Mathematical modelling of TB with the effects of case detection and treatment

Abstract

In this paper a non-linear mathematical model of tuberculosis with case detection and treatment is proposed and analyzed. The whole population under consideration is divided into four compartments e.g. susceptible, exposed, infected and Recovered to study the transmission dynamics of the tuberculosis. Based on the immunity level, susceptible individuals move to exposed class or directly to infected class once they come into contact with an infective. This has been incorporated through progression rate which could be fast or slow. The equilibria of the model and the basic reproduction number $$\mathcal{R }_0$$ are computed. It is observed that the disease free equilibrium of the model is locally asymptotically stable when $$\mathcal{R }_0 < 1$$ . The model exhibits backward bifurcation under certain restriction on parameters, which gives rise to existence of multiple endemic equilibria for $$\mathcal{R }_0 < 1$$ . This suggests that an accurate estimation of parameters and the level of control measures are required to reduce the infection prevalence of TB in endemic region and just $$\mathcal{R }_0 < 1$$ is not enough to eliminate the disease from the population. $$\mathcal{R }_0$$ needs to be lowered much below one to confirm the global stability of the disease free equilibrium. Numerical simulation is performed to demonstrate the analytical results. It is found that the increase in the rate of case detection shifts the bacward bifurcation diagram towards right which leads to increase in the threshold value of $$\mathcal{R }_0$$ . It is also shown that the treatment is reducing the equilibrium level of infective population. Numerical simulations have been carried out to support the analytic results.