This paper aims to study the saturation effect on the infection and recovery process within a Susceptible-Vaccination-Infected model featuring an imperfect vaccine efficacy. First, we nondimensionalized the model under the assumption of a constant population, resulting in the reduction of the model from three to two-dimensional differential equations. The analysis indicates the presence of a disease-free equilibrium (DFE) and potentially multiple endemic equilibria (EE) within the model. The calculation of the basic reproduction number further explains the model's solution conditions. In particular, we discovered that a backward bifurcation is possible under specific saturation effect values. Dulac's criterion confirmed the absence of a closed orbit in the solution region, suggesting the global stability of the endemic equilibrium when the basic reproduction number exceeds one. To supplement the analytical study, a numerical simulation was conducted to generate a bifurcation diagram, autonomous simulation, and global sensitivity analysis. The global sensitivity analysis revealed that changing the vaccination rate or recovery rate could significantly impact the basic reproduction number. Moreover, the bifurcation diagram depicting the relationship between the transmission rate and vaccination rate demonstrated that increasing the vaccination rate while maintaining the transmission rate can reduce the proportion of infected individuals within the population.