Fractional-order model of the disease Psoriasis: A control based mathematical approach

Cao, Xianbing; Datta, Abhirup; Basir, Fahad Al; Roy, Priti Kumar

doi:10.1007/s11424-016-5198-x

Cited by 39 publications

(25 citation statements)

References 14 publications

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“…Using results from [1,16,19,20], parameters γ 1 , γ 2 , λ, µ, ν of system (2.1) satisfy the following constraints:…”

Section: Control Modelmentioning

confidence: 99%

“…We note that the optimal control problem (2.1),(3.5) differs from problems that are typically considered in the literature on the control of psoriasis models ( [1,3,19]) in that the functional of (3.5) does not include an integral of the square of the control u(t), which is responsible for the total cost of the drug dosage. In psoriasis therapy, in most cases, either a skin cream or an oral medication, such as cyclosporine, are used.…”

Section: Optimal Control Problemmentioning

confidence: 99%

“…Therefore the total cost of psoriasis treatment in the meaning of "harm" to a patient and that usually mathematically is described by an integral of the square of the control, can be ignored. Moreover, using the terminal functional in (3.5) instead of the corresponding integral functional (see [1,19]), simplifies the subsequent analytical arguments. The existence in problem (2.1),(3.5) of the optimal control u * (t) and the corresponding optimal solutions l * (t), m * (t), k * (t) for system (2.1), follows from Lemma 2.1 and Theorem 4 ( [13], Chapter 4).…”

Section: Optimal Control Problemmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Treatment Strategies for Control Model of Psoriasis

Grigorieva¹,

Хайлов²,

Deignan³

2017

2017 Proceedings of the Conference on Control and Its Applications

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Psoriasis is an autoimmune disorder with the symptom of chronic inflammation of the skin. Psoriasis makes skin cells build up quickly on the skin surface which causes the skin to appear red, dry, and with scaly patches. Adequate treatment for psoriasis is very challenging and a total cure of the disease still does not exit. Mathematical models have long been effective means to predict cellular behaviors of skin regulation under normal or pathological circumstances. In this paper, a proposed control model of psoriasis is described on a given time interval by a nonlinear control system of three differential equations involving the concentrations of dendritic cells (tissues macrophages), T-lymphocytes, and keratinocytes with medication intake as a control function. An optimal control problem of minimizing the release of keratinocytes at the end of the time interval is stated and studied using the Pontryagin maximum principle. Different types of optimal control dependent on model parameters are obtained analytically. Possible applications to an optimal drug therapy are discussed.

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“…Using results from [1,16,19,20], parameters γ 1 , γ 2 , λ, µ, ν of system (2.1) satisfy the following constraints:…”

Section: Control Modelmentioning

confidence: 99%

Section: Optimal Control Problemmentioning

confidence: 99%

Section: Optimal Control Problemmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Treatment Strategies for Control Model of Psoriasis

Grigorieva¹,

Хайлов²,

Deignan³

2017

2017 Proceedings of the Conference on Control and Its Applications

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“…Recently it is showed that the fractional model perfectly fits the test data of memory phenomena in different disciplines [46] they have found that a possible physical meaning of the fractional order is an index of memory. From this viewpoint fractional calculus has found many applications in new research on physics of biological structures and living organisms, from DNA dynamics [47][48][49] to protein folding [50], cancer cells [51], tumor-immune system [52], modeling of some human autoimmune diseases such as psoriasis [53], bioimpedance [54], spiking neurons [55], and also the transport of drugs across biological materials and human skin [56] and electrical impedance applied to human skin [57] and even modeling of HIV dynamics [58].…”

Section: Fractional Calculus In Bioscience and Biomedicinementioning

confidence: 99%

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

Nasrolahpour

2017

Preprint

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Almost all phenomena and structures in nature exhibit some degrees of fractionality or fractality. Fractional calculus and fractal theory are two interrelated concepts. In this article we study the memory effects in nature and particularly in biological structures. Based on this fact that natural way to incorporate memory effects in the modeling of various phenomena and dealing with complexities is using of fractional calculus, in this article we present different examples in various branch of science from cosmology to biology and we investigate this idea that are we able to describe all of such these phenomena using the well-know and powerful tool of fractional calculus. In particular we focus on fractional calculus approach as an effective tool for better understanding of physics of living systems and organism and especially physics of cancer.Keywords: Fractional Dynamics; Memory Effect; Biological Structures; Physics of Cancer IntroductionThe concept of memory effect plays an important role in a large number of phenomena in different contexts and systems from biological structures to cosmological phenomena. For instance: memory effects in nanoscale systems [1,2], optical memory effect [3], gravitational and cosmological memory effect (which means: the induction of a permanent change in the relative separation of the test particles when gravitational radiation passes through a configuration of test particles that make up a gravitational wave detector) [4], memory effects in gene regulatory networks [5], memory effects in economics [6] have been investigated. In addition other kinds of memory effect including: shape memory effect [7][8][9][10], magnetic shape memory effect [11] and temperature memory effect [12] have been also considered in recent years. In all above mentioned references authors have used the standard calculus, however nowadays it is well-known that a natural way to incorporate memory effects in the modeling of various phenomena is using of fractional calculus. On the other hand almost all phenomena and structures in nature exhibit some degrees and levels of fractionality or fractality (low or high level or something between them), also nowadays it is well-known that there is a close relation between fractality and fractionality. In this work we investigate this idea that are we able to describe all of such these phenomena using the well-know and powerful tool of fractional calculus. Therefore for this purpose in the following, concepts of fractality and fractional dynamics are briefly reviewed respectively in Sec. 2.Then in Sec. 3 we introduce fractional calculus as a powerful tool for modeling of memory effects in different context and we present some important applications. At last, in Sec. 4, we will present some conclusions.

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“…A biochemical reaction with integer order derivatives does not include the effect of memory [58,59]. This can be incorporated by introduction of fractional order (α ∈ (0, 1]) derivatives as an index of memory [60][61][62].…”

Section: Introductionmentioning

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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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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Fractional-order model of the disease Psoriasis: A control based mathematical approach

Cited by 39 publications

References 14 publications

Optimal Treatment Strategies for Control Model of Psoriasis

Optimal Treatment Strategies for Control Model of Psoriasis

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study

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