  The behavior at criticality of spatial SIR (susceptible/infected/recovered)
epidemic models in dimensions two and three is investigated. In these models,
finite populations of size N are situated at the vertices of the integer
lattice, and infectious contacts are limited to individuals at the same or at
neighboring sites. Susceptible individuals, once infected, remain contagious
for one unit of time and then recover, after which they are immune to further
infection. It is shown that the measure-valued processes associated with these
epidemics, suitably scaled, converge, in the large-N limit, either to a
standard Dawson-Watanabe process (super-Brownian motion) or to a
Dawson-Watanabe process with location-dependent killing, depending on the size
of the the initially infected set. A key element of the argument is a proof of
Adler's 1993 conjecture that the local time processes associated with branching
random walks converge to the local time density process associated with the
limiting super-Brownian motion.
