Global stability of a two-patch cholera model with fast and slow transmissions

A two-patch model for a waterborne disease, such as cholera, is considered, with the aim of investigating the impact of human population movements between two cities (patches). We derive the reproduction number R0, which depends on human movement rates. It is shown that the disease-free equilibrium is globally asymptotically stable whenever R0≤1. Three types of equilibria are explored: boundary endemic equilibria (patch-1 disease-free equilibrium and patch-2 disease-free equilibrium); interior endemic equilibrium (both patches endemic). They depend on four threshold parameters. The global asymptotic stability of equilibria is established using Lyapunov functions that combine quadratic, Volterra-type and linear functions. The theory is supported by numerical simulations, which further suggest that the human movement can increase or reduce the spread of the disease in one patch.