A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions

Toledo-Hernandez, Rasiel; Rico-Ramı́rez, Vicente; Iglesias‐Silva, Gustavo A.; Diwekar, Urmila M.

doi:10.1016/j.ces.2014.06.034

Cited by 106 publications

(64 citation statements)

References 16 publications

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“…To use standard initial conditions, the RL derivatives must be redefined as Caputo's fractional derivatives. The relation between RL and Caputo's derivatives is as follows [67]:…”

Section: Memory Induced Systemmentioning

confidence: 99%

“…This phenomena is known as the effect of memory of the system [66,67]. Applicability of the fractional order equations is not considered ad hoc but since it is earlier used to model biological signals by Cole 1933 andHodgkin 1946 of the electrical properties of nerve cell membranes and the propagation of electrical signals [68,69].…”

Section: Memory Induced Systemmentioning

confidence: 99%

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How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

Ghosh

Pal

Roy

2017

Int. J. Appl. Comput. Math

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Drug resistance is one of the adaptive evolutions exhibited by bacteria with the advent of the antibiotics. Different bacterial population expressed a variety of antibacterial resistance in relation to the prescribed drug. Oftentimes, it has been successful to demonstrate full deactivation and hydrolyzing the drug in the event of its application to the affected ailment. Such resistance is witnessed in cases where the drug falls in the β-lactam family of antibiotics. New biological insights now reveal a responsive functional memory driven with primitive cross-talk among cells and other density dependent cellular signaling. Pathogenesis of a particular strain of bacterial cell manifest through this instantaneous memory function. In this article, we formulate a mathematical model to study "memory effect" towards bacterial cell wall biosynthesis in presence of β-lactam/β-lactamase inhibitor drug combination. We have introduced fractional order derivative to the system as an index of memory. Dynamics of the drug is studied when it is introduced in an impulsive way. The memory induced system is solved numerically. Memory of the system leads to quick conversion of immature bacteria to mature bacteria. Impulsively administered antibiotics show a quick recovery of the disease. Comparison of different dosing time interval establishes that the disease is cured promptly for shorter interval. Variation in the dose of β-lactam directs a quick recovery of the pathogenic infection for high doses.Keywords β-Lactam · Bacterial resistance · β-Lactamase inhibitor · Enzyme kinetics · Memory effect · Fractional order differential equations Mathematics Subject Classification 92B05 · 92C45 · 68T05 · 91E40 · 26A33

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“…To use standard initial conditions, the RL derivatives must be redefined as Caputo's fractional derivatives. The relation between RL and Caputo's derivatives is as follows [67]:…”

Section: Memory Induced Systemmentioning

confidence: 99%

Section: Memory Induced Systemmentioning

confidence: 99%

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

Ghosh

Pal

Roy

2017

Int. J. Appl. Comput. Math

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“…[22][23][24]. The fractional derivative has also been used in chemistry [25], biology [26][27][28], and psychology [29], etc. Nevertheless, some fundamental problems have prevented the popularization of fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Wang

2015

Acta Mech. Sin.

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Fractional differential equations are more and more used in modeling memory (history-dependent, nonlocal, or hereditary) phenomena. Conventional initial values of fractional differential equations are defined at a point, while recent works define initial conditions over histories. We prove that the conventional initialization of fractional differential equations with a Riemann-Liouville derivative is wrong with a simple counter-example. The initial values were assumed to be arbitrarily given for a typical fractional differential equation, but we find one of these values can only be zero. We show that fractional differential equations are of infinite dimensions, and the initial conditions, initial histories, are defined as functions over intervals. We obtain the equivalent integral equation for Caputo case. With a simple fractional model of materials, we illustrate that the recovery behavior is correct with the initial creep history, but is wrong with initial values at the starting point of the recovery. We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.

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“…However, the fractional calculus had been stayed in the pure theories of mathematics during almost 300 years [16]. After holding the First Conference on Fractional Calculus and its Applications on June 1974 [17], the fractional calculus has been widely applied in various fields [18][19][20][21][22][23][24][25][26], because the fractional models are more accurate than the integer ones. In the field of remote sensing, fractional calculus is mainly used for image enhancement and feature extraction [27][28][29][30].…”

Section: Fractional Differentialmentioning

confidence: 99%

Influence of Fractional Differential on Correlation Coefficient between EC_1:5 and Reflectance Spectra of Saline Soil

Xia¹,

Tiyip²,

Kelimu³

et al. 2017

Journal of Spectroscopy

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Soil salinization is one of the most serious environmental issues in arid and semiarid area with severe social, economic, and ecological problems. At present, most inversion models are based on raw reflectance spectra or integer differential transform. In this study, we measured the hyperspectral reflectance and EC 1:5 of soil samples collected form Ebinur Lake to analyze the influence of fractional differential on correlation coefficient between EC 1:5 and reflectance spectra. The results showed that the fractional differential increased sensibly the accuracy for the analysis of the reflectance spectra. The study might provide a new insight for monitoring soil salinity using hyperspectral data, and further researches should be concentrated on physical meaning of fractional differential in hyperspectral data to provide theoretical basis to building, describing, and spreading inversion models.

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A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions

Cited by 106 publications

References 16 publications

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Influence of Fractional Differential on Correlation Coefficient between EC_1:5 and Reflectance Spectra of Saline Soil

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A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions

Cited by 106 publications

References 16 publications

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Influence of Fractional Differential on Correlation Coefficient between EC1:5 and Reflectance Spectra of Saline Soil

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Product

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Influence of Fractional Differential on Correlation Coefficient between EC_1:5 and Reflectance Spectra of Saline Soil