IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v129y2019icp25-39.html
   My bibliography  Save this article

Modelling immune systems based on Atangana–Baleanu fractional derivative

Author

Listed:
  • Al-khedhairi, A.
  • Elsadany, A.A.
  • Elsonbaty, A.

Abstract

Memory effects play a critical role in complex immune systems. In this paper, the recent efficient and realistic Atangana–Baleanu fractional order derivative, with non-local and non-singular kernel, was employed in two mathematical models for immune systems having multiple immune effectors. For each model, we derive the conditions under which a unique set of exact solutions exists. Stability analysis of equilibrium points of the two systems is carried out where the effects of model’s parameters and fractional derivatives are examined. Furthermore, a recent numerical scheme is utilized to solve each model numerically and to compare theoretical results with those of numerical experiments. Results depict that memory influences induce stabilization of immune systems such that the solution trajectories of the model always converge to either a single immune effector or a persistent immune effector/antigen equilibrium states.

Suggested Citation

  • Al-khedhairi, A. & Elsadany, A.A. & Elsonbaty, A., 2019. "Modelling immune systems based on Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 25-39.
    • Handle: RePEc:eee:chsofr:v:129:y:2019:i:c:p:25-39
    DOI: 10.1016/j.chaos.2019.07.053
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077919303054
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2019.07.053?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Hai Zhang & Daiyong Wu & Jinde Cao & Hui Zhang, 2014. "Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-8, April.
    2. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 285-294.
    3. Atangana, Abdon & Khan, Muhammad Altaf, 2019. "Validity of fractal derivative to capturing chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 50-59.
    4. Owolabi, Kolade M. & Gómez-Aguilar, J.F., 2018. "Numerical simulations of multilingual competition dynamics with nonlocal derivative," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 175-182.
    5. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "A new derivative with normal distribution kernel: Theory, methods and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 476(C), pages 1-14.
    6. Fatmawati, & Khan, Muhammad Altaf & Azizah, Muftiyatul & Windarto, & Ullah, Saif, 2019. "A fractional model for the dynamics of competition between commercial and rural banks in Indonesia," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 32-46.
    7. Owolabi, Kolade M. & Hammouch, Zakia, 2019. "Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1072-1090.
    8. Atangana, Abdon, 2018. "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 688-706.
    9. Ghanbari, Behzad & Gómez-Aguilar, J.F., 2018. "Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 114-120.
    10. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    11. Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Reyes-Reyes, J. & Adam-Medina, M., 2016. "Modeling diffusive transport with a fractional derivative without singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 467-481.
    12. Owolabi, Kolade M. & Atangana, Abdon, 2019. "Mathematical analysis and computational experiments for an epidemic system with nonlocal and nonsingular derivative," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 41-49.
    13. Ullah, Saif & Altaf Khan, Muhammad & Farooq, Muhammad, 2018. "A fractional model for the dynamics of TB virus," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 63-71.
    14. Coronel-Escamilla, A. & Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Guerrero-Ramírez, G.V., 2016. "Triple pendulum model involving fractional derivatives with different kernels," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 248-261.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kamal, F.M. & Elsonbaty, A. & Elsaid, A., 2021. "A novel fractional nonautonomous chaotic circuit model and its application to image encryption," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    2. Skwara, Urszula & Mozyrska, Dorota & Aguiar, Maira & Stollenwerk, Nico, 2024. "Dynamics of vector-borne diseases through the lens of systems incorporating fractional-order derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    3. Dalal Yahya Alzahrani & Fuaada Mohd Siam & Farah A. Abdullah, 2023. "Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives," Mathematics, MDPI, vol. 11(7), pages 1-21, April.
    4. Muñoz-Vázquez, Aldo Jonathan & Sánchez-Torres, Juan Diego & Defoort, Michael & Boulaaras, Salah, 2021. "Predefined-time convergence in fractional-order systems," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

    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. Owolabi, Kolade M., 2019. "Mathematical modelling and analysis of love dynamics: A fractional approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 849-865.
    2. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    3. Owolabi, Kolade M. & Pindza, Edson, 2019. "Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 146-157.
    4. Singh, C.S. & Singh, Harendra & Singh, Somveer & Kumar, Devendra, 2019. "An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1440-1448.
    5. Kumar, Sachin & Pandey, Prashant, 2020. "A Legendre spectral finite difference method for the solution of non-linear space-time fractional Burger’s–Huxley and reaction-diffusion equation with Atangana–Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    6. Yavuz, Mehmet & Bonyah, Ebenezer, 2019. "New approaches to the fractional dynamics of schistosomiasis disease model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 373-393.
    7. Wang, Wanting & Khan, Muhammad Altaf & Fatmawati, & Kumam, P. & Thounthong, P., 2019. "A comparison study of bank data in fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 369-384.
    8. Vázquez-Guerrero, P. & Gómez-Aguilar, J.F. & Santamaria, F. & Escobar-Jiménez, R.F., 2019. "Synchronization patterns with strong memory adaptive control in networks of coupled neurons with chimera states dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 167-175.
    9. Ma, Chaoqun & Ma, Zonggang & Xiao, Shisong, 2019. "A closed-form pricing formula for vulnerable European options under stochastic yield spreads and interest rates," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 59-68.
    10. Wei, Q. & Yang, S. & Zhou, H.W. & Zhang, S.Q. & Li, X.N. & Hou, W., 2021. "Fractional diffusion models for radionuclide anomalous transport in geological repository systems," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    11. Qureshi, Sania & Bonyah, Ebenezer & Shaikh, Asif Ali, 2019. "Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    12. Khan, Hasib & Gómez-Aguilar, J.F. & Khan, Aziz & Khan, Tahir Saeed, 2019. "Stability analysis for fractional order advection–reaction diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 737-751.
    13. Ishtiaq Ali & Sami Ullah Khan, 2023. "A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method," Mathematics, MDPI, vol. 11(6), pages 1-16, March.
    14. Khennaoui, Amina-Aicha & Ouannas, Adel & Bendoukha, Samir & Grassi, Giuseppe & Lozi, René Pierre & Pham, Viet-Thanh, 2019. "On fractional–order discrete–time systems: Chaos, stabilization and synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 150-162.
    15. Kubeka, Amos S. & Doungmo Goufo, Emile F. & Khumalo, Melusi, 2018. "On the quasi-normal modes of a Schwarzschild white hole for the lower angular momentum and perturbation by non-local fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 348-357.
    16. Shah, Syed Azhar Ali & Khan, Muhammad Altaf & Farooq, Muhammad & Ullah, Saif & Alzahrani, Ebraheem O., 2020. "A fractional order model for Hepatitis B virus with treatment via Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    17. Owolabi, Kolade M. & Karaagac, Berat, 2020. "Dynamics of multi-pulse splitting process in one-dimensional Gray-Scott system with fractional order operator," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    18. Gatabazi, P. & Mba, J.C. & Pindza, E., 2019. "Modeling cryptocurrencies transaction counts using variable-order Fractional Grey Lotka-Volterra dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 283-290.
    19. Pritam, Kocherlakota Satya & Sugandha, & Mathur, Trilok & Agarwal, Shivi, 2021. "Underlying dynamics of crime transmission with memory," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    20. Chatibi, Y. & El Kinani, E.H. & Ouhadan, A., 2019. "Variational calculus involving nonlocal fractional derivative with Mittag–Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 117-121.

    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:129:y:2019:i:c:p:25-39. 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.