The theory of learning in games has extensively studied situations where agents respond dynamically to each other in light of a fixed utility function. However, in many settings of interest, agent utility functions themselves vary as a result of past agent choices. The ongoing COVID-19 pandemic has highlighted the need to formulate and analyze such models which feature game-environment feedback. For instance, a highly prevalent virus may incentivize individuals to wear masks, but extensive adoption of mask-wearing reduces virus prevalence which in turn reduces individual incentives for mask-wearing. What is the interplay between epidemic severity and the behaviors of a victim population? For initial answers, we develop a general framework using probabilistic coupling methods that can be used to derive the stochastically stable states of log-linear learning in certain games which feature such game-environment feedback. We then apply this framework to a simple dynamic game-theoretic model of social precautions in an epidemic and give conditions under which maximallycautious social behavior in this model is stochastically stable.