Application of LAPM and ABM methods to a fractional SCIR model of pneumonia diseases

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.