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. 2022 Dec 21:1–26. Online ahead of print. doi: 10.1007/s00199-022-01475-9

Optimal vaccination in a SIRS epidemic model

Salvatore Federico 1,✉,#, Giorgio Ferrari 2,#, Maria-Laura Torrente 1,#
PMCID: PMC9770565  PMID: 36573250

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

Acknowledgements

We thank two anonymous reviewers and the guest editor for valuable comments and suggestions. Salvatore Federico was partially supported by the Italian Ministry of University and Research (MUR), in the framework of PRIN project 2017FKHBA8 001 “The Time-Space Evolution of Economic Activities: Mathematical Models and Empirical Applications”.

Declarations

Conflict of interest

The authors certify that they have no affiliations with or involvement in any organization or entity with any financial or non-financial interest in the subject matter or materials discussed in this manuscript.

Footnotes

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Salvatore Federico, Giorgio Ferrari and Maria-Laura Torrente have contributed equally to this work.

Contributor Information

Salvatore Federico, Email: salvatore.federico@unige.it.

Giorgio Ferrari, Email: giorgio.ferrari@uni-bielefeld.de.

Maria-Laura Torrente, Email: marialaura.torrente@economia.unige.it.

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