Global Dynamics of a Multi-Population Water Pollutant Model with Distributed Delays

This paper presents a comprehensive mathematical analysis of a novel compartmental model describing the dynamics of dispersed water pollutants and their interaction with two distinct host populations. The model is formulated as a system of integro-differential equations that incorporates multiple distributed delays to realistically account for time lags in the infection process and pollutant transport. We rigorously establish the biological well-posedness of the model by proving the non-negativity and ultimate boundedness of solutions, confirming the existence of a positively invariant feasible region. The analysis characterizes the long-term behavior of the system through the derivation of the basic reproduction number R0d, which serves as a sharp threshold determining the system’s fate. For the model without delays, we prove the global asymptotic stability of the infection-free equilibrium (IFE) when R0≤1 and of the endemic equilibrium (EE) when R0>1. These stability results are extended to the distributed-delay model by using sophisticated Lyapunov functionals, demonstrating that R0d universally governs the global dynamics: the IFE (E0d) is globally asymptotically stable (GAS) if R0d≤1, while the EE (Ed∗) is GAS if R0d>1. Numerical simulations validate the theoretical findings and provide further insights. Sensitivity analysis identifies the most influential parameters on R0d, highlighting the recruitment rate of susceptible individuals, exposure rate, and pollutant shedding rate as key intervention targets. Furthermore, we investigate the impact of control measures, showing that treatment efficacy exceeding a critical value is sufficient for disease eradication. The analysis also reveals the inherent mitigating effect of the maturation delay, demonstrating that a delay longer than a critical duration can naturally suppress the outbreak. This work provides a robust mathematical framework for understanding and managing dispersed water pollution, emphasizing the critical roles of multi-source contributions, time delays, and targeted interventions for environmental sustainability.