Analysis of Streptococcus Pyogenes Transmission with Pseudo-Recovery Dynamics Using Fractal–Fractional Caputo–Fabrizio Derivative

Streptococcus pyogenes is a bacterial pathogen responsible for a wide range of infections, from the mildest throat conditions to the most severe invasive diseases, and it remains one of the major global health concerns. A new fractal–fractional mathematical model for its transmission dynamics is hereby proposed using the Caputo–Fabrizio (CF) derivative. Compared to the classical integer-order models, the introduced model characterizes memory and hereditary effects, hence it becomes more realistic in describing the pattern of disease spread and the effectiveness of interventions. The model presents a system of fractal–fractional differential equations in time that describes dependencies associated with epidemic spreading. The mathematical well-posedness of the model is established by proving the existence and uniqueness of solution utilizing fixed-point theory. Furthermore, the disease-free equilibrium (DFE), endemic equilibrium (EE), and basic reproduction number $R_0$ are derived to investigate the transmission dynamics and persistence of the disease. The derived condition allows us to prove the Ulam–Hyers stability of the system, hence, the robustness and resilience of the dynamics against small perturbations. For the numerical analysis, the modified Adams–Bashforth scheme has been adapted, which fits perfectly in handling the memory properties encoded in the CF operator. Furthermore, calibration of the model is done with real epidemiological data, which highlights strong agreement between the simulated trends and the real data observed.The numerical results demonstrate that the fractional-order parameters play an important role in the progression of the disease. Variations in these parameters influence the timing of peak infections, the duration of disease persistence, and the overall transmission pattern. Compared with the classical model, the proposed fractal--fractional framework provides a more flexible description of the epidemic process and offers additional insight into the underlying transmission dynamics. These observations suggest that fractal--fractional modeling can serve as a useful approach for studying complex infectious diseases and supporting the evaluation of disease control measures.