Critical scaling of stochastic epidemic models

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p−coin tosses. Spatial variants of these models are proposed, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a sizedependent drift. 1. Stochastic epidemic models 1.1. Mean-field models The simplest and most thoroughly studied stochastic models of epidemics aremeanfield models, in which all individuals of a finite population interact in the same manner. In these models, a contagious disease is transmitted among individuals of a homogeneous population of size N . In the simple SIS epidemic, individuals are at any time either infected or susceptible; infected individuals remain infected for one unit of time and then become susceptible. In the simple SIR epidemic (more commonly known as the Reed-Frost model), individuals are either infected, susceptible, or recovered ; infected individuals remain infected for one unit of time, after which they recover and acquire permanent immunity from future infection. In both models, the mechanism by which infection occurs is random: At each time, for any pair (i, s) of an infected and a susceptible individual, the disease is transmitted from i to s with probability p = pN . These transmission events are mutually independent. Thus, in both the SIR and the SIS model, the number Jt+1 = J N t of infected individuals at time t+ 1 is given by