- Open Access
Dynamics of a delayed SEIRS-V model on the transmission of worms in a wireless sensor network
© Zhang and Si; licensee Springer. 2014
Received: 30 September 2014
Accepted: 11 November 2014
Published: 25 November 2014
A delayed SEIRS-V model on the transmission of worms in a wireless sensor network is considered. Choosing delay as a bifurcation parameter, the existence of the Hopf bifurcation of the model is investigated. Furthermore, we use the normal form method and the center manifold theorem to determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions. Finally, some numerical simulations are presented to verify the theoretical results.
where , , , and represent the numbers of sensor nodes at time t in states susceptible, exposed, infectious, recovered and vaccinated, respectively. A is the inclusion of new nodes to the wireless sensor network, β is the transmission coefficient. α, γ, δ, η and p are state transition rates. ε and μ are the crashing rates of the sensor nodes due to the attack of worms and the reason other than the attack of worms, respectively. Mishra and Keshri  studied the stability of system (1).
This paper is organized as follows. In Section 2, we investigate local stability of the positive equilibrium and obtain sufficient conditions for the existence of local Hopf bifurcation. In Section 3, we determine direction and stability of the Hopf bifurcation by using the normal form theory and the center manifold theorem. In order to testify the theoretical analysis, a numerical example is presented in Section 4. Section 5 concludes the paper and indicates future directions for research.
2 Local stability of positive equilibrium and existence of local Hopf bifurcation
According to , we consider the following two cases.
If all the parameters of system (3) are given, we can calculate the roots of Eq. (15) by Matlab software package. Therefore, we make the following assumption in order to give the main results in this paper.
Then, when , Eq. (5) has a pair of purely imaginary roots .
Obviously, if condition (H3) holds, then . Therefore, by the Hopf bifurcation theorem in , we have the following results.
Theorem 1 For system (3), if conditions (H1)-(H3) hold, then the positive equilibrium of system (3) is asymptotically stable for , and system (3) undergoes a Hopf bifurcation at the positive equilibrium when .
3 Direction and stability of the Hopf bifurcation
where δ is the Dirac delta function.
such that , .
In conclusion, we have the following results.
Theorem 2 For system (3), if (), then the Hopf bifurcation is supercritical (subcritical). If (), then the bifurcating periodic solutions are stable (unstable). If (), then the bifurcating periodic solutions increase (decrease).
4 Numerical simulation
This paper is concerned with a delayed SEIRS-V model on the transmission of worms in a wireless sensor network. The main results are given in terms of local stability and local Hopf bifurcation. By choosing the delay as a bifurcation parameter, sufficient conditions for local stability of the positive equilibrium and existence of the Hopf bifurcation of system (3) are obtained. We have proven that when the conditions are satisfied, there exists a critical value of the delay below which system (3) is stable and above which system (3) is unstable. Especially, system (3) undergoes a Hopf bifurcation at the positive equilibrium when . The occurrence of Hopf bifurcation means that the state of worms prevalence in a wireless sensor network changes from a positive equilibrium to a limit cycle, which is not welcomed in a wireless sensor network. Hence, we should control the occurrence of Hopf bifurcation by combining some bifurcation control strategies, and we leave this as the future work. Further, the properties of Hopf bifurcation are studied by using the normal form method and the center manifold theorem. Finally, a numerical example is given to support our theoretical results.
The authors would like to thank the editor and the anonymous referees for their work on the paper. This work was supported by the Natural Science Foundation of Higher Education Institutions of Anhui Province (KJ2014A005).
- Kephart JO, White SR: Measuring and modeling computer virus prevalence. In IEEE Computer Security Symposium on Research in Security and Privacy. IEEE Press, New York; 1993:2-15.Google Scholar
- Kephart JO, White SR: Directed-graph epidemiological models of computer viruses. IEEE Symposium on Security and Privacy 1991, 343-361.Google Scholar
- Yuan H, Chen G: Network virus epidemic model with the point-to-group information propagation. Appl. Math. Comput. 2008, 206: 357-367. 10.1016/j.amc.2008.09.025MathSciNetView ArticleMATHGoogle Scholar
- Yang LX, Yang XF, Wen LS, Liu JM: A novel computer virus propagation model and its dynamics. Int. J. Comput. Math. 2012, 89: 2307-2314. 10.1080/00207160.2012.715388MathSciNetView ArticleMATHGoogle Scholar
- Huang CY, Lee CL, Wen TH, Sun CT: A computer virus spreading model based on resource limitations and interaction costs. J. Syst. Softw. 2013, 86: 801-808. 10.1016/j.jss.2012.11.027View ArticleGoogle Scholar
- Mishra BK, Pandey SK: Fuzzy epidemic model for the transmission of worms in computer network. Nonlinear Anal., Real World Appl. 2010, 11: 4335-4341. 10.1016/j.nonrwa.2010.05.018View ArticleMATHGoogle Scholar
- Mishra BK, Pandey SK: Dynamic model of worms with vertical transmission in computer network. Appl. Math. Comput. 2011, 217: 8438-8446. 10.1016/j.amc.2011.03.041MathSciNetView ArticleMATHGoogle Scholar
- Yang LX, Yang XF: Propagation behavior of virus codes in the situation that infected computers are connected to the Internet with positive probability. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 693695Google Scholar
- Yang MB, Zhang ZF, Li Q, Zhang G: An SLBRS model with vertical transmission of computer virus over the Internet. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 925648Google Scholar
- Yang LX, Yang XF, Zhu QY, Wen LS: A computer virus model with graded cure rates. Nonlinear Anal., Real World Appl. 2013, 14: 414-422. 10.1016/j.nonrwa.2012.07.005MathSciNetView ArticleMATHGoogle Scholar
- Mishra BK, Pandey SK: Dynamic model of worm propagation in computer network. Appl. Math. Model. 2014, 38: 2173-2179. 10.1016/j.apm.2013.10.046MathSciNetView ArticleGoogle Scholar
- Gan CQ, Yang XF, Liu WP, Zhu QY: A propagation model of computer virus with nonlinear vaccination probability. Commun. Nonlinear Sci. Numer. Simul. 2014, 19: 92-100. 10.1016/j.cnsns.2013.06.018MathSciNetView ArticleGoogle Scholar
- Akyildiz I, Su W, Sankarasubramaniam Y, Cayirci E: A survey on sensor networks. IEEE Commun. Mag. 2002, 40: 102-114.View ArticleGoogle Scholar
- Mishra BK, Keshri N: Mathematical model on the transmission of worms in wireless sensor network. Appl. Math. Model. 2013, 37: 4103-4111. 10.1016/j.apm.2012.09.025MathSciNetView ArticleMATHGoogle Scholar
- Feng LP, Liao XF, Li HQ, Han Q: Hopf bifurcation analysis of a delayed viral infection model in computer networks. Math. Comput. Model. 2012, 56: 167-179. 10.1016/j.mcm.2011.12.010MathSciNetView ArticleMATHGoogle Scholar
- Ren JG, Yang XF, Yang LX, Xu YH, Yang FZ: A delayed computer virus propagation model and its dynamics. Chaos Solitons Fractals 2012, 45: 74-79. 10.1016/j.chaos.2011.10.003MathSciNetView ArticleMATHGoogle Scholar
- Dong T, Liao XF, Li HQ: Stability and Hopf bifurcation in a computer virus model with multistate antivirus. Abstr. Appl. Anal. 2012., 2012: Article ID 841987Google Scholar
- Hassard BD, Kazarinoff ND, Wan YH: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge; 1981.MATHGoogle Scholar
- Bianca C, Ferrara M, Guerrini L: The Cai model with time delay: existence of periodic solutions and asymptotic analysis. Appl. Math. Inf. Sci. 2013, 7: 21-27. 10.12785/amis/070103MathSciNetView ArticleGoogle Scholar
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