An SEITR Model for Tuberculosis Incorporating Migration, General Non-Linear Incidence, and Treatment Effectiveness

Tuberculosis (TB) remains a major global public health challenge, particularly in populations affected by migration and limited access to healthcare. In this study, we formulate and analyze a compartmental SEITR model (Susceptible–Exposed–Infectious–Treated–Recovered) for tuberculosis transmission, incorporating a general nonlinear incidence function of the form f1(S)f2(I)+f3(T). The model accounts for migration-related population influx and distinguishes between scenarios with and without influx into disease-related compartments. For the model with migration, we establish the existence and global asymptotic stability of a unique endemic equilibrium under suitable conditions. For the reduced model without direct influx into disease-related compartments, we derive the basic reproduction number R0 using the next-generation matrix method and establish the global asymptotic stability of the disease-free equilibrium for R0<1 and the endemic equilibrium for R0>1, using suitable Lyapunov functions and LaSalle’s invariance principle. We further employ the center manifold approach of Castillo-Chavez and Song to demonstrate a forward transcritical bifurcation at the threshold R0=1. The model explicitly accounts for incomplete and fully effective treatment, allowing us to investigate how treatment outcomes influence disease persistence and control. A global sensitivity analysis based on Sobol indices assesses the relative influence of model parameters on R0. Notably, we identify a critical treatment-efficacy threshold pc : when pc>1, even perfect treatment (p=1) cannot drive R0 below unity, implying that treatment alone cannot eliminate TB in high-burden regimes. Finally, numerical simulations investigate the effects of migration-related influx, bilinear versus saturated incidence, and treatment effectiveness on the epidemic dynamics. These findings contribute to the mathematical understanding of tuberculosis transmission and highlight the importance of treatment outcomes and migration in assessing disease control strategies.