Optimal control for a multi-group reaction-diffusion SIR model with heterogeneous incidence rates

This paper is devoted to the study of an optimal control problem for a generalized multi-group reaction-diffusion SIR epidemic model, with heterogeneous nonlinear incidence rates. The proposed model incorporates a wide range of spatiotemporal epidemic models. Primarily, the ones used to describe the propagation of zoonotic and sexually transmitted diseases, as well as epidemics that propagate disparately within populations. For the aforementioned diseases, dividing the susceptible, infected and recovered populations into several subpopulations is necessary in order to capture all the possible ways of the disease transmission. This makes the problem of finding the possible optimal control strategies and the division of the available control resources complicated. To address this problem mathematically, for each subpopulation, we introduce two types of control variables, namely vaccination for the susceptible and treatment for the infected. The existence and uniqueness of a biologically feasible solution to the proposed model, for fixed controls, is derived by means of a truncation technique and a semigroup approach. Moreover, first-order necessary optimality conditions for the introduced optimal control problem are obtained using the adjoint state method. Finally, numerical simulations are performed for a two-group epidemic model with particular incidence rates and by considering three cases in the maximal control resources allowed for each subpopulation.