In this paper, we analyze and characterize the set ${\mathcal{A}}_T$ which consists of all possible profiles at a fixed time of the entropy solution of the elementary wave interaction problem in a bounded domain for a convex scalar conservation law. The elementary wave interaction problem is the initial and boundary value problem for a scalar conservation law, where the flux is a strictly convex function, and the initial and boundary data are constant functions. In the first main result of the article, we state and prove that ${\mathcal{A}}_T$ is a subset of the set of piecewise functions that are constant on each subdomain, or there is a subdomain where the function is strictly increasing. We prove the result by applying the method of characteristics in three steps: the Riemann problem solution, the entropy solution of the interaction of two Riemann problems, and restriction of the entropy solution to the spatial bounded domain. Moreover, we characterize the strictly increasing part of the solution's profile regarding the flux function. In the second result, which is stated as an application of the first result, we introduce the conditions for ill-posedness and local flux identification from the knowledge of the entropy solution's profile.

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.