In this chapter, we examine a COVID-19 epidemic model that involves a continual influx of susceptible individuals, coupled with a non-monotonic disease transmission rate. Vaccination to susceptible individuals in terms of recovery is the control strategy that is considered in the model to reduce the effect of the disease. We conduct both local and global stability analyses for the disease-free equilibrium point and the endemic equilibrium point, guided by the values of the basic reproduction number, \(R_0\) . Consequently, the disease is eradicated from the population when \(R_0\) is less than unity, and it persists when \(R_0\) exceeds unity. The center manifold theorem is employed to examine the dynamic behavior of the disease-free equilibrium point at \(R_0=1\) . We examine various types of bifurcations along with their epidemiological interpretations. Numerical simulations validate the theoretical findings presented in the manuscript. Sensitivity analysis is conducted to pinpoint the influential model parameters that exert the most significant impact on the basic reproduction number of the proposed model.