J. Appl. Prob. 43, 892–898 (2006) Printed in Israel  Applied Probability Trust 2006 CRITICAL SCALING FOR THE SIS STOCHASTIC EPIDEMIC R. G. DOLGOARSHINNYKH, ∗ Columbia University STEVEN P. LALLEY, ∗∗ University of Chicago Abstract We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of √ N infected and N - √ N susceptible individuals, then when the time and the number currently infected are both scaled by √ N , the resulting process converges, as N →∞, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled 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-Löf (1998) and Aldous (1997) for the simple SIR epidemic. Keywords: Stochastic epidemic model; SIS; SIR; Feller diffusion; Ornstein–Uhlenbeck process 2000 Mathematics Subject Classification: Primary 60K30 Secondary 92D30 1. Introduction Among the most thoroughly studied stochastic epidemic models are the simple SIR and SIS epidemics (see [13] for the origin of the SIS model); among the many problems associated with these models, perhaps the most basic and most interesting are to do with the duration and size of the epidemic. When the epidemic is either subcritical or supercritical, the large-population behavior of the duration and size is reasonably well understood: in the subcritical case the epidemic is stochastically dominated by a subcritical Galton–Watson process (see below), and in the supercritical case the epidemic may become endemic (see [7] and [12]). For critical epidemics, large-population asymptotics are more delicate. It was shown in [9] (see also [1]) that the size S = S N has an interesting and nontrivial asymptotic behavior as the population size N tends to ∞. If the number X 0 of individuals initially infected is of order bN 1/3 , then the total infection time S N (i.e. the sum of the total infection times over the whole population) has a limit distribution S N N 2/3 d -→ T ∗ b , (1) where T ∗ b is the first passage time of W(t) + t 2 /2 to the level b, W(t) is a standard Wiener process, and ‘ d -→’ denotes convergence in distribution. Furthermore, the critical exponent is equal to 1 3 , i.e. the quadratic drift is not felt if X 0 is much smaller than N 1/3 . If X(0) ∼ bN α Received 17 January 2006. ∗ Postal address: Department of Statistics, Columbia University, NewYork, NY 10027, USA. Email address: regina@stat.columbia.edu ∗∗ Postal address: Department of Statistics, University of Chicago, Chicago, IL 60637, USA. Email address: lalley@galton.uchicago.edu 892 https://doi.org/10.1239/jap/1158784956 Published online by Cambridge University Press