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Volume 17,     Number 1,     Spring 2009

 

MATHEMATICAL ANALYSIS OF A TB
TRANSMISSION MODEL WITH DOTS
S. O. ADEWALE, C. N. PODDER,
AND A. B. GUMEL

Abstract. The paper presents a deterministic model for the transmission dynamics of Mycobacterium tuberculosis (TB) in a population in the presence of Directly Observed Therapy Short-course (DOTS). The model, which allows for the detection and treatment of individuals with symptoms and uses standard incidence function for the infection rate, is rigorously analysed to gain insight into its dynamical features. The analysis reveals that the model undergoes a backward bifurcation, where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium when the associated reproduction threshold (Rd ) is less than unity. This phenomenon resulted due to the exogenous re-infection property of TB disease. It is shown that, in the absence of such re-infection, the model has a globally-asymptotically stable DFE when Rd is less than unity. Further, the model has a unique endemic equilibrium, for a special case, whenever the associated threshold quantity exceeds unity. This endemic equilibrium is shown to be globally-asymptotically stable, for a special case, using a non-linear Lyapunov function of Goh-Volterra type. The model provides a reasonable fit to a data set for the TB transmission data in Nigeria. Additional numerical simulations show that exogenous re-infection increases the cumulative number of new TB cases regardless of whether or not the DOTS program is implemented (but the corresponding cumulative number of new cases decreases for the case with DOTS program, in comparison to the case without DOTS).

 

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