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Critical epidemics, random graphs, and Brownian motion with a parabolic drift. (English) Zbl 1206.92039

Summary: We investigate the final size distribution of a SIR (susceptible-infected-recovered) epidemic model in the critical regime. Using the integral representation of A. Martin-Löf [J. Appl. Probab. 35, No. 3, 671–682 (1998; Zbl 0919.92034)] for the hitting time of a Brownian motion with parabolic drift, we derive asymptotic expressions for the final size distribution that capture the effect of the initial number of infectives and the closeness of the reproduction number to zero. These asymptotics shed light on the bimodularity of the limiting density of the final size observed by Martin-Löf. We also discuss the connection to the largest component in the Erdős-Rényi random graph, and, using this connection, find an integral expression of the Laplace transform of the normalized Brownian excursion area in terms of Airy functions.

MSC:

92D30 Epidemiology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
05C80 Random graphs (graph-theoretic aspects)

Citations:

Zbl 0919.92034
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References:

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