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**Stability and Hopf bifurcation in a computer virus model with multistate antivirus.**
*(English)*
Zbl 1237.37067

Summary: By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

### MSC:

37N35 | Dynamical systems in control |

68M10 | Network design and communication in computer systems |

92D30 | Epidemiology |

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\textit{T. Dong} et al., Abstr. Appl. Anal. 2012, Article ID 841987, 16 p. (2012; Zbl 1237.37067)

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### References:

[1] | J. E. Sawyer, M. C. Kernan, D. E. Conlon, and H. Garland, “Responses to the Michelangelo computer virus threat: the role of information sources and risk homeostasis theory,” Journal of Applied Social Psychology, vol. 29, no. 1, pp. 23-51, 1999. |

[2] | B. K. Mishra and D. K. Saini, “SEIRS epidemic model with delay for transmission of malicious objects in computer network,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1476-1482, 2007. · Zbl 1118.68014 |

[3] | B. K. Mishra and D. Saini, “Mathematical models on computer viruses,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 929-936, 2007. · Zbl 1120.68041 |

[4] | B. K. Mishra and N. Jha, “Fixed period of temporary immunity after run of anti-malicious software on computer nodes,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1207-1212, 2007. · Zbl 1117.92052 |

[5] | E. Gelenbe, “Dealing with software viruses: a biological paradigm,” Information Security Technical Report, vol. 12, no. 4, pp. 242-250, 2007. |

[6] | E. Gelenbe, “Keeping viruses under control,” in Proceedings of the 20th International Symposium Computer and Information Sciences (ISCIS ’05), vol. 3733 of Lecture Notes in Computer Science, Springer, 2005. |

[7] | W. O. Kermack and A. G. McKendrick, “Contributions of mathematical theory to epidemics,” Proceedings of the Royal Society of London Series A, vol. 115, pp. 700-721, 1927. · JFM 53.0517.01 |

[8] | W. O. Kermack and A. G. McKendrick, “Contributions of mathematical theory to epidemics,” Proceedings of the Royal Society of London Series A, vol. 138, pp. 55-83, 1932. · Zbl 0005.30501 |

[9] | W. O. Kermack and A. G. McKendrick, “Contributions of mathematical theory to epidemics,” Proceedings of the Royal Society of London Series A, vol. 141, pp. 94-122, 1933. · Zbl 0007.31502 |

[10] | W. O. Kermack and A. G. McKendrick, “Contributions of mathematical theory to epidemics,” Proceedings of the Royal Society of London Series A, vol. 115, pp. 700-721, 1927. · JFM 53.0517.01 |

[11] | W. O. Kermack and A. G. McKendrick, “Contributions of mathematical theory to epidemics,” Proceedings of the Royal Society of London Series A, vol. 138, pp. 55-83, 1932. · Zbl 0005.30501 |

[12] | W. O. Kermack and A. G. McKendrick, “Contributions of mathematical theory to epidemics,” Proceedings of the Royal Society of London Series A, vol. 141, pp. 94-122, 1933. · Zbl 0007.31502 |

[13] | W. T. Richard and J. C. Mark, “Modeling virus propagation in peer-to-peer networks,” in Proceedings of the IEEE International Conference on Information, Communications and Signal Processing (ICICS ’05), pp. 981-985, 2005. |

[14] | Y. Yao, X. Xie, and H. Gao, “Hopf bifurcation in an Internet worm propagation model with time delay in quarantine,” Mathematical and Computer Modelling. In press. · Zbl 1286.92049 |

[15] | H. Yuan and G. Chen, “Network virus-epidemic model with the point-to-group information propagation,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 357-367, 2008. · Zbl 1162.68404 |

[16] | B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981. · Zbl 0474.34002 |

[17] | M. Y. Li and J. S. Muldowney, “Global stability for the SEIR model in epidemiology,” Mathematical Biosciences, vol. 125, no. 2, pp. 155-164, 1995. · Zbl 0821.92022 |

[18] | Y. Song, M. Han, and J. Wei, “Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays,” Physica D, vol. 200, no. 3-4, pp. 185-204, 2005. · Zbl 1062.34079 |

[19] | S. Ruan and J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” IMA Journal of Mathemathics Applied in Medicine and Biology, vol. 18, no. 1, pp. 41-52, 2001. · Zbl 0982.92008 |

[20] | X. Li and J. Wei, “On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 519-526, 2005. · Zbl 1098.37070 |

[21] | H. Hu and L. Huang, “Stability and Hopf bifurcation analysis on a ring of four neurons with delays,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 587-599, 2009. · Zbl 1175.34092 |

[22] | D. Fan, L. Hong, and J. Wei, “Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 305-319, 2010. · Zbl 1259.34081 |

[23] | S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems Series A, vol. 10, no. 6, pp. 863-874, 2003. · Zbl 1068.34072 |

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