Analyzing pneumonia transmission with a fractional SVEIR model: the role of vaccination in disease eradication

In recent years, pneumonia has emerged as a significant global health concern, particularly affecting children under five and adults over 65. This study employs the Caputo-Fabrizio fractional derivative to analyze the dynamics of pneumonia using a deterministic Susceptible-Vaccinated-Exposed-Infected-Recovered (SVEIR) model. The research focuses on stability analysis, the basic reproduction number, and equilibrium points within the framework of dynamical systems theory. The study examines disease-free equilibrium states and establishes conditions for their local asymptotic stability. The Lagrange interpolation method is applied to obtain numerical solutions, and MATLAB simulations are conducted to assess the impact of various parameters on disease progression. The reproduction numbers, both with and without vaccination, provide crucial insights into population responses to immunization. A comparative analysis demonstrates that if the vaccination rate surpasses a critical threshold, the prevalence of pneumonia declines, ultimately leading to disease eradication. This study underscores the vital role of vaccination in pneumonia control and presents a mathematical framework for predicting disease behavior under different epidemiological conditions.