Modeling remote healthcare adoption with fractional-order reaction-diffusion: stability and policy insights.

This study develops a fractional-order (FO) reaction-diffusion (RD) model to capture the spatiotemporal adoption dynamics of Remote Healthcare Services (RHS) in geographically constrained regions. The model integrates sociological mechanisms social contagion, external influence, relapse, and spatial diffusion within a framework enhanced by Caputo fractional derivatives to account for memory effects and historical dependencies in adoption behavior. A numerical scheme based on Grunwald-Letnikov (GL) discretization in time and finite differences method (FDM) in space is implemented and analyzed for stability using the von Neumann method adapted to FO systems. Simulation results demonstrate the emergence of spatially heterogeneous adoption patterns, damped responses to perturbations, and the influence of memory on long-term dynamics. The present work focuses on the mathematical formulation, numerical discretization, and stability analysis of the FO-RD model. Empirical calibration using field data from the Chittagong Hill Tracts (a region characterized by natural barriers and limited infrastructure) and scenario-based policy evaluations are identified as essential next steps but lie outside the scope of this paper. The model provides a robust predictive tool for evaluating policy interventions, optimizing resource allocation, and supporting equitable healthcare deployment in remote and underserved areas.