A shape theorem for an epidemic model in dimension $d\ge 3$

We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${\mathbb Z}^d,d\ge 3$, 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.