One of the principal ways nations are responding to the COVID-19 pandemic is by locking down portions of their economies to reduce infectious spread. This is expensive in terms of lost jobs, lost economic productivity, and lost freedoms. So it is of interest to ask: What is the optimal intensity with which to lockdown, and how should that intensity vary dynamically over the course of an epidemic? This paper explores such questions with an optimal control model that recognizes the particular risks when infection rates surge beyond the healthcare system’s capacity to deliver appropriate care. The analysis shows that four broad strategies emerge, ranging from brief lockdowns that only “smooth the curve” to sustained lockdowns that prevent infections from spiking beyond the healthcare system’s capacity. Within this model, it can be optimal to have two separate periods of locking down, so returning to a lockdown after initial restrictions have been lifted is not necessarily a sign of failure. Relatively small changes in judgments about how to balance health and economic harms can alter dramatically which strategy prevails. Indeed, there are constellations of parameters for which two or even three of these distinct strategies can all perform equally well for the same set of initial conditions; these correspond to so-called triple Skiba points. The performance of trajectories can be highly nonlinear in the state variables, such that for various times $t$, the optimal unemployment rate could be low, medium, or high, but not anywhere in between. These complex dynamics emerge naturally from modeling the COVID-19 epidemic and suggest a degree of humility in policy debates. Even people who share a common understanding of the problem’s economics and epidemiology can prefer dramatically different policies. Conversely, favoring very different policies is not evident that there are fundamental disagreements.

各国应对 COVID-19 大流行病的主要方式之一是封锁部分经济体以减少传染病的传播。就失去工作、失去经济生产力和失去自由而言，这是代价高昂的。因此，有兴趣问：锁定的最佳强度是多少，以及该强度在流行病过程中应如何动态变化？本文通过最佳控制模型探讨此类问题，该模型在感染率激增超出医疗保健系统提供适当护理的能力时识别特定风险。分析表明，出现了四种广泛的策略，从仅“平滑曲线”的短暂封锁到防止感染激增超出医疗保健系统能力的持续封锁。在这个模型中，最好有两个单独的锁定期，因此在解除初始限制后重新锁定不一定是失败的迹象。关于如何平衡健康和经济危害的判断的相对较小的变化可能会显着改变哪种策略占上风。事实上，有一些参数群，其中两个甚至三个不同的策略都可以在同一组初始条件下表现得同样好；这些对应于所谓的三重 Skiba 点。轨迹的性能在状态变量中可能是高度非线性的，这样对于不同的时间 关于如何平衡健康和经济危害的判断的相对较小的变化可能会显着改变哪种策略占上风。事实上，有一些参数群，其中两个甚至三个不同的策略都可以在同一组初始条件下表现得同样好；这些对应于所谓的三重 Skiba 点。轨迹的性能在状态变量中可能是高度非线性的，这样对于不同的时间 关于如何平衡健康和经济危害的判断的相对较小的变化可能会显着改变哪种策略占上风。的确，有一些参数群，其中两种甚至三种不同的策略在同一组初始条件下都可以表现得同样好；这些对应于所谓的三重 Skiba 点。轨迹的性能在状态变量中可能是高度非线性的，这样对于不同的时间$吨$，最佳失业率可能是低、中或高，但不会介于两者之间。这些复杂的动态从对 COVID-19 流行病的建模中自然而然地出现，并表明在政策辩论中要保持一定程度的谦逊。即使对问题的经济学和流行病学有共同理解的人也可能更喜欢截然不同的政策。相反，支持截然不同的政策并不代表存在根本分歧。