We develop a mathematical model for the transmission and spread of infections caused by the severe acute respiratory syndrome coronavirus 2 (SAR-CoV-2) which causes the coronavirus disease 2019 (COVID-19), a disease that has since been classified, by the World Health Organization, as a global pandemic. We focus attention on virus transmissions in a closed population and hence use a compartmental epidemic model to study the inherent dynamics of infections between the various subgroups of the population. We assume random interactions between members from different subgroups and hence we employ stochastic modelling techniques. In the absence of a vaccine for this novel coronavirus, governments worldwide have put in place various intervention strategies, including travel bans, lockdowns, screening, testing, quarantine, etc. in order to reduce and hopefully eliminate the transmission and spread of the COVID-19 virus. These interventions are built into our model and we investigate their effects and effectiveness. In particular, we observe that the two subgroups containing infectious individuals, namely; the subgroup comprising of unidentified asymptomatic individuals as well as; the subgroup comprising of un-quarantined symptomatic individuals; pose the greatest risk of the transmission and spread of infections. We therefore also observe from our results that; rapid (realtime) mass testing as well as effective (or mandatory) quarantine of all infected individuals are the fundamentally critical and necessary steps in reducing the internal transmissions and spread of the COVID-19 virus. In particular, our results indicate that lockdowns are only important to the extent that, if implemented effectively, they help in reducing the rate of transmissions (i.e. help to 'flatten' the transmission curves) and hence allow policy makers and healthcare practitioners to put in place the important processes of realtime mass testing and mandatory quarantine, including hospitalization and treatment. Our results are presented over a single wave (or part of a single wave) of COVID-19 infections in a given population. If conditions are repeated, the results would correspondingly extend to multiple waves of infection.