The study of epidemic models is one of the central topics of mathematical biology. This volume presents in monograph form the rigorous mathematical ... Show synopsis The study of epidemic models is one of the central topics of mathematical biology. This volume presents in monograph form the rigorous mathematical theory developed to analyze the asymptotic behaviour of certain types of epidemic models. The main model discussed is the so-called spatial deterministic epidemic in which infected individuals are not allowed to again become susceptible, and infection is spread by means of contact distributions. Results concern the existence of travelling wave solutions, the asymptotic speed of propagation and the spatial final size. A central result for radially symmetric contact distributions is that the speed of propagation is the minimum wave speed. Further results are obtained using a saddle point method, suggesting that this result also holds for more general situations. Methodology, used to extend the analysis from one-type to multi-type models, is likely to prove useful when analyzing other multi-type systems in mathematical biology. This methodology is applied to two other areas in the monograph, namely epidemics with return to the susceptible state and contact branching processes.