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 , disease persistence can occur if saturation effects and vaccine leakiness exceed critical levels. The advanced numerical simulations demonstrate that fractional order ϑ = 0.7 prolongs outbreak duration by approximately 30% compared to ϑ = 1.0, while spatial diffusion creates traveling wavefronts with speeds of 0.25 km per day that maintain coherence despite memory effects. These findings provide new mathematical insights into measles control, showing that elimination requires not only reducing but also addressing the interaction between behavioral saturation and vaccine quality through combined intervention strategies.