Threshold dynamics of a reaction–advection–diffusion schistosomiasis epidemic model with seasonality and spatial heterogeneity

Most water-borne disease models ignore the advection of water flows in order to simplify the mathematical analysis and numerical computation. However, advection can play an important role in determining the disease transmission dynamics. In this paper, we investigate the long-term dynamics of a periodic reaction–advection–diffusion schistosomiasis model and explore the joint impact of advection, seasonality and spatial heterogeneity on the transmission of the disease. We derive the basic reproduction number \({\mathcal {R}}_0\) ${\mathcal{R}}_0$ and show that the disease-free periodic solution is globally attractive when \({\mathcal {R}}_0<1\) ${\mathcal{R}}_0<1$ whereas there is a positive endemic periodic solution and the system is uniformly persistent in a special case when \({\mathcal {R}}_0>1\) ${\mathcal{R}}_0>1$ . Moreover, we find that \({\mathcal {R}}_0\) ${\mathcal{R}}_0$ is a decreasing function of the advection coefficients which offers insights into why schistosomiasis is more serious in regions with slow water flows.