# Article

 Title: Absorption in stochastic epidemics (English) Author: Štěpán, Josef Author: Staněk, Jakub Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 45 Issue: 3 Year: 2009 Pages: 458-474 Summary lang: English . Category: math . Summary: A two dimensional stochastic differential equation is suggested as a stochastic model for the Kermack–McKendrick epidemics. Its strong (weak) existence and uniqueness and absorption properties are investigated. The examples presented in Section 5 are meant to illustrate possible different asymptotics of a solution to the equation. (English) Keyword: SIR epidemic models Keyword: stochastic epidemic models Keyword: stochastic differential equation Keyword: strong solution Keyword: weak solution Keyword: absorption Keyword: Kermack–McKendrick model MSC: 37N25 MSC: 60H10 MSC: 92D25 MSC: 92D30 idZBL: Zbl 1165.92319 idMR: MR2543134 . Date available: 2010-06-02T18:44:11Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/140009 . Reference: [1] L. J. S. Allen and N. Kirupaharan: Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens.Internat. J. Numer. Anal. Modeling 3 (2005), 2, 329–344. MR 2112651 Reference: [2] A. N. Borodin and P. Salminen: Handbook of Brownian Motion-Facts and Formulae.Birkh$\ddot{{\rm a}}$user Verlag, Basel – Boston – Berlin 2002. MR 1912205 Reference: [3] S. Busenberg and C. Kenneth: Vertically Transmitted Diseases – Models and Dynamics.Springer-Verlag, Berlin – Heidelberg – New York 1993. MR 1206227 Reference: [4] D. J. Daley and J. Gani: Epidemic Modelling: An Introduction.Cambridge University Press, Cambridge 1999. MR 1688203 Reference: [5] P. Greenwood, L. F. Gordillo, and R. Kuske: Autonomous stochastic resonance produces epidemic oscillations of fluctuating Size.In: Proc. Prague Stochastics 2006 (M. Hušková and M. Janžura, eds.), Matfyzpress, Praha 2006. Reference: [6] N. Ikeda and S. Watanabe: Stochastic Differential Equation and Diffusion Processes.North-Holland, Amsterdam 1981. MR 1011252 Reference: [7] : .J. Kalas and Z. Pospíšil: Continuous Models in Biology (in Czech).Masarykova Univerzita v Brně, Brno 2001. Reference: [8] O. Kallenberg: Foundations of Modern Probability.Second edition. Springer, New York 2002. Zbl 0996.60001, MR 1876169 Reference: [9] W. O. Kermack and A. G. McKendrick: A contribution to the mathematical theory of epidemics.Proc. Roy. Soc. London A 155 (1927), 700–721. Reference: [10] L. C. G. Rogers and D. Williams: Diffusions, Markov Processes and Martingales.Cambridge University Press, Cambridge 2006. Reference: [11] J. Štěpán and D. Hlubinka: Kermack–McKendrick epidemic model revisited.Kybernetika 43 (2007), 4, 395–414. MR 2377919 Reference: [12] T. Wai-Yuan and W. Hulin: Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention.World Scientific, Singapore 2005. MR 2169300 .

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