Optimal control of a reaction-diffusion epidemic model with non-compliance

Abstract In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural 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 non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance 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 non-compliance 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 minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.