The stochastic epidemic is a classic example of probabilistic modelling dating back to work of Daniel Bernoulli in 1760, if not earlier. The problem becomes much harder when the model is "spatial" and the disease results in future immunity/death.

Inspired in part by a personal encounter with Covid-19, we consider two models for the spread of infection about a population of diffusing individuals in Euclidean space. The emphasis is upon the case of post-infection immunity. Partial results for the existence (or not) of a pandemic may sometimes be proved via comparisons with branching processes and percolation processes. The principal difficulties lie in the combination of movement and immunity.

The presentation will include summaries of the percolation and contact models, and an outline of their uses in epidemic theory. (Joint work with Zhongyang Li.)