A fractional-order model for MINMOD Millennium

Cho, Yongjin; Kim, Imbunm; Sheen, Dongwoo

doi:10.1016/j.mbs.2014.11.008

Cited by 21 publications

(9 citation statements)

References 38 publications

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“…3by the Caputo fractional derivatives of orders q 1 , q 2 ∈ (0, 1], respectively, the resulting equations still have the problem in which the units of the left-hand side and the right-hand side of the resulting system mismatch, i.e., the units of the left-hand side are time q 1 and time q 2 while the units of the right-hand side of the system have the dimension time -1 . Hence, we must preserve units as described in [21,43] on both sides of each equation in the resulting system by adjusting some of the original parameters or multiplying some original parameters by a fractional time constant β. After taking care of the units for both sides of the equations, we obtain the delayed fractionalorder model of glucose-insulin interaction with incommensurate orders as follows:…”

Section: Consider a General Delayed Fractional-order Systemmentioning

confidence: 99%

“…Especially, we call the parameters α 1 and α 2 the fractional tissue glucose uptake rate and the fractional disappearance rate constant for insulin, respectively. As mentioned in the generalized MINMOD Millennium model [21], the terms β 1-q 1 and β 1-q 2 can be physiologically considered as the parameters describing the rheological behavior in enhancing the muscular and liver sensibility to the action of insulin. Roughly speaking, the new fractional-order model (29) is the generalization of the first-order system (3).…”

Section: Consider a General Delayed Fractional-order Systemmentioning

confidence: 99%

“…For example, Ertürk et al [20] proposed and solved numerically the fractional-order differential system for a giving up smoking model using the multi-step generalized differential transform method (MSGDTM). Cho et al [21] developed a fractional-order version of the minimal model for glucose-insulin dynamics. They also investigated local stability of the model.…”

Section: Introductionmentioning

confidence: 99%

“…They also concluded that for physiologic plausibility, their fractional-order models provide a reasonable rate of movement of glucose from the blood into the environment. In 2015, Kim et al [21] proposed a generalized MINMOD Millennium model using the Caputo fractional-order derivative of order α ∈ (0, 1]. They discovered from the model that the value of β 1-α is inversely proportional to the value of S I , where β is a fractional time constant for preserving units and S I is the insulin sensitivity.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

2019

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This paper proposes a delayed fractional-order model of glucose-insulin interaction in the sense of the Caputo fractional derivative with incommensurate orders. This fractional-order model is developed from the first-order model of glucose-insulin interaction. Firstly, we investigate the non-negativity and the boundedness of solutions of the fractional-order model. Secondly, the stability and the bifurcation of the model are studied by separating the associated characteristic equation of the model into its real and imaginary parts and taking a time delay as the bifurcation parameter. The asymptotic stability and the Hopf bifurcation are discussed via the condition of creation of the bifurcation. Furthermore, it is shown that the onset of the bifurcation is related to the fractional orders of the model. Finally, some numerical simulations of the model using the Adam-Bashforth-Moulton predictor corrector scheme are demonstrated to support our obtained theoretical results.

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Section: Consider a General Delayed Fractional-order Systemmentioning

confidence: 99%

Section: Consider a General Delayed Fractional-order Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

2019

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“…They typically convert the problem of model parameter estimation into an optimization problem, and then, formulate a suitable estimator by minimizing an objective function. These methods are stretched over but no limited to perturbation methods [60], weighted least squares approach [12,15,25], nonlinear regression [34], and Levenberg-Marquardt method [20,13,63,64]. We develop a bi-level FSEM-based parameter estimation method in order to construct fractional models, in a sense that the method obtains model coefficients in one level, and then searches for estimate of fractional indices in the next level.…”

mentioning

confidence: 99%

Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Kharazmi

Zayernouri

2019

J Sci Comput

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Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further construct a gradient-based optimization algorithm to compute an accurate parameter estimation in fractional model construction. We develop a fast, stable, and convergent Petrov-Galerkin spectral method to numerically solve the coupled system of original fractional model and its corresponding adjoint fractional sensitivity equations.Key words. sensitive fractional orders, model error, logarithmic-power law kernel, Petrov-Galerkin spectral method, iterative algorithm, parameter estimation.Fractional Sensitivity Analysis. Sensitivity assessment approaches are commonly categorized as, finite difference, continuum and discrete derivatives, and computational or automatic differentiation, where the sensitivity coefficients are generally defined as partial derivative of corresponding functions (model output) with respect to design/analysis parameters of inter- †

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Variational approach for the Kirchhoff problem involving the p$$ p $$‐Laplace operator and the ψ$$ \psi $$‐Hilfer derivative

Alsaedi

Ghanmi

2023

Math Methods in App Sciences

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This work aims to develop the variational framework for some Kirchhoff problems involving both the p$$ p $$‐Laplace operator and the ψ$$ \psi $$‐Hilfer derivative. Precisely, we use the mountain pass theorem to prove the existence of nontrivial solutions. Moreover, the multiplicity of solutions is proved by the use of the Z2$$ {Z}_2 $$‐symmetry mountain pass theorem. Our main results generalize the paper of Torres (J Fract Calculus Appli. 2014;5(1):1‐10) and the work of Sousa et al. (Comp Appl Math. 2019;38:4).

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A fractional-order model for MINMOD Millennium

Cited by 21 publications

References 38 publications

Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Variational approach for the Kirchhoff problem involving the p$$ p $$‐Laplace operator and the ψ$$ \psi $$‐Hilfer derivative

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