This study applies dynamical and statistical modeling techniques to quantify the proliferation and popularity of trending hashtags on Twitter. Using time-series data reflecting actual tweets in New York City and San Francisco, we present estimates for the dynamics (i.e., rates of infection and recovery) of several hundred trending hashtags using an epidemic modeling framework coupled with Bayesian Markov Chain Monte Carlo (MCMC) methods. This methodological strategy is an extension of techniques traditionally used to model the spread of infectious disease. We demonstrate that in some models, hashtags can be grouped by infectiousness, possibly providing a method for quantifying the trendiness of a topic.