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In this paper, a mathematical model describing tuberculosis transmission with
incomplete treatment and continuous age structure for latently infected and
infectious individuals is investigated. It is assumed in the model that the treated
individuals may enter either the latent compartment due to the remainder of
Mycobacterium tuberculosis or the infectious compartment due to the treatment
failure. It is shown that the global transmission dynamics of the disease is fully
determined by the basic reproduction number. The asymptotic smoothness of the
semi-flow generated by the system is established. By analyzing the corresponding
characteristic equations, the local stability of a disease-free steady state and an
endemic steady state of the model is established. By using the persistence theory for
infinite dimensional system, the uniform persistence of the system is established
when the basic reproduction number is greater than unity. By means of suitable
Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic
reproduction number is less than unity, the disease-free steady state is globally
asymptotically stable; if the basic reproduction number is greater than unity, the
endemic steady state is globally asymptotically stable.