Bifurcation in a Fractional SIR Model With Normalized Caputo–Fabrizio Derivative

Models of fractional-order epidemics more accurately represent memory and hereditary effects compared to traditional integer-order models. However, many Caputo–Fabrizio (CF) SIR models utilize a non-normalized kernel, which can distort the scaling and understanding of memory. This research introduces a thorough formulation and analysis of an SIR model that uses the normalized CF (NCF) derivative, ensuring both dimensional consistency and meaningful contributions from memory. We identify important theoretical properties, such as existence, uniqueness, positivity, boundedness, and conservation, and we create an efficient time-stepping method specifically designed for the normalized exponential kernel. Next, we examine the dynamics of the model when influenced by seasonal factors. The numerical results show that normalization significantly impacts outbreak patterns, leading to delayed and reduced infection peaks compared to both classical and unnormalized CF models, as well as altering bifurcation thresholds. Across the parameter ranges we examined, the long-term behavior aligns into periodic patterns: Poincaré sets are finite, the dispersion of stroboscopic samples diminishes with increasing forcing amplitude, and the leading Lyapunov exponent remains nonpositive, suggesting no sensitive dependence on initial conditions. Therefore, in our investigations, the normalized kernel alters the periodic structure while avoiding the onset of chaos.