We develop a fractional SCIR (susceptible-carrier-infected-recovered) model for pneumococcal pneumonia using Caputo derivatives of order $0<\varrho \le 1$ to capture memory effects from long carriage, waning immunity, and reinfection. The force of infection explicitly accounts for carriers' transmissibility. Using a next-generation approach, we derive the basic reproduction number ${\mathcal{R}}_0$ and prove the global asymptotic stability of the disease-free equilibrium when ${\mathcal{R}}_0<1$ and of the endemic equilibrium when ${\mathcal{R}}_0>1$ via Lyapunov functionals and a fractional LaSalle principle. Numerically, we combine the Laplace-Adomian-Padé method (LAPM) with a fractional Adams-Bashforth-Moulton scheme (ABM) to capture memory-driven transients. A sensitivity analysis identifies transmission intensity and routing into carriage as the dominant epidemic drivers, while treatment and mortality exert mitigating effects. A control extension yields a closed-form, control-adjusted ${\mathcal{R}}_0$; a minimal vaccination threshold; and an optimal control problem solved numerically. Finally, we outline a calibration workflow linking the model-predicted incidence to surveillance data, permitting a statistical estimation of the fractional order. Altogether, incorporating carriers and fractional memory modifies the thresholds and persistence conditions, producing dynamics that are more consistent with pneumococcal epidemiology.

We developed a fractional-order glucose–insulin regulatory model in the Caputo sense to encode memory effects in metabolic dynamics. The three-equation nonlinear system employed component-wise fractional orders to represent heterogeneous memory depths across plasma glucose, insulin action, and secretion. We established well-posedness (existence, uniqueness), positivity, and boundedness, and assess local stability; oscillatory regimes were further examined via discrete-time Hopf conditions for the discretized dynamics. For computation, we implement the successive approximation method (SAM) and a fractional Adams–Bashforth–Moulton (ABM) predictor–corrector scheme. In head-to-head tests, ABM achieved lower residuals, better stability, and higher efficiency than SAM, with validation against frequently sampled intravenous glucose tolerance test (FSIGT) data and a global sensitivity analysis highlighting insulin responsiveness and glucose-threshold parameters as most influential. Residual analysis indicated that increasing the fractional order(s) toward the integer case reduced numerical error—for example, the representative state error $|\mathrm{\Delta }u|$ decreased from $129.6$ at $\nu =0.5$ to $34.1$ at $\nu =0.9$. These results supported the clinical relevance of fractional-order modeling for improved diabetes management, parameter tuning, and control strategy design.