Classification of spatial dynamics of a vector–host epidemic model in advective heterogeneous environment

Abstract

In this paper, we propose and analyze a reaction–diffusion vector–host disease model with advection effect in an one-dimensional domain. We introduce the basic reproduction number (BRN) ${\Re }_0$ and establish the threshold dynamics of the model in terms of ${\Re }_0$. When there are no advection terms, we revisit the asymptotic behavior of ${\Re }_0$ w.r.t. diffusion rate and the monotonicity of ${\Re }_0$ under certain conditions. Furthermore, we obtain the asymptotic behavior of ${\Re }_0$ under the influence of advection effects. Our results indicate that when the advection rate is large enough relative to the diffusion rate, ${\Re }_0$ tends to be the value of local basic reproduction number (LBRN) at the downstream end, which enriches the asymptotic behavior results of the BRN in nonadvection heterogeneous environments. In addition, we explore the level set classification of ${\Re }_0$, that is, there exists a unique critical surface indicating that the disease-free equilibrium is globally asymptotically stable on one side of the surface, while it is unstable on the other side. Our results also reveal that the aggregation phenomenon will occur, namely, when the ratio of advection rate to diffusion rate is large enough, infected individuals will gather at the downstream end.