This study focuses on the prevalence of COVID-19 disease along with vaccination in the United States. We have considered the daily total infected cases of COVID-19 with total vaccinated cases as exogenous input and modeled them using light/heavy tailed auto-regressive with exogenous input model based on the innovations that belong to the flexible class of the two-piece scale mixtures of normal (TP-SMN) family. We have shown that the prediction of COVID-19 spread is affected by the rate of vaccine injection. In fact, the presence of exogenous input variables in time series models not only increases the accuracy of modeling, but also causes better and closer approximations in some issues including predictions. An Expectation-Maximization (EM) type algorithm has been considered for finding the maximum likelihood (ML) estimations of the model parameters, and modeling as well as predicting the infected numbers of COVID-19 in the presence of the vaccinated cases in the US.