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

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 R-0. 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 R-0 > 1, there exists a constant c* > 0 such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for c > c*, and when R-0 > 1 and c < c*, the model has no positive traveling wave solutions connecting them.