Solution of an infection model near threshold

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.