Mathematical modeling is essential for understanding infectious disease transmission and evaluating control strategies. In this study, we investigate the classical susceptible-infected-recovered (SIR) epidemic model using both analytical and numerical approaches. The novelty of this work lies in the integrated presentation of analytical stability analysis alongside explicit numerical quantification of how vaccination and recovery rates affect epidemic peaks, supported by phase-plane interpretations. (Vaccination is incorporated as an external control parameter ν in the modified SIR model, where susceptible individuals are vaccinated at a constant rate ν and move directly to the recovered class). Unlike previous studies that focus solely on theoretical thresholds, we provide a practical, simulation-driven framework that directly links control parameters to outbreak outcomes. We analyze the equilibrium structure and stability properties of the disease-free equilibrium. The analysis shows that the basic reproduction number R 0 = β / γ acts as a threshold parameter governing disease spread. Using the numerical values β = 0.5, γ = 0.1, total population N = 1000, and initial infected I 0 = 1, we obtain R 0 = 5. When R 0 1, an outbreak occurs. Numerical simulations illustrate the temporal evolution and the influence of vaccination and recovery rates on epidemic progression. Our results quantify that increasing the vaccination rate from 0 to 0.3 reduces the infection peak by approximately 65%. Additionally, increasing the recovery rate by 50% reduces the infection peak by approximately 40%. These findings demonstrate the practical value of the model for public health planning.
Alnafisah et al. (Mon,) studied this question.