A general epidemic model with variable-order fractional derivatives and Lévy noise: Dynamical analysis and application to historical influenza data

Abstract

This paper introduces a novel fractional-stochastic epidemic model that integrates $\alpha$-stable Lévy jumps with time-dependent fractional derivatives, providing a comprehensive framework for capturing the complex dynamics of epidemiological systems. We rigorously establish the existence and uniqueness of solutions, with a particular emphasis on Ulam–Hyers (UH) stability to assess the system’s resilience to fluctuations and parameter variations. To support numerical investigations, we develop a MATLAB-based simulation framework that combines the Adams–Bashforth–Moulton (ABM) and Chambers–Mallows–Stuck (CMS) algorithms. This computational approach enables a detailed examination of how the stability index $\alpha$ and the fractional-order function $h\left(t\right)$ influence the system’s behavior. Furthermore, we conduct an extensive series of numerical tests to analyze the statistical properties of the generated distribution and perform a sensitivity analysis of key deterministic parameters governing transmission and vaccination. To assess the reliability of our approach, we validate the model using historical data from the 1919 Spanish flu outbreak in Sydney. We also perform stochastic model fitting to generate forward-looking predictions. These analyses demonstrate the model’s applicability and effectiveness in real-world epidemiological scenarios. Additionally, this study advances the mathematical modeling of infectious diseases and establishes a foundation for further developments in epidemiological research.