Spatial Contact Models for Ecological and Epidemic Spread

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.

Inferences from Spatial Population Genetics

“…The question that one must address prior to statistical analysis is how important such long-distance immigrants are for population processes. For example, the speed of range expansions is known to be affected by the most extreme long-distance migrants in a way generally not characterized by the σ parameter (Mollison, 1977;Clark et al, 2001), so if one is interested in characterizing such processes, not only it will be difficult to estimate σ but this may be irrelevant. One the other hand, some processes of local adaptation (as may lead to allele frequency clines, for example) are not very sensitive to long-distance migrants, and then approximations ignoring them are not only adequate but required to formulate good statistical inferences.…”

Spread of invading organisms

“…For heterogeneous landscapes, detailed simulation models have proven useful adjuncts to the basic theory, as in Murray's discussion of the potential spread of rabies in England (Murray 1986(Murray , 1987. Two-phase models such as those implicit in Mollison's scheme (Mollison 1977) or others that take into account higher-order moments in the movements of individuals, seem to hold promise when knowledge is available on two scales of movement. For example, for diseases such as influenza, it seems reasonable to use models such as those of Rvachev (see Rvachev and Longini 1985) to explain inter-city transport, and couple them with diffusion models to describe the spread from points of introduction.…”

Modelling Spread in Invasion Ecology: A Synthesis

“…the birth rate can be seasonal and locational dependent and further infl uenced by fi re and fl oods). In general, Mollison (1977Mollison ( , 1991 suggests that stochastic models predict the same rate of spread for density -independent populations as the rate of spread in deterministic models, but could alter the rate of spread in density -dependent populations. Although recent work on modelling the spread of invasive species often includes environmental stochasticity, or uses stochastic models (Schreiber & Lloyd -Smith 2009 ), further attention needs to be given to clarifying measures of stochasticity and the relationship between the stochasticity and rates of spread.…”