The SIR model in a moving population: propagation of infection and herd immunity

In a collection of particles performing independent random walks on ${\mathbb{Z}}^d$ we study the spread of an infection with SIR dynamics. Susceptible particles become infected when they meet an infected particle. Infected particles heal and are removed at rate $\nu$. We show that when $\nu$ is small, with positive probability the infection survives forever and grows linearly. Furthermore, after the infection reaches a region, it quickly passes through and leaves behind a $\mathit{\text{herd}}\mathit{\text{immunity}}$ regime consisting of recovered particles, a small positive density of susceptible particles, and no infected particles. One notable feature of this model is the simultaneously existence of supercritical and subcritical phases on either side of an infection front of $O(1)$ width.