Optimal regional control of a spatial diffusion Cholera model with environmental pathogen transmission

In this paper, we propose a regional control strategy to address the challenge of cholera transmission, which is characterized by significant spatial heterogeneity due to its waterborne nature. A spatiotemporal dynamic model was formulated using reaction-diffusion equations, coupled with a four-dimensional optimal control framework encompassing awareness education, expanded vaccination, water chlorination, and treatment. The core mathematical analysis proceeded in three stages: First, the well-posedness of the state system was established, proving the existence and uniqueness of strong solutions. Second, the existence of an optimal control solution was rigorously demonstrated using the method of minimizing sequences. Finally, the first-order necessary optimality conditions were derived by constructing an adjoint system and applying Pontryagin's maximum principle, which explicitly characterized the optimal control quadruple. Numerical simulations validated the theoretical framework, with direct comparisons between regional and uniform control strategies demonstrating the distinct advantage of our approach in cost-effectiveness and outbreak containment efficacy.