Optimal control analysis in a reaction-diffusion SIRC model with cross-immune class

In this paper, we propose a reaction-diffusion SIRC model with cross-immunization. The model includes the implementation of vaccination measures for susceptible, and treatment and quarantine measures for infected in order to control the spread of the disease. The optimal control strategies for the spread of disease are mainly investigated. Our research focuses on four main aspects. Firstly, we prove the existence and uniqueness of the global positive strong solution to the control system by employing the theories of semigroups of operators. Secondly, we demonstrate the existence of optimal control through the utilization of functional analysis techniques. Thirdly, we establish the first-order necessary optimality conditions that the optimal control must satisfy, employing the methods of convex perturbation. Lastly, we provide numerical examples and simulations to verify the feasibility of optimal control.