Exploring wave dynamics and disease progression in a fractional-order β-derivative multiple sclerosis model

This study explores analytical solutions of a simplified β-derivative Multiple Sclerosis (MS) model via the generalized Riccati equation method, focusing on their dynamic behaviors and the influence of fractional-order dynamics on disease progression. A variety of analytical solutions are derived and presented through 2D graphical representations, illustrating different wave structures, including kink, anti-kink, bell, anti-bell, and singular-shaped waves. These structures capture the nonlinear, localized, and memory-driven behaviors of brain cell populations affected by MS. In particular, singular solutions indicate potential instability points that may correspond to critical pathological events during disease progression. The analysis demonstrates that both the fractional order and wave velocity significantly influence the shape, dispersion, and stability of these solutions. Moreover, the model parameters regulate key biological mechanisms that modulate immune activation, drive neurodegeneration, control immune-inflammatory dynamics, and facilitate neural repair. Collectively, these parameters shape the interactions among healthy, infected, and pathogenic brain cell populations. By incorporating fractional-order dynamics, this study enriches the MS model, bridging mathematical analysis with biological complexity and providing a more accurate representation of MS pathology. These findings emphasize the importance of fractional-order models in providing deeper insight into disease progression mechanisms and highlight potential directions for developing therapeutic strategies.