The global COVID-19 pandemic has underscored the importance of understanding and managing infectious disease dynamics. This paper focuses on the numerical solution of the Susceptible-Infectious-Recovered (SIR) model, a widely used mathematical framework for studying disease transmission. The numerical solution is implemented through the Adams–Bashforth method, known for its efficiency in solving ordinary differential equations. The project incorporates the element of vaccination into the classical SIR model to explore the impact of vaccination campaigns on disease dynamics. This project contributes to the ongoing efforts to understand and mitigate the effects of COVID-19 by offering a computational perspective on disease dynamics and vaccination strategies. The combined utilization of the Adams–Bashforth and RK4 methods enhances the accuracy and efficiency of the numerical solution, making it a valuable tool for researchers and policymakers alike in addressing the challenges posed by infectious diseases. The results analysis shows that the vaccinated model outperformed the unvaccinated model, with fewer estimated infections in general, indicating that the Adams–Bashforth method is efficient in approximating the further behavior of vaccine and number of infected people interaction.