DYNAMICAL ANALYSIS AND OPTIMAL CONTROL STRATEGIES OF A GENERAL REACTION-DIFFUSION WATERBORNE PATHOGEN MODEL

Dynamical Analysis of a Waterborne Pathogen Model

Résumé

This paper investigates the global stability analysis and optimal control strategies of a
reaction–diffusion waterborne pathogen model with a general incidence rate, incorporating both direct
and indirect transmission pathways. First, we establish the well-posedness of solutions for the model.
Then, we analyze the threshold dynamics in terms of the basic reproduction number R0: the diseasefree
equilibrium is globally asymptotically stable if R0 ≤ 1, while the endemic equilibrium is globally
asymptotically stable if R0 > 1. The model is subsequently extended by introducing control intervention
strategies such as vaccination, treatment, and water purification, with the aim of minimizing disease
spread at the lowest possible cost. We further prove the existence of an optimal control. Finally, we
derive the first-order necessary conditions for optimality and characterize the optimal controls in terms
of the state and adjoint variables. Numerical simulations are performed to confirm and illustrate the
different theoretical results.