Stochastic extinction of epidemics: how long does would it take for Sars-Cov-2 to die out?

The SARS-Cov-2 pandemic has seen the challenge of controlling novel zoonotic diseases that have high infection fatality rates, including a natural capacity for the evolution of variants that transmit more easily and evade immunity. In dealing with current and future similar pandemics, the question arises: what is the optimum strategy to control infections. Although a complex question, a key neglected component to appraise the elimination strategy is simple theory predicting the expected timescales of elimination. We use simple random walk and branching process theory to provide new insights on the process of elimination using non-pharmaceutical interventions. Our central achievement is a full theory of the distribution of extinction times -- which we find is an extreme value Gumbel distribution -- for any value of the reproductive number including some degree of population immunity. 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 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. Looking to the future, our results would suggest that elimination using NPIs is a more optimal strategy to control -- and potentially eradicate -- a Sars-like pandemic, in its early stages before the evolution of variants with greater transmissibility.