We study the problem of a policymaker who aims at taming the spread of an epidemic while minimizing its associated social costs. The main feature of our model lies in the fact that the disease’s transmission rate is a diffusive stochastic process whose trend can be adjusted via costly confinement policies. We provide a complete theoretical analysis, as well as numerical experiments illustrating the structure of the optimal lockdown policy. In all our experiments the latter is characterized by three distinct periods: the epidemic is first let freely evolve, then vigorously tamed, and finally a less stringent containment should be adopted. Moreover, the optimal containment policy is such that the product “reproduction number × percentage of susceptible” is kept after a certain date strictly below the critical level of one, although the reproduction number is let oscillate above one in the last more relaxed phase of lockdown. Finally, an increase in the fluctuations of the transmission rate is shown to give rise to an earlier beginning of the optimal lockdown policy, which is also diluted over a longer period of time.

中文翻译：

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通过封锁政策遏制流行病的蔓延

我们研究了政策制定者的问题，他的目标是遏制流行病的传播，同时将其相关的社会成本降至最低。我们模型的主要特征在于疾病的传播率是一个扩散的随机过程，其趋势可以通过昂贵的限制政策进行调整。我们提供了完整的理论分析，以及说明最佳锁定策略结构的数值实验。在我们所有的实验中，后者的特点是三个不同的时期：首先是让流行病自由发展，然后是大力驯服，最后应该采取不那么严格的遏制。此外，最佳遏制政策是这样的，即产品“繁殖数量×易感百分比”在某个日期后严格保持在临界水平以下，尽管在锁定的最后一个更宽松的阶段让复制数在 1 以上振荡。最后，传输速率波动的增加被证明会导致最佳锁定策略的更早开始，该策略也会在更长的时间内被稀释。