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 efficiency of treatments is subject to uncertainty. We consider two different 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 100the 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.

COVID-19 大流行说明了人群中与治疗相关的决策制定的重要性。本文考虑了疾病传播率和治疗效率存在不确定性的情况。我们考虑随机 SIR 模型的两种不同状态或子模型，其中人口由三组组成：易感者、感染者、康复者和死亡者。在第一个方案中，感染比例很低，易感比例非常接近 100，感染比例适中，但不可忽略。我们表明，第一种制度几乎完全对应于金融中的一个众所周知的问题，即均值回归回报下的投资组合和消费决策问题（Wachter，JFQA 2002），最优控制具有分析解决方案。我们为第二个问题开发了一个微扰解决方案。据我们所知，与随机 SIR 模型的数值解相反，本文代表了开发分析/微扰解的首次尝试之一。