  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 these 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 size-dependent drift. The
corresponding scaling limits are super-Brownian motions and Dawson-Watanabe
processes with killing, respectively.
