The spread of a parasitic infection in a spatially distributed host population

Starting from a stochastic model for the spread of a parasitic infection in a spatially distributed host population we describe the way to a continuum formulation by a deterministic model in terms of a nonlinear partial differential equation and an integro-differential equation. The hosts are assumed to occupy fixed spatial positions, whereas the parasites are mobile, however can propagate only within the hosts. To perform the continuum limit we suppose that the size Nh of the host population, the size Np of the parasite population, and the ratio Np/Nh tend to infinity. Accordingly, the parameters determining the time evolution of the host and parasite populations are rescaled suitably.