Abstract
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.
Highlights
Tuberculosis (TB) is a bacterial disease caused by Mycobacterium tuberculosis, and is usually acquired through airborne infection from active TB cases [ ]
A complete mathematical analysis has been performed to show that the global dynamics of system ( . ) with boundary conditions ( . ) is completely determined by the basic reproduction number
By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, it has been shown that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable and the disease dies out; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease persists
Summary
Tuberculosis (TB) is a bacterial disease caused by Mycobacterium tuberculosis, and is usually acquired through airborne infection from active TB cases [ ]. According to the World Health Organization, one third of the world’s population is infected, either latently or actively, with tuberculosis. Tuberculosis infection remains a leading cause of death from an infectious disease [ ]. It is well known that, for tuberculosis, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class. This recurrence of disease is an important feature of tuberculosis, including human and bovine [ , ], and herpes [ , ]. Mathematical models of tuberculosis have contributed to the understanding of tubercu-
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