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Mathematical modeling approach to the fractional Bergman's model

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

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

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


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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 $\alpha_3-\beta$ .

    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.

    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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