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  4. Fractional derivative model for tumor cells and immune system competition

Fractional derivative model for tumor cells and immune system competition

MMC.
2023;
Volume 10, Number 2
: pp. 288–298
https://doi.org/10.23939/mmc2023.02.288
Received: August 14, 2022
Revised: January 31, 2023
Accepted: February 01, 2023

Mathematical Modeling and Computing, Vol. 10, No. 2, pp. 288–298 (2023)

Authors:
  1. M. Elkaf
  2. K. Allali
1
Laboratory of Mathematics, Computer science and Applications, Faculty of Sciences and Techniques Mohammedia, University Hassan-II Casablanca, Mohammedia, Morocco
2
Laboratory of Mathematics, Computer science and Applications, Faculty of Sciences and Techniques Mohammedia, University Hassan-II Casablanca, Mohammedia, Morocco

Modeling a dynamics of complex biologic disease such as cancer still present a complex dealing.  So, we try in our case to study it by considering the system of normal cells, tumor cells and immune response as mathematical variables structured in fractional-order derivatives equations which express the dynamics of cancer's evolution under immunity of the body.  We will analyze the stability of the formulated system at different equilibrium points.  Numerical simulations are carried out to get more helpful and specific outcome about the variations of the cancer's dynamics.

cancer modeling
immune response
fractional-order
stability
numerical solution
  1. Pearson–Stuttard J., Zhou B., Kontis V., Bentham J., Gunter M. J., Ezzati M.  Retracted: Worldwide burden of cancer attributable to diabetes and high body-mass index: a comparative risk assessment.  The Lancet Diabetes & Endocrinology.  6 (2),  95–104 (2018).
  2. Addi R. A., Benksim A., Cherkaoui M.  Vulnerability of people with cancer and the potential risks of COVID-19 Pandemic: A perspective in Morocco.  Signa Vitae.  16 (1), 207–208 (2020).
  3. Gabriel J. A.  The Biology of Cancer. John Wiley & Sons (2007).
  4. Solís–Pérez J., Gómez–Aguilar J., Atangana A.  A fractional mathematical model of breast cancer competition model.  Chaos, Solitons & Fractals.  127, 38–54 (2019).
  5. El Alami laaroussi A., El Hia M., Rachik M., Ghazzali R.  Analysis of a Multiple Delays Model for Treatment of Cancer with Oncolytic Virotherapy.  Computational and Mathematical Methods in Medicine.  2019, 1732815  (2019).
  6. Lai X., Friedman A.  Exosomal miRs in lung cancer: A mathematical model.  PLoS One.  11 (12), e0167706 (2016).
  7. Kang H.-W., Crawford M., Fabbri M., Nuovo G., Garofalo M., Nana-Sinkam S. P., Friedman A.  A mathematical model for microRNA in lung cancer.  PloS One.  8 (1), e53663 (2013).
  8. Danane J., Hammouch Z., Allali K., Rashid S., Singh J.  A fractional-order model of coronavirus disease 2019 (COVID-19) with governmental action and individual reaction.  Mathematical Methods in the Applied Sciences. 1–14 (2021).
  9. Danane J., Allali K., Hammouch Z.  Mathematical analysis of a fractional differential model of HBV infection with antibody immune response.  Chaos, Solitons & Fractals.  136, 109787 (2020).
  10. Kumar P., Erturk V. S., Yusuf A., Kumar S.  Fractional time-delay mathematical modeling of Oncolytic Virotherapy.  Chaos, Solitons & Fractals.  150, 111123 (2021).
  11. Pawar D. D., Patil W. D., Raut D. K.  Fractional-order mathematical model for analysing impact of quarantine on transmission of COVID-19 in India.  Mathematical Modeling and Computing.  8 (2), 253–266 (2021).
  12. Miller K. S., Ross B.  An Introduction to the Fractional Calculus and Fractional Differential Equations.  Wiley–Interscience (1993).
  13. Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M.  A fractional-order model for drinking alcohol behaviour leading to road accidents and violence.  Mathematical Modeling and Computing.  9 (3), 501–518 (2022).
  14. Allali K.  Stability analysis and optimal control of HPV infection model with early-stage cervical cancer.  Biosystems.  199, 104321 (2021).
  15. Bretti G., De Ninno A., Natalini R., Peri D., Roselli N.  Estimation Algorithm for a Hybrid PDE–ODE Model Inspired by Immunocompetent Cancer-on-Chip Experiment.  Axioms.  10 (4), 243 (2021).
  16. Paterson C., Clevers H., Bozic I.  Mathematical model of colorectal cancer initiation.  Proceedings of the National Academy of Sciences.  117 (34), 20681–20688 (2020).
  17. Fadugba S. E., Ali F., Abubakar A. B.  Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order.  Mathematical Modeling and Computing. 8 (3), 537–548 (2021).
  18. Özdemir N., Uçar E.  Investigating of an immune system-cancer mathematical model with Mittag–Leffler kernel.  AIMS Mathematics.  5 (2), 1519–1531 (2020).
  19. Amine S., Hajri Y., Allali K.  A delayed fractional-order tumor virotherapy model: Stability and Hopf bifurcation.  Chaos, Solitons & Fractals.  161, 112396 (2022).
  20. Alharbi S. A., Rambely A. S.  A New ODE-Based Model for Tumor Cells and Immune System Competition.  Mathematics.  8 (8), 1285 (2020).
  21. Lin W.  Global existence theory and chaos control of fractional differential equations.  Journal of Mathematical Analysis and Applications.  332 (1), 709–726 (2007).
  22. Matignon D.  Stability results for fractional differential equations with applications to control processing.  Computational Engineering in Systems Applications.  2, 963–968 (1996).
  23. Atangana A., Owolabi K. M.  New numerical approach for fractional differential equations.  Mathematical Modelling of Natural Phenomena.  13 (1), 3 (2018).
  24. Garrappa R.  On linear stability of predictor–corrector algorithms for fractional differential equations.  International Journal of Computer Mathematics.  87 (10), 2281–2290 (2010).