The spread of infectious diseases in spatially structured populations: An invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R0, in a lattice population. On lattices, R0 should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R0 with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R0 that remains bounded above, and also predicts R0 values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.