Rich dynamics in a delayed water borne pathogen model with overexposure

In this paper, we formulate a delayed water borne pathogen model incorporating overexposure, investigate threshold dynamics, and analyze the impact of overexposure and delay on disease transmission. Threshold dynamics are characterized by the basic reproduction number R₀. The model exhibits backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium for R₀ values below 1, indicating that the disease may persist even when R₀<1. Furthermore, we theoretically and numerically examined the existence of Hopf bifurcation in absence of time delay and the results reveal that the overexposure induces rich dynamics, including stability switches, endemic bubble and multiple limits cycles. For the delayed model, by regarding time delay as the bifurcation parameter, the local and global Hopf bifurcation have been carried out to show influence of time delay on model dynamics. Numerical simulations confirm that delay induces stability switches and coexistence of multiple periodic solutions. Our findings indicate that overexposure and time delay are responsible for the model's complex dynamics, complicating disease control efforts.