Global dynamics of discrete mathematical models of tuberculosis.

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional $R_0$ which is based on the disease-free equilibrium, and a new net reproduction number $R_0\left(E^{\ast }\right)$ based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if $R_0\le 1$ and unstable if $R_0>1$. Moreover, the endemic equilibrium is locally asymptotically stable if $R_0\left(E^{\ast }\right)<1<R_0$.