Mathematical modeling approach to the fractional Bergman's model

Victor Fabian Morales-Delgado; José Francisco Gómez-Aguilar; Marco Antonio Taneco-Hernández

doi:10.3934/dcdss.2020046

Article Contents

2020, Volume 13, Issue 3: 805-821. Doi: 10.3934/dcdss.2020046

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

Received: June 2018

Revised: August 2018

Published: December 2019

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Abstract

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.

Keywords:

Bergman's model,
fractional conformable derivative,
Atangana-Baleanu fractional derivative,
Laplace transform,
modified homotopy analysis transform method.

Mathematics Subject Classification: Primary: 34A34, 65M12; Secondary: 26A33, 34A08, 65C20, 65P20.

Citation:

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Figure 2. 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 $\alpha_3-\beta$ .

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Figure 1. 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.

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Table 1. 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

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