Spatial epidemics: Critical behavior in one dimension (Q2391167)

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 considered, 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 epidemic is \textit{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 a branching random walk on the integer lattice which is called the branching envelope of the epidemic. At the threshold it evolves like a branching process with a size-dependent drift. The scaling limits are super-Brownian motions and Dawson-Watanabe processes with killing, respectively.