Fractional-Order Reaction-Diffusion SVEIQR Model for Measles With Saturation, Vaccination, and a Backward Bifurcation Threshold

This study develops a comprehensive fractional-order reaction-diffusion measles model that uniquely integrates three critical epidemiological features often studied in isolation: memory effects via Caputo fractional derivatives, spatial heterogeneity through Fickian diffusion, and nonlinear transmission dynamics via a saturated incidence function. The primary novelty lies in the simultaneous incorporation of these elements within a unified SVEIQR framework, enabling rigorous analysis of how their interactions influence disease persistence and control. The main mathematical contributions include explicit derivation of the basic reproduction number, proof of existence-uniqueness of solutions using fixed-point theory, and establishment of a novel threshold condition for backward bifurcation that quantifies the synergistic effects of vaccination imperfections and transmission saturation. Specifically, we derive the necessary condition for backward bifurcation occurrence, revealing that even when R0