A Fractional-Order Complexity Framework for Misinformation Propagation With Fact-Checking Saturation and Social Vulnerability Dynamics

Misinformation spreads through communities in ways that resemble infectious diseases, but existing mathematical models often miss three real-world complexities: People remember past information (memory effects), some folks are more easily fooled than others (vulnerability), and fact-checkers can get overwhelmed during big outbreaks (saturation). This study builds a fresh mathematical framework that puts all these pieces together using fractional calculus, a tool that naturally captures how past exposures shape current beliefs. The model divides people into six groups: susceptible folks, vulnerable individuals (who spread falsehoods faster), active carriers of misinformation, corrected people who accepted fact-checks, resistant individuals with immunity, and a variable tracking fact-checking intensity. We analyzed this system thoroughly, proving that solutions stay positive and bounded, and we calculated the critical “virality threshold†Rv that determines whether misinformation dies out or becomes endemic. Our sensitivity analysis pinpointed the transmission rate to vulnerable folks and fact-checking effectiveness as the most powerful levers for control. Using optimal control theory, we designed three intervention strategies—preventative education, active fact-checking, and targeted support for vulnerable populations and simulated them. The key finding is clear: no single approach works best alone; the real magic happens when we combine all three simultaneously, which nearly eliminates misinformation entirely. This work provides researchers and policymakers with a rigorous tool to understand, predict, and combat the spread of false narratives in our increasingly complex information environment.