# Optimal Vaccination in a SIRS Epidemic Model

*Salvatore Federico*,
*Giorgio Ferrari* and
*Maria-Laura Torrente*

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Salvatore Federico: Center for Mathematical Economics, Bielefeld University

Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University

Maria-Laura Torrente: Center for Mathematical Economics, Bielefeld University

No 667, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

**Abstract:**
We propose and solve an optimal vaccination problem within a deterministic compart-mental 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 colosed-loop equation admits a solution. Coditions 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 quadrativ 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 soltuion; nonsmooth verification theorem: epidemic; optimal vaccination (search for similar items in EconPapers)

**Pages:** 18

**Date:** 2022-06-08

**New Economics Papers:** this item is included in nep-dge and nep-hea

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https://pub.uni-bielefeld.de/download/2963714/2963716 First Version, 2022 (application/pdf)

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**Persistent link:** https://EconPapers.repec.org/RePEc:bie:wpaper:667

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