Discrete Hepatitis C virus model with local dynamics, chaos and bifurcations

Abdul Qadeer Khan; Ayesha Yaqoob; Ateq Alsaadi; Abdul Qadeer Khan; Ayesha Yaqoob; Ateq Alsaadi

doi:10.3934/math.20241390

AIMS Mathematics

2024, Volume 9, Issue 10: 28643-28670. doi: 10.3934/math.20241390

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Research article

Discrete Hepatitis C virus model with local dynamics, chaos and bifurcations

1.
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan
2.
Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia

Received: 10 July 2024 Revised: 08 August 2024 Accepted: 14 August 2024 Published: 09 October 2024
MSC : 35B35, 39A10, 40A05, 70K50, 92D25

Mathematical models play a crucial role in understanding the dynamics of epidemic diseases by providing insights into how they spread and be controlled. In biomathematics, mathematical modeling is a powerful tool for interpreting the experimental results of biological phenomena related to disease transmission, offering precise and quantitative insights into the processes involved. This paper focused on a discrete mathematical model of the Hepatitis C virus (HCV) to analyze its dynamical behavior. Initially, we examined the local dynamics at steady states, providing a foundation for understanding the system's stability under various conditions. We then conducted a detailed bifurcation analysis, revealing that the discrete HCV model undergoes a Neimark-Sacker bifurcation at the uninfected steady state. Notably, our analysis showed that no period-doubling or fold bifurcations occur at this state. Further investigation at the infected steady state demonstrated the presence of both period-doubling and Neimark-Sacker bifurcations, which are characterized using explicit criteria. By employing a feedback control strategy, we explored chaotic behavior within the HCV model, highlighting the complex dynamics that can arise under certain conditions. Numerical simulations were conducted to verify the theoretical results, illustrating the model's validity and applicability. From a biological perspective, the insights gained from this analysis enhance our understanding of HCV transmission dynamics and potential intervention strategies. The presence of Neimark-Sacker bifurcation at the uninfected steady state implies that small perturbations could lead to oscillatory behavior, which may correspond to fluctuations in the number of infections over time. This finding suggests that maintaining stability at this steady state is critical for preventing outbreaks. The period-doubling and Neimark-Sacker bifurcations at the infected steady state indicate the potential for more complex oscillatory patterns, which could represent persistent cycles of infection and remission in a population. Finally, exploration of chaotic dynamics through feedback control highlights the challenges in predicting disease spread and the need for careful management strategies to avoid chaotic outbreaks.
- HCV model,
- bifurcations and hybrid control,
- numerical simulation,
- center manifold theorem
Citation: Abdul Qadeer Khan, Ayesha Yaqoob, Ateq Alsaadi. Discrete Hepatitis C virus model with local dynamics, chaos and bifurcations[J]. AIMS Mathematics, 2024, 9(10): 28643-28670. doi: 10.3934/math.20241390

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Abstract

Mathematical models play a crucial role in understanding the dynamics of epidemic diseases by providing insights into how they spread and be controlled. In biomathematics, mathematical modeling is a powerful tool for interpreting the experimental results of biological phenomena related to disease transmission, offering precise and quantitative insights into the processes involved. This paper focused on a discrete mathematical model of the Hepatitis C virus (HCV) to analyze its dynamical behavior. Initially, we examined the local dynamics at steady states, providing a foundation for understanding the system's stability under various conditions. We then conducted a detailed bifurcation analysis, revealing that the discrete HCV model undergoes a Neimark-Sacker bifurcation at the uninfected steady state. Notably, our analysis showed that no period-doubling or fold bifurcations occur at this state. Further investigation at the infected steady state demonstrated the presence of both period-doubling and Neimark-Sacker bifurcations, which are characterized using explicit criteria. By employing a feedback control strategy, we explored chaotic behavior within the HCV model, highlighting the complex dynamics that can arise under certain conditions. Numerical simulations were conducted to verify the theoretical results, illustrating the model's validity and applicability. From a biological perspective, the insights gained from this analysis enhance our understanding of HCV transmission dynamics and potential intervention strategies. The presence of Neimark-Sacker bifurcation at the uninfected steady state implies that small perturbations could lead to oscillatory behavior, which may correspond to fluctuations in the number of infections over time. This finding suggests that maintaining stability at this steady state is critical for preventing outbreaks. The period-doubling and Neimark-Sacker bifurcations at the infected steady state indicate the potential for more complex oscillatory patterns, which could represent persistent cycles of infection and remission in a population. Finally, exploration of chaotic dynamics through feedback control highlights the challenges in predicting disease spread and the need for careful management strategies to avoid chaotic outbreaks.

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