The threshold dynamics of a waterborne pathogen model with seasonality in a polluted environment

This paper concentrates on the dynamics of a waterborne pathogen periodic PDE model with environmental pollution. For this model, we derive the basic reproduction number \(\mathcal{R}_{0}\) $${\mathcal{R}}_0$$ and establish a threshold type result on its global dynamics in terms of \(\mathcal{R}_{0}\) $${\mathcal{R}}_0$$ , which predicts the extinction or persistence of diseases. More precisely, the disease-free steady state is globally attractive if \(\mathcal{R}_{0}<1\) $${\mathcal{R}}_0<1$$ , while the system admits at least one positive periodic solution and the disease is uniformly persistent if \(\mathcal{R}_{0}>1\) $${\mathcal{R}}_0>1$$ . Moreover, we carry out some numerical simulations to illustrate the long-term behaviors of solutions and explore the influence of environmental pollution and seasonality on the spread of waterborne diseases.