if() holds, then .Summarizing the analysis above, we have the following.
Theorem 1. For system (4), let be specified by (18). If the conditions are satisfied, then is asymptotically stable when and a Hopf bifurcation occurs at when .
Case 3. When and , (6) becomesMultiplying by on both sides of (23), we haveLet () be the root of (24). Then,where
According to the analysis in [11], we obtain the expressions of and when and we denote and . Then, we obtain a function regarding :Next, we suppose thatEquation (27) has at least one positive root.Then, there exists which makes (24) have roots . For ,
Similarly, one can also obtain expressions of and for and we denote and . Then,Obviously, if (29) has a root , then (24) has roots . For ,Letand are the roots of (24) when . Differentiating (24) regarding , one can obtainwithDefine
Obviously, if the condition() holds, then .Therefore, we can otain the following according to the analysis above.
Theorem 2. For system (4), let be specified by (31). If the conditions are satisfied, then is asymptotically stable when and a Hopf bifurcation occurs at when .
Case 4 (, , and ). Let () be the root of (6); then, one can obtainwhereThus, we can obtainwhere
Suppose thatEquation (37) has at least one positive root.Then, there exists a positive root for (37) and (6) has roots . For , we haveDifferentiating (6) with respect to , one can obtainwherewithDefine
Obviously, if the condition() holds, then .Thus, we have the following based on the analysis above.
Theorem 3. For system (4), let be specified by (39) and . If the conditions are satisfied, then is asymptotically stable when and a Hopf bifurcation occurs at when .
Case 5 (, , and ). Multiplying by , (6) becomesLet () be a root of (44). Then,where
Similar to Case 3, we can obtain the expressions of and when and we denote and . Then, one can obtainIn order to give the main results in this paper, we suppose thatEquation (47) has at least one positive root.
If the condition holds, then there exists such that (6) has a pair of purely imaginary roots . For , we have
Similarly, one can obtain expressions of and when and we denote and . Then, one can obtainIf (49) has one positive root , then (6) has roots . For , we haveLetand be the roots of (6) with . Differentiating (6) with respect to , we getwherewithDefine
Obviously, if the condition() holds, then .Thus, we have the following based on the analysis above.
Theorem 4. For system (4), let be specified by (51) and . If the conditions are satisfied, then is asymptotically stable when and a Hopf bifurcation occurs at when .
3. Properties of the Hopf Bifurcation
We will investigate properties of the Hopf bifurcation of system (4) with respect to and in this section. Let with and , . By the transformation , , , , , and , system (4) can be written aswhere , andwhere
According to the Riesz representation theorem, we know that there exists a matrix function such thatIn fact, we chooseFor , we defineThen, system (56) is equivalent towhere for .
Then, we define :and a bilinear formwhere .
Let be the eigenvector of with and let be the eigenvector of with . Then, we obtainFrom (64), we obtainThen, we can choosesuch that and .
Following the same algorithms introduced in [17] and the similar computation used in [18–22], we obtainwithwhere
Consequently, we can compute the following values:
According to the analysis about properties of the Hopf bifurcation in [17], we have the following for system (4).
Theorem 5. Let , , and be specified by (71). determines the direction of the Hopf bifurcation (supercritical if , subcritical if ); determines the stability of the bifurcating periodic solutions (stable if , unstable if ); determines the period of the bifurcating periodic solutions (increasing if , decreasing if ).
4. A Numerical Example
In this section we give a numerical example of system (4) to validate the analysis above. By extracting the same values from [11], we get the following system:
It is easy to get and system (72) has a unique positive equilibrium . Also we get that the condition holds.
Firstly, we can obtain and by some computations. According to Theorem 1, we can conclude that the positive equilibrium is asymptotically stable for . In this case, the propagation of worms can be predicted and controlled easily. However, once the value of is above , a Hopf bifurcation will occur which implies that the propagation of worms will be out of control. This property can be illustrated by the bifurcation diagram with respect to in Figure 1. Similarly, we have and . The corresponding bifurcation diagram with respect to is as shown in Figure 2.
Secondly, we may obtain and for and by computing. It follows from Theorem 3 that when , a Hopf bifurcation occurs, which can be shown by the bifurcation diagram regarding and in Figure 3. Moreover, we have , , , and . Thus, It follows from Theorem 5 that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable and decreasing. Since the bifurcating periodic solutions are stable, the numbers of every class of nodes in system (72) may coexist in an oscillatory mode, which is not welcome in networks.
Lastly, by computing, we obtain and for and . By Theorem 4, we know that the positive equilibrium is asymptotically stable for , which makes it possible for the propagation of worms to be predicted and controlled by taking effective measures. However, the positive equilibrium loses stability and a Hopf bifurcation occurs, when passes through . In this case, it is difficult to take measures to control the propagation of worms. The Hopf bifurcation phenomenon can be illustrated by Figure 4.
5. Conclusions
In the present paper, the generalization of the delayed SEIRS-V model describing worms spreading in a wireless sensor network investigated in [11] by inserting the latent period delay in the exposed sensor nodes has been considered. We find that not only but also can influence stability of system (4) and make system (4) undergo a Hopf bifurcation under some certain conditions. It has been shown that characteristics of the propagation of worms in system (4) can be easily predicted and eliminated when the value of delay is below the corresponding critical value and the propagation of worms in system (4) may be out of control once the delay passes through the corresponding critical value. Thus, we can conclude that the worms propagation can be controlled by postponing occurrence of a Hopf bifurcation. Subsequently, we use the normal form approach theory and center manifold theory introduced in [17] to deal with properties of the Hopf bifurcation when and .
According to the numerical example, we can conclude that the dynamics of system (4) is more complicated than that of the system considered in [11]. However, it should be pointed out that we assume that the exposed sensor nodes can not infect other nodes, which is not consistent with reality, because the exposed nodes can also infect other nodes through vulnerability seeking or other methods. Consequently, it is more realistic to investigate dynamics of the proposed model in this paper with graded infection rate. We leave this as our future work.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by Natural Science Foundation of Anhui Province (nos. 1608085QF145 and 1608085QF151) and Natural Science Foundation of the Higher Education Institutions of Anhui Province (no. KJ2015A144).