Optimal control problem and reaction identification term for carrier-borne epidemic spread with a general infection force and diffusion

In this paper, we simultaneously study reaction identification and the optimal control problem for a reaction-diffusion system modeling carrier-borne epidemics with a general transmission function and vaccination. The state equations are given by a susceptible-infected-recovered reaction-diffusion system with zero-flux boundary conditions and initial conditions. The reaction is modeled by three terms: a general transmission function modeling the force of the infection or the effective contact between susceptible and infected individuals, a linear function for transition between susceptible and infected individuals, and a function for control of vaccination of susceptible individuals. The cost function consists of two parts: two terms related to parameter identification, comprising a regularized least squares cost function, and five terms related to the control of the population through vaccination. The optimal control problem is analyzed by applying the Dubovitskii and Milyutin formalism. In the main results, we deduce the well-posedness of the state equation, the existence of the optimal control problem, the existence of solutions of the adjoint state, and a first-order optimality condition. We develop a numerical approximation for the optimal control problem by employing an IMEX method to approximate the state equations. In this approach, the coefficients of the reaction terms and the control functions depend on a finite set of parameters. We provide two numerical examples to demonstrate the agreement of our numerical solution with the measurement observations.