Mathematical analysis of neurological disorder under fractional order derivative

Nadeem Khan; Amjad Ali; Aman Ullah; Zareen A. Khan; Nadeem Khan; Amjad Ali; Aman Ullah; Zareen A. Khan

doi:10.3934/math.2023959

AIMS Mathematics

2023, Volume 8, Issue 8: 18846-18865. doi: 10.3934/math.2023959

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Research article

Mathematical analysis of neurological disorder under fractional order derivative

1.
Department of Basic Sciences & Islamiat, University of Engineering and Technology, Peshawar, Pakistan
2.
Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 14 March 2023 Revised: 25 April 2023 Accepted: 27 April 2023 Published: 05 June 2023
MSC : 26A33, 34K37

Multiple sclerosis (MS) is a common neurological disorder that affects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for different parameter values.
- fractional operator,
- existence theory,
- numerical method,
- unique solution
Citation: Nadeem Khan, Amjad Ali, Aman Ullah, Zareen A. Khan. Mathematical analysis of neurological disorder under fractional order derivative[J]. AIMS Mathematics, 2023, 8(8): 18846-18865. doi: 10.3934/math.2023959

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Abstract

Multiple sclerosis (MS) is a common neurological disorder that affects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for different parameter values.

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