Statistics and Data Science Seminar
Steven P. Lalley
University of Chicago
Critical behavior in stochastic models of spatial epidemics
Abstract: I will survey of some recent work in scaling limits of stochastic spatial SIR epidemics. In these models,
colonies of $N$ individuals are located at lattice points of $\mathbb{Z}^d$. Each individual is, at any time, susceptible, infected,
or recovered (and immune to future infection). Infected individuals recover at rate 1, and infect susceptibles
in the same or neighboring colonies at rate (say) $\lambda/N$. When $\lambda = 1/(2d + 1)$, where $d$ is the dimension of the
lattice, the epidemic is critical: the mean number of new infections produced by a single infected individual
when all other individuals in the same or neighboring colonies are susceptible is 1. The questions of natural
interest center on the duration and spatial extent of a critical epidemic initiated by a large number $N^\alpha$
(where $0<\alpha <1$) of infected individuals all located at the colony at the origin of the lattice. The main results are
large$-N$ scaling laws for the epidemic.
The stochastic epidemic models can be reformulated as percolation processes on graphs that are, in a natural
sense, hybrids of the standard regular lattices and the complete graph on $N$ vertices. The spread of the epidemic
is determined by the geometry of the connected clusters of the associated percolation process. The scaling laws
for epidemic processes consequently translate to scaling laws for percolation clusters.
Wednesday April 11, 2012 at 4:00 PM in SEO 636