Parameter Estimation of an Epidemic Model with State Constraints

We consider a mathematical model with five compartments relevant to depict the feature of a certain type of epidemic transmission. We aim to identify some system parameters by means of a minimization problem for a functional involving available measurements for observable compartments, which we treat by an optimal control technique with a state constraint imposed by realistic considerations. The proof of the maximum principle is done by passing to the limit in the conditions of optimality for an appropriate approximating problem. The proof of the estimates for the dual approximating system requires a more challenging treatment since the trajectories are not absolutely continuous, but only with bounded variation. These allow to pass to the limit to obtain the conditions of optimality for the primal problem and lead to a singular dual backward system with a generalized solution in the sense of measure. As far as we know, this approach developed here for the identification of parameters in an epidemic model considering a state constraint related to the actions undertaken for the disease containment was not addressed in the literature and represents a novel issue in this paper.