Enhancing Disease Dynamics: A Comprehensive Analysis of SIR and SIRV Models with Vaccination Impact

Abstract

This study presents a refined mathematical framework for modeling infectious disease dynamics using classical SIR and extended SIRV mod
els, with a particular focus on the role of vaccination. We incorporate nonlinear vaccination effects and natural birth–death processes and de
rive a vaccination-adjusted reproduction number Rv to assess epidemic thresholds. Analytical results include new conditions for local and global
stability of disease-free and endemic equilibria, supported by Lyapunov based proofs. Sensitivity analysis quantifies the influence of key parame
ters such as infection rate, recovery rate, and vaccine coverage. Numerical simulations validate the theoretical findings and illustrate how increasing
vaccination rates can suppress outbreaks and shift the system toward a stable endemic state. The results provide actionable insights for public
health planning and epidemic control strategies.