Abstract
A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton–Jacobi–Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach.
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Acknowledgements
The authors warmly thank the Referees and the Associate Editors for their careful scrutiny which brought to a substantially improved version of the paper.
Funding
Open access funding provided by Luiss University within the CRUI-CARE Agreement. F. Gozzi and F. Lippi acknowledge financial support from the ERC grant 101054421-DCS. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. A. Calvia, F. Gozzi, G. Zanco are supported by the Italian Ministry of University and Research (MIUR), in the framework of PRIN project 2017FKHBA8 001 (The Time-Space Evolution of Economic Activities: Mathematical Models and Empirical Applications).
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Calvia, A., Gozzi, F., Lippi, F. et al. A simple planning problem for COVID-19 lockdown: a dynamic programming approach. Econ Theory (2023). https://doi.org/10.1007/s00199-023-01493-1
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DOI: https://doi.org/10.1007/s00199-023-01493-1
Keywords
- Controlled SIRD model
- Optimal lockdown policies
- Optimal control with state space constraints
- Optimality conditions
- Viscosity solutions