A simple planning problem for COVID-19 lockdown: a dynamic programming approach

• Published in: Economic theory Vol. 77; no. 1-2; pp. 169 - 196
• Main Authors: Calvia, Alessandro, Gozzi, Fausto, Lippi, Francesco, Zanco, Giovanni
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024; Springer; Springer Nature B.V
• Subjects: COVID-19; Disease transmission; Dynamic programming; Economic theory; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Economics and Finance; Game Theory; Microeconomics; Optimization; Prevention programs; Public Finance; Research Article; Shelter in place; Social and Behav. Sciences; Value; Viscosity; Controlled SIRD model; E23; I12; Optimal lockdown policies; I15; Viscosity solutions; Optimality conditions; I18; 49L20; Optimal control with state space constraints; 49K15; 49L25; C61
• ISSN: 0938-2259; 1432-0479; 1432-0479
• DOI: 10.1007/s00199-023-01493-1

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.