Probability of extinction and peak time for multi-type epidemics with application to COVID-19 variants of concern

During the COVID-19 pandemic, the emergence of novel variants of concern (VoCs) prompted different responses from governments across the world aimed at mitigating the impacts of more transmissible or more harmful strains. We model the invasion of a novel VoC into a population with heterogeneous vaccine- and infection-acquired immunity using a multi-type branching process framework with immigration. We define the number of cases needed to be reached to ensure stochastic extinction of this strain is unlikely and, therefore, the strain has become established in the population. To estimate the first-passage time distribution to reach this number of cases we use a mixture of stochastic simulations and analytic results. The first-passage time distribution gives a time window that is useful for policymakers planning interventions aimed at suppressing or delaying the introduction of novel VoC. We apply our method to a model of COVID-19 in the United Kingdom, though our results are applicable to other pathogens and settings.