This paper studies an optimal growth model where there is an infectious disease with $SIR$ dynamics which can lead to mortality. Health expenditures (alternatively intensity of lockdowns) can be made to reduce infectivity of the disease. We study implications of two different ways to model the disease related mortality – early and late in infection mortality – on the equilibrium health and economic outcomes. In the former, increasing mortality reduces infections by decreasing the fraction of infectives in the population, while in the latter the fraction of infectives increases. We characterize the steady states and the outcomes depend in the way mortality is modeled. With early mortality, increasing mortality leads to higher equilibrium per capita output and consumption while in the late mortality model these decrease. We establish sufficiency conditions and provide the first results in economic models with $SIR$ dynamics with and without disease related mortality — a class of models which are non-convex and have endogenous discounting so that no existing results are applicable.

本文研究了具有传染病的最优生长模型。 $\mathrm{小号}\mathrm{一世}\mathrm{[R}$动态可能导致死亡。可以进行医疗保健支出（或者锁定的强度）以减少疾病的传染性。我们研究了两种与疾病相关的死亡率建模的不同方法的影响-感染死亡率的早期和晚期-对均衡健康和经济结果的影响。在前者中，增加死亡率通过减少人群中传染性的比例来减少感染，而在后者中，传染性的比例则增加。我们表征稳态，其结果取决于死亡率建模的方式。随着早期死亡率的增加，死亡率提高了人均产出和消费的均衡水平，而在晚期死亡率模型中，死亡率下降了。我们建立充分条件，并在经济模型中提供第一个结果，$\mathrm{小号}\mathrm{一世}\mathrm{[R}$ 具有和不具有疾病相关死亡率的动力学—一类非凸且具有内生折现的模型，因此没有适用的现有结果。