Figure 1.
The plot shows the dynamics of the model (1), when $ \omega = 1 $
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October
2021, 14(10): 3557-3575.
doi: 10.3934/dcdss.2020429
Application of Caputo-Fabrizio derivative to a cancer model with unknown parameters
| 1. | Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia |
| 2. | Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt |
| 3. | Informetrics Research Group Ton Duc Thang University Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City, Vietnam |
The present work explore the dynamics of the cancer model with fractional derivative. The model is formulated in fractional type of Caputo-Fabrizio derivative. We analyze the chaotic behavior of the proposed model with the suggested parameters. Stability results for the fixed points are shown. A numerical scheme is implemented to obtain the graphical results in the sense of Caputo-Fabrizio derivative with various values of the fractional order parameter. Further, we show the graphical results in order to study that the model behave the periodic and quasi periodic limit cycles as well as chaotic behavior for the given set of parameters.
Mathematics Subject Classification:
Primary:26A33;Secondary:92J15.
Citation:
M. M. El-Dessoky, Muhammad Altaf Khan. Application of Caputo-Fabrizio derivative to a cancer model with unknown parameters. Discrete and Continuous Dynamical Systems - S,
2021, 14
(10)
: 3557-3575.
doi: 10.3934/dcdss.2020429
![]()
References:
| [1] |
A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp.
doi: 10.1051/mmnp/2018010. |
| [2] |
National cancer institute, https://www.cancer.gov/about-cancer/causes-prevention/genetics, Accessed, Nov, 2018, 1–3. |
| [3] |
M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 1-13.
|
| [4] |
D. Dingli, M. D. Cascino, K. Josic, S. J. Russell and Z. Bajzer,
Mathematical modeling of cancer radiovirotherapy, Math Biosci., 199 (2006), 55-78.
doi: 10.1016/j.mbs.2005.11.001. |
| [5] |
M. M. El-Dessoky and M. A. Khan, Application of fractional calculus to combined modified function projective synchronization of different systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013107, 9 pp.
doi: 10.1063/1.5079955. |
| [6] |
A. El-Gohary and F. A. Bukhari,
Optimal control of stochastic prey-predator models, Appl. Math. Comput., 146 (2003), 403-415.
doi: 10.1016/S0096-3003(02)00592-1. |
| [7] |
A. El-Gohary,
Chaos and optimal control of cancer self-remission and tumor system steady states, Chaos, Solitons & Fractals, 37 (2008), 1305-16.
doi: 10.1016/j.chaos.2006.10.060. |
| [8] |
A. El-Gohary and A. S. Al-Ruzaiza,
Chaos and adaptive control in two prey, one predator system with nonlinear feedback, Chaos, Solitons & Fractals, 34 (2007), 443-453.
doi: 10.1016/j.chaos.2006.03.101. |
| [9] |
A. El-Gohary and I. A. Alwasel,
The chaos and optimal control of cancer model with complete unknown parameters, Chaos, Solitons and Fractals, 42 (2009), 2865-2874.
doi: 10.1016/j.chaos.2009.04.028. |
| [10] |
E. A. Gohary,
Optimal control of the genital herpes epidemic, Chaos, Solitons & Fractals, 12 (2001), 1817-1822.
|
| [11] |
M. A. Khan, S. Ullah and M. Farhan,
The dynamics of Zika virus with Caputo fractional derivative, Aims Mathematics, 4 (2019), 134-146.
doi: 10.3934/Math.2019.1.134. |
| [12] |
M. A. Khan, Neglecting nonlocality leads to unrealistic numerical scheme for fractional differential equation: Fake and manipulated results, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013144.
doi: 10.1063/1.5085661. |
| [13] |
Y. Kuang, J. D. Nagy and J. J. Elser,
Biological stoichiometry of tumor dynamics: Mathematical models and analysis, Discrete Continuous Dyn. Syst. Ser. B, 4 (2004), 221-240.
doi: 10.3934/dcdsb.2004.4.221. |
| [14] |
J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015) 87–92. |
| [15] |
J. E. Escalante-Martínez, J. F. Gómez-Aguilar, C. Calderø'n-Ramø'n, A. Aguilar-Meléndez and P. Padilla-Longoria, Synchronized bioluminescence behavior of a set of fireflies involving fractional operators of Liouville Caputo type, Int. J. Biomath., 11 (2018), 1850041, 25 pp.
doi: 10.1142/S1793524518500419. |
| [16] |
J. C. Misra and A. Mitra,
Synchronization among tumour-like cell aggregations coupled by quorum sensing: A theoretical study, Comput. Math. Appl., 55 (2008), 1842-1853.
doi: 10.1016/j.camwa.2007.06.027. |
| [17] |
V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khan and P. Agarwal,
Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Physica A: Statistical Mechanics and its Applications, 523 (2019), 48-65.
doi: 10.1016/j.physa.2019.02.018. |
| [18] |
G. S. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993. |
| [19] |
R. R. Sarkar and S. Banerjee,
Cancer and self-remsission and tumor stability, a stochastic approach, Math. Biosci., 196 (2005), 65-81.
doi: 10.1016/j.mbs.2005.04.001. |
| [20] |
J. E. Satulovky and T. Tome, Stochastic Lattice gas model for a predator-prey system, Phys. Rev. E, 49 (1994), 5073.
doi: 10.1103/PhysRevE.49.5073. |
| [21] |
J. Singh J, D. Kumar, M. A. Qurashi and D. Baleanu, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press, 1999. |
| [22] |
N. H. Sweilam, S. M. AL-Mekhlafi and D. Baleanu,
Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains, Journal of Advanced Research, 17 (2019), 125-137.
doi: 10.1016/j.jare.2019.01.007. |
| [23] |
S. Ullah, M. A. Khan and M. Farooq,
A fractional model for the dynamics of TB virus, Chao. Solit. Fract., 116 (2018), 63-71.
doi: 10.1016/j.chaos.2018.09.001. |
show all references
References:
| [1] |
A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp.
doi: 10.1051/mmnp/2018010. |
| [2] |
National cancer institute, https://www.cancer.gov/about-cancer/causes-prevention/genetics, Accessed, Nov, 2018, 1–3. |
| [3] |
M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 1-13.
|
| [4] |
D. Dingli, M. D. Cascino, K. Josic, S. J. Russell and Z. Bajzer,
Mathematical modeling of cancer radiovirotherapy, Math Biosci., 199 (2006), 55-78.
doi: 10.1016/j.mbs.2005.11.001. |
| [5] |
M. M. El-Dessoky and M. A. Khan, Application of fractional calculus to combined modified function projective synchronization of different systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013107, 9 pp.
doi: 10.1063/1.5079955. |
| [6] |
A. El-Gohary and F. A. Bukhari,
Optimal control of stochastic prey-predator models, Appl. Math. Comput., 146 (2003), 403-415.
doi: 10.1016/S0096-3003(02)00592-1. |
| [7] |
A. El-Gohary,
Chaos and optimal control of cancer self-remission and tumor system steady states, Chaos, Solitons & Fractals, 37 (2008), 1305-16.
doi: 10.1016/j.chaos.2006.10.060. |
| [8] |
A. El-Gohary and A. S. Al-Ruzaiza,
Chaos and adaptive control in two prey, one predator system with nonlinear feedback, Chaos, Solitons & Fractals, 34 (2007), 443-453.
doi: 10.1016/j.chaos.2006.03.101. |
| [9] |
A. El-Gohary and I. A. Alwasel,
The chaos and optimal control of cancer model with complete unknown parameters, Chaos, Solitons and Fractals, 42 (2009), 2865-2874.
doi: 10.1016/j.chaos.2009.04.028. |
| [10] |
E. A. Gohary,
Optimal control of the genital herpes epidemic, Chaos, Solitons & Fractals, 12 (2001), 1817-1822.
|
| [11] |
M. A. Khan, S. Ullah and M. Farhan,
The dynamics of Zika virus with Caputo fractional derivative, Aims Mathematics, 4 (2019), 134-146.
doi: 10.3934/Math.2019.1.134. |
| [12] |
M. A. Khan, Neglecting nonlocality leads to unrealistic numerical scheme for fractional differential equation: Fake and manipulated results, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013144.
doi: 10.1063/1.5085661. |
| [13] |
Y. Kuang, J. D. Nagy and J. J. Elser,
Biological stoichiometry of tumor dynamics: Mathematical models and analysis, Discrete Continuous Dyn. Syst. Ser. B, 4 (2004), 221-240.
doi: 10.3934/dcdsb.2004.4.221. |
| [14] |
J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015) 87–92. |
| [15] |
J. E. Escalante-Martínez, J. F. Gómez-Aguilar, C. Calderø'n-Ramø'n, A. Aguilar-Meléndez and P. Padilla-Longoria, Synchronized bioluminescence behavior of a set of fireflies involving fractional operators of Liouville Caputo type, Int. J. Biomath., 11 (2018), 1850041, 25 pp.
doi: 10.1142/S1793524518500419. |
| [16] |
J. C. Misra and A. Mitra,
Synchronization among tumour-like cell aggregations coupled by quorum sensing: A theoretical study, Comput. Math. Appl., 55 (2008), 1842-1853.
doi: 10.1016/j.camwa.2007.06.027. |
| [17] |
V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khan and P. Agarwal,
Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Physica A: Statistical Mechanics and its Applications, 523 (2019), 48-65.
doi: 10.1016/j.physa.2019.02.018. |
| [18] |
G. S. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993. |
| [19] |
R. R. Sarkar and S. Banerjee,
Cancer and self-remsission and tumor stability, a stochastic approach, Math. Biosci., 196 (2005), 65-81.
doi: 10.1016/j.mbs.2005.04.001. |
| [20] |
J. E. Satulovky and T. Tome, Stochastic Lattice gas model for a predator-prey system, Phys. Rev. E, 49 (1994), 5073.
doi: 10.1103/PhysRevE.49.5073. |
| [21] |
J. Singh J, D. Kumar, M. A. Qurashi and D. Baleanu, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press, 1999. |
| [22] |
N. H. Sweilam, S. M. AL-Mekhlafi and D. Baleanu,
Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains, Journal of Advanced Research, 17 (2019), 125-137.
doi: 10.1016/j.jare.2019.01.007. |
| [23] |
S. Ullah, M. A. Khan and M. Farooq,
A fractional model for the dynamics of TB virus, Chao. Solit. Fract., 116 (2018), 63-71.
doi: 10.1016/j.chaos.2018.09.001. |
Figure 2.
The plot shows the dynamics of the model (1), when $ \omega = 0.95 $
Figure 3.
The plot shows the dynamics of the model (1), when $ \omega=0.9 $
Figure 4.
The plot shows the dynamics of the model (1), when $ \omega=0.85 $
Figure 5.
The plot shows the dynamics of the model (1), when $ \omega = 0.5 $
Figure 6.
The plot shows the dynamics of the model (1), when $ \omega = 1 $
Figure 7.
The plot shows the dynamics of the model (1), when $ \omega=0.9 $
Figure 8.
The plot shows the dynamics of the model (1), when $ \omega = 0.8 $
Figure 9.
The plot shows the dynamics of the model (1), when $ \omega = 1 $
Figure 10.
The plot shows the dynamics of the model (1), when $ \omega=0.9 $
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