  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 $N^{2/3}$, with the average scaling as $N^{1/3}$, with an explicit form for
the scaling function.
