Numerical Simulation of a Fractional-Order Pneumonia Transmission Model with Carrier Compartment via Homotopy Analysis Method

This study develops a fractional-order VSCIR(Vaccinated-Susceptible-Carrier-Infected-Recovered) compartmental model to analyze pneumonia transmission transmission dynamics in Mississippi, USA. By employing the Caputo fractional derivative, the framework captures memory and hereditary effects that are often overlooked in classical integer-order epidemiological models. The resulting nonlinear system is solved analytically using the Homotopy Analysis Method (HAM), which provides a convergent series solution with explicit control over approximation accuracy. Model parameters are estimated via HAM calibration against recent pneumonia transmission incidence data from Mississippi (2019–2023). Our results demonstrate that the fractional order significantly modulates outbreak trajectories: lower values (e.g., 0.7) delay and reduce peak infections, effectively “flattening the curve” and reflecting real-world sub-diffusive transmission patterns. The estimated basic reproduction number confirms substantial epidemic potential, with carriers contributing approximately 47% of secondary transmissions. Sensitivity analysis identifies the recovery rate of symptomatic individuals and transmission rates as the most influential parameters. This work provides public health officials with a refined, memory-aware predictive tool and underscores the necessity of combining vaccination with active carrier detection for effective pneumonia transmission control in high-burden regions.