We introduce a recursive algorithm which enables the computation of the distribution of epidemic size in a stochastic SIR model for very large population sizes. In the important parameter region where the model is just slightly supercritical, the distribution of epidemic size is decidedly bimodal. We find close agreement between the distribution for large populations and the limiting case where the distribution is that of the time a Brownian motion hits a quadratic curve. The model includes the possibility of vaccination during the epidemic. The effects of the parameters, including vaccination level, on the form of the epidemic size distribution are explored.
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Gordillo, L.F., Marion, S.A., Martin-Löf, A. et al. Bimodal Epidemic Size Distributions for Near-Critical SIR with Vaccination. Bull. Math. Biol. 70, 589–602 (2008). https://doi.org/10.1007/s11538-007-9269-y