Stability, chaos and bifurcations by explicit criterions of a discrete tuberculosis epidemic model

In this paper, we investigated the local stability, chaotic dynamics, and bifurcations of a discrete tuberculosis (TB) epidemic model, which provides crucial insights into the complex behavior of infectious disease spread in discrete-time settings. Specifically, we established the existence of a disease-free fixed point for all parametric values and demonstrated that an endemic fixed point emerges under specific parametric condition. The local stability at these fixed points was examined using the linear stability theory, revealing important thresholds for disease persistence or eradication. A comprehensive bifurcation analysis was conducted, showing that while no flip bifurcation occurred at the disease-free fixed point, both Neimark-Sacker and flip bifurcations took place at the endemic fixed point, and we studied said bifurcations by the explicit criterions. To further understand the model's nonlinear dynamics, we explored chaotic behavior by a feedback control strategy to mitigate chaotic oscillations. Numerical simulations are presented to validate our theoretical findings, demonstrating the model's capacity to capture real-world epidemiological patterns. Finally, theoretical results are also interpreted biologically/medically.