Modelling optimal lockdowns with waning immunity

Aditya Goenka, Lin Liu, and Manh-Hung Nguyen


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

JEL codes

  • E13: Neoclassical
  • E22: Capital • Investment • Capacity
  • D50: General
  • D63: Equity, Justice, Inequality, and Other Normative Criteria and Measurement
  • I10: General
  • I15: Health and Economic Development
  • I18: Government Policy • Regulation • Public Health
  • O41: One, Two, and Multisector Growth Models
  • C61: Optimization Techniques • Programming Models • Dynamic Analysis


Aditya Goenka, Lin Liu, and Manh-Hung Nguyen, Modeling optimal quarantines with waning immunity, TSE Working Paper, n. 21-1206, May 2021, revised July 2022.


Aditya Goenka, Lin Liu, and Manh-Hung Nguyen, Modelling optimal lockdowns with waning immunity, Economic Theory, November 2022.

Published in

Economic Theory, November 2022