Nonlinear Dynamics and Hopf Bifurcation in a TB Model with Saturated Incidence and Limited Treatment for High-Risk Latent Cases

This study introduces and analyzes a complex tuberculosis vaccination model that includes a saturated incidence rate and treatment for high-risk latent individuals. The latent individuals are classified into early and late latent. The model also accounts for exogenous reinfection, relapse, and treatment failure. A comprehensive stability analysis of the equilibrium points is conducted. The global stability of both the Disease-Free Equilibrium (DFE) and the Endemic Equilibrium Point (EEP) is investigated by constructing appropriate Lyapunov functions. The forward and backward bifurcations are observed at reproduction number under certain conditions. The model also exhibits a Hopf bifurcation under specific, well-established conditions. By employing the normal form of the system, the first Lyapunov coefficient is calculated to identify the different kinds of Hopf bifurcations that may exist. Extensive numerical simulations are conducted to validate these findings. It is noticed that when the transmission rate and treatment saturation increase gradually, the unique EEP remains asymptotically stable within a certain range. However, as these parameters continue to rise, the EEP loses stability, leading to the emergence of periodic solutions through Hopf bifurcation. Further, increasing these parameters beyond the Hopf-bifurcation threshold restores the stability of the EEP, creating an interesting structure in the bifurcation diagram known as an endemic bubble. Furthermore, the occurrence of a subcritical Hopf bifurcation is observed, leading to the emergence of two concentric limit cycles. It is noted that when [Formula: see text], the system has two EEPs: one is always a saddle, while the other loses stability via a Hopf bifurcation. Additionally, the emergence of a generalized Hopf (Bautin) bifurcation is identified for the same set of parameters. For the Bautin bifurcation, two feasible pairs of Bautin points are present and the nature of the bifurcation is specified by calculating the second Lyapunov coefficient at the Bautin point.