Fixed period of temporary immunity after run of anti-malicious software on computer nodes. (English) Zbl 1117.92052

Mishra, Bimal Kumar; Jha, Navnit

Appl. Math. Comput. 190, No. 2, 1207-1212 (2007).

Summary: An SIRS epidemic model has been developed with a fixed period of temporary immunity, following temporary recovery from the infection of malicious objects in place of an exponentially distributed period of temporary immunity. When a node is recovered from the infected class, it recovers temporarily, acquiring temporary immunity with probability \(p\) \((0 \leq p\leq 1)\) and dies from the attack of malicious objects with probability \((1 - p)\). Temporary immunity is observed in the computer network when anti-malicious software is run after a node gets affected by malicious object(s).
The model consists of a set of integro-differential equations. The stability of the results is stated in terms of threshold parameters. It has been observed that the endemic equilibrium for this model may be unstable thus giving an example of a generalization which leads to new possibilities for the behavior of the model. Numerically it has been verified that the endemic equilibrium is not asymptotically stable for all parameter values.

MSC:

92D30	Epidemiology
45J05	Integro-ordinary differential equations
45M10	Stability theory for integral equations

Keywords:

SIRS epidemic model; malicious objects; computer network; temporary immunity; endemic equilibrium; time-delay

References:

[3]	Beretta, E.; Takeuchi, Y., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47, 4107-4115 (2001) · Zbl 1042.34585
[4]	Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33, 250-260 (1995) · Zbl 0811.92019
[5]	Brauer, Fred; Castillo-Chavez, Carlos, Mathematical Models in Population Biology and Epidemiology (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0967.92015
[6]	Hethcote, H. W., Qualitative analyses of communicable disease models, J. Math. Biosci., 28, 335-356 (1976) · Zbl 0326.92017
[7]	Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033
[8]	Hethcote, H. W., Three basic epidemiological models, (Gross, L.; Hallam, T. G.; Levin, S. A., Applied Mathematical Ecology (1989), Springer: Springer Berlin), 119-144
[10]	Kermack, W. O.; McKendrick, A. G., Contributions to the mathematical theory of epidemics, Proc. R. Soc. A, 115, 700-721 (1927) · JFM 53.0517.01