Research Article

Year 2024,
Volume: 4 Issue: 3, 256 - 279, 30.09.2024
### Abstract

### References

- [1] Daley, D.J. and Gani, J.M. Epidemic Modelling: An Introduction. Cambridge University Press: New York, (1999).
- [2] Duarte, J., Januário, C., Martins, N., Ramos, C.C., Rodrigues, C. and Sardanyés, J. Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells. Numerical Algorithms, 77, 261–88, (2018).
- [3] Boshoff, C. and Weiss, R. AIDS-related malignancies. Nature Reviews Cancer, 2, 373–382, (2002).
- [4] Callaway, D.S. and Perelson, A.S. HIV-1 infection and low steady state viral loads. Bulletin of Mathematical Biology, 64(1), 29–64, (2002).
- [5] Naik, P.A., Yeolekar, B.M., Qureshi, S., Yeolekar, M. and Madzvamuse, A. Modeling and analysis of the fractional-order epidemic model to investigate mutual influence in HIV/HCV co-infection. Nonlinear Dynamics, 112, 11679–11710, (2024).
- [6] Saeed, T., Djeddi, K., Guirao, J.L., Alsulami, H.H. and Alhodaly, M.S. A discrete dynamics approach to a tumor system. Mathematics, 10(10), 1774, (2022).
- [7] Rogosin, S. and Dubatovskaya, M. Fractional calculus in Russia at the end of XIX century. Mathematics, 9(15), 1736, (2021).
- [8] Miller, K.S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience: New York, (1993).
- [9] Samko, S.G., Kilbas, A.A. and Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers: Philadelphia, USA, (1993).
- [10] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier: Tokyo, (2006).
- [11] Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73-85, (2015).
- [12] Atangana, A. and Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science, 20(2), 763-769, (2016).
- [13] Debbouche, N., Ouannas, A., Grassi, G., Al-Hussein, A.B.A., Tahir, F.R., Saad, K.M. and Aly, A.A. Chaos in cancer tumor growth model with commensurate and incommensurate fractional-order derivatives. Computational and Mathematical Methods in Medicine, 2022, 5227503, (2022).
- [14] Chamgoué, A.C., Ngueuteu, G.S.M., Yamapi, R. and Woafo, P. Memory effect in a self-sustained birhythmic biological system. Chaos, Solitons & Fractals, 109, 160-169, (2018).
- [15] Gholami, M., Ghaziani, R.K. and Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41-47, (2022).
- [16] Naik, P.A., Yavuz, M., Qureshi, S., Zu, J. and Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135, 795, (2020).
- [17] Naik, P.A. Global dynamics of a fractional-order SIR epidemic model with memory. International Journal of Biomathematics, 13(08), 2050071, (2020).
- [18] Yapışkan, D. and Eroğlu, B.B.İ. Fractional-order brucellosis transmission model between interspecies with a saturated incidence rate. Bulletin of Biomathematics, 2(1), 114-132, (2024).
- [19] Atede, A.O., Omame, A. and Inyama, S.C. A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110, (2023).
- [20] Omame, A., Onyenegecha, I.P., Raezah, A.A. and Rihan, F.A. Co-dynamics of COVID-19 and viral hepatitis B using a mathematical model of non-integer order: impact of vaccination. Fractal and Fractional, 7(7), 544, (2023).
- [21] Nwajeri, U.K., Omame, A. and Onyenegecha, C.P. Analysis of a fractional order model for HPV and CT co-infection. Results in Physics, 28, 104643, (2021).
- [22] Omame, A. and Zaman, F.D. Analytic solution of a fractional order mathematical model for tumour with polyclonality and cell mutation. Partial Differential Equations in Applied Mathematics, 8, 100545, (2023).
- [23] Munir, S., Omame, A. and Zaman, F.D. Mathematical analysis of a time-fractional coupled tumour model using Laplace and finite Fourier transforms. Physica Scripta, 99(2), 025241, (2024).
- [24] Holm, M. The Theory of Discrete Fractional Calculus: Development and Application. Ph.D. Thesis, Department of Mathematics, The University of Nebraska-Lincoln, (2011). [https://digitalcommons.unl.edu/mathstudent/27/]
- [25] Podlubny, I. Fractional Differential Equations (Vol. 198). Academic Press: San Diego, (1999).
- [26] Diaz, J.B. and Osler, T.J. Differences of fractional order. Mathematics of Computation, 28(125), 185-202, (1974).
- [27] Atici, F.M. and Eloe, P. Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, 3, 1-12, (2009).
- [28] Abdeljawad, T. On Riemann and Caputo fractional differences. Computers & Mathematics with Applications, 62(3), 1602-1611, (2011).
- [29] Abdeljawad, T., Baleanu, D., Jarad, F. and Agarwal, R.P. Fractional sums and differences with binomial coefficients. Discrete Dynamics in Nature and Society, 2013, 104173, (2013).
- [30] Andreichenko, K.P., Smarun, A.B. and Andreichenko, D.K. Dynamical modelling of linear discrete-continuous systems. Journal of Applied Mathematics and Mechanics, 64(2), 177-188, (2000).
- [31] Goodrich, C. and Peterson, A.C. Discrete Fractional Calculus. Springer Cham: Switzerland, (2015).
- [32] Devaney, R. An Introduction to Chaotic Dynamical Systems. CRC Press: USA, (2003).
- [33] Strogatz, S.H. Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press: USA, (2018).
- [34] Anastassiou, G.A. Principles of delta fractional calculus on time scales and inequalities. Mathematical and Computer Modelling, 52(3-4), 556-566, (2010).
- [35] Cermák, J., Gyori, I. and Nechvátal, L. On explicit stability conditions for a linear fractional difference system. Fractional Calculus and Applied Analysis, 18, 651-672, (2015).
- [36] Lou, J., Ruggeri, T. and Tebaldi, C. Modeling cancer in HIV-1 infected individuals: equilibria, cycles and chaotic behavior. Mathematical Biosciences and Engineering, 3(2), 313-324, (2006).
- [37] Naik, P.A., Owolabi, K.M., Yavuz, M. and Zu, J. Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272, (2020).
- [38] Cafagna, D. and Grassi, G. Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization. Chinese Physics B, 24(8), 080502, (2015).
- [39] Wu, G.C. and Baleanu, D. Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps. Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 95-100, (2015).
- [40] Sun, K.H., Liu, X. and Zhu, C.X. The 0-1 test algorithm for chaos and its applications. Chinese Physics B, 19(11), 110510, (2010).
- [41] En-Hua, S., Zhi-Jie, C. and Fan-Ji, G. Mathematical foundation of a new complexity measure. Applied Mathematics and Mechanics, 26, 1188-1196, (2005).
- [42] Pincus, S.M. Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences, 88(6), 2297-2301, (1991).

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using $C_{0}$ complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper.

- [1] Daley, D.J. and Gani, J.M. Epidemic Modelling: An Introduction. Cambridge University Press: New York, (1999).
- [2] Duarte, J., Januário, C., Martins, N., Ramos, C.C., Rodrigues, C. and Sardanyés, J. Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells. Numerical Algorithms, 77, 261–88, (2018).
- [3] Boshoff, C. and Weiss, R. AIDS-related malignancies. Nature Reviews Cancer, 2, 373–382, (2002).
- [4] Callaway, D.S. and Perelson, A.S. HIV-1 infection and low steady state viral loads. Bulletin of Mathematical Biology, 64(1), 29–64, (2002).
- [5] Naik, P.A., Yeolekar, B.M., Qureshi, S., Yeolekar, M. and Madzvamuse, A. Modeling and analysis of the fractional-order epidemic model to investigate mutual influence in HIV/HCV co-infection. Nonlinear Dynamics, 112, 11679–11710, (2024).
- [6] Saeed, T., Djeddi, K., Guirao, J.L., Alsulami, H.H. and Alhodaly, M.S. A discrete dynamics approach to a tumor system. Mathematics, 10(10), 1774, (2022).
- [7] Rogosin, S. and Dubatovskaya, M. Fractional calculus in Russia at the end of XIX century. Mathematics, 9(15), 1736, (2021).
- [8] Miller, K.S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience: New York, (1993).
- [9] Samko, S.G., Kilbas, A.A. and Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers: Philadelphia, USA, (1993).
- [10] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier: Tokyo, (2006).
- [11] Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73-85, (2015).
- [12] Atangana, A. and Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science, 20(2), 763-769, (2016).
- [13] Debbouche, N., Ouannas, A., Grassi, G., Al-Hussein, A.B.A., Tahir, F.R., Saad, K.M. and Aly, A.A. Chaos in cancer tumor growth model with commensurate and incommensurate fractional-order derivatives. Computational and Mathematical Methods in Medicine, 2022, 5227503, (2022).
- [14] Chamgoué, A.C., Ngueuteu, G.S.M., Yamapi, R. and Woafo, P. Memory effect in a self-sustained birhythmic biological system. Chaos, Solitons & Fractals, 109, 160-169, (2018).
- [15] Gholami, M., Ghaziani, R.K. and Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41-47, (2022).
- [16] Naik, P.A., Yavuz, M., Qureshi, S., Zu, J. and Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135, 795, (2020).
- [17] Naik, P.A. Global dynamics of a fractional-order SIR epidemic model with memory. International Journal of Biomathematics, 13(08), 2050071, (2020).
- [18] Yapışkan, D. and Eroğlu, B.B.İ. Fractional-order brucellosis transmission model between interspecies with a saturated incidence rate. Bulletin of Biomathematics, 2(1), 114-132, (2024).
- [19] Atede, A.O., Omame, A. and Inyama, S.C. A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110, (2023).
- [20] Omame, A., Onyenegecha, I.P., Raezah, A.A. and Rihan, F.A. Co-dynamics of COVID-19 and viral hepatitis B using a mathematical model of non-integer order: impact of vaccination. Fractal and Fractional, 7(7), 544, (2023).
- [21] Nwajeri, U.K., Omame, A. and Onyenegecha, C.P. Analysis of a fractional order model for HPV and CT co-infection. Results in Physics, 28, 104643, (2021).
- [22] Omame, A. and Zaman, F.D. Analytic solution of a fractional order mathematical model for tumour with polyclonality and cell mutation. Partial Differential Equations in Applied Mathematics, 8, 100545, (2023).
- [23] Munir, S., Omame, A. and Zaman, F.D. Mathematical analysis of a time-fractional coupled tumour model using Laplace and finite Fourier transforms. Physica Scripta, 99(2), 025241, (2024).
- [24] Holm, M. The Theory of Discrete Fractional Calculus: Development and Application. Ph.D. Thesis, Department of Mathematics, The University of Nebraska-Lincoln, (2011). [https://digitalcommons.unl.edu/mathstudent/27/]
- [25] Podlubny, I. Fractional Differential Equations (Vol. 198). Academic Press: San Diego, (1999).
- [26] Diaz, J.B. and Osler, T.J. Differences of fractional order. Mathematics of Computation, 28(125), 185-202, (1974).
- [27] Atici, F.M. and Eloe, P. Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, 3, 1-12, (2009).
- [28] Abdeljawad, T. On Riemann and Caputo fractional differences. Computers & Mathematics with Applications, 62(3), 1602-1611, (2011).
- [29] Abdeljawad, T., Baleanu, D., Jarad, F. and Agarwal, R.P. Fractional sums and differences with binomial coefficients. Discrete Dynamics in Nature and Society, 2013, 104173, (2013).
- [30] Andreichenko, K.P., Smarun, A.B. and Andreichenko, D.K. Dynamical modelling of linear discrete-continuous systems. Journal of Applied Mathematics and Mechanics, 64(2), 177-188, (2000).
- [31] Goodrich, C. and Peterson, A.C. Discrete Fractional Calculus. Springer Cham: Switzerland, (2015).
- [32] Devaney, R. An Introduction to Chaotic Dynamical Systems. CRC Press: USA, (2003).
- [33] Strogatz, S.H. Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press: USA, (2018).
- [34] Anastassiou, G.A. Principles of delta fractional calculus on time scales and inequalities. Mathematical and Computer Modelling, 52(3-4), 556-566, (2010).
- [35] Cermák, J., Gyori, I. and Nechvátal, L. On explicit stability conditions for a linear fractional difference system. Fractional Calculus and Applied Analysis, 18, 651-672, (2015).
- [36] Lou, J., Ruggeri, T. and Tebaldi, C. Modeling cancer in HIV-1 infected individuals: equilibria, cycles and chaotic behavior. Mathematical Biosciences and Engineering, 3(2), 313-324, (2006).
- [37] Naik, P.A., Owolabi, K.M., Yavuz, M. and Zu, J. Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272, (2020).
- [38] Cafagna, D. and Grassi, G. Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization. Chinese Physics B, 24(8), 080502, (2015).
- [39] Wu, G.C. and Baleanu, D. Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps. Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 95-100, (2015).
- [40] Sun, K.H., Liu, X. and Zhu, C.X. The 0-1 test algorithm for chaos and its applications. Chinese Physics B, 19(11), 110510, (2010).
- [41] En-Hua, S., Zhi-Jie, C. and Fan-Ji, G. Mathematical foundation of a new complexity measure. Applied Mathematics and Mechanics, 26, 1188-1196, (2005).
- [42] Pincus, S.M. Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences, 88(6), 2297-2301, (1991).

There are 42 citations in total.

Primary Language | English |
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Subjects | Dynamical Systems in Applications |

Journal Section | Research Articles |

Authors | |

Publication Date | September 30, 2024 |

Submission Date | May 16, 2024 |

Acceptance Date | July 15, 2024 |

Published in Issue | Year 2024 Volume: 4 Issue: 3 |

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