
-
Previous Article
Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative
- DCDS-S Home
- This Issue
-
Next Article
Compactness results for linearly perturbed Yamabe problem on manifolds with boundary
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |










[1] |
Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021007 |
[2] |
Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021001 |
[3] |
Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020432 |
[4] |
Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269 |
[5] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021 doi: 10.3934/nhm.2021003 |
[6] |
Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 |
[7] |
Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020453 |
[8] |
Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020106 |
[9] |
Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180 |
[10] |
Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2021001 |
[11] |
Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020382 |
[12] |
Ravi Anand, Dibyendu Roy, Santanu Sarkar. Some results on lightweight stream ciphers Fountain v1 & Lizard. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020128 |
[13] |
Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231 |
[14] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289 |
[15] |
Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 |
[16] |
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 |
[17] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[18] |
Urszula Ledzewicz, Heinz Schättler. On the role of pharmacometrics in mathematical models for cancer treatments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 483-499. doi: 10.3934/dcdsb.2020213 |
[19] |
Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050 |
[20] |
Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020466 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]