How long does it take to eliminate an epidemic without herd immunity?

Abstract

The global response to the SARS-Cov-2 pandemic has consisted of two main strategies both involving non-pharmaceutical interventions to control spread: mitigation, ultimately relying on herd immunity from vaccination, and elimination of infections locally. While simple theory for controlling an epidemic through herd immunity exist, there is no corresponding simple theory for the strategy of elimination with non-pharmaceutical interventions. Here we quantify an important aspect of the elimination strategy: the time to extinction without herd immunity, based solely on non-pharmaceutical interventions. Using a simple well-mixed stochastic SIR model, we find two new results: 1) using random walk theory we calculate a simple approximation of the mean extinction time and 2) using branching process theory the full distribution of times to extinction, which we show is given by the extreme value Gumbel distribution. We compare these results against complex spatially-resolved stochastic simulations to show very good quantitative agreement, demonstrating the validity of this simple approach. Overall, for SARS-Cov-2 our results predict rapid extinction — of order months — of an epidemic or pandemic if the reproductive number is kept to Re< 0.5; in a counterfactual scenario with global adoption of an elimination strategy in June 2020, SARS-Cov-2 could have been eliminated world-wide by early January 2021.