We formulate a mathematical model has been proposed to describe the stochastic influence of SARS-CoV-2 virus with various sources of randomness and vaccination. We assume the various sources of ran-domness in each population groups by different Brownian motion. We develop the correlated stochastic model by taking into account the various sources of randomness by different Brownian motions and distributed the total human population in three groups of susceptible, infected and recovered with reservoir class. Because reservoir play a significant role in the transmission of SARS-CoV-2 virus spreading. Moreover, the vaccination of susceptible are also accorded. Once we formulate the correlated stochastic model, the existence and uniqueness of positive solution will be discussed to show the problem feasibility. The SARS-CoV-2 extinction as well as persistency will be also discussed and we will obtain the sufficient conditions for it. At the last all the theoretical results will be supported via numerical/graphical findings.