We consider several models of propagation arising in evolutionary
epidemiology. We aim at performing a rigorous mathematical analysis
leading to new biological insights. At first we investigate the spread
of an epidemic in a population of homogeneously distributed hosts on a
straight line. An underlying mutation process can shift the virulence of
the pathogen between two values, causing an interaction between
epidemiology and evolution. We study the propagation speed of the
epidemic and the influence of some biologically relevant quantities,
like the effects of stochasticity caused by the hosts' finite population
size (numerical explorations), on this speed. In a second part we take
into account a periodic heterogeneity in the hosts' population and study
the propagation speed and the existence of pulsating fronts for the
associated (non-cooperative) reaction-diffusion system. Finally, we
consider a model in which the pathogen is allowed to shift between a
large number of different phenotypes, and construct possibly singular
traveling waves for the associated nonlocal equation, thus modelling
concentration on an optimal trait.