Sparse Optimal Control of Pattern Formations for an SIR Reaction-Diffusion Epidemic Model

Pattern formation arising from the reaction-diffusion epidemic model is a space-time depiction of the distribution and transmission of infectious diseases. Disease control can be achieved by controlling the associated pattern formations. For an SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates under the case of a constant recovery rate. To control pattern formations of the SIR epidemic model, we introduce a regulator as a control function of the recovery rate. For a perfect control strategy, it not only leads to a desired pattern formation, but also has a small support in space-time domains. In order to obtain such a control strategy, we propose a sparse optimal control problem governed by the SIR epidemic model. We study the existence of optimal solutions, derive the first order necessary optimality system, obtain the sparsity structure of the control function, and numerically solve the control problem. Numerical results demonstrate the feasibility and effectivity of our method.