Supercritical spatial SIR epidemics: Spreading speed and herd immunity

Abstract

We study supercritical spatial SIR epidemics on ${\mathbb{Z}}^2\times \{1,2,\dots ,\mathit{N}\}$, where each site in ${\mathbb{Z}}^2$ represents a village and N stands for the village size. We establish several asymptotic results as $\mathit{N}\to \infty$. In particular, we derive the probability that the epidemic will last forever if the epidemic is started by one infected individual. Moreover, we show that, conditional on that the epidemic lasts forever, the epidemic spreads out linearly in all directions and derive an explicit formula for the spreading speed. Furthermore, we prove that the accumulated proportion of infection converges to a number that is constant over space and find its explicit value. An important message is that if there is no vaccination, then the accumulated proportion of infection can be much higher than the vaccination proportion required to prevent sustained spread of infection.