Many studies have indicated the potential of using Social Networks for the early detection of public health events, such as epidemic outbreaks, so that a faster response can take place. Anyhow, the most of these studies are focused on one or two diseases, and consequently to date it is not clear if and how different outbreaks give rise to different temporal dynamics of the messages. Furthermore, it is not clear if it is possible to define a single generic Data Mining solution for the detection of epidemic outbreaks from this Big Data, or if specifically tailored approaches should be implemented for each disease. To get an insight on this issue, we collected a massive dataset of Twitter messages to extract relevant information regarding different outbreaks from different countries in 2011. The manual analysis we conducted allowed us to define some macro-classes of diseases. Results show that there is a considerable variability in the temporal dynamics of Twitter messages from different diseases, and that the identification of a suitable source of information, to define a ground truth suitable for the assessment of time series analysis algorithms, is a challenging task. Finally we also report on a special case we found, highlighting that a lot of research has still to be done in this field.