Global stability for a tuberculosis model with isolation and incomplete treatment

Abstract

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with isolation and incomplete treatment. The model exhibits the traditional threshold behavior. We prove that when the basic reproductive number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproductive number is greater than unity, the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is locally asymptotically stable and globally asymptotically stable when the disease-induced death rate is equal to zero. The stability of disease-free equilibrium is derived by using Lyapunov stability theory and LaSalle’s invariant set theorem. The global stability of endemic equilibrium is proved by generalized Dulac–Bendixson criterion when the disease-induced death rate is equal to zero. Numerical simulations support our analytical results.