Global stability of an SI epidemic model with feedback controls in a patchy environment

In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n(n ≥ 2) patches. We derive the basic reproduction number R₀ and prove that the disease-free equilibrium is globally asymptotically stable if R₀ ≤ 1. In the case of R₀ > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results.