Optimal vaccination in a SIRS epidemic model

Salvatore Federico (), Giorgio Ferrari () and Maria-Laura Torrente ()
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Salvatore Federico: Università di Genova
Giorgio Ferrari: Bielefeld University
Maria-Laura Torrente: Università di Genova

Economic Theory, 2024, vol. 77, issue 1, No 3, 49-74

Abstract: Abstract We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton–Jacobi–Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.

Keywords: SIRS model; Optimal control; Viscosity solution; Non-smooth verification theorem; Epidemic; Optimal vaccination; 93C15; 49K15; 49L25; 92D30 (search for similar items in EconPapers)
JEL-codes: C61 I12 I18 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00199-022-01475-9

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