    ## Mathematical modeling approach to the fractional Bergman's model

 1 Facultad de Matemáticas. Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo, Guerrero, México 2 CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, México

* Corresponding authors: J. F. Gómez-Aguilar and M. A. Taneco-Hernández

Received  June 2018 Revised  August 2018 Published  March 2019

This paper presents the solution for a fractional Bergman's minimal blood glucose-insulin model expressed by Atangana-Baleanu-Caputo fractional order derivative and fractional conformable derivative in Liouville-Caputo sense. Applying homotopy analysis method and Laplace transform with homotopy polynomial we obtain analytical approximate solutions for both derivatives. Finally, some numerical simulations are carried out for illustrating the results obtained. In addition, the calculations involved in the modified homotopy analysis transform method are simple and straightforward.

Citation: Victor Fabian Morales-Delgado, José Francisco Gómez-Aguilar, Marco Antonio Taneco-Hernández. Mathematical modeling approach to the fractional Bergman's model. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020046
##### References:
  B. S. Alkahtani, O. J. Algahtani, R. S. Dubey and P. Goswami, The solution of modified fractional bergman's minimal blood glucose-insulin model, Entropy, 19 (2017), 114. doi: 10.3390/e19050114. A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, New York, 2016. doi: 10.1016/B978-0-08-100644-3.00001-5.  A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. A. Atangana and K. M. Owolabi., New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), 1-21. doi: 10.1051/mmnp/2018010.  A. Atangana, Blind in a commutative world: Simple illustrations with functions and chaotic attractors, Chaos, Solitons & Fractals, 114 (2018), 347-363. doi: 10.1016/j.chaos.2018.07.022.  A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056.  A. Atangana and J. F. Gómez Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 1-23.  A. Atangana and E. F. D. Goufo, On the mathematical analysis of Ebola hemorrhagic fever: Deathly infection disease in West African countries, BioMed Research International, 2014 (2014), Article ID 261383, 7 pages. doi: 10.1155/2014/261383. A. Atangana and B. S. T. Alkahtani, Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter, Complexity, 21 (2016), 442-451. doi: 10.1002/cplx.21704.  A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm Sci., 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. R. N. Bergman, Y. Z. Ider, C. R. Bowden and C. Cobelli, Quantitative estimation of insulin sensitivity, American Journal of Physiology-Endocrinology And Metabolism, 236 (1979), 667-677. doi: 10.1152/ajpendo.1979.236.6.E667. A. Caumo, C. Cobelli and M. Omenetto, Overestimation of minimal model glucose effectiveness in presence of insulin response is due to under modeling, American Journal of Physiology, 278 (1999), 481-488.  A. De Gaetano and O. Arino, Mathematical modelling of the intravenous glucose tolerance test, Journal of Mathematical Biology, 40 (2000), 136-168. doi: 10.1007/s002850050007.  L. C. Gatewood, E. Ackerman, J. W. Rosevear, G. D. Molnar and T. W. Burns, Tests of a mathematical model of the blood-glucose regulatory system, Computional Biomedical Research, 2 (1968), 1-14. doi: 10.1016/0010-4809(68)90003-7. A. Fabre and J. Hristov, On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity, Heat and Mass Transfer, 53 (2017), 177-204. doi: 10.1007/s00231-016-1806-5. J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, Thermal Science, 1 (2016), 115-115.  R. Jain, K. Arekar and R. Shanker Dubey, Study of Bergman's minimal blood glucose-insulin model by Adomian decomposition method, Journal of Information and Optimization Sciences, 38 (2017), 133-149. doi: 10.1080/02522667.2016.1187919.  F. Jarad, E. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Advances in Difference Equations, 2017 (2017), Paper No. 247, 16 pp. doi: 10.1186/s13662-017-1306-z.  R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002.  S. Kumar, A. Kumar and I. K. Argyros, A new analysis for the Keller-Segel model of fractional order, Numerical Algorithms, 75 (2017), 213-228. doi: 10.1007/s11075-016-0202-z.  S. Kumar, A new analytical modelling for telegraph equation via Laplace transform, Appl. Math. Modell, 38 (2014), 3154-3163. doi: 10.1016/j.apm.2013.11.035.  S. Kumar and M. M. Rashidi, New analytical method for gas dynamic equation arising in shock fronts, Comput. Phy. Commun, 185 (2014), 1947-1954. doi: 10.1016/j.cpc.2014.03.025.  G. A. Losa, On the fractal design in human brain and nervous tissue, Applied Mathematics, 5 (2014), 1725-1732. doi: 10.4236/am.2014.512165. V. F. Morales-Delgado, J. F. Gómez-Aguilar, S. Kumar and M. A. Taneco-Hernández, Analytical solutions of the Keller-Segel chemotaxis model involving fractional operators without singular kernel, The European Physical Journal Plus, 133 (2018), 200. doi: 10.1140/epjp/i2018-12038-6. Z. Odibat and A. S. Bataineh, An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: Construction of homotopy polynomials, Math. Meth. Appl. Sci, 38 (2015), 991-1000. doi: 10.1002/mma.3136.  K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317. doi: 10.1016/j.cnsns.2016.08.021.  K. M. Owolabi and A. Atangana, Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative, The European physical Journal Plus, 131 (2016), 335. doi: 10.1140/epjp/i2016-16335-8. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Academic Press, an Diego, California, USA, 1999. show all references

##### References:
  B. S. Alkahtani, O. J. Algahtani, R. S. Dubey and P. Goswami, The solution of modified fractional bergman's minimal blood glucose-insulin model, Entropy, 19 (2017), 114. doi: 10.3390/e19050114. A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, New York, 2016. doi: 10.1016/B978-0-08-100644-3.00001-5.  A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. A. Atangana and K. M. Owolabi., New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), 1-21. doi: 10.1051/mmnp/2018010.  A. Atangana, Blind in a commutative world: Simple illustrations with functions and chaotic attractors, Chaos, Solitons & Fractals, 114 (2018), 347-363. doi: 10.1016/j.chaos.2018.07.022.  A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056.  A. Atangana and J. F. Gómez Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 1-23.  A. Atangana and E. F. D. Goufo, On the mathematical analysis of Ebola hemorrhagic fever: Deathly infection disease in West African countries, BioMed Research International, 2014 (2014), Article ID 261383, 7 pages. doi: 10.1155/2014/261383. A. Atangana and B. S. T. Alkahtani, Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter, Complexity, 21 (2016), 442-451. doi: 10.1002/cplx.21704.  A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm Sci., 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. R. N. Bergman, Y. Z. Ider, C. R. Bowden and C. Cobelli, Quantitative estimation of insulin sensitivity, American Journal of Physiology-Endocrinology And Metabolism, 236 (1979), 667-677. doi: 10.1152/ajpendo.1979.236.6.E667. A. Caumo, C. Cobelli and M. Omenetto, Overestimation of minimal model glucose effectiveness in presence of insulin response is due to under modeling, American Journal of Physiology, 278 (1999), 481-488.  A. De Gaetano and O. Arino, Mathematical modelling of the intravenous glucose tolerance test, Journal of Mathematical Biology, 40 (2000), 136-168. doi: 10.1007/s002850050007.  L. C. Gatewood, E. Ackerman, J. W. Rosevear, G. D. Molnar and T. W. Burns, Tests of a mathematical model of the blood-glucose regulatory system, Computional Biomedical Research, 2 (1968), 1-14. doi: 10.1016/0010-4809(68)90003-7. A. Fabre and J. Hristov, On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity, Heat and Mass Transfer, 53 (2017), 177-204. doi: 10.1007/s00231-016-1806-5. J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, Thermal Science, 1 (2016), 115-115.  R. Jain, K. Arekar and R. Shanker Dubey, Study of Bergman's minimal blood glucose-insulin model by Adomian decomposition method, Journal of Information and Optimization Sciences, 38 (2017), 133-149. doi: 10.1080/02522667.2016.1187919.  F. Jarad, E. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Advances in Difference Equations, 2017 (2017), Paper No. 247, 16 pp. doi: 10.1186/s13662-017-1306-z.  R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002.  S. Kumar, A. Kumar and I. K. Argyros, A new analysis for the Keller-Segel model of fractional order, Numerical Algorithms, 75 (2017), 213-228. doi: 10.1007/s11075-016-0202-z.  S. Kumar, A new analytical modelling for telegraph equation via Laplace transform, Appl. Math. Modell, 38 (2014), 3154-3163. doi: 10.1016/j.apm.2013.11.035.  S. Kumar and M. M. Rashidi, New analytical method for gas dynamic equation arising in shock fronts, Comput. Phy. Commun, 185 (2014), 1947-1954. doi: 10.1016/j.cpc.2014.03.025.  G. A. Losa, On the fractal design in human brain and nervous tissue, Applied Mathematics, 5 (2014), 1725-1732. doi: 10.4236/am.2014.512165. V. F. Morales-Delgado, J. F. Gómez-Aguilar, S. Kumar and M. A. Taneco-Hernández, Analytical solutions of the Keller-Segel chemotaxis model involving fractional operators without singular kernel, The European Physical Journal Plus, 133 (2018), 200. doi: 10.1140/epjp/i2018-12038-6. Z. Odibat and A. S. Bataineh, An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: Construction of homotopy polynomials, Math. Meth. Appl. Sci, 38 (2015), 991-1000. doi: 10.1002/mma.3136.  K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317. doi: 10.1016/j.cnsns.2016.08.021.  K. M. Owolabi and A. Atangana, Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative, The European physical Journal Plus, 131 (2016), 335. doi: 10.1140/epjp/i2016-16335-8. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Academic Press, an Diego, California, USA, 1999.  Numerical simulations for the blood glucose concentration $G(t)$ , the effect of active insulin $X(t)$ and the blood insulin concentration $I(t)$ for several values of $\alpha_1-\beta$ , $\alpha_2-\beta$ and $\alpha_3-\beta$ . Numerical simulations for the blood glucose concentration G(t), the effect of active insulin X(t) and the blood insulin concentration I(t) for several values of α, β and γ.
Description of parameters in system (4)
 Parameter Description Unit $G_b$ Basal blood glucose concentration mg/dL $I_b$ Basal blood insuline concentration mU/L $p_1$ Insulin-independent glucose clearance rate 1/min $p_2$ Active insulin clearance rate 1/min $p_3$ Increase in uptake ability caused by insulin L/min $^{2}$ $\cdot$ mU $p_4$ Decay rate of blood insulin 1/min $p_5$ The target glucose level mg/dL $p_6$ Pancreatic release rate after glucose bolus mU $\cdot$ dL/L $\cdot$ mg $\cdot$ min
 Parameter Description Unit $G_b$ Basal blood glucose concentration mg/dL $I_b$ Basal blood insuline concentration mU/L $p_1$ Insulin-independent glucose clearance rate 1/min $p_2$ Active insulin clearance rate 1/min $p_3$ Increase in uptake ability caused by insulin L/min $^{2}$ $\cdot$ mU $p_4$ Decay rate of blood insulin 1/min $p_5$ The target glucose level mg/dL $p_6$ Pancreatic release rate after glucose bolus mU $\cdot$ dL/L $\cdot$ mg $\cdot$ min
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