In the following section we give an overview of the modelling approach. We then present new analytical results for the simple models studied previously, which ignored spatial host population structure. We use these expressions to provide useful answers to important questions: if we knew how a zoonotic pathogen would adapt to human physiology, could we anticipate its emergence? How reliable would such predictions be? Furthermore, can we predict which zoonoses will cause outbreaks which do not turn into epidemics? Next, we ask: how large does a single, finite host population have to be, for population size to have a negligible effect? We then incorporate spatial heterogeneity by separating the human host population into communities. We present a model in which a small village is connected, by human travel, with a large city as an example of the general case of two interconnected communities. We use this model to ask: how strong do community interconnections have to be for us to safely ignore the separation of a population into spatially structured communities, such as cities and villages? We review available commuting data to ask how these thresholds compare with typical human mobility patterns? We close with a discussion of implications for public health.