We introduce and study a model stemming from game theory for the spread of an epidemic throughout a given population. Each agent is allowed to choose an action whose value dictates to what extent they limit their social interactions, if at all. Each of them is endowed with a certain amount of immunity such that if the viral risk/exposure is more than that they get infected. We consider a discrete-time stochastic process where, at the beginning of each epoch, a randomly chosen agent is allowed to update their action, which they do with the aim of maximizing a utility function that is a function of the state in which the process is currently in. The state itself is determined by the subset of infected agents at the beginning of that epoch, and the most recent action profile of all the agents. Our main results are concerned with the limiting distributions of both the cardinality of the subset of infected agents and the action profile as time approaches infinity, considered under various settings (such as the initial action profile we begin with, the value of each agent's immunity etc.). We also provide some simulations to show that the final asymptotic distributions for the cardinality of infected set are almost always achieved within the first few epochs.