Mathematical Analysis of an Infection Age-Structured Model for Tuberculosis Disease Dynamics Incorporating Control Measures

An infection-age-structured mathematical model for tuberculosis (TB) disease dynamics and treatment, vaccination and public health education campaign as control parameters is developed and analyzed. Here, both the latent and infectious classes are structured according to time and age-of-infection. An explicit threshold value for the effective reproduction number is obtained in terms of the demographic and epidemiological parameters of the model. Using the method of linearization, the disease-free equilibrium was found to be locally asymptotically stable when the effective reproduction number is less than unity. By constructing a suitable Lyapunov functional, the disease-free equilibrium state was found to be globally asymptotically stable whenever the associated effective reproduction number is less or equal to unity. This means that tuberculosis could be put under control in the population when the associated effective reproduction number is less than unity. Sensitivity analysis was also carried out on the effective reproduction number, in order to ascertain the parameters of the model that are most sensitive and that should be targeted by way of intervention strategies. It was found out that the, infection rate , recruitment number due to birth , treatment rates [both for latent and active TB ], vaccination and public health education campaign are the most sensitive parameters. From the effective reproduction number , the basic reproduction number and the reproduction numbers with single double and triple control strategies were obtained. Results from the numerical simulations of the various reproduction numbers revealed that, the more the application of the control strategies, the faster the control/eradication of TB is achieved