Optimal control of the SIR model in the presence of transmission and treatment uncertainty.

The COVID-19 pandemic illustrates the importance of treatment-related decision making in populations. This article considers the case where the transmission rate of the disease as well as the eciency of treatments is subject to uncertainty. We consider two dierent regimes, or submodels, of the stochastic SIR model, where the population consists of three groups: susceptible, infected and recovered and dead. In the first regime the proportion of infected is very low, and the proportion of susceptible is very close to 100 with few deaths and where recovered individuals do not acquire immunity. In a second regime, the proportion of infected is moderate, but not negligible. We show that the first regime corresponds almost exactly to a well-known problem in finance, the problem of portfolio and consumption decisions under mean-reverting returns (Wachter, JFQA 2002), for which the optimal control has an analytical solution. We develop a perturbative solution for the second problem. To our knowledge, this paper represents one of the first attempts to develop analytical/perturbative solutions, as opposed to numerical solutions to stochastic SIR models.