Host-parasite interactions between the local and the mean-field: how and when does spatial population structure matter?

The assumption that populations are completely mixed is reasonable for many populations, but there is likely to be some degree of local interaction whether spatially or socially in many systems. An important question is therefore how strong these local interactions need to be before there are significant effects on the dynamics of the system. Here, our approach is to use a multi-scale pair-approximation model to move between completely local and completely mixed host-parasite interactions. We show that systems dominated by near neighbour effects have less persistence of disease, and a greater possibility of parasite driven extinction and limit cycles. Furthermore this reduction in persistence occurs over a wide range of infection scales and is still significant in predominantly mixed host populations. Deterministic extinctions are only likely in highly spatial SI systems while oscillations also persist over a wide range of infection ranges, but only in hosts that reproduce mostly locally. In general the mean-field may well be a good approximation for many systems, even when there are a significant proportion of near neighbour events, but this depends crucially on the ecological context.