Backward Bifurcation in a Mathematical Model of Drug-Resistance Tuberculosis

Abstract

This study focuses on the proliferation of tuberculosis, a contagious condition caused by Bacillus Mycobacterium, with particular emphasis on its effects on drug-resistant individuals. Tuberculosis treatment typically lasts 6–8 months for newly infected individuals and can extend up to 2.5 years for patients with multidrug resistance. Despite decades of research, the widespread use of a vaccine, and the seeming attempt by the WHO to support a single worldwide management approach in recent years, TB is the second most common infectious killer, behind COVID-19. A projected 10.8 million persons worldwide contracted tuberculosis (TB) in 2023, either latently or actively. The dynamics of tuberculosis transmission among the human population are examined using a mathematical model that considers two subgroups: primary infectious and drug-resistant individuals. The basic reproduction number was established, and the model underwent sensitivity analysis to identify the primary variables affecting the disease's transmission. The findings of this analysis can aid in proposing effective intervention strategies. This study investigated the TB endemic equilibrium point and evaluated the global and local stability associated with the TB disease-free equilibrium point. First, we examined how the transmission rate impacted the model's backward bifurcation. According to our findings, the model experiences a backward bifurcation as the transmission rate increases. Complete eradication of TB becomes unattainable within the range of a scaling factor between 2.53 and 2.13. Second, we investigated the effect of the recovery rate among individuals who developed drug resistance on the backward bifurcation of the model. Our results show that the model exhibits a back bifurcation when the recovery rate rises. The bifurcation changes backward to forward when the transmission rate's scaling factor values go from 0.5 to 1. Stable disease-free equilibrium (DFE) and stable endemic equilibria coexist globally. Based on these observations, we conclude that the drug-resistance compartment is a more significant concern than the infected class, highlighting the need for vigilance among health workers and government agencies in monitoring this silent source of tuberculosis transmission.