# Article

 Title: Kermack-McKendrick epidemic model revisited (English) Author: Štěpán, Josef Author: Hlubinka, Daniel Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 43 Issue: 4 Year: 2007 Pages: 395-414 Summary lang: English . Category: math . Summary: This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale $N_t$ that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size $N_t$. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed. (English) Keyword: SIR epidemic models Keyword: stochastic differential equations Keyword: weak solution Keyword: simulation MSC: 34F05 MSC: 37N25 MSC: 60H10 MSC: 60H35 MSC: 92D25 idZBL: Zbl 1137.37338 idMR: MR2377919 . Date available: 2009-09-24T20:24:56Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/135783 . Reference: [1] Allen E. J.: Stochastic differential equations and persistence time for two interacting populations.Dynamics of Continuous, Discrete and Impulsive Systems 5 (1999), 271–281 Zbl 0946.60058, MR 1678255 Reference: [2] Allen L. J. S., Kirupaharan N.: Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens.Internat. J. Num. Anal. Model. 2 (2005), 329–344 Zbl 1080.34033, MR 2112651 Reference: [3] Andersson H., Britton T.: Stochastic Epidemic Models and Their Statistical Analysis.(Lecture Notes in Statistics 151.) Springer–Verlag, New York 2000 Zbl 0951.92021, MR 1784822 Reference: [4] Bailey N. T. J.: The Mathematical Theory of Epidemics.Hafner Publishing Comp., New York 1957 MR 0095085 Reference: [5] Ball F., O’Neill P.: A modification of the general stochastic epidemic motivated by AIDS modelling.Adv. in Appl. Prob. 25 (1993), 39–62 Zbl 0777.92018, MR 1206532 Reference: [6] Becker N. G.: Analysis of infectious disease data.Chapman and Hall, London 1989 Zbl 0782.92015, MR 1014889 Reference: [7] Daley D. J., Gani J.: Epidemic Modelling; An Introduction.Cambridge University Press, Cambridge 1999 Zbl 0964.92035, MR 1688203 Reference: [8] Greenhalgh D.: Stochastic Processes in Epidemic Modelling and Simulation.In: Handbook of Statistics 21 (D. N. Shanbhag and C. R. Rao, eds.), North–Holland, Amsterdam 2003, pp. 285–335 Zbl 1017.92030, MR 1973547 Reference: [9] Hurt J.: Mathematica$^{®}$ program for Kermack–McKendrick model.Department of Probability and Statistics, Charles University in Prague 2005 Reference: [10] Kallenberg O.: Foundations of Modern Probability.Second edition. Springer–Verlag, New York 2002 Zbl 0996.60001, MR 1876169 Reference: [11] Kendall D. G.: Deterministic and stochastic epidemics in closed population.In: Proc. Third Berkeley Symp. Math. Statist. Probab. 4, Univ. of California Press, Berkeley, Calif. 1956, pp. 149–165 MR 0084936 Reference: [12] Kermack W. O., McKendrick A. G.: A contribution to the mathematical theory of epidemics.Proc. Roy. Soc. London Ser. A 115 (1927), 700–721 Reference: [13] Kirupaharan N.: Deterministic and Stochastic Epidemic Models with Multiple Pathogens.PhD Thesis, Texas Tech. Univ., Lubbock 2003 Zbl 1080.34033, MR 2704799 Reference: [14] Kirupaharan N., Allen L. J. S.: Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality.Bull. Math. Biol. 66 (2004), 841–864 MR 2255779 Reference: [15] Rogers L. C. G., Williams D.: Diffusions, Markov Processes and Martingales.Vol. 1: Foundations. Cambridge University Press, Cambridge 2000 Zbl 0977.60005, MR 1796539 Reference: [16] Rogers L. C. G., Williams D.: Diffusions, Markov Processes and Martingales.Vol. 2: Itô Calculus. Cambridge University Press, Cambridge 2000 Zbl 0977.60005, MR 1780932 Reference: [17] Štěpán J., Dostál P.: The $dX(t)=Xb(X)dt + X\sigma (X)\,dW$ equation and financial mathematics I.Kybernetika 39 (2003), 653–680 MR 2035643 Reference: [18] Štěpán J., Dostál P.: The $dX(t)=Xb(X)dt + X\sigma (X)dW$ equation and financial mathematics II.Kybernetika 39 (2003), 681–701 MR 2035644 Reference: [19] Subramaniam R., Balachandran, K., Kim J. K.: Existence of solution of a stochastic integral equation with an application from the theory of epidemics.Nonlinear Funct. Anal. Appl. 5 (2000), 23–29 MR 1795707 .

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