Hybrid Fuzzy Fractional for Multi-Phasic Epidemics: The Omicron–Malaria Case Study

This study introduces a novel Fuzzy Piecewise Fractional Derivative (FPFD) framework to enhance epidemiological modeling, specifically for the multi-phasic co-infection dynamics of Omicron and malaria. We address the limitations of traditional models by incorporating two key realities. First, we use fuzzy set theory to manage the inherent uncertainty in biological parameters. Second, we employ piecewise fractional operators to capture the dynamic, phase-dependent nature of epidemics. The framework utilizes a fuzzy classical derivative for initial memoryless spread and transitions to a fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivative to capture post-intervention memory effects. We establish the mathematical rigor of the FPFD model through proofs of positivity, boundedness, and stability of equilibrium points, including the basic reproductive number (R0). A hybrid numerical scheme, combining Fuzzy Runge–Kutta and Fuzzy Fractional Adams–Bashforth–Moulton algorithms, is developed for solving the system. Simulations show that the framework successfully models dynamic shifts while propagating uncertainty. This provides forecasts that are more robust and practical, directly informing public health interventions.