Epidemics, such as COVID-19, can have profound and far-reaching impacts on societies and individuals worldwide. Infectious disease outbreaks can result in millions of lives lost and place a tremendous burden on public health resources. Additionally, the opioid epidemic in the U.S. has been further fueled by COVID-19, leading to record drug overdose deaths in 2021. In the face of these epidemics, policy makers face dicult decisions in allocating limited resources to improve population health. To address these challenges, this dissertation focuses on the development of mathematical models to inform critical decisions in public policy, specifically optimizing resource allocation to control epidemics.

Chapter 2 centers on the opioid epidemic. We develop a dynamic model to assess the e↵ectiveness of interventions for controlling the US opioid epidemic. We show that reductions in opioid prescriptions are necessary but may lead to a short-term increase in heroin overdose deaths, and thus must be combined with scale up of treatment for addicted individuals – but that even with immediate policy changes, significant morbidity and mortality will still occur. Our analysis provides critically needed evidence-informed recommendations for reducing opioid-related morbidity and mortality in the U.S.

Chapters 3, 4 and 5 focus on developing interpretable models to guide the allocation of limited vaccines to control the spread of an infectious disease. In Chapter 3 we first consider an SIR (susceptible, infected, recovered) model with interacting population groups and a single allocation of vaccine. By approximating the disease dynamics, we derive intuitive analytical conditions characterizing the optimal solution for four di↵erent objectives:​ minimize infections, deaths, life years lost and QALYs lost due to deaths. We extend the work in Chapter 4 to a dynamic setting and develop a method for allocating vaccines over time. In Chapter 5, we further extend the work to an endemic setting, considering vaccine booster doses to take into account waning immunity from vaccination. We show that the approximated optimal solution is an all-or-nothing allocation based on a prioritized list of population groups given by the analytical conditions. Numerical simulations show that the analytical solutions achieve near-optimal results with objective function values significantly better than would be obtained using simple allocation rules such as allocation proportional to population group size.

By leveraging interdisciplinary approaches, this dissertation aims to aid in decision making in the areas of opioid abuse, COVID-19, and epidemic control. Importantly, the work provides general theoretical frameworks that can be adapted to other public health challenges for epidemic control.