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

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