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Within only a few years after the launch of video sharing platforms, viral videos have become a pervasive Internet phenomenon. Yet, notwithstanding growing scholarly interest, the suitability of the viral metaphor seems not to have been studied so far. In this paper, we therefore investigate the attention dynamics of viral videos from the point of view of mathematical epidemiology. We introduce a novel probabilistic model of the progression of infective diseases and use it to analyze time series of YouTube view counts and Google searches. Our results on a data set of almost 800 videos show that their attention dynamics are indeed well accounted for by our epidemic model. In particular, we find that the vast majority of videos considered in this study show very high infection rates.