A simple stochastic theory of extinction shows rapid elimination of a Sars-like pandemic

Department of Life Sciences, Imperial College London - Silwood Park - Ascot, SL5 7PY, UK and The Francis Crick Institute - Midland Road, London, NW1 1AT, UK

Received: 15 July 2025
Accepted: 26 September 2025

Abstract

The SARS-Cov-2 pandemic demonstrated the challenge of controlling a zoonotic disease with high infection fatality rates, including the evolution of more transmissible variants. In dealing with future similar pandemics, the question of elimination vs. mitigation is still an open question. Although a complex question, a key neglected component to appraise the elimination strategy is a simple theory predicting the timescales of elimination. Using simple random walk and branching process theory we provide insights on the process of elimination using non-pharmaceutical interventions. We find the distribution of times is an extreme-valued Gumbel distribution with a new stochastic threshold of infections, below which random changes in infection dominate. We also determine that there are two regimes for the effective reproductive number R_e —a weak and strong immunity regime— delineated by a new critical value , describing the role of population immunity in the decline of infections. Overall, for the original SARS-Cov-2 variant our results predict rapid extinction —of order months— of an epidemic or pandemic if the reproductive number is kept to ; 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.