Supercritical Spatial SIR Epidemics: Spreading Speed and Herd Immunity

TL;DR

This paper analyzes supercritical spatial SIR epidemic models on a lattice, deriving probabilities of indefinite spread, spreading speeds, and final infection proportions, highlighting the impact of vaccination thresholds.

Contribution

It provides explicit asymptotic formulas for epidemic duration, spread speed, and final infection size in a spatial SIR model as village size grows.

Findings

Probability of indefinite epidemic spread is derived.

Explicit spreading speed formula is established.

Final infection proportion converges to a constant value.

Abstract

We study supercritical spatial SIR epidemics on $\mathbb{Z}^2\times \{1,2,\ldots, N\}$, where each site in $\mathbb{Z}^2$ represents a village and $N$ stands for the village size. We establish several key asymptotic results as $N\to\infty$. In particular, we derive the probability that the epidemic will last forever if the epidemic is started by one infected individual. Moreover, conditional on that the epidemic lasts forever, we show that the epidemic spreads out linearly in all directions and derive an explicit formula for the spreading speed. Furthermore, we prove that the ultimate 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 ultimate proportion of population who will be infected can be \emph{much higher} than the vaccination proportion that is needed in order to prevent sustained spread of the infection.