Fake news has emerged as a pervasive problem within Online Social Networks, leading to a surge of research interest in this area. Understanding the dissemination mechanisms of fake news is crucial in comprehending the propagation of disinformation/misinformation and its impact on users in Online Social Networks. This knowledge can facilitate the development of interventions to curtail the spread of false information and inform affected users to remain vigilant against fraudulent/malicious content. In this paper, we specifically target the Twitter platform and propose a Multivariate Hawkes Point Processes model that incorporates essential factors such as user networks, response tweet types, and user stances as model parameters. Our objective is to investigate and quantify their influence on the dissemination process of fake news. We derive parameter estimation expressions using an Expectation Maximization algorithm and validate them on a simulated dataset. Furthermore, we conduct a case study using a real dataset of fake news collected from Twitter to explore the impact of user stances and tweet types on dissemination patterns. This analysis provides valuable insights into how users are influenced by or influence the dissemination process of disinformation/misinformation, and demonstrates how our model can aid in intervening in this process.