  Characterizing the spatial extent of epidemics at the outbreak stage is key
to controlling the evolution of the disease. At the outbreak, the number of
infected individuals is typically small, so that fluctuations around their
average are important: then, it is commonly assumed that the
susceptible-infected-recovered (SIR) mechanism can be described by a stochastic
birth-death process of Galton-Watson type. The displacements of the infected
individuals can be modelled by resorting to Brownian motion, which is
applicable when long-range movements and complex network interactions can be
safely neglected, as in case of animal epidemics. In this context, the spatial
extent of an epidemic can be assessed by computing the convex hull enclosing
the infected individuals at a given time. We derive the exact evolution
equations for the mean perimeter and the mean area of the convex hull, and
compare them with Monte Carlo simulations.

Populations and Evolution;Statistical Mechanics