Pneumonia is often considered a common respiratory infection, yet its burden remains substantial, particularly among vulnerable populations. The disease is caused by various microorganisms, with Streptococcus pneumoniae being a major bacterial pathogen. Although medical interventions have mitigated its impact, outbreaks and persistent transmission continue in many regions. This reality motivated our research: to understand pneumonia dynamics better using a mathematical framework that reflects real-world biological complexities.
Traditional epidemiological models typically rely on integer-order derivatives, which assume that disease progression depends solely on the current state of the system. However, biological processes often exhibit memory effects, where past exposures, immune responses, and interventions influence future outcomes. To address this limitation, we adopted a fractional-order modeling approach using the Caputo-Fabrizio fractional derivative, which allows for the inclusion of memory effects without mathematical singularities.
We developed a Susceptible-Vaccinated-Exposed-Infected-Recovered (SVEIR) model to represent the dynamics of pneumonia transmission. Vaccination was explicitly included as a key intervention, allowing us to analyze its direct impact on disease spread. This model was designed not only to simulate infection trends but also to rigorously study the stability properties and long-term behavior of the disease.
A central component of our analysis is the basic reproduction number. This threshold parameter measures the average number of secondary infections per infected individual. We derived the reproduction number both with and without vaccination, enabling a direct comparison of intervention scenarios. To complement the theoretical analysis, we obtained numerical solutions using the Lagrange interpolation method and performed simulations in MATLAB. These simulations allowed us to observe how variations in vaccination rates, transmission parameters, and recovery rates affect disease dynamics. The numerical results consistently supported the analytical findings, showing an apparent reduction in infection spread as vaccination coverage increased.
A key insight emerged from the comparative analysis between vaccinated and unvaccinated populations. Even a modest increase in vaccination rates significantly reduced the peak and duration of infections. Once vaccination reached a critical threshold, pneumonia transmission gradually subsided. This highlights the importance of mathematical modeling in setting quantitative targets for public health interventions.
Beyond the technical results, this study emphasizes the importance of bridging mathematical theory with public health practice. Fractional calculus is often considered complex, yet our work demonstrates its practical relevance in epidemiology. By incorporating memory effects, fractional models provide a more realistic description of disease dynamics and intervention outcomes.
This research was also a collaborative effort that combined expertise in mathematical modeling and epidemiological interpretation. One challenge we faced was ensuring that the results remained understandable and meaningful to applied researchers and policymakers while maintaining mathematical rigor. Achieving this balance is crucial for translating mathematical insights into real-world impact.
Moving forward, this modeling framework can be extended to incorporate additional features such as waning immunity, seasonality, or stochastic effects. We hope this work will foster further interdisciplinary collaborations and demonstrate how advanced mathematics can contribute to effective disease control strategies. By sharing the story behind this paper, we aim to demonstrate how mathematical innovation and public health priorities can work together to address serious global challenges, such as pneumonia.