The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${mathbb Z}^d,dge 3$, when there is no extinction of the infection. For this, we derive percolation estimates (
using dynamic renormalization techniques) for a locally dependent random graph in correspondence with the epidemic model.
