Neimark-Sacker bifurcation, chaos, and local stability of a discrete Hepatitis C virus model

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

doi:10.3934/math.20241537

AIMS Mathematics

2024, Volume 9, Issue 11: 31985-32013. doi: 10.3934/math.20241537

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Research article

Neimark-Sacker bifurcation, chaos, and local stability of a discrete Hepatitis C virus model

1.
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan
2.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 15 August 2024 Revised: 18 October 2024 Accepted: 25 October 2024 Published: 11 November 2024
MSC : 92D25, 40A05, 70K50

In this paper, we explore the bifurcation, chaos, and local stability of a discrete Hepatitis C virus infection model. More precisely, we studied the local stability at fixed points of a discrete Hepatitis C virus model. We proved that at a partial infection fixed point, the discrete HCV model undergoes Neimark-Sacker bifurcation, but no other local bifurcation exists at this fixed point. Moreover, it was also proved that period-doubling bifurcation does not occur at liver-free, disease-free, and total infection fixed points. Furthermore, we also examined chaos control in the understudied discrete HCV model. Finally, obtained theoretical results were confirmed numerically.
- HCV model,
- chaos,
- bifurcations,
- numerical simulation,
- stability
Citation: Abdul Qadeer Khan, Ayesha Yaqoob, Ateq Alsaadi. Neimark-Sacker bifurcation, chaos, and local stability of a discrete Hepatitis C virus model[J]. AIMS Mathematics, 2024, 9(11): 31985-32013. doi: 10.3934/math.20241537

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Abstract

In this paper, we explore the bifurcation, chaos, and local stability of a discrete Hepatitis C virus infection model. More precisely, we studied the local stability at fixed points of a discrete Hepatitis C virus model. We proved that at a partial infection fixed point, the discrete HCV model undergoes Neimark-Sacker bifurcation, but no other local bifurcation exists at this fixed point. Moreover, it was also proved that period-doubling bifurcation does not occur at liver-free, disease-free, and total infection fixed points. Furthermore, we also examined chaos control in the understudied discrete HCV model. Finally, obtained theoretical results were confirmed numerically.

References

[1]	S. Pan, S. P. Chakrabarty, Hopf bifurcation and stability switches induced by humoral immune delay in hepatitis C, Indian J. Pure Ap. Mat., 51 (2020), 1673–1695. https://doi.org/10.1007/s13226-020-0489-2 doi: 10.1007/s13226-020-0489-2
[2]	A. Mojaver, H. Kheiri, Dynamical analysis of a class of hepatitis C virus infection models with application of optimal control, Int. J. Biomath., 9 (2016), 1650038. https://doi.org/10.1142/S1793524516500388 doi: 10.1142/S1793524516500388
[3]	M. Gümüş, K. Türk, Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme, J. Appl. Math. Comput., 70 (2024), 3767–3788. https://doi.org/10.1007/s12190-024-02103-6 doi: 10.1007/s12190-024-02103-6
[4]	H. Kong, G. Zhang, K. Wang, Stability and Hopf bifurcation in a virus model with self-proliferation and delayed activation of immune cells, Math. Biosci., 17 (2020), 4384–4405. https://doi.org/10.3934/mbe.2020242 doi: 10.3934/mbe.2020242
[5]	X. Jiang, J. Li, B. Li, W. Yin, L. Sun, X. Chen, Bifurcation, chaos, and circuit realisation of a new four-dimensional memristor system, Int. J. Nonlin. Sci. Num., 24 (2023), 2639–2648. https://doi.org/10.1515/ijnsns-2021-0393 doi: 10.1515/ijnsns-2021-0393
[6]	Q. Chen, B. Li, W. Yin, X. Jiang, X. Chen, Bifurcation, chaos and fixed-time synchronization of memristor cellular neural networks, Chaos Soliton. Fract., 171 (2023), 113440. https://doi.org/10.1016/j.chaos.2023.113440 doi: 10.1016/j.chaos.2023.113440
[7]	Q. He, P. Xia, C. Hu, B. Li, Public information, actual intervention and inflation expectations, Transform. Bus. Econ., 21 (2022), 644.
[8]	H. D. Qu, M. U. Rahman, M. Arfan, M. Salimi, S. Salahshour, A. Ahmadian, Fractal-fractional dynamical system of Typhoid disease including protection from infection, Eng. Comput., 39 (2023), 1553–1562. https://doi.org/10.1007/s00366-021-01536-y doi: 10.1007/s00366-021-01536-y
[9]	X. Zhu, P. Xia, Q. He, Z. Ni, L. Ni, Ensemble classifier design based on perturbation binary salp swarm algorithm for classification, Comput. Model. Eng. Sci., 135 (2023), 653–671. https://doi.org/10.32604/cmes.2022.022985 doi: 10.32604/cmes.2022.022985
[10]	M. Chong, M. Shahrill, L. Crossley, A. Madzvamuse, The stability analyses of the mathematical models of hepatitis C virus infection, Mod. Appl. Sci., 9 (2015), 250. https://doi.org/10.5539/mas.v9n3p250 doi: 10.5539/mas.v9n3p250
[11]	H. Dahari, A. Lo, R. M. Ribeiro, A. S. Perelson, Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy, J. Theor. Biol., 247 (2007), 371–381. https://doi.org/10.1016/j.jtbi.2007.03.006 doi: 10.1016/j.jtbi.2007.03.006
[12]	T. C. Reluga, H. Dahari, A. S. Perelson, Analysis of hepatitis C virus infection models with hepatocyte homeostasis, SIAM J. Appl. Math., 69 (2009), 999–1023. https://doi.org/10.1137/080714579 doi: 10.1137/080714579
[13]	E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, New York: Chapman and Hall/CRC, 2004. https://doi.org/10.1201/9781420037722
[14]	A. Wikan, Discrete dynamical systems with an introduction to discrete optimization problems, London, 2013. http://10.6.20.12:80/handle/123456789/30732
[15]	M. R. Kulenović, G. Ladas, Dynamics of second-order rational difference equations: With open problems and conjectures, New York: Chapman and Hall/CRC, 2001. https://doi.org/10.1201/9781420035384
[16]	E. Camouzis, G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, New York: Chapman and Hall/CRC, 2007. https://doi.org/10.1201/9781584887669
[17]	J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[18]	Y. A. Kuznetsov, Elements of applied bifurcation theorey, New York: Springer Science & Business Media, 2004. https://doi.org/10.1007/978-1-4757-3978-7
[19]	S. M. Rana, Chaotic dynamics and control of discrete ratio-dependent predator-prey system, Discrete Dyn. Nat. Soc., 2017 (2017), 4537450. https://doi.org/10.1155/2017/4537450 doi: 10.1155/2017/4537450
[20]	A. Rahman, D. Blackmore, Neimark-Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers, Chaos Soliton. Fract., 91 (2016), 339–349. https://doi.org/10.1016/j.chaos.2016.06.016 doi: 10.1016/j.chaos.2016.06.016
[21]	K. S. A. Basyouni, A. Q. Khan, Discrete-time predator-prey model with bifurcations and chaos, Math. Probl. Eng., 2020 (2020), 8845926. https://doi.org/10.1155/2020/8845926 doi: 10.1155/2020/8845926
[22]	H. N. Agiza, E. M. Elabbssy, H. E. Metwally, A. A. Elsadany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real, 10 (2009), 116–129. https://doi.org/10.1016/j.nonrwa.2007.08.029 doi: 10.1016/j.nonrwa.2007.08.029
[23]	X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Soliton. Fract., 32 (2007), 80–94. https://doi.org/10.1016/j.chaos.2005.10.081 doi: 10.1016/j.chaos.2005.10.081
[24]	P. Chakraborty, U. Ghosh, S. Sarkar, Stability and bifurcation analysis of a discrete prey-predator model with square-root functional response and optimal harvesting, J. Biol. Syst., 28 (2020), 91–110. https://doi.org/10.1142/S0218339020500047 doi: 10.1142/S0218339020500047
[25]	W. Liu, D. Cai, Bifurcation, chaos analysis and control in a discrete-time predator-prey system, Adv. Differ. Equ., 2019, (2019), 11. https://doi.org/10.1186/s13662-019-1950-6
[26]	A. Q. Khan, J. Ma, D. Xiao, Bifurcations of a two-dimensional discrete time plant-herbivore system, Commun. Nonlinear Sci., 39 (2016), 185–198. https://doi.org/10.1016/j.cnsns.2016.02.037 doi: 10.1016/j.cnsns.2016.02.037
[27]	A. Q. Khan, J. Ma, D. Xiao, Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect, J. Biol. Dyn., 11 (2017), 121–146. https://doi.org/10.1080/17513758.2016.1254287 doi: 10.1080/17513758.2016.1254287
[28]	E. Ott, C. Grebogi, J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196. https://doi.org/10.1103/PhysRevLett.64.1196 doi: 10.1103/PhysRevLett.64.1196
[29]	D. Blackmore, A. Rahman, J. Shah, Discrete dynamical modeling and analysis of the R-S flip-flop circuit, Chaos Soliton. Fract., 42 (2009), 951–963. https://doi.org/10.1016/j.chaos.2009.02.032 doi: 10.1016/j.chaos.2009.02.032
[30]	A. Rahman, Y. Joshi, D. Blackmore, Sigma map dynamics and bifurcations, Regul. Chaotic Dyn., 22 (2017), 740–749. https://doi.org/10.1134/S1560354717060107 doi: 10.1134/S1560354717060107
[31]	A. Rahman, D. Blackmore, Interesting bifurcations in walking droplet dynamics, Commun. Nonlinear Sci., 90 (2020), 105348. https://doi.org/10.1016/j.cnsns.2020.105348 doi: 10.1016/j.cnsns.2020.105348
[32]	Y. Joshi, D. Blackmore, A. Rahman, Generalized attracting horseshoes and chaotic strange attractors, arXiv Preprint, 2016, 1–8. https://doi.org/10.48550/arXiv.1611.04133
[33]	K. Murthy, I. Jordan, P. Sojitra, A. Rahman, D. Blackmore, Generalized attracting horseshoe in the Rössler attractor, Symmetry, 13 (2020), 30. https://doi.org/10.3390/sym13010030 doi: 10.3390/sym13010030

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