Fuzzy Fractal–Fractional Differential Framework for Pneumonia Transmission Dynamics

Abstract

Pneumonia caused by Streptococcus pneumoniae remains a major public health concern, especially in vulnerable populations. Its transmission is affected by uncertain and irregular factors such as environmental changes, contact variability, and imprecise clinical data. This study introduces a novel fuzzy fractal–fractional mathematical model incorporating epidemiological compartments: susceptible, exposed, infected, vaccinated, and recovered groups. To handle uncertainties in biological parameters, fuzzy logic is applied, while the fractal–fractional Atangana–Baleanu operator captures memory and hereditary effects in disease transmission. Theoretical analysis
confirms the existence, uniqueness, and Ulam–Hyers stability of the model. Numerical solutions are derived using Lagrange polynomial interpolation. Simulation results show a decline in exposed and infected populations within 20–30 time units, while vaccinated and recovered classes steadily increase and stabilize. The susceptible population also reaches a steady state, indicating potential disease control under uncertain conditions. Compared to classical and standard fractional models, the proposed model achieves improved convergence and better accommodates parameter imprecision. Overall, this integrated fuzzy fractal–fractional approach enhances the accuracy and robustness of pneumonia modeling, offering a valuable framework for analyzing infectious diseases under uncertainty in public health settings.