The effects of the mean infection (incubation plus infectious) period on the dynamics of infectious diseases are well understood. We examine the dynamics and persistence of epidemics when the distribution of the infection period also is modelled, using the well-documented childhood disease measles as a test case. We pay particular attention to the differences between exponentially distributed and constant periods. The use of constant periods increases the persistence of epidemics by reducing the individual stochasticity. The infection period distribution is also shown to have a significant effect on the spatial spread of disease.