Global dynamics of a tuberculosis model with sensitivity of the smear microscopy

Abstract

Sputum smear microscopy and chest X-ray are the key TB diagnosis methods available in resource-constrained health settings of many developing countries worldwide. The test has moderate sensitivity towards the detection of pulmonary tuberculosis (PTB) cases. However, the undetected cases are also capable of transmitting the disease with a reduced transmission possibility. In this work, a five-dimensional compartmental model is formulated considering the infectivity of both smear-positive and negative individuals. The next-generation matrix method yields the expression of basic reproduction number . The global asymptotic stability of the disease-free equilibrium point for and that of endemic equilibrium point for are established with suitably constructed Lyapunov functions. The sensitivity indices of the associated parameters of are obtained with a suitable choice of parameter values. It has been found that neglecting the transmission capacity of smear-negative individuals underestimates the value of whereas ignoring the smear-negative compartment overestimates the same quantity.