A tuberculosis model with the impact of sputum smear microscopy

This paper proposes and analyzes a five-dimensional tuberculosis model incorporating slow-fast progression, endogenous reactivation, exogenous reinfection, and the assumption that infectives pass the smear microscopy test. This study disseminates information regarding how the presence and infectiousness of smear-negative patients in a tuberculosis epidemic significantly affect the threshold epidemic quantity \({\mathcal {R}_0}\) ${\mathcal{R}}_0$ (basic reproduction number). Stability and bifurcation analysis is carried out, and we find that the exogenous reinfection causes backward bifurcation and multiple endemic steady states in the system. We analytically and numerically explored the case of the periodic oscillations in the population via Hopf bifurcation for \({\mathcal {R}_0<1}\) ${\mathcal{R}}_0<1$ as well as \({\mathcal {R}_0>1}.\) ${\mathcal{R}}_0>1.$