Tuberculosis has been an endemic disease leading to high level of mortality. A model of TB epidemic incorporating the asymptomatic and symptomatic infection stages of individuals are studied and introducing treatments for these stages. The local behavior of the model is analyzed using linearization for non endemic equilibrium and using bifurcation at R0 = 1 for endemic equilibrium. Non endemic state and endemic state are proven locally asymptotically stable. Numerical simulations show the effectiveness of treatment in asymptomatic and symptomatic infection stages can reduce the rate of spread of TB.
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.
Tuberculosis (TB) is an infectious disease becoming a serious health problem that causes mortality and morbidity. A model of Tuberculosis spread by incorporating the effect of relapse and susceptible from recovery individuals, as well as treatment is studied. The existence of endemic equilibrium is shown through the basic reproduction ratio. The analysis results show that the non endemic equilibrium is global stable if the ratio value is less than unity, the endemic equilibrium the global stable if the ratio value is greater than unity. The results of study show that the relapse factor affects in the decrease of healthy individuals, and treatment of chemoprophylaxis is more significant in preventing the spread of TB disease compared to therapy.
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