The combination of health social data analytics and machine learning led to the so called big geo-social data. However, analyzing such data brings significant challenges. Therefore, new methods and high computing tools are required to effectively analyze the big data from social media platforms. In this direction, we propose an approach for gathering health-related information from social networks and combine such data with geographical, social and temporal data normally embedded in social media. Our research may offer insights into diseases, uncover social dynamics that lead to broad phenomena and forecast disease outbreaks. Social media played also a key role in the management of COVID-19 especially in tracking disease spreading as users self-report their health-related issues. In this paper we present a case study of the proposed methodology to monitor COVID-19 spreading characteristics. Our aim is to understand spreading dynamics of the virus in US, through the analysis of people movements between US states by exploiting official surveillance data and geo-tagged tweets related to COVID-19. In the analysis we investigated human mobility patterns by considering COVID-19 related tweets posted in regions exhibiting high correlation with the official number of COVID-19 cases. As first step of the methodology, we mine users trajectories from the collected geo-tagged posts and build a mobility map including the most frequent movements. After that we extracted a set of different spatial-temporal features characterizing the trajectories, including the frequency of visit in a specific locations, the direction of movements among locations, the frequency of movements. The approach gives us the possibility of monitoring the spread of the epidemics and detecting outbreak locations.