Stability analysis of a tuberculosis epidemic model with nonlinear incidence rate and treatment effects

Abstract

We present a tuberculosis epidemic model with nonlinear incidence rates. The mathematical model consists of five variables that are susceptible, exposed, infectious, and recovered. Where infectious is divided into two categories, the first is latent infectious and the second categories is MDR (Resistant). The parameters on infectious describe the level of tuberculosis’s treatments are the treatment for the prevention of epidemic tuberculosis is by chemoprohylaxis for the the exposed individuals. Whereas treatment for infected individuals uses anti-tuberculosis drug theraphy with the directly observed treatment short course strategy(DOTS). The research method uses analytical (using the MAPLE) and numerical (using the MATLAB application) analysis. The steps in the analytical analysis include making a tuberculosis disease model, determining the point of equilibrium, and analyzing stability. Meanwhile, numerical analysis is used to explain the dynamic simulation of the spread of tuberculosis and the effectiveness of the treatment. The results of this research obtained are two equilibrium points (endemic and non-endemic) with a condition of conditional stability for each point. The stability will apply if the conditions proposed are met, namely local stability at a point of non-endemic equilibrium (ε ₀) is stable if ℜ₀ less than 1 and endemic equilibrium point (ε ^*) will be stable if ℜ₀ more than 1. From the results of analytic calculations and numerical simulations, by using Ruth-Hurwitz Method ℜ₀ = 0.312 at the non-endemic point and Centre Manifold method on endemic point is ℜ₀ = 0.312. So it can be concluded that the treatment on the first stage is more important to protect on TB spread.