Fractional-order modeling of pneumonia transmission with vaccination, reinfection, and data fitting to England cases

Pneumonia remains a major global health concern, particularly in low- and middle-income countries, where it contributes substantially to morbidity and mortality. In this study, a fractional-order mathematical model was developed to investigate the transmission dynamics of pneumonia, incorporating key control measures such as vaccination, treatment of infected individuals, reinfection dynamics, and environmental interventions. The model employed three types of non-integer order differential operators—Caputo, Caputo–Fabrizio, and Atangana–Baleanu. The existence and uniqueness of solutions were established using fixed-point theory. Model calibration was performed using monthly pneumonia hospitalization data for children aged 0–14 years in England from January 2021 to March 2024, representing a post-COVID phase characterized by increased respiratory illness activity. Parameters were estimated through least-squares optimization under biologically realistic constraints, and the simulated results showed strong agreement with observed hospitalization trends, confirming the model’s validity. Numerical simulations demonstrated that higher fractional orders reduce transmission speed, lower the peak number of cases, and extend outbreak duration, reflecting the memory-dependent nature of fractional systems. Sensitivity analysis further revealed that effective environmental interventions and prompt treatment of infected individuals play a crucial role in reducing disease transmission. The study provided a robust and flexible approach to understanding pneumonia dynamics and highlights the importance of integrating vaccination, timely treatment, and environmental improvements as complementary strategies to sustainably reduce the pneumonia burden.