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 diseasefree 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
vaccinationfree model, the optimal treatment rate needed for
disease eradication is determined. The vaccinationfree model
is then extended to monitor the effect of exogenous reinfection
in the transmission dynamics of TB. Our analysis shows that
the vaccinationfree model with exogenous reinfection may undergo
the phenomenon of bistability where a stable endemic
equilibrium can coexist with the stable diseasefree equilibrium
when R0 < 1. This shows that, unlike in the case without
exogenous reinfection, reducing R0 to values less than unity
may fail to eradicate the disease.
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