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.

Novel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

This paper is concerned with modelling the dynamics of social outbursts of activity, such as protests or riots. In this sequel to our work in Berestycki et al. (Networks and Heterogeneous Media, vol. 10, no. 3, 1–34), written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore its impact on the existence of upheaval waves. The system involves the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in ${\mathbb R}^d$ as well as the existence of traveling wave solutions in certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which lead to pulsating waves in certain cases. We analyse the effects of periodic domains as well as the barrier problem with the help of numerical simulations. The barrier problem refers to the potential blockage of a wavefront due to a spatial heterogeneity in the system which leads to an area of low excitability (referred to as the barrier). We conclude with a variety of open problems.