Analysis of a mathematical model for tuberculosis with diagnosis

This work presents a new mathematical model that investigates the impact of diagnosis and treatment of both latent tuberculosis infections and active cases on the transmission dynamics of the disease in a population. Mathematical analysis reveal that the model undergoes the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exist with a stable endemic (positive) equilibrium when the associated reproduction number is less than unity. It is shown that this phenomenon does not exist in the absence of exogenous re-infection. In the absence of exogenous re-infection, the disease-free solution of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. It is further shown that a special case of the model has a unique endemic equilibrium point, which is globally asymptotically stable when the associated reproduction number exceeds unity. Uncertainty and sensitivity analysis is carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population using the reproduction number of the model, incidence of the disease and the total number of infected individuals in the various infective classes as output responses. The analysis shows that the top three parameters of the model that have the most influence on the reproduction number of the model are the transmission rate, the fraction of fast disease progression and the rate of detection of active TB cases, with other key parameters influencing the outcomes of the other output responses. Numerical simulations of the model show that the treatment rates for latent and active TB cases significantly determines the impact of the fraction of new latent TB cases diagnosed (and the fraction of active TB cases that promptly receives treatment) on the burden of the disease in a population. The simulations suggest that, with availability of treatment for both latent and active TB cases, increasing the fraction of latent TB cases that are diagnosed and treated (even with a small fraction of active TB cases promptly receiving treatment) will result in a reduction in the TB burden in the population.