We study the Susceptible-Infected-Recovered model of epidemics in the
vicinity of the threshold infectivity. We derive the distribution of total
outbreak size in the limit of large population size N. This is accomplished
by mapping the problem to the first passage time of a random walker subject to
a drift that increases linearly with time. We recover the scaling results of
Ben-Naim and Krapivsky that the effective maximal size of the outbreak scales
as N2/3, with the average scaling as N1/3, with an explicit form for
the scaling function