Optimal Control of a Reaction-Diffusion Epidemic Model with Noncompliance

TL;DR

This paper develops an optimal control framework for a reaction-diffusion epidemic model incorporating human behavioral noncompliance, analyzing how nonpharmaceutical interventions can be optimized despite partial adherence.

Contribution

It introduces a novel reaction-diffusion SIR model accounting for noncompliance spread and establishes existence, regularity, and optimality conditions for control strategies.

Findings

Existence of global-in-time solutions for fixed controls.

Optimal control solutions exist under general objective functions.

Simulations demonstrate the impact of noncompliance on epidemic dynamics.

Abstract

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioral effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this noncompliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of noncompliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of noncompliance, and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first--order optimality conditions are obtained via a Lagrangian based stationarity system. We conclude with a discussion regarding minimization of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behavior of the model.