Critical scaling for the SIS stochastic epidemic

We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of root N infected and N - root N susceptible individuals, then when the time and the number currently infected are both scaled by root N, the resulting process converges, as N -> infinity, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the resealed size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Lof (1998) and Aldous (1997) for the simple SIR epidemic.