# American Institute of Mathematical Sciences

## 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
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##### References:
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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