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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

M. M. El-Dessoky 1,2, and  Muhammad Altaf Khan 3,,

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

* Corresponding author: Muhammad Altaf Khan (muhammad.altaf.khan@tdtu.edu.vn)

Received  November 2019 Revised  February 2020 Published  October 2021 Early access  September 2020

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.

Keywords: A new cancer model, Caputo-Fabrizio derivative, stability results, numerical results.
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 & 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.  Google Scholar

[2]

National cancer institute, https://www.cancer.gov/about-cancer/causes-prevention/genetics, Accessed, Nov, 2018, 1–3. Google Scholar

[3]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 1-13.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. A. Gohary, Optimal control of the genital herpes epidemic, Chaos, Solitons & Fractals, 12 (2001), 1817-1822.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015) 87–92. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. S. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

National cancer institute, https://www.cancer.gov/about-cancer/causes-prevention/genetics, Accessed, Nov, 2018, 1–3. Google Scholar

[3]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 1-13.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. A. Gohary, Optimal control of the genital herpes epidemic, Chaos, Solitons & Fractals, 12 (2001), 1817-1822.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015) 87–92. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. S. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The plot shows the dynamics of the model (1), when $ \omega = 1 $
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Figure 2.  The plot shows the dynamics of the model (1), when $ \omega = 0.95 $
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Figure 3.  The plot shows the dynamics of the model (1), when $ \omega=0.9 $
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Figure 4.  The plot shows the dynamics of the model (1), when $ \omega=0.85 $
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Figure 5.  The plot shows the dynamics of the model (1), when $ \omega = 0.5 $
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Figure 6.  The plot shows the dynamics of the model (1), when $ \omega = 1 $
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Figure 7.  The plot shows the dynamics of the model (1), when $ \omega=0.9 $
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Figure 8.  The plot shows the dynamics of the model (1), when $ \omega = 0.8 $
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Figure 9.  The plot shows the dynamics of the model (1), when $ \omega = 1 $
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Figure 10.  The plot shows the dynamics of the model (1), when $ \omega=0.9 $
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Figure 11.  The plot shows the dynamics of the model (1), when $ h = 0.2 $ and $ \omega = 0.8 $
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