Mathematical analysis of Immune Response Dynamics in Parkinson's Disease with Immunotherapeutic Intervention

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Siksha `O' Anusandhan (Deemed to be University), Khandagiri Square, Bhubaneswar- 751030, Odisha, India.

Abstract

We develop a fractional-order, time-delay mathematical model of Parkinson’s disease that
couples neuronal compartments, extracellular α-synuclein, microglial activation and adap
tive immune responses. The fractional derivative models long-memory processes (e.g., slow
protein aggregation and persistent inflammation) while the discrete delay represents bi
ologically observed lags in immune activation. Parameters were estimated by digitizing
published time-series, and the fitted model reproduces the qualitative dynamics reported in
the literature. We compute the basic reproduction number ℜ0 using the next-generation
matrix and perform linear stability and Hopf bifurcation analyses; the first Lyapunov coefficient is negative, indicating a supercritical Hopf bifurcation and the emergence of stable
oscillations for sufficiently large delays. Numerical experiments show that increasing the
α-synuclein clearance efficacy (ϵ1) reduces ℜ0 and stabilizes the system, whereas microglial
and T-cell suppression (ϵ2,ϵ3) mainly attenuate oscillation amplitude. Our results support
immunotherapeutic strategies that prioritize clearance of pathological α-synuclein to limit
disease progression.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 16 May 2026

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History

  • Receive Date: 18 September 2025
  • Revise Date: 01 May 2026
  • Accept Date: 13 May 2026

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