Optimal control of pattern formations for an SIR reaction–diffusion epidemic model

Patterns arising from the reaction-diffusion epidemic model provide insightful aspects into the transmission of infectious diseases. For a classic SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates. A quantitative indicator, "normal serious prevalent area (NSPA)", is introduced to characterize the relationship between patterns and the extent of the epidemic. The extent of epidemic is positively correlated to NSPA. To effectively reduce NSPA of patterns under the large transmission rates, taken removed (recovery or isolation) rate as a control parameter, we consider the mathematical formulation and numerical solution of an optimal control problem for the SIR reaction-diffusion model. Numerical experiments demonstrate the effectiveness of our method in terms of control effect, control precision and control cost.