Open Access Open Access  Restricted Access Subscription or Fee Access

A Distributed Time Delay Model of Worms in Computer Network

Samir Kumar Pandey, Bimal Kumar Mishra, Prasanna Kumar Satpathy

Abstract


An e-epidemic dynamical model for the transmission of worms in the computer network has been developed. In this model, we have applied antivirus software and distributed time delay with vertical transmission. We have used the discrete dynamical system to obtain the exact worm – free periodic solution of the impulsive epidemic system and observed that the solution is globally attractive, if the rate of use of antivirus software is larger enough and the solution is uniformly persistent if the rate is less than some critical value. We have also analyzed the permanence of the model analytically and study the contribution of vertical transmission in the system and then investigate that the large use of antivirus software is enough for worm eradication. Numerical methods and MATLAB are used to solve and simulate the system of equations developed and analyze the behavior of different classes of computer nodes in the network.

Keywords


Antivirus Software, Computer Network, Distributed Time Delay, Epidemic Model, Vertical Transmission, Worms.

Full Text:

PDF

References


Bimal Kumar Mishra and D. K. Saini, SEIRS epidemic model with delay for transmission of malicious objects in computer network, Applied Mathematics and Computation, 188 (2) (2007), 1476-1482.

Bimal Kumar Mishra and Dinesh Saini, Mathematical models on computer viruses, Applied Mathematics and Computation, 187 (2) (2007), 929-936.

Bimal Kumar Mishra and Navnit Jha, Fixed period of temporary immunity after run of anti-malicious software on computer nodes, Applied Mathematics and Computation, 190 (2) (2007) 1207 – 1212.

Bimal K. Mishra, Navnit Jha, SEIQRS model for the transmission of malicious objects in computer network, Applied Mathematical Modelling, 34 (2010), 710-715 .

Bimal Kumar Mishra, Samir Kumar Pandey, Fuzzy epidemic model for the transmission of worms in Computer network, Nonlinear Analysis: Real World Applications, 11 (2010) 4335-4341.

Shujing Gao, Zhidong Teng, Juan J. Nieto, Angela Torres, Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, Journal of Biomedicine and Biotechnology, doi:10.1155/2007/64870.

W.O. Kermack, A.G. McKendrick, Contributions of mathematical theory to epidemics, Proc. Royal Soc. London – Series A 115 (1927) 700–721.

W.O. Kermack, A.G. McKendrick, Contributions of mathematical theory to epidemics, Proc. Roy. Soc. London – Series A 138 (1932) 55 – 83.

W.O. Kermack, A.G. McKendrick, Contributions of mathematical theory to epidemics, Proc. Royal Soc. London – Series A 141 (1933) 94 – 122.

Z. Ma, Y. Zhou, W. Wang, Z. Jin, Mathematical models and dynamics of infectious diseases, China Science Press, Beijing, China, 2004.

E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis: Theory, methods and applications, vol. 47, no. 6, pp. 4107-4115, 2001.

X. Z. Li, G. Gupur, G. T. Zhu, Threshold and stability results for an age-structured SEIR epidemic model, Computers and Mathematics with Applications, vol.42, no. 6-7, pp. 883-907, 2001.

W. Wang, Global behavior of an SEIRS epidemic model with time delays, Applied Mathematics Lectures, vol. 15, no. 4, pp. 423-428, 2002.

A. D’Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Mathematical Biosciences, vol. 179, no. 1, pp.57-72, 2002.

E. Beretta and Y. Tkeuchi, Global stability of an SIR epidemic model with time delays, Journal of Mathematical Biology, vol. 33, no. 3, pp. 250-260, 1995.

C. Zhang, M. Liu, B. Zheng, Hopf bifurcation in numerical approximation of a class delay differential equation, Applied Mathematics and computation, vol. 146, no. 2-3, pp. 335-349, 2003.

D. D. Bainov, P.S. Simeonov, Impulsive differential equations: Periodic solutions and applications, Longman Scientific and Technical press, New York, NY, USA, 1993.

Bimal Kumar Mishra, Samir Kumar Pandey, Fuzzy e-epidemic SEIRS model for the transmission of worms in Computer network, International Journal of Mathematical Modelling, Simulation and Applications, 4(1), 2011.


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.