Stochastic models for epidemic scenario planning

The COVID-19 pandemic has highlighted the need for mathematical modelling that captures the uncertainty that is inherent to disease dynamics. There has also been an emphasis on modelling scenarios and projections, rather than true predictions, which are useful for understanding the impacts that different policies and interventions can have on the spread of disease. We develop stochastic models to quantify the uncertainty in outcomes in different settings, which is essential for designing interventions and preparing for reasonable worst-case scenarios. In Chapters 2-4, we use branching processes to model the early growth of outbreaks, with a particular focus on COVID-19. In Chapters 2 and 3 we investigate the temporal uncertainty in the early growth phase of an epidemic and develop analytic expressions for the approximate distribution in times taken for an outbreak to reach a given number of cases. Our results quantify this uncertainty in seconds, making our methods much faster than stochastic simulations, and provide mathematical insights into the nature of the peak-timing distribution for an epidemic. We show that, in general, the peak-timing distribution can be thought of as well-approximated by the inverse of a non-central chi-squared distribution with zero degrees of freedom. In Chapter 4, we use a well-studied branching process with a negative binomial offspring distribution in order to estimate the overdispersion in early clusters of infections during the COVID-19 pandemic. We augment this process by explicitly accounting for case under-ascertainment, reducing bias in the estimates of overdispersion. We also develop stochastic models for operational use in prison and hospital settings during the pandemic. In Chapter 5, we develop models aimed at understanding the risk of ingress of COVID-19 into prisons and the impact of policies aimed at mitigating this risk. In Chapter 6, we develop a multi-state survival model in order to estimate the length of stay for COVID-19 patients in intensive care, and use stochastic simulations to predict future bed capacity in hospitals. In both of these settings, we demonstrate the utility of stochastic models in conveying uncertainty to policymakers planning for different scenarios. This thesis contains pieces of work that contributed to the broader modelling effort during the COVID-19 pandemic. It also provides a novel framework for thinking about temporal uncertainty in disease outbreaks that has the potential to be incorporated into a number of different models in mathematical epidemiology and in biology more broadly.