Analysis of a diffusive vector-borne disease model with nonlinear incidence and nonlocal delayed transmission

In this paper, we formulate a nonlocal and time-delayed reaction-diffusion vector-borne model with the saturated incidence with respect to infectious terms and derive the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{0}$$\end{document}, which serves as a threshold parameter that predicts whether the vector-borne disease will spread. Furthermore, the global attractivity of the endemic equilibrium is obtained by Lyapunov functionals if all the parameters are positive constants. Also, we can get the globally asymptotical stability for the corresponding time-delayed ODE system. In the case of an unbounded spatial habitat, we show the existence and non-existence of travelling wave solutions for the spatially homogeneous model.