Deterministic compartmental model for optimal control strategies of Giardiasis infection with saturating incidence and environmental dynamics

This study develops a deterministic compartmental model that tracks Giardiasis’s direct and indirect transmission dynamics. The study begins by constructing a model incorporating four constant controls: health education, screening, hospitalization, and sanitation. The analytical results of the model are investigated and presented. The positivity of the solutions and the existence of invariant regions were established. The model exhibits a unique disease-free equilibrium and multiple endemic equilibria. The effective reproduction number was derived using the Next-Generation Matrix (NGM) approach, and its implications for the stability of the equilibria were explored. Local stability of the disease-free equilibrium was confirmed using the Routh–Hurwitz criteria, while global stability results were also presented. Sensitivity analysis was conducted based on the effective reproduction number, identifying the most influential parameters. We introduce an optimal control problem to curb the spread of Giardiasis. We rigorously establish the existence of optimal control solutions and analytically characterize these solutions using Pontryagin’s Maximum Principle. We conduct numerical simulations to evaluate the effectiveness of various control strategies. The results are promising, showing that the simultaneous implementation of all four control measures, education, screening, treatment, and sanitation, can lead to a significant reduction in disease cases, thereby offering a reassuring solution to the spread of Giardiasis.