An important task of public health officials is to keep track of spreading epidemics, and the locations and speed with which they appear. Furthermore, there is interest in understanding how concerned the population is about a disease outbreak. Twitter can serve as an important data source to provide this information in real time. In this paper, we focus on sentiment classification of Twitter messages to measure the Degree of Concern (DOC) of the Twitter users. In order to achieve this goal, we develop a novel two-step sentiment classification workflow to automatically identify personal tweets and negative tweets. Based on this workflow, we present an Epidemic Sentiment Monitoring System (ESMOS) that provides tools for visualizing Twitter users' concern towards different diseases. The visual concern map and chart in ESMOS can help public health officials to identify the progression and peaks of concern for a disease in space and time, so that appropriate preventive actions can be taken. The DOC measure is based on the sentiment-based classifications. We compare clue-based and different Machine Learning methods to classify sentiments of Twitter users regarding diseases, first into personal and neutral tweets and then into negative from neutral personal tweets. In our experiments, Multinomial Naïve Bayes achieved overall the best results and took significantly less time to build the classifier than other methods.