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An introduction to stochastic epidemic models. (English) Zbl 1206.92022
Bauer, Fred (ed.) et al., Mathematical epidemiology. Berlin: Springer (ISBN 978-3-540-78910-9/pbk). Lecture Notes in Mathematics 1945. Mathematical Biosciences Subseries, 81-130 (2008).
Summary: A brief introduction to the formulation of various types of stochastic epidemic models is presented based on the well-known deterministic SIS and SIR epidemic models. Three different types of stochastic model formulations are discussed: discrete time Markov chain, continuous time Markov chain and stochastic differential equations. Properties unique to the stochastic models are presented: probability of disease extinction, probability of disease outbreak, quasistationary probability distribution, final size distribution, and expected duration of an epidemic. The chapter ends with a discussion of two stochastic formulations that cannot be directly related to the SIS and SIR epidemic models. They are discrete time Markov chain formulations applied in the study of epidemics within households (chain binomial models) and in the prediction of the initial spread of an epidemic (branching processes).
MSC:
92D30Epidemiology
60J20Applications of Markov chains and discrete-time Markov processes on general state spaces
60H10Stochastic ordinary differential equations
92C60Medical epidemiology
92-04Machine computation, programs (appl. to natural sciences)