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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Chaos in a Discrete Cancer Model

Journal of Applied Nonlinear Dynamics 11(2) (2022) 297--308 | DOI:10.5890/JAND.2022.06.003

Kenneth Dukuza

University of Pretoria, Department of Mathematics and Applied Mathematics, P/Bag X 20, Pretoria, Republic

of South Africa

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Abstract

In this paper, we construct and analyse a discrete cancer mathematical model. Essential dynamic properties such as positivity and boundedness of solutions are discussed. Using the Lyapunov stability theorem, we prove that one of the tumor-free equilibria is globally asymptotically stable. Furthermore, the discrete model exhibits chaos for certain parameter values and this is supported by the existence of a positive Lyapunov exponent. Numerical simulations are performed to demonstrate our analytical results.

Acknowledgments

The authors acknowledge the University of Pretoria, Department of Mathematics and Applied Mathematics for the provision of research facilities.

References

  1. [1]  World health statistics (2018), Monitoring health for the SDGs, sustainable development goals. Geneva: World Health Organization; Licence: CC BY-NC-SA 3.0 IGO.
  2. [2] Ames, B.N., Gold, L.S., and Willett, W.C. (1995), The causes and prevention of cancer, Proceedings of the National Academy of Sciences, 92(12), 5258-5265.
  3. [3] Ames, B.N., Shigenaga, M.K., and Hagen, T.M. (1993), Oxidants, antioxidants, and the degenerative diseases of aging, Proceedings of the National Academy of Sciences, 90(17), 7915-7922.
  4. [4] Cohen, S.M., Purtilo, D.T., and Ellwein, L.B. (1991), Ideas in pathology. Pivotal role of increased cell proliferation in human carcinogenesis, Modern pathology: an official journal of the United States and Canadian Academy of Pathology, Inc, 4(3), 371-382.
  5. [5] Hollstein, M., Sidransky, D., Vogelstein, B., and Harris, C.C. (1991), p53 mutations in human cancers, Science, 253(5015), 49-53.
  6. [6] Vogelstein, B., Fearon, E.R., Kern, S.E., Preisinger, A.C., Nakamura, Y., and White, R. (1989), Allelotype of colorectal carcinomas, Science, 244(4901), 207-211.
  7. [7] Boveri, T. (2008), Concerning the origin of malignant tumours by Theodor Boveri. Translated and annotated by Henry Harris, Journal of cell science, 121(Supplement 1), 1-84.
  8. [8] Cantley, L.C., Dalton, W.S., DuBois, R.N., Finn, O.J., Futreal, P.A., Golub, T.R., Hait, W.N., Lozano, G., Maris, J.M., Nelson, W.G., and Sawyers, C.L. (2012), AACR Cancer progress report 2012, Clinical cancer research: an official journal of the American Association for Cancer Research, 18(Supplement 21), S1-S100.
  9. [9] Zakiryanova, G.K., Wheeler, S., and Shurin, M.R. (2018), Oncogenes in immune cells as potential therapeutic targets, ImmunoTargets and therapy, 7, 21.
  10. [10] Manchester, K.L. (1995), Theodor Boveri and the origin of malignant tumours, Trends in cell biology, 5(10), 384-387.
  11. [11] Tsygvintsev, A., Marino, S., and Kirschner, D.E. (2016), A mathematical model of gene therapy for the treatment of cancer, In Mathematical Methods and Models in Biomedicine, 367-385, Springer, New York, NY.
  12. [12] Yasir, M., Ahmad, S., Ahmed, F., Aqeel, M., and Akbar, M.Z. (2017), Improved numerical solutions for chaotic-cancer-model, AIP Advances, 7(1), 015110.
  13. [13] Burton, A., Billingham, L.J., and Bryan, S. (2007), Cost-effectiveness in clinical trials: using multiple imputation to deal with incomplete cost data, Clinical Trials, 4(2), 154-161.
  14. [14] Mery, B., Rancoule, C., Guy, J.B., Espenel, S., Wozny, A.S., Battiston-Montagne, P., Ardail, D., Beuve, M., Alphonse, G., Rodriguez-Lafrasse, C., and Magn, N. (2017), Preclinical models in HNSCC: A comprehensive review, Oral oncology, 65, 51-56.
  15. [15] Aranda, D.F., Trejos, D.Y., and Valverde, J.C. (2017), A discrete epidemic model for bovine Babesiosis disease and tick populations, Open Physics, 15(1), 360-369.
  16. [16] Garba, S.M., Gumel, A.B., and Lubuma, J.S. (2011), Dynamically-consistent non-standard finite difference method for an epidemic model, Mathematical and Computer Modelling, 53(1-2), 131-150.
  17. [17] Ndii, M.Z., Tambaru, D., and Djahi, B.S. (2019), A nonstandard finite difference scheme for water-related disease mathematical model, Appl Math Inf Sci, 13(4), 545-551.
  18. [18] Supriatna, A.K., Soewono, E., and van Gils, S.A. (2008), A two-age-classes dengue transmission model, Mathematical Biosciences, 216, 114-121 .
  19. [19] Singh, J.P. and Roy, B.K. (2016), A novel hyperchaotic system with stable and unstable line of equilibria and sigma shaped poincare map, IFAC-PapersOnLine, 49(1), 526-531.
  20. [20] Zhang, L. (2017), A novel 4-D butterfly hyperchaotic system, Optik, 131, 215-220.
  21. [21] Chan, M.H. and Kim, P.S. (2013), Modelling a Wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75(9) 1501-1523.
  22. [22] Sample, C., Fryxell, J.M., Bieri, J.A., Federico, P., Earl, J.E., Wiederholt, R., Mattsson, B.J., Flockhart, D.T., Nicol, S., Diffendorfer, J.E., and Thogmartin, W.E. (2018), A general modeling framework for describing spatially structured population dynamics, Ecology and Evolution, 8(1), 493-508.
  23. [23] Bunimovich-Mendrazitsky, S., Byrne, H., and L. Stone, L. (2008), Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70, 2055-2076.
  24. [24] de Pillis, L.G. and Radunskaya, A.E. (2003), A mathematical model of immune response to tumor invasion, Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of the Second M.I.T. Conference on Computational Fluid Dynamics and Solid Mechanics, Boston, 17-20 June 2003, USA, (Elsevier, Amsterdam), 1661-1668.
  25. [25] Itik, M. and Banks, S.P. (2010), Chaos in a three-dimensional cancer model, International Journal of Bifurcation and Chaos, 20(01), 71-79.
  26. [26] Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A., and Perelson, A.S. (1994), Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56, 295-321.
  27. [27]  Wei, H.C. (2018), A mathematical model of tumour growth with Beddington-DeAngelis functional response: a case of cancer without disease, Journal of biological dynamics, 12(1), 194-210.
  28. [28]  Borges, F.S., Iarosz, K.C., Ren, H.P., Batista, A.M., Baptista, M.S., Viana, R.L., Lopes, S.R., and Grebogi, C. (2014), Model for tumour growth with treatment by continuous and pulsed chemotherapy, Biosystems, 116, 43-48.
  29. [29]  Iarosz, K.C., Borges, F.S., Batista, A.M., Baptista, M.S., Siqueira, R.A., Viana, R.L., and Lopes, S.R. (2015), Mathematical model of brain tumour with glia–neuron interactions and chemotherapy treatment, Journal of theoretical biology, 368, 113-121.
  30. [30]  L{o}pez, {A}.G., Iarosz, K.C., Batista, A.M., Seoane, J.M., Viana, R.L., and Sanju{a}n, M.A. (2019), Nonlinear cancer chemotherapy: Modelling the Norton-Simon hypothesis, Communications in Nonlinear Science and Numerical Simulation, 70, 307-317.
  31. [31]  N'Doye, I., Voos, H., and Darouach, M. (2014), Chaos in a fractional-order cancer system, In 2014 European Control Conference (ECC), 171-176, IEEE.
  32. [32]  Anderson, A.R. and Chaplain, M.A.J. (1998), Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of mathematical biology, 60(5), 857-899.
  33. [33]  Kamel, D. (2019), Dynamics in a Discrete-Time Three Dimensional Cancer System, IAENG International Journal of Applied Mathematics, 49(4), 1-7.
  34. [34]  Lyu, J., Cao, J., Zhang, P., Liu, Y., and Cheng, H. (2016), Coupled hybrid continuum-discrete model of tumor angiogenesis and growth, PloS one, 11(10), e0163173.
  35. [35]  Nguyen Edalgo, Y.T., Zornes, A.L., and Ford Versypt, A.N. (2019), A hybrid discrete-continuous model of metastatic cancer cell migration through a remodeling extracellular matrix, AIChE Journal, 65(9), e16671.
  36. [36]  Solis, F.J. and Delgadillo, S.E. (2013), Discrete modeling of aggressive tumor growth with gradual effect of chemotherapy, Mathematical and Computer Modelling, 57(7-8), 1919-1926.
  37. [37]  Solis, F.J. and Delgadillo, S.E. (2009), Discrete mathematical models of an aggressive heterogeneous tumor growth with chemotherapy treatment, Mathematical and computer modelling, 50(5-6), 646-652.
  38. [38] Anguelov, R., Dumont, Y., Lubuma, J., and Mureithi, E. (2013), Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model, Mathematical Population Studies, 20(2), 101-122.
  39. [39] Kumar Verma, A. and Kayenat, S. (2019), On the stability of Micken's type NSFD schemes for generalized Burgers Fisher equation, Journal of Difference Equations and Applications, 25(12), 1706-1737.
  40. [40] Suryanto, A., Kusumawinahyu, W.M., Darti, I., and Yanti, I. (2013), Dynamically consistent discrete epidemic model with modified saturated incidence rate, Computational and Applied Mathematics, 32(2), 373-383.
  41. [41] Anguelov, R., Dukuza, K., and Lubuma, J.M.S. (2018), Backward bifurcation analysis for two continuous and discrete epidemiological models, Mathematical Methods in the Applied Sciences, 41(18), 8784-8798.
  42. [42] Hu, Z., Teng, Z., and Zhang, L. (2014), Stability and bifurcation analysis in a discrete SIR epidemic model, Mathematics and computers in Simulation, 97, 80-93.
  43. [43] Dang, Q.A. and Hoang, M.T. (2020), Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses, Journal of Computational and Applied Mathematics, 374, 112753.
  44. [44] Dang, Q.A. and Hoang, M.T. (2019), Complete global stability of a metapopulation model and its dynamically consistent discrete models, Qualitative theory of dynamical systems, 18(2), 461-475.
  45. [45] Memarbashi, R., Alipour, F., and Ghasemabadi, A. (2017), A nonstandard finite difference scheme for a SEI epidemic model, Punjab Univ. J. Math., 49, 133-147.
  46. [46] Hu, Z., Teng, Z., and Jiang, H. (2012), Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Analysis: Real World Applications, 13(5), 2017-2033.
  47. [47]  Banks, J., Brooks, J., Cairns, G., Davis, G., and Stacey, P. (1992), On Devaney's definition of chaos, The American mathematical monthly, 99(4), 332334.
  48. [48]  Effah-Poku, S., Obeng-Denteh, W., and Dontwi, I.K. (2018), A study of chaos in dynamical systems, Journal of Mathematics.
  49. [49]  Hunt, B.R. and Ott, E. (2015), Defining chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(9), 097618.
  50. [50]  Wang, L., Liu, H., and Gao, Y. (2013), Chaos for Discrete Dynamical System, Journal of Applied Mathematics.
  51. [51]  Ekstr\"{o}m, M., Vergo, M.T., Ahmadi, Z., and Currow, D.C. (2016), Prevalence of sudden death in palliative care: data from the Australian Palliative Care Outcomes Collaboration, Journal of pain and symptom management, 52(2), 221-227.
  52. [52]  Hui, D. (2015), Unexpected death in palliative care: what to expect when you are not expecting, Current opinion in supportive and palliative care, 9(4), 369.
  53. [53]  Mercadante, S., Ferrera, P., and Casuccio, A. (2016), Unexpected Death on an Acute Palliative Care Unit, Journal of pain and symptom management, 51(1), e1-e2.
  54. [54] Valle, P.A., Coria, L.N., Gamboa, D., and Plata, C. (2018), Bounding the dynamics of a chaotic-cancer mathematical model, Mathematical Problems in Engineering.
  55. [55] Groscurth, P. (1989), Cytotoxic effector cells of the immune system, Anatomy and embryology, 180(2), 109-119.
  56. [56] Moreno, E. (2008), Is cell competition relevant to cancer? Nature Reviews Cancer, 8(2), 141-147.
  57. [57] Paul, S. and Lal, G. (2017), The molecular mechanism of natural killer cells function and its importance in cancer immunotherapy, Frontiers in immunology, 8, 1124.
  58. [58] Anguelov, R. and Lubuma, J.M.S. (2001), Contributions to the mathematics of the nonstandard finite difference method and applications, Numerical Methods for Partial Differential Equations: An International Journal, 17(5), 518-543.
  59. [59] Mickens, R.E. (2007), Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations: An International Journal, 23(3), 672-691.
  60. [60] Mickens, R.E. (2005), Advances in the applications of nonstandard finite difference schemes, World Scientific.
  61. [61] Mickens, R.E. (2002), Nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications, 8(9), pp.823-847.
  62. [62] Wiggins, S. (2003), Introduction to applied nonlinear dynamical systems and chaos, 2, Springer Science \& Business Media.
  63. [63] Fadali, M.S. and Visioli, A. (2013), Digital control engineering: analysis and design, 2, Academic Press.
  64. [64] Wang, C., Fan, C., and Ding, Q. (2018), Constructing discrete chaotic systems with positive Lyapunov exponents, International Journal of Bifurcation and Chaos, 28(07), 1850084.
  65. [65]  Moreno, E. and Basler, K. (2004), dMyc transforms cells into super-competitors, Cell, 117(1), 117-129.
  66. [66]  Dang, C.V. (2012), MYC on the path to cancer, Cell, 149(1), 22-35.
  67. [67]  de la Cova, C., Abril, M., Bellosta, P., Gallant, P., and Johnston, L.A. (2004), Drosophila myc regulates organ size by inducing cell competition, Cell, 117(1), 107-116.
  68. [68]  Jak{o}bisiak, M., Lasek, W., and Go{\l}\c{a}b, J. (2003), Natural mechanisms protecting against cancer, Immunology letters, 90(2-3), 103-122.
  69. [69]  McMahon, S.B. (2014), MYC and the control of apoptosis, Cold Spring Harbor perspectives in medicine, 4(7), a014407. %
  70. [70]  Thevenon, D., Seffouh, I., Pillet, C., Crespo-Yanez, X., Fauvarque, M.O., and Taillebourg, E. (2020), A nucleolar isoform of the Drosophila ubiquitin specific protease dUSP36 regulates MYC-dependent cell growth, Frontiers in cell and developmental biology, 8, 506. %

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