A Comparative Study of Caputo and Caputo-Fabrizio Models for Measles Dynamics Using AB2 and AB3 Numerical Schemes

Fractional-order differential equations have gained increasing attention in epidemiological modeling due to their ability to incorporate memory and hereditary effects. In this paper, we investigate the comparative behavior of Caputo and Caputo-Fabrizio fractional derivatives in a measles transmission model using Adams-Bashforth (AB2 and AB3) multistep methods. The model incorporates Susceptible, Infected, Recovered, and Pathogen compartments and is simulated under both fractional frameworks. Our findings show that Caputo's power-law kernel leads to faster epidemic spread, sharper peaks in infection and recovery, and a rapid decline in pathogen concentration, indicating short but intense outbreaks. In contrast, Caputo-Fabrizio's exponential kernel produces smoother dynamics with prolonged infections and sustained recoveries, highlighting the role of recent events in epidemic progression. Numerical results underline the importance of memory choice in forecasting and control, with Caputo-Fabrizio systems showing greater numerical stability under AB2/AB3 schemes. This study offers a clear entry point for beginners exploring fractional epidemiological models.