An Introduction to Stochastic Epidemic Models

Linda Allen (June 19, 2012)

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Abstract

A brief introduction is presented to modeling in stochastic epidemiology. Several
useful epidemiological concepts such as the basic reproduction number and the nal size
of an epidemic are de ned. Three well-known stochastic modeling formulations are in-
troduced: discrete-time Markov chains, continuous-time Markov chains, and stochastic
di erential equations. Methods for derivation, analysis and numerical simulation of the
three types of stochastic epidemic models are presented. Emphasis is placed on some of
the di erences between the three stochastic modeling formulations as illustrated in the
classic SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered)
epidemic models. In addition, some of the unique properties of stochastic epidemic
models, such as the probability of an outbreak, nal size distribution, critical commu-
nity size, and expected duration of an epidemic are demonstrated in various models of
diseases impacting humans and wildlife.