Models of contagion dynamics, originally developed for infectious diseases, have proven relevant to the study of information, news, and political opinions in online social systems. Modelling diffusion processes and predicting viral information cascades are important problems in network science. Yet, many studies of information cascades neglect the variation in infectivity across different pieces of information. Here, we employ early-time observations of online cascades to estimate the infectivity of distinct pieces of information. Using simulations and data from real-world Twitter retweets, we demonstrate that these estimated infectivities can be used to improve predictions about the virality of an information cascade. Developing our simulations to mimic the real-world data, we consider the effect of the limited effective time for transmission of a cascade and demonstrate that a simple model of slow but non-negligible decay of the infectivity captures the essential properties of retweet distributions. These results demonstrate the interplay between the intrinsic infectivity of a tweet and the complex network environment within which it diffuses, strongly influencing the likelihood of becoming a viral cascade.