Spatial Contact Models for Ecological and Epidemic Spread

TLDR

Geographical spread is modeled by growth mechanisms and spatial contact distributions, with most work focusing on asymptotic behavior and facing challenges in short‑term dynamics and inference of contact patterns. The paper surveys spatial contact models, comparing stochastic and deterministic formulations and emphasizing the need for nonlinear stochastic models. Simulations show nonlinear stochastic models exhibit richer non‑asymptotic behavior than linear ones, yet linear models can sometimes approximate them adequately.

Abstract

Summary A wide variety of phenomena of geographical spread can be described in terms of a mechanism of “growth” (e.g. birth, infection) and a “contact distribution” which describes how the locations of the individual(s) involved in a migratory move, or infection at a distance, are spatially related. I shall survey work on such models, beginning with an examination of the relations between stochastic and deterministic models; it emerges that both linear and nonlinear deterministic models have close connections with the less interesting “linear” (exponential growth) stochastic models. More realistic models must be nonlinear as well as stochastic; some results are now available for such models. As in the linear case, these deal mainly with asymptotic behaviour. Simulations reveal that nonlinear stochastic processes have a richer spectrum of non-asymptotic behaviour than linear models, though in some circumstances the simpler models may provide an adequate approximation. Thus theoretical study of the short-term behaviour of such processes may be difficult, but should prove rewarding. The other main outstanding problems are those of inference for such models, especially the estimation of contact distributions.