IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v195y2025ics0960077925002206.html
   My bibliography  Save this article

Deterministic and stochastic dynamic analysis of a Parkinson’s disease model

Author

Listed:
  • Bi, Yuanhong
  • Zhang, Xiaoqi
  • Liu, Quansheng

Abstract

Parkinson’s disease (PD) is closely related to high level of reactive oxygen species (ROS) and misfolded α-synaptic protein (αSYN∗). A deterministic model of ROS and αSYN∗ was proposed by Cloutier et al, who analyzed the effect of different stress signals on a switch from low level to high one for ROS. In this paper, we further investigate the existence and stability of a positive equilibrium of the deterministic model and derive the conditions on which the model experiences saddle–node bifurcation inducing a bistability with low and high levels. Then, a stochastic model of ROS and αSYN∗ is formulated through considering Gaussian white noise into the deterministic one. The existence of global unique positive solution is analyzed and sufficient conditions for the existence of stationary distribution are provided for the stochastic model. Furthermore, noise-induced transition between the bistability is explored through confidence ellipse for the same noise intensity and the average number of alternations between the bistability and the average dominance duration that the model spends on a stable steady state for different noise intensity. Our results reveal that ROS displays bistability with low and high levels under moderate stress. In the presence of noise, the decreasing of stress and the increasing of noise intensity easily induce the transition from high stable steady state to low one to relieve the disease. In addition, smaller stress is an important factor in suppressing the transition from low stable steady state to high one, which also can be prevented by decreasing noise intensity for larger stress. Therefore, disease state can switch to healthy state through regulating noise intensity. Our results may provide a new idea to control noise to alleviate PD through physical therapy.

Suggested Citation

  • Bi, Yuanhong & Zhang, Xiaoqi & Liu, Quansheng, 2025. "Deterministic and stochastic dynamic analysis of a Parkinson’s disease model," Chaos, Solitons & Fractals, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:chsofr:v:195:y:2025:i:c:s0960077925002206
    DOI: 10.1016/j.chaos.2025.116207
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925002206
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.116207?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Xu, Chaoqun & Yuan, Sanling & Zhang, Tonghua, 2018. "Sensitivity analysis and feedback control of noise-induced extinction for competition chemostat model with mutualism," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 891-902.
    2. Song, Yi & Xu, Wei, 2021. "Asymmetric Lévy noise changed stability in a gene transcriptional regulatory system," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    3. Din, Anwarud & Khan, Amir & Baleanu, Dumitru, 2020. "Stationary distribution and extinction of stochastic coronavirus (COVID-19) epidemic model," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    4. Zhao, Yongqiang & Tang, Yanbin, 2024. "Critical behavior of a semilinear time fractional diffusion equation with forcing term depending on time and space," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    5. Bashkirtseva, I. & Ryashko, L., 2020. "Analysis of noise-induced phenomena in the nonlinear tumor–immune system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    6. Zahra Faghani & Sajad Jafari & Chao-Yang Chen & Fahimeh Nazarimehr, 2020. "Investigating bifurcation points of neural networks: application to the epileptic seizure," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 93(12), pages 1-18, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yu, Haiyan & Liu, Quansheng & Bi, Yuanhong, 2023. "Lévy noise-induced phase transition in p53 gene regulatory network near bifurcation points," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    2. Zai-Yin He & Abderrahmane Abbes & Hadi Jahanshahi & Naif D. Alotaibi & Ye Wang, 2022. "Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity," Mathematics, MDPI, vol. 10(2), pages 1-18, January.
    3. Nikolaos P. Rachaniotis & Thomas K. Dasaklis & Filippos Fotopoulos & Platon Tinios, 2021. "A Two-Phase Stochastic Dynamic Model for COVID-19 Mid-Term Policy Recommendations in Greece: A Pathway towards Mass Vaccination," IJERPH, MDPI, vol. 18(5), pages 1-21, March.
    4. Singh, Harendra, 2021. "Analysis of drug treatment of the fractional HIV infection model of CD4+ T-cells," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    5. Saha, Pritam & Mondal, Bapin & Ghosh, Uttam, 2023. "Dynamical behaviors of an epidemic model with partial immunity having nonlinear incidence and saturated treatment in deterministic and stochastic environments," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    6. Mondal, Bapin & Mandal, Sayan & Tiwari, Pankaj Kumar & Upadhyay, Ranjit Kumar, 2025. "How predator harvesting affects prey-predator dynamics in deterministic and stochastic environments?," Applied Mathematics and Computation, Elsevier, vol. 498(C).
    7. Irina Bashkirtseva, 2021. "Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information," Mathematics, MDPI, vol. 9(24), pages 1-12, December.
    8. Mandal, Sayan & Sk, Nazmul & Tiwari, Pankaj Kumar & Chattopadhyay, Joydev, 2024. "Bistability in modified Holling II response model with harvesting and Allee effect: Exploring transitions in a noisy environment," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    9. Ogunmiloro, Oluwatayo Michael, 2021. "Mathematical analysis and approximate solution of a fractional order caputo fascioliasis disease model," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    10. Amouch, Mohamed & Karim, Noureddine, 2021. "Modeling the dynamic of COVID-19 with different types of transmissions," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    11. Din, Anwarud & Li, Yongjin & Yusuf, Abdullahi, 2021. "Delayed hepatitis B epidemic model with stochastic analysis," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    12. Yassine Sabbar & Asad Khan & Anwarud Din, 2022. "Probabilistic Analysis of a Marine Ecological System with Intense Variability," Mathematics, MDPI, vol. 10(13), pages 1-19, June.
    13. Xu, Chaoqun, 2020. "Probabilistic mechanisms of the noise-induced oscillatory transitions in a Leslie type predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    14. Parsamanesh, Mahmood & Erfanian, Majid, 2021. "Stability and bifurcations in a discrete-time SIVS model with saturated incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    15. Mengnan Chi & Wencai Zhao, 2019. "Dynamical Analysis of Two-Microorganism and Single Nutrient Stochastic Chemostat Model with Monod-Haldane Response Function," Complexity, Hindawi, vol. 2019, pages 1-13, March.
    16. Xuan Leng & Asad Khan & Anwarud Din, 2023. "Probability Analysis of a Stochastic Non-Autonomous SIQRC Model with Inference," Mathematics, MDPI, vol. 11(8), pages 1-18, April.
    17. Bashkirtseva, I. & Ryashko, L., 2019. "Stochastic sensitivity analysis of chaotic attractors in 2D non-invertible maps," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 78-84.
    18. Liu, Qun & Jiang, Daqing, 2020. "Threshold behavior in a stochastic SIR epidemic model with Logistic birth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    19. Hao, Mengli & Jia, Wantao & Wang, Liang & Li, Fuxiao, 2022. "Most probable trajectory of a tumor model with immune response subjected to asymmetric Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    20. Marsa Gholamzadeh & Hamidreza Abtahi & Reza Safdari, 2021. "Suggesting a framework for preparedness against the pandemic outbreak based on medical informatics solutions: a thematic analysis," International Journal of Health Planning and Management, Wiley Blackwell, vol. 36(3), pages 754-783, May.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:195:y:2025:i:c:s0960077925002206. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.