Transmission of Tuberculosis and Reproduction Number Using Lyapunov Function of SIR Model

This paper studies the mechanism of transmission of Tuberculosis (TB) caused by the mycobacterium tuberculosis bacteria. TB is an airborne disease that endangering human population globally. Mathematical modelling of tuberculosis was developed using the Susceptible-Infected-Recovered (SIR) model in this analysis. The model generated by the four-dimensional, nonlinear dynamic system is reduced to three dimensions, with some assumptions. Then, the model will be analyzed by creating a mathematical theorem that will prove the existence of a TB event, the disease-free equilibrium phase, and the TB endemic disease stage. By using the Lyapunov function method, the three theorems can be proved. The basic reproduction number R0 also can be obtained from the model. If the basic reproduction number R0 ≤ 1, then the disease-free equilibrium is global asymptotically stable and if R0 > 1, then the endemic equilibrium is asymptotically stable globally. Carrying out a simulation using data in a selected region, the result of the model successfully describes and forecast the number of TB cases and it may also be used for assessing the TB disease status