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Volume 10,     Number 3,     Fall 2002

 

ANALYSIS OF A MODEL FOR TRANSMISSION DYNAMICS OF TB
S. M. MOGHADAS AND A. B. GUMEL

Abstract. A population model for the transmission dynamics of tuberculosis (TB) which incorporates a preventive vaccine and an effective therapeutic treatment is analysed qualitatively. The existence and uniqueness of the associated endemic equilibrium are discussed. By constructing a Lyapunov function, the global stability of the disease-free equilibrium of the model is established. The stability analysis of the diseasefree equilibrium reveals that the disease dynamics depends on a threshold quantity called the basic reproduction number (R0) such that the disease will be eradicated from the community if R0 < 1 and it persists if R0 > 1. The model is adapted and used to study the case where no preventive vaccine is administered. In addition to establishing the global stability of the vaccination-free model, the optimal treatment rate needed for disease eradication is determined. The vaccination-free model is then extended to monitor the effect of exogenous re-infection in the transmission dynamics of TB. Our analysis shows that the vaccination-free model with exogenous re-infection may undergo the phenomenon of bistability where a stable endemic equilibrium can co-exist with the stable disease-free equilibrium when R0 < 1. This shows that, unlike in the case without exogenous re-infection, reducing R0 to values less than unity may fail to eradicate the disease.

 

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