A spatial echinococcosis transmission model with time delays: Stability and traveling waves

Abstract

In this paper, we derive a time-delayed and diffusive echinococcosis transmission model. We first address the well-posedness to the initial-value problem for the model and give the basic reproduction number [Formula: see text]. In the case of a bounded spatial domain, we establish the local stability as well as the global stability of the disease-free and disease equilibria of the model. The methods to prove the local and the global stability are to analyze the corresponding characteristic equations and construct Lyapunov functionals, respectively. In the case of an unbounded spatial domain, by applying Schauder’s fixed point theorem and the limiting arguments, we show that when [Formula: see text], there exists a constant [Formula: see text] such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for [Formula: see text], and when [Formula: see text] and [Formula: see text], the model has no positive traveling wave solutions connecting them.