Optimal Vaccination in a SIRS Epidemic Model

TL;DR

This paper develops an optimal vaccination strategy within an SIRS epidemic model, combining theoretical guarantees with numerical case studies to show long-term disease eradication under certain conditions.

Contribution

It introduces a non-smooth verification theorem for the Hamilton-Jacobi-Bellman equation and analyzes the well-posedness of the closed-loop system using Regular Lagrangian Flows.

Findings

Optimal vaccination can eliminate infection in the long run when parameters are favorable.

Theoretical framework guarantees the identification of the minimal cost function.

Numerical case study demonstrates practical implementation and results.

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 \emph{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.