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

Abdul Qadeer Khan; Raja Ramiz Ahmed Khan; Saud Fahad Aldosary; Abdul Qadeer Khan; Raja Ramiz Ahmed Khan; Saud Fahad Aldosary

doi:10.3934/math.2025762

AIMS Mathematics

2025, Volume 10, Issue 7: 16957-16993. doi: 10.3934/math.2025762

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Research article

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

1.
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan
2.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia

Received: 26 March 2025 Revised: 13 June 2025 Accepted: 23 June 2025 Published: 29 July 2025
MSC : 40A05, 70K50, 92D25

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.
- tuberculosis model,
- explicit criterions,
- numerical simulation,
- bifurcation
Citation: Abdul Qadeer Khan, Raja Ramiz Ahmed Khan, Saud Fahad Aldosary. Stability, chaos and bifurcations by explicit criterions of a discrete tuberculosis epidemic model[J]. AIMS Mathematics, 2025, 10(7): 16957-16993. doi: 10.3934/math.2025762

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Abstract

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.

References

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