Social media and streaming platforms have reshaped music consumption, enabling songs to go viral and achieve commercial success. In this paper, we explore the use of epidemic models to represent, explain, and forecast music popularity on streaming platforms. We introduce a wave-based approach that captures multiple independent bursts of popularity, which is not possible using classic epidemic models. Using streaming data from Spotify for more than 1000 songs, we evaluate our approach’s ability to fit and forecast virality over time, comparing its performance with traditional time-series forecasting methods. Our findings show that our approach effectively captures viral dynamics, whereas it is less adapted to other aspects of popularity, such as long-term success. Moreover, it offers forecast accuracy comparable to conventional time series algorithms, with the additional benefit of providing interpretable parameters that shed light on the underlying diffusion processes.