Bifurcation analysis in a delayed computer virus model with the effect of external computers
- Zizhen Zhang^{1}Email author and
- Dianjie Bi^{1}
https://doi.org/10.1186/s13662-015-0652-y
© Zhang and Bi 2015
Received: 21 August 2015
Accepted: 30 September 2015
Published: 14 October 2015
Abstract
A delayed Susceptible-Infected-External (SIE) computer virus propagation model is investigated in the present paper. The linear stability conditions are obtained with characteristic root method. The Hopf bifurcation is demonstrated. Furthermore, some explicit formulae for determining the stability and the direction of the Hopf bifurcation are derived by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out to support the theoretical predictions.
Keywords
1 Introduction
The organization of this paper is as follows. Section 2 considers stability of the positive equilibrium and existence of the Hopf bifurcation. Section 3 is devoted to the properties of the Hopf bifurcation. Some numerical simulations are carried out to verify the theoretical results in Section 4, and this work is summarized in Section 5.
2 Stability of the positive equilibrium and existence of Hopf bifurcation
If the condition (H_{1}) holds, then there exists one positive root \(v_{0}\) of Eq. (6) such that Eq. (3) has a pair of purely imaginary roots \(\pm i\omega _{0} =\pm i\sqrt{v_{0}}\).
It is clear that if the condition (H_{3}) \(f'({v_{0}})\ne0\) holds, then \(\operatorname{Re} [{\frac{d\lambda}{d\tau}} ]_{\tau=\tau_{0} }^{-1}\ne0\). Therefore, according to the Hopf bifurcation theorem in [20], we have the following results.
Theorem 1
For system (2), we assume that the conditions (H_{1})-(H_{3}) hold for the parameters. Then the positive equilibrium \(E_{\ast}(S_{\ast}, I_{\ast}, E_{\ast})\) is asymptotically stable for \(\tau\in[0, \tau_{0} )\) and the positive equilibrium \(E_{\ast}(S_{\ast}, I_{\ast}, E_{\ast})\) becomes unstable for τ staying in some right neighborhood of \(\tau_{0} \), with a Hopf bifurcation occurring when \(\tau =\tau_{0}\).
3 Direction and stability of the Hopf bifurcation
In this section, we shall obtain the explicit formulae for determining the direction, stability and period of these periodic solutions bifurcating from the positive equilibrium \(E_{\ast}(S_{\ast},I_{\ast},E_{\ast})\) of system (2) at the critical value \(\tau_{0} \). For convenience, let \(\tau=\tau_{0} +\upsilon\), \(\upsilon\in R\). Then \(\upsilon=0\) is the Hopf bifurcation value of system (2).
Theorem 2
For system (2), we assume that the conditions (H_{1})-(H_{3}) hold for the parameters. Then, if \(\mu_{2} >0\) (\(\mu_{2} <0\)), then the Hopf bifurcation is supercritical (subcritical); if \(\beta_{2}<0\) (\(\beta_{2} >0\)), then the bifurcating periodic solutions are stable (unstable); if \(T_{2}>0\) (\(T_{2}<0\)), then the bifurcating periodic solutions increase (decrease).
4 Numerical simulation
5 Conclusions
Considering that some computer viruses may purposely lay dormant for a period of time prior to infecting other computers, we incorporate the latent period delay of the computer viruses into the model considered in the literature [15] and propose a delayed SIE computer virus propagation model in this paper. Compared with the work in [15], we mainly consider the effect of the latent period delay on system (2). It is shown that the latent period delay plays an important role on the stability of system (2). When \(\tau <\tau_{0} \), system (2) is locally asymptotically stable and the characteristics of computer viruses propagation can be easily predicted and eliminated. However, when \(\tau\ge\tau_{0} \), a Hopf bifurcation occurs and the computer viruses propagation is unstable and may be out of control. Furthermore, the properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. It should be pointed out that the assumptions for the parameters of system (2) in this paper are only technical and we do not take the specific meanings of them into account. Namely, our study is restricted only to the theoretical analysis of the Hopf bifurcation phenomena of system (2). It may be helpful for field investigation or experimental studies on the propagation of computer viruses in networks. In addition, the other behaviors of system (2) out of the assumptions on the parameters have been disregarded. We leave these as our future work.
Declarations
Acknowledgements
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, KJ2014A006).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Yuan, H, Chen, G, Wu, J, Xiong, H: Towards controlling virus propagation in information systems with point-to-group information sharing. Decis. Support Syst. 48(1), 57-68 (2009) View ArticleGoogle Scholar
- Luiijf, E: Understanding cyber threats and vulnerabilities. Lect. Notes Comput. Sci. 7130(1), 52-67 (2012) View ArticleGoogle Scholar
- Huang, CY, Lee, CL, Wen, TH, Sun, CT: A computer virus spreading model based on resource limitations and interaction costs. J. Syst. Softw. 86(3), 801-808 (2013) View ArticleGoogle Scholar
- Kephart, JO, White, SR: Directed-graph epidemiological models of computer viruses. In: Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343-359 (1991) Google Scholar
- Wierman, JC, Marchette, DJ: Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction. Comput. Stat. Data Anal. 45(1), 3-23 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Piqueira, JRC, Araujo, VO: A modified epidemiological model for computer viruses. Appl. Math. Comput. 213(2), 355-360 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Ren, JG, Yang, XF, Zhu, QY, Yang, LX, Zhang, CM: A novel computer virus model and its dynamics. Nonlinear Anal., Real World Appl. 13(1), 376-384 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Gan, CQ, Yang, XF, Liu, WP, Zhu, QY, Zhang, XL: Propagation of computer virus under human intervention: a dynamical model. Discrete Dyn. Nat. Soc. 2012, Article ID 106950 (2012) MathSciNetView ArticleGoogle Scholar
- Yuan, H, Chen, GQ: Network virus-epidemic model with the point-to-group information propagation. Appl. Math. Comput. 206(1), 357-367 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Dong, T, Liao, XF, Li, HQ: Stability and Hopf bifurcation in a computer virus model with multistate antivirus. Abstr. Appl. Anal. 2012, Article ID 841987 (2012) MathSciNetGoogle Scholar
- Zhang, CM, Zhao, Y, Wu, YJ, Deng, SW: A stochastic dynamic model of computer viruses. Discrete Dyn. Nat. Soc. 2012, Article ID 264874 (2012) MathSciNetGoogle Scholar
- Mishra, BK, Pandey, SK: Dynamics model of worms with vertical transmission in computer network. Appl. Math. Comput. 217(21), 8438-8446 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Mishra, BK, Saini, DK: SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput. 188(2), 1476-1482 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Mishra, BK, Jia, N: SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Model. 34(3), 710-715 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Chen, JY, Yang, XF, Gan, CQ: Propagation of computer virus under the influence of external computers: a dynamical model. J. Inf. Comput. Sci. 10(16), 5275-5282 (2013) View ArticleGoogle Scholar
- Bianca, C, Guerrini, L: On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity. Acta Appl. Math. 128(1), 39-48 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Bianca, C, Guerrini, L, Riposoj, J: A delayed mathematical model for the acute inflammatory response to infection. Appl. Math. Inf. Sci. 9(6), 2775-2782 (2015) Google Scholar
- Xu, CJ, He, XF: Stability and bifurcation analysis in a class of two-neuron networks with resonant bilinear terms. Abstr. Appl. Anal. 2011, Article ID 697630 (2011) MathSciNetGoogle Scholar
- Bianca, C, Guerrini, L: Hopf bifurcations in a delayed microscopic model of credit risk contagion. Appl. Math. Inf. Sci. 9(3), 1493-1497 (2015) Google Scholar
- Hassard, BD, Kazarinoff, ND, Wan, YH: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar