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Home  >  Volume 27 (July 2014)

18. Some Global Properties of a Tuberculosis Mathematical Model Involving Case Detection and Waning Immunity by Daniel Okuonghae. Volume27, (July, 2014), pp 137 – 142
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This work is concerned with providing an insight into some global behaviours of a tuberculosis mathematical model that included the effect of case detection and waning immunity acquired after previous treatment of the disease, using density dependent incidence. Using appropriately formulated Lyapunov functions, we show that the disease free equilibrium (DFE) of the model is globally asymptotically stable whenever a threshold quantity larger than the reproduction number is less than unity. It is suggested that this bound on the reproduction number was due to the presence of exogenous re-infection which  further suggests the possibility of a backward bifurcation (whereby the DFE will co-exists with a stable endemic equilibrium) since the reproduction number will have to be less than the calculated threshold for the DFE to be globally stable. When the exogenous re-infection terms are removed and we assume that treatment confers long term immunity, the endemic equilibrium was shown to be globally asymptotically stable whenever it exists.