Local Stability of the Effects of Early Detection and Treatment on the Dynamics of Tuberculosis Using Lyapunov Function Method

Abstract

This study aims to use Lyapunov function method to build a SEIR model in the analysis of early detection and treatment. The SEIR model is a system of ordinary differential equations of six dimension developed from our compartment then building a mathematical theorem which guarantees the existence of a case of TB, the disease free-equilibrium and the total eradication of the disease from its host community that is disease endemic TB. Three theorems were proved using Lyapunov function method. With these, we concluded that in this research work have given a complete stability analysis of a tuberculosis model with two differential infectivity classes of early detected infected individual and late detected infected individual. By analysing this model, we found that it is locally asymptotically stable and possesses the only locally stable equilibrium state depending on the basic reproductive ratio R0 this steady state is either the endemic or the disease-free. The local stability of the infection-free equilibrium state implies that for an initial level of infection the disease will eventually fade out from the population when the condition for the stability, number R0 ≤1, hold. The condition R0>1, implies that the disease will persist in a population.