Optimal control of diseases in prison populations through screening policies of new inmates

TL;DR

This paper develops an optimal control model for managing communicable diseases in prisons by screening new inmates, balancing screening costs and disease management, using advanced mathematical tools to derive an effective bang-bang control strategy.

Contribution

It introduces a novel optimal control framework for disease management in prisons, incorporating screening policies and disease dynamics modeled by SIS, with explicit control strategies derived.

Findings

Optimal bang-bang control strategy with at most two switching times.

Complete synthesis of feedback control based on model parameters.

Characterization of different control profiles depending on parameters.

Abstract

In this paper, we study an optimal control problem of a communicable disease in a prison population. In order to control the spread of the disease inside a prison, we consider an active case-finding strategy, consisting on screening a proportion of new inmates at the entry point, followed by a treatment depending on the results of this procedure. The control variable consists then in the coverage of the screening applied to new inmates. The disease dynamics is modeled by a SIS (susceptible-infected-susceptible) model, typically used to represent diseases that do not confer immunity after infection. We determine the optimal strategy that minimizes a combination between the cost of the screening/treatment at the entrance and the cost of maintaining infected individuals inside the prison, in a given time horizon. Using the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman equation, among other tools, we provide the complete synthesis of an optimal feedback control, consisting in a bang-bang strategy with at most two switching times and no singular arc trajectory, characterizing different profiles depending on model parameters.