Nonlinear dynamics of Streptococcus Pneumoniae: A modified Atangana-Baleanu-Caputo fractional model with vaccination, asymptomatic carriage, and saturated hospital capacity

Abstract

Mathematical modelling of Streptococcus pneumoniae ($\p$) transmission is essential for understanding infection dynamics and evaluating intervention strategies. This study develops a comprehensive seven-compartment fractional-order model incorporating susceptible, vaccinated, asymptomatic carriers, exposed, symptomatic infectious, hospitalized, and recovered individuals ($\mathbb{SVAEIHR}$). The model employs the modified Atangana-Baleanu-Caputo ($\m$) fractional derivative to capture memory effects and hereditary properties inherent in disease transmission dynamics. Nonlinear interactions are incorporated through a saturated incidence function that accounts for behavioral changes at high infection levels, and a saturated hospitalization rate reflecting treatment capacity limitations. We establish the positivity and boundedness of solutions, demonstrating that all state variables remain non-negative and biologically meaningful. The disease-free equilibrium is derived analytically and validated numerically. Using fixed-point theory, we prove the existence and uniqueness of solutions under Lipschitz continuity conditions. Ulam-Hyers stability analysis confirms the robustness of the model to small perturbations. A predictor-corrector numerical scheme based on fractional calculus is developed for computational implementation. Comprehensive sensitivity analysis with fractional order $\kappa = 0.9$ reveals the influence of key epidemiological parameters—including the saturation parameter $k$, recovery rates $\zeta$ and $\zeta_H$, hospitalization rate $r$, and disease progression rate $\varsigma$—on infection and hospitalization dynamics. The results demonstrate that fractional-order modeling provides enhanced flexibility in capturing complex transmission patterns compared to classical integer-order models, offering valuable insights for pneumococcal disease control strategies and vaccination policy design.