This work examines the global stability of the disease equilibria of three tuberculosis mathematical models that considered the effect of case detection vis a vis the implementation of the direct observation therapy strategy, factors that enhances the case detection rate and effect of heterogeneity in susceptibility and disease progression. Both linear and non-linear Lyapunov functions are constructed and used to show that the disease-free equilibrium is globally asymptotically stable when the corresponding effective reproduction number is less than or equal to one. However, under some special cases where the disease-induced death is insignificant, the endemic equilibrium is globally asymptotically stable when the effective reproduction number is greater than one.
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Lyapunov functions and global properties of some tuberculosis models