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This study analyzes the SARS-CoV-2 genome sequence mutations by modeling its nucleotide mutations as a stochastic process in both the time-series and spatial domain of the gene sequence. In the time-series model, a Markov Chain embedded Poisson random process characterizes the mutation rate matrix, while the spatial gene sequence model delineates the distribution of mutation inter-occurrence distances. Our experiment focuses on five key variants of concern that had become a global concern due to their high transmissibility and virulence. The time-series results reveal distinct asymmetries in mutation rate and propensities among different nucleotides and across different strains, with a mean mutation rate of approximately 2 mutations per month. In particular, our spatial gene sequence results reveal some novel biological insights on the characteristic distribution of mutation inter-occurrence distances, which display a notable pattern similar to other natural diseases. Our findings contribute interesting insights to the underlying biological mechanism of SARS-CoV-2 mutations, bringing us one step closer to improving the accuracy of existing mutation prediction models. This research could also potentially pave the way for future work in adopting similar spatial random process models and advanced spatial pattern recognition algorithms in order to characterize mutations on other different kinds of virus families.