Dynamical analysis of a reaction–diffusion vector-borne disease model incorporating age-space structure and multiple transmission routes

Understanding the role of age structure and the spatial heterogeneity on disease spreading and vanishing is a vital question in the transmission of diseases. In this paper we construct a reaction–diffusion vector-borne disease model on a bounded domain subject to the no-flux boundary condition, with two novel features: age-space structure and multiple transmission routes. The contribution of mathematical analysis in this paper is two-fold: (i) in a bounded domain, by using the Banach fixed point theory to solve the existence of the local solution, and then guaranteeing the existence of the global solution through the eventual boundedness, and (ii) in a homogenous case, we establish the formula of the basic reproduction number, denoted by ℜ0, predicting whether the disease persist or not. In particular, by analyzing the characteristic equation and constructing new Lyapunov functionals, we achieve that the local and global stability of constant equilibria strictly depends on the sign of ℜ0−1. Our threshold results were verified by numerical simulations in one and two-dimensional bounded domain. This paper also provides a framework of theoretical analysis in dealing with age-space structure disease model.