Chaos of calcium diffusion in Parkinson's infectious disease model and treatment mechanism via Hilfer fractional derivative

Hardik Joshi; Brajesh Kumar Jha

doi:10.53391/mmnsa.2021.01.008

EN

Chaos of calcium diffusion in Parkinson's infectious disease model and treatment mechanism via Hilfer fractional derivative

Abstract

Calcium is a vital element in our body and plays a crucial role to moderate the calcium signalling process. Calcium-dependent protein and flux through the sodium-calcium exchanger are also involved in signalling process to perform and execute necessary cellular activities. The loss or alteration in this cellular activity starts the early progress of Parkinson’s disease. A mathematical calcium model is developed in the form of the Hilfer fractional reaction-diffusion equation to examine the calcium diffusion in the cells. The effect of calcium-dependent protein and flux through the sodium-calcium exchanger is incorporated in the model. The solution of the Hilfer fractional calcium model is obtained by using the Sumudu transform technique in the form of the Wright function and Mittag-Leffler function. The graphical results are obtained for the different amounts of proteins, presence, and absence of sodium-calcium exchanger, and various orders of Hilfer derivative. The obtained results show that the modified calcium model is a function of time, position, and Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a rich physical interpretation of a calcium model as compared to the classical calcium model.

Keywords

Calcium, Parkinson’s disease, Hilfer fractional derivative, Sumudu transform, sodium-calcium exchanger

References

Squire L.R., Berg D., Bloom F. E., Du Lac S., Ghosh A., and Spitzer N. C., Fundamental Neuroscience, (Vol. 4). Elsevier Inc., (2012).
Verkhratsky A. & Butt A. Glial neurobiology: a textbook, John Wiley and Sons, (2007).
Petersen, O.H., Michalak, M. & Verkhratsky A. Calcium signalling: past, present and future. Cell Calcium, 38(3-4), 161–169, (2005).
Clapham, D.E. Calcium Signaling. Cell, 131(6), 1047–1058, (2007).
Panday, S. & Pardasani, K.R. Finite element model to study effect of advection diffusion and Na +/Ca2+ exchanger on Ca2+ distribution in oocytes. Journal of Medical Imaging and Health Informatics, 3(3), 374–379, (2013).
Tewari, S.G. & Pardasani, K.R. Modeling effect of sodium pump on calcium oscillations in neuron cells. Journal of Multiscale Modelling, 4(3), 1250010, (2012).
Jha, A., Adlakha, N. & Jha, B.K. Finite element model to study effect of Na+ - Ca2+ exchangers and source geometry on calcium dynamics in a neuron cell. Journal of Mechanics in Medicine and Biology, 16(2), 1–22, (2015).
Mattson, M.P. Calcium and neurodegeneration. Aging Cell, 6(3), 337–350, (2007).
Bezprozvanny, I. Calcium signaling and neurodegenerative diseases. Trends in molecular medicine, 15(3), 89–100, (2009).
Surmeier, D.J. Calcium, ageing, and neuronal vulnerability in Parkinson’s disease. The Lancet Neurology, 6(10), 933–938, (2007).

Zaichick, S.V., McGrath, K.M. & Caraveo, G. The role of Ca2+ signaling in Parkinson’s disease. Disease Models & Mechanisms, 10(5), 519–535, (2017).
Calì, T., Ottolini, D. & Brini, M. Calcium signaling in Parkinson’s disease. Cell and tissue research, 357(2), 439–454, (2014).
Blaustein, M.P. & Lederer, W.J. Sodium/calcium exchange: Its physiological implications. Physiological Reviews, 79(3), 763–854, (1999).
Sato, D., Despa, S. & Bers, D.M. Can the sodium-calcium exchanger initiate or suppress calcium sparks in cardiac myocytes? Biophysical journal, 102(8), L31–L33, (2012).
Philipson, K.D. & Nicoll, D.A. Sodium-calcium exchange: a molecular perspective. Annual review of physiology, 62(1), 111–133, (2000).
Noble, D., Noble, S.J., Bett, G.C.L., Earm, Y.E., Ho, W.K. & So, I.K. The Role of Sodium - Calcium Exchange during the Cardiac Action Potential a. Annals of the New York Academy of Sciences, 639(1), 334–353, (1991).
Jha, B.K., Adlakha, N. & Mehta, M.N. Two-Dimensional Finite Element Model To Study Calcium Distribution in Astrocytes in Presence of VGCC and Excess Buffer. International Journal of Modeling, Simulation, and Scientific Computing, 4(2), 1250030, (2013).
Jha, B.K. & Jha, A. Two dimensional finite element estimation of calcium ions in presence of NCX and Buffers in Astrocytes. Boletim da Sociedade Paranaense de Matemática, 36(1), 151–160, (2018).
Jha, B.K., Jha, A. & Adlakha, N. Three-Dimensional Finite Element Model to Study Calcium Distribution in Astrocytes in Presence of VGCC and Excess Buffer. Differential Equations and Dynamical Systems, 28(3), 603–616, (2020).
Jha, B.K., Adlakha, N. & Mehta, M.N. Two-dimensional finite element model to study calcium distribution in astrocytes in presence of excess buffer. International Journal of Biomathematics, 7(03), 1450031, (2014).
Gill, V., Singh, Y., Kumar, D. & Singh, J. Analytical study for fractional Order mathematical model of concentration of Ca2+ in astrocytes cell with a composite fractional derivative. Journal of Multiscale Modelling, 11(3), 2050005, (2020).
Devi, A. & Jakhar, M. Analysis of Concentration of Ca2 + Arising in Astrocytes Cell. International Journal of Applied and Computational Mathematics, 7(1), 1–9, (2021).
Jha, A. & Adlakha, N. Two-dimensional finite element model to study unsteady state Ca2+ diffusion in neuron involving ER LEAK and SERCA. International Journal of Biomathematics, 8(01), 1550002, (2015).
Joshi, H. & Jha, B.K. Fractionally delineate the neuroprotective function of calbindin-28k in Parkinson’s disease. International Journal of Biomathematics, 11(08), 1850103, (2018).
Joshi H. and Jha B. K. Generalized Diffusion Characteristics of Calcium Model with Concentration and Memory of Cells: A Spatiotemporal Approach. Iranian Journal of Science and Technology, Transactions A: Science, 1–14, (2021).
Joshi, H. & Jha, B.K. On a reaction–diffusion model for calcium dynamics in neurons with Mittag–Leffler memory. The European Physical Journal Plus, 136(6), 1–15, (2021).
Joshi, H. & Jha, B.K. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects. International Journal of Nonlinear Sciences and Numerical Simulation, (2021).
Joshi, H. & Jha, B.K. Fractional-order mathematical model for calcium distribution in nerve cells. Computational and Applied Mathematics, 39(2), 1–22, (2020).
Jha, B.K., Joshi, H. & Dave, D.D. Portraying the Effect of Calcium-Binding Proteins on Cytosolic Calcium Concentration Distribution Fractionally in Nerve Cells. Interdisciplinary Sciences: Computational Life Sciences, 10(4), 674–685, (2018).
Dave, D.D. & Jha, B.K. Mathematical Modeling of Calcium Oscillatory Patterns in a Neuron. Interdisciplinary Sciences: Computational Life Sciences, 13(1), 12–24, (2021).
Naik, P.A. & Pardasani, K.R. Finite element model to study calcium distribution in oocytes involving voltage gated Ca2 + channel, ryanodine receptor and buffers. Alexandria Journal of Medicine, 52(1), 43–49, (2016).
Naik, P.A. & Pardasani, K.R. Three-dimensional finite element Model to Study Effect of RyR Calcium Channel, ER Leak and SERCA Pump on calcium distribution in oocyte cell. International Journal of Computational Methods, 16(01), 1850091, (2019).
Pathak, K. & Adlakha, N. Finite element model to study two dimensional unsteady state calcium distribution in cardiac myocytes. Alexandria Journal of Medicine, 52(3), 261–268, (2016).
Chen, W., Aistrup, G., Wasserstrom J.A. & Shiferaw, Y. A mathematical model of spontaneous calcium release in cardiac myocytes. American Journal of Physiology-Heart and Circulatory Physiology, 300(5), H1794-H1805, (2011).
Singh, N. & Adlakha, N. A mathematical model for interdependent calcium and inositol 1,4,5-trisphosphate in cardiac myocyte. Network Modeling Analysis in Health Informatics and Bioinformatics, 8(1), 1–15, (2019).
Jagtap, Y. & Adlakha, N. Numerical study of one-dimensional buffered advection–diffusion of calcium and IP 3 in a hepatocyte cell. Network Modeling Analysis in Health Informatics and Bioinformatics, 8(1), 1–9, (2019).
Naik, P.A. & Zu, J. Modeling and simulation of spatial-temporal calcium distribution in T lymphocyte cell by using a reaction-diffusion equation. Journal of bioinformatics and computational biology, 18(2), 2050013, (2020).
Naik, P.A. Modeling the mechanics of calcium regulation in T lymphocyte: A finite element method approach. International Journal of Biomathematics, 13(05), 2050038, (2020).
Yavuz, M., Coşar, F.Ö., Günay F., & Özdemir, F.N. A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign. Open Journal of Modelling and Simulation, 9(3), 299–321, (2021).
Özkose, F. & Yavuz M. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in biology and medicine, 105044, (2021).
Abboubakar, H., Kumar, P., Erturk V.S. & Kumar, A. A mathematical study of a tuberculosis model with fractional derivatives. International Journal of Modeling, Simulation, and Scientific Computing, 2150037, (2021).
Kumar, P. & Erturk, V.S. Environmental persistence influences infection dynamics for a butterfly pathogen via new generalised Caputo type fractional derivative. Chaos, Solitons & Fractals, 144, 110672, (2021).
Abboubakar, H., Kumar, P., Rangaig, N.A. & Kumar, S. A malaria model with Caputo-Fabrizio and Atangana-Baleanu derivatives. International Journal of Modeling, Simulation, and Scientific Computing, 12(2), 2150013, (2021).
Abu-Shady, M. & Kaabar, M.K. A Generalized Definition of the Fractional Derivative with Applications. Mathematical Problems in Engineering, 2021, (2021).
Debbouche, N., Ouannas, A., Batiha, I.M., Grassi, G., Kaabar, M.K., Jahanshahi, H., ... & Aljuaid, A.M. Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System. Complexity, 2021, (2021).
Mohammadi, H., Kaabar, M.K.A, Alzabut, J., Selvam, A. & Rezapour, S. A Complete Model of Crimean-Congo Hemorrhagic Fever (CCHF) Transmission Cycle with Nonlocal Fractional Derivative. Journal of Function Spaces, 2021, (2021).
Kumar, P., Erturk, V.S., Banerjee, R., Yavuz, M. & Govindaraj, V. Fractional modeling of plankton-oxygen dynamics under climate change by the application of a recent numerical algorithm. Physica Scripta, 96(12), 124044, (2021).
Bonyah, E., Yavuz, M., Baleanu, D. & Kumar, S. A robust study on the listeriosis disease by adopting fractal-fractional operators. Alexandria Engineering Journal, 61(3), 2016-2028, (2021).
Veeresha, P. A Numerical Approach to the Coupled Atmospheric Ocean Model Using a Fractional Operator. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 1–10, (2021).
Erturk, V.S., Godwe, E., Baleanu, D., Kumar, P., Asad, J. & Jajarmi, A. Novel Fractional-Order Lagrangian to Describe Motion of Beam on Nanowire. Acta Physica Polonica, A., 140(3), 265–272, (2021).
Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A. & Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(2), 56–66, (2021).
Hammouch, Z., Yavuz, M. & Özdemir N. Numerical Solutions and Synchronization of a Variable-Order Fractional Chaotic System. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 11–23, (2021).
Naik, P.A., Yavuz, M., Qureshi, S., Zu, J. & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).
Dasbasi, B. Stability Analysis of an Incommensurate Fractional-Order SIR Model. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 44–55, (2021).
Yokuş, A. Construction of Different Types of Traveling Wave Solutions of the Relativistic Wave Equation Associated with the Schrödinger Equation. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 24–31, (2021).
Kumar, P., Erturk, V.S., Yusuf, A. & Kumar, S. Fractional time-delay mathematical modeling of Oncolytic Virotherapy. Chaos, Solitons & Fractals, 150, 111123, (2021).
Kumar, P., Ertürk, V.S. & Nisar, K.S. Fractional dynamics of huanglongbing transmission within a citrus tree. Mathematical Methods in the Applied Sciences, 44(14), 11404–11424, (2021).
Kumar, P., Erturk, V.S., Yusuf, A., Nisar, K.S. & Abdelwahab, S.F. A study on canine distemper virus (CDV) and rabies epidemics in the red fox population via fractional derivatives. Results in Physics, 25, 104281, (2021).
Odibat, Z., Erturk, V.S., Kumar, P. & Govindaraj, V. Dynamics of generalized Caputo type delay fractional differential equations using a modified Predictor-Corrector scheme. Physica Scripta, 96(12), 125213, (2021).
Kumar, P., Erturk, V.S. & Almusawa, H. Mathematical structure of mosaic disease using microbial biostimulants via Caputo and Atangana–Baleanu derivatives. Results in Physics, 24, 104186, (2021).
Hilfer, R. Applications of Fractional Calculus in Physics, World Scientific: Singapore, (2000).
Oldham, K.B. & Spanier, J. The Fractional Calculus: Theory and Applications of differentiation and integration of arbitrary Order, Elsevier, (2006).
Podlubny, I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (Vol.1), Academic Press, Elsevier, (1998).
Watugala, G.K. Sumudu transform: A new integral transform to solve differential equations and control engineering problems. International Journal of Mathematical Education in Science and Technology, 24(1), 35–43, (1993).
Crank, J. The Mathematics of Diffusion, (Vol.2), Oxford University Press: Oxford, (1975).
Sherman, A., Smith, G.D., Dai, L. & Miura, R.M. Asymptotic analysis of buffered calcium diffusion near a point source. SIAM Journal on Applied Mathematics, 61(5), 1816–1838, (2001).

Details

Primary Language

English

Subjects

Bioinformatics and Computational Biology, Applied Mathematics

Journal Section

Research Article

Authors

Hardik Joshi ^*
0000-0003-2014-3765
India

Brajesh Kumar Jha This is me
0000-0001-9370-2949
India

Publication Date

December 30, 2021

Submission Date

November 6, 2021

Acceptance Date

December 7, 2021

Published in Issue

Year 2021 Volume: 1 Number: 2

DOI

https://doi.org/10.53391/mmnsa.2021.01.008

IZ

https://izlik.org/JA74LY23NA

APA

Joshi, H., & Jha, B. K. (2021). Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation With Applications, 1(2), 84-94. https://doi.org/10.53391/mmnsa.2021.01.008

AMA

1.Joshi H, Jha BK. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. MMNSA. 2021;1(2):84-94. doi:10.53391/mmnsa.2021.01.008

Chicago

Joshi, Hardik, and Brajesh Kumar Jha. 2021. “Chaos of Calcium Diffusion in Parkinson’s Infectious Disease Model and Treatment Mechanism via Hilfer Fractional Derivative”. Mathematical Modelling and Numerical Simulation With Applications 1 (2): 84-94. https://doi.org/10.53391/mmnsa.2021.01.008.

EndNote

Joshi H, Jha BK (December 1, 2021) Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation with Applications 1 2 84–94.

IEEE

[1]H. Joshi and B. K. Jha, “Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative”, MMNSA, vol. 1, no. 2, pp. 84–94, Dec. 2021, doi: 10.53391/mmnsa.2021.01.008.

ISNAD

Joshi, Hardik - Jha, Brajesh Kumar. “Chaos of Calcium Diffusion in Parkinson’s Infectious Disease Model and Treatment Mechanism via Hilfer Fractional Derivative”. Mathematical Modelling and Numerical Simulation with Applications 1/2 (December 1, 2021): 84-94. https://doi.org/10.53391/mmnsa.2021.01.008.

JAMA

1.Joshi H, Jha BK. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. MMNSA. 2021;1:84–94.

MLA

Joshi, Hardik, and Brajesh Kumar Jha. “Chaos of Calcium Diffusion in Parkinson’s Infectious Disease Model and Treatment Mechanism via Hilfer Fractional Derivative”. Mathematical Modelling and Numerical Simulation With Applications, vol. 1, no. 2, Dec. 2021, pp. 84-94, doi:10.53391/mmnsa.2021.01.008.

Vancouver

1.Hardik Joshi, Brajesh Kumar Jha. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. MMNSA. 2021 Dec. 1;1(2):84-9. doi:10.53391/mmnsa.2021.01.008