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Modeling the dynamics of tumor–immune cells interactions via fractional calculus

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Abstract

The immune response in the tumor micro-environment is a complicated biological phenomenon that needs to be investigated further. It is eminent that tumor is murderous and almost effects all the sectors of humans life. It is reported that patients, families, businesses, and community as a whole lose economic resources due to the treatment of cancer. These losses include financial loss, disease, a worse quality of life, and early death. Therefore, it is of great importance to investigate the dynamical behavior of tumor–immune cells interactions to point out the critical factors of the system to the researchers and health officials for the prevention of infection. Here, we structure the interactions of tumor–immune in the framework of fractional derivative. Moreover, we focused on the qualitative analysis and dynamical behavior of tumor–immune cells interactions in our work. The uniqueness and existence of the solution of the proposed model of tumor system are investigated through Banach’s and Schaefer’s fixed point theorem. We determine adequate conditions for the Ulam–Hyers stability of our proposed fractional model. The solution pathways are scrutinized with the help of a novel numerical scheme to highlight the impact of the parameters on the dynamics of tumor–immune. To be more specific, we highlighted the dynamical and chaotic behavior of the proposed system with the variation of fractional order and other parameters of the system. The most important aspects of tumor–immune cell dynamics are identified and recommended to the policymakers. The result of this dynamic is a powerful predictor of clinical success, and immune cell penetration into, or exclusion from, a tumor is also crucial to immunotherapy effectiveness. Hence, it is valuable to look into the dynamical behavior of tumor–immune cell interactions in order to alert policymakers and health authorities to the system’s critical components for management and prevention.

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Data Availability Statement

No new data were analyzed and created to this research study; therefore, data sharing is not applicable to this work.

References

  1. A. Lahrouz, L. Omari, A. Settati, A. Belmati, Comparison of deterministic and stochastic SIRS epidemic model with saturating incidence and immigration. Arabian J. Math. 4(2), 101–116 (2015)

    Article  MathSciNet  Google Scholar 

  2. R. Jan, Y. Xiao, Effect of partial immunity on transmission dynamics of dengue disease with optimal control. Math. Method. Appl. Sci. 42(6), 1967–1983 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  3. X. Zhou, Z. Guo, Analysis of an influenza A (H1N1) epidemic model with vaccination. Arabian J. Math. 1(2), 267–282 (2012)

    Article  MathSciNet  Google Scholar 

  4. P. Das, R.K. Upadhyay, P. Das, D. Ghosh, Exploring dynamical complexity in a time-delayed tumor-immune model. Chaos: Interdiscip. J. Nonlinear Sci. 30(12), 123118 (2020)

    Article  MathSciNet  Google Scholar 

  5. M.A. Chaplain, Multiscale mathematical modelling in biology and medicine. IMA J. Appl. Math. 76(3), 371–388 (2011)

    Article  MathSciNet  Google Scholar 

  6. R.B. Martin, M.E. Fisher, R.F. Minchin, K.L. Teo, A mathematical model of cancer chemotherapy with an optimal selection of parameters. Math. Biosci. 99(2), 205–230 (1990)

    Article  MathSciNet  Google Scholar 

  7. T.S. Deisboeck, Z. Wang, P. Macklin, V. Cristini, Multiscale cancer modeling. Annu. Rev. Biomed. Eng. 13, 127–155 (2011)

    Article  Google Scholar 

  8. A. d’Onofrio, F. Gatti, P. Cerrai, L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction. Math. Comput. Model. 51(5–6), 572–591 (2010)

    Article  MathSciNet  Google Scholar 

  9. C. Letellier, F. Denis, L.A. Aguirre, What can be learned from a chaotic cancer model? J. Theor. Biol. 322, 7–16 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. A.M.A. Rocha, M.F.P. Costa, E.M. Fernandes, On a multiobjective optimal control of a tumor growth model with immune response and drug therapies. Int. Trans. Oper. Res. 25(1), 269–294 (2018)

    Article  MathSciNet  Google Scholar 

  11. C. Tripathi, B.N. Tewari, R.K. Kanchan, K.S. Baghel, N. Nautiyal, R. Shrivastava, H. Kaur, M.L.B. Bhatt, S. Bhadauria, Macrophages are recruited to hypoxic tumor areas and acquire a pro-angiogenic M2-polarized phenotype via hypoxic cancer cell derived cytokines Oncostatin M and Eotaxin. Oncotarget 5(14), 5350 (2014)

    Article  Google Scholar 

  12. P. Italiani, D. Boraschi, From monocytes to M1/M2 macrophages: phenotypical vs. functional differentiation. Front. Immunol. 5, 514 (2014)

    Article  Google Scholar 

  13. D. Laoui, K. Movahedi, E. Van Overmeire, J. Van den Bossche, E. Schouppe, C. Mommer, A. Nikolaou, Y. Morias, P. De Baetselier, J.A. Van Ginderachter, Tumor-associated macrophages in breast cancer: distinct subsets, distinct functions. Int. J. Develop. Biol. 55, 861–867 (2011)

    Article  Google Scholar 

  14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations Elsevier (North-Holland, Mathematics studies, 2006)

  15. H.M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments. Kyungpook Math. J. 60(1), 73–116 (2020)

    MathSciNet  MATH  Google Scholar 

  16. M.S. Abdo, K. Shah, H.A. Wahash, S.K. Panchal, On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative. Chaos, Solitons & Fractals 135, 109867 (2020)

    Article  MathSciNet  Google Scholar 

  17. M. Mandal, S. Jana, S.K. Nandi, T.K. Kar, Modelling and control of a fractional-order epidemic model with fear effect. Energy, Ecol. Environ. 5(6), 421–432 (2020)

    Article  Google Scholar 

  18. M. Sher, K. Shah, Z.A. Khan, H. Khan, A. Khan, Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler power law. Alex. Eng. J. 59(5), 3133–3147 (2020)

    Article  Google Scholar 

  19. A. Ali, M. ur Rahman, M. Arfan, Z. Shah, P. Kumam, W. Deebani, Investigation of time-fractional SIQR Covid-19 mathematical model with fractal-fractional Mittage-Leffler kernel. Alexandria Engineering Journal (2022)

  20. K. Shah, M. Arfan, I. Mahariq, A. Ahmadian, S. Salahshour, M. Ferrara, Fractal-fractional mathematical model addressing the situation of corona virus in Pakistan. Results Phys. 19, 103560 (2020)

    Article  Google Scholar 

  21. Z. Lin, H. Wang, Modeling and application of fractional-order economic growth model with time delay. Fractal Fract. 5(3), 74 (2021)

    Article  Google Scholar 

  22. A. Dzielinski, D. Sierociuk, G. Sarwas, Some applications of fractional order calculus (Technical Sciences, Bulletin of the Polish Academy of Sciences, 2010), pp. 583–592

  23. F. Fatmawati, R. Jan, M.A. Khan, Y. Khan, S. Ullah, A new model of dengue fever in terms of fractional derivative. Math. Biosci. Eng.: MBE 17(5), 5267–5287 (2020)

    Article  MathSciNet  Google Scholar 

  24. H.M. Srivastava, R. Jan, A. Jan, W. Deebani, M. Shutaywi, Fractional-calculus analysis of the transmission dynamics of the dengue infection. Chaos: Interdiscip. J. Nonlinear Sci. 31(5), 053130 (2021)

    Article  MathSciNet  Google Scholar 

  25. S. Qureshi, R. Jan, Modeling of measles epidemic with optimized fractional order under Caputo differential operator. Chaos, Solitons Fractals 145, 110766 (2021)

    Article  MathSciNet  Google Scholar 

  26. Z. Shah, R. Jan, P. Kumam, W. Deebani, M. Shutaywi, Fractional dynamics of HIV with source term for the supply of New CD4+ T-cells depending on the viral load via Caputo-Fabrizio derivative. Molecules 26(6), 1806 (2021)

    Article  Google Scholar 

  27. A. Jan, R. Jan, H. Khan, M.S. Zobaer, R. Shah, Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination. Commun. Math. Biol. Neurosci., 2020, Article-ID

  28. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Elsevier, Amsterdam, 1998)

    MATH  Google Scholar 

  29. A. Granas, J. Dugundji, Elementary fixed point theorems. In Fixed Point Theory (pp. 9-84). Springer, New York, NY (2003)

  30. Hilfer, R. ed. Applications of fractional calculus in physics. World scientific (2020)

  31. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations (Vol. 204). Elsevier, Amsterdam

  32. V. Lakshmikantham, S. Leela, J.V. Devi, Theory of fractional dynamic systems. CSP (2009)

  33. Z.U. Khan, A. Ali, F.M. Khan, K. Shah, Existence theory and numerical simulation of ebola model corresponding to nonlocal fractional order derivative. Arab J. Basic Appl. Sci. 27(1), 375–388 (2020)

    Article  Google Scholar 

  34. T.Q. Tang, Z. Shah, E. Bonyah, R. Jan, M. Shutaywi, N. Alreshidi, Modeling and Analysis of Breast Cancer with Adverse Reactions of Chemotherapy Treatment through Fractional Derivative. Computational and Mathematical Methods in Medicine, (2022)

  35. S.M. Ullam, Problems in modern mathematics (Chapter VI) (Wiley, New York, 1940)

    Google Scholar 

  36. D.H. Hyers, On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27(4), 222 (1941)

    Article  ADS  MathSciNet  Google Scholar 

  37. T.M. Rassias, On the stability of the linear mapping in Banach spaces. Proceed. Am. Math. Soc. 72(2), 297–300 (1978)

    Article  MathSciNet  Google Scholar 

  38. Z. Ali, A. Zada, K. Shah, On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations. Bull. Malaysian Math. Sci. Soc. 42(5), 2681–2699 (2019)

    Article  MathSciNet  Google Scholar 

  39. A. Benkerrouche, M.S. Souid, S. Etemad, A. Hakem, P. Agarwal, S. Rezapour, S.K. Ntouyas, J. Tariboon, Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam-Hyers-Rassias Stability. Fractal Fract. 5(3), 108 (2021)

    Article  Google Scholar 

  40. P.S. Kim, J.J. Crivelli, I.K. Choi, C.O. Yun, J.R. Wares, Quantitative impact of immunomodulation versus oncolysis with cytokine-expressing virus therapeutics. Math. Biosci. Eng. 12(4), 841 (2015)

    Article  MathSciNet  Google Scholar 

Download references

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Funding is not applicable to this research work.

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Authors and Affiliations

  1. International Intercollegiate Ph.D. Program, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Tao-Qian Tang

  2. Department of Internal Medicine, E-Da Hospital, Kaohsiung, 82445, Taiwan

    Tao-Qian Tang

  3. School of Medicine, College of Medicine, I-Shou University, Kaohsiung, 82445, Taiwan

    Tao-Qian Tang

  4. Department of Family and Community Medicine, E-Da Hospital, Kaohsiung, 82445, Taiwan

    Tao-Qian Tang

  5. Department of Engineering and System Science, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Tao-Qian Tang

  6. Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat, 28420, KPK, Pakistan

    Zahir Shah

  7. Department of Mathematics, University of Swabi, Swabi, 23561, Pakistan

    Rashid Jan

  8. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ebraheem Alzahrani

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  1. Tao-Qian Tang
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Correspondence to Zahir Shah.

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Tang, TQ., Shah, Z., Jan, R. et al. Modeling the dynamics of tumor–immune cells interactions via fractional calculus. Eur. Phys. J. Plus 137, 367 (2022). https://doi.org/10.1140/epjp/s13360-022-02591-0

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