Fake news can have a significant negative impact on society because of the growing use of mobile devices and the worldwide increase in Internet access. It is therefore essential to develop a simple mathematical model to understand the online dissemination of fake news. In this study, we propose a point process model of the spread of fake news on Twitter. The proposed model describes the spread of a fake news item as a two-stage process: initially, fake news spreads as a piece of ordinary news; then, when most users start recognizing the falsity of the news item, that itself spreads as another news story. We validate this model using two datasets of fake news items spread on Twitter. We show that the proposed model is superior to the current state-of-the-art methods in accurately predicting the evolution of the spread of a fake news item. Moreover, a text analysis suggests that our model appropriately infers the correction time, i.e., the moment when Twitter users start realizing the falsity of the news item. The proposed model contributes to understanding the dynamics of the spread of fake news on social media. Its ability to extract a compact representation of the spreading pattern could be useful in the detection and mitigation of fake news.