Modeling optimal quarantines with waning immunity

This paper studies continuing optimal quarantines (can also be interpreted as lockdowns or self-isolation) in the long run if a disease (Covid-19) is endemic and immunity can fail, i.e. the disease has SIRS dynamics. We model how the disease related mortality affects optimal choices in a dynamic general equilibrium neoclassical growth framework. An extended welfare function that incorporates loss from mortality is used. Without welfare loss from mortality, in the long run even if there is continuing mortality, it is not optimal to impose a quarantine. We characterize the optimal decision of quarantines and how the disease endemic steady state changes with effectiveness of quarantine, productivity of working from home, rate of mortality from the disease, and failure of immunity. We also give the sufficiency conditions for economic models with SIRS dynamics - a class of models which are non-convex and have endogenous discounting so that no existing results are applicable.