Epidemiological Modeling of Pneumococcal Pneumonia: Insights from ABC Fractal-Fractional Derivatives

This study investigates the dynamics of pneumococcal pneumonia using a novel fractal-fractional Susceptible-Carrier-Infected-Recovered model formulated with the Atangana-Baleanu in Caputo (ABC) sense. Unlike traditional epidemiological models that rely on classical or Caputo fractional derivatives, the proposed model incorporates nonlocal memory effects, hereditary properties, and complex transmission dynamics through fractal-fractional calculus. The Atangana-Baleanu operator, with its non-singular Mittag-Leffler kernel, ensures a more realistic representation of disease progression compared to classical integer-order models and singular kernel-based fractional models. The study establishes the existence and uniqueness of the proposed system and conducts a comprehensive stability analysis, including local and global stability. Furthermore, numerical simulations illustrate the effectiveness of the ABC operator in capturing long-memory effects and nonlocal interactions in disease transmission. The results provide valuable insights into public health interventions, particularly in optimizing vaccination strategies, treatment approaches, and mitigation measures. By extending epidemiological modeling through fractal-fractional derivatives, this study offers an advanced framework for analyzing infectious disease dynamics with enhanced accuracy and predictive capabilities.