Propagation direction of traveling waves for a class of bistable epidemic models

Traveling waves of a reaction–diffusion (RD) system connecting two spatially uniform stable equilibria are termed as bistable waves. Due to the uniqueness of a bistable wave in RD systems, it is difficult to determine its propagation direction, and there are very few analytical results on this subject. In this study, we propose an approach to give a complete characterization of the propagation direction of bistable waves for a class of bistable epidemic models arising from the spread of a cholera epidemic. Moreover, this characterization also gives a parameter threshold above which the epidemic disease eventually tends to extinction, and below which the epidemic outbreak happens.