On an interacting particle system modeling an epidemic

We consider an interacting particle system on Zd to model an epidemic. Each site of Zd can be in either one of three states: empty, healthy or infected. Healthy and infected individuals give birth at different rates to healthy individuals on empty sites. Healthy individuals get infected by infected individuals. Infected and healthy individuals die at different rates. We prove that in dimension 1 and with nearest-neighbor interactions the epidemic may persist forever if and only if the rate at which infected individuals give birth to healthy individuals is high enough. This is in sharp contrast with models analysed by Andjel and Schinazi (1994) and Sato et al. (1994) where infected individuals do not give birth. We also show that some results in the latter reference can be obtained easily and rigorously using probabilistic coupling to the contact process.