Contact tracing is a key tool for managing epidemic diseases like HIV, tuberculosis, and COVID-19. Manual investigations by human contact tracers remain a dominant way in which this is carried out. This process is limited by the number of contact tracers available, who are often overburdened during an outbreak or epidemic. As a result, a crucial decision in any contact tracing strategy is, given a set of contacts, which person should a tracer trace next? In this work, we develop a formal model that articulates these questions and provides a framework for comparing contact tracing strategies. Through analyzing our model, we give provably optimal prioritization policies via a clean connection to a tool from operations research called a "branching bandit". Examining these policies gives qualitative insight into trade-offs in contact tracing applications.