The contact process is an interesting model for the spread of disease through space. Susceptibles are placed at the vertices of a lattice, and infection is introduced at the origin. The process has a rich theory, which is greatly facilitated by the fact that it is stochastically monotone in the rate of infection. Minor variants of the contact model lack this property. A simple such extension is obtained by allowing "removal periods"' before a cured individual becomes available for reinfection. Such systems are much more challenging.

We discuss the phase diagram of such a model, together with some applications of techniques first developed for percolation. There are open problems, such as to prove the uniqueness of the phase transition. The analysis includes an "essential" application of Reimer's inequality.