Fractional Reaction–Diffusion Modelling of Immune-Mediated Demyelination in Multiple Sclerosis Under IFN-Beta and Glatiramer Acetate Therapy

We propose a dimensionally consistent fractional spatio-temporal PDE framework for modelling immune-mediated demyelination in multiple sclerosis (MS). The system couples effector and regulatory T cells, M1/M2 macrophage polarisation, pro- and anti-inflammatory cytokines, oligodendrocyte dynamics, and time-dependent therapeutic controls within a unified distributed-parameter structure. In contrast to ad hoc replacements of integerorder derivatives by Caputo fractional derivatives, the fractional extension proposed here is derived from an underlying continuous-time random walk (CTRW) process with Mittag–Leffler-distributed residence times. This stochastic derivation yields a governing system in which a single commensurate fractional order α∈(0,1], together with a characteristic memory timescale τ0, ensures dimensional consistency and mass balance across all coupled components. The model is formulated as a system of nonlinear reaction–diffusion equations with cross-regulatory and multiplicative interaction terms governing immune amplification, cytokine feedback, and the demyelination–remyelination balance. Analytical interpretation shows how non-Markovian residence times induce Mittag–Leffler-type relaxation and thereby modify effective growth, decay, and stability properties. Numerical simulations compare classical and fractional dynamics, revealing that memory-driven kinetics prolong effector T-cell and M1-macrophage activity, attenuate reparative M2 and oligodendrocyte responses, and extend the effective action of bang–bang therapy inputs representing IFN-β and glatiramer acetate beyond their dosing windows. The results indicate that integer-order models may underestimate chronic inflammatory persistence and demyelination severity, while providing a mathematically and physically well-posed platform for memory-aware immune modelling and therapy evaluation in MS.