Critical epidemics, random graphs, and Brownian motion with a parabolic drift

We investigate the final size distribution of the SIR (susceptible-infected-recovered) epidemic model in the critical regime. Using the integral representation of Martin-Lof (1998) 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 in Martin-Lof (1998). We also discuss the connection to the largest component in the Erdos-Renyi 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.