Social media is an appropriate source for analyzing public attitudes towards the COVID-19 vaccine and various brands. Nevertheless, there are few relevant studies. In the research, we collected tweet posts by the UK and US residents from the Twitter API during the pandemic and designed experiments to answer three main questions concerning vaccination. To get the dominant sentiment of the civics, we performed sentiment analysis by VADER and proposed a new method that can count the individual's influence. This allows us to go a step further in sentiment analysis and explain some of the fluctuations in the data changing. The results indicated that celebrities could lead the opinion shift on social media in vaccination progress. Moreover, at the peak, nearly 40\% of the population in both countries have a negative attitude towards COVID-19 vaccines. Besides, we investigated how people's opinions toward different vaccine brands are. We found that the Pfizer vaccine enjoys the most popular among people. By applying the sentiment analysis tool, we discovered most people hold positive views toward the COVID-19 vaccine manufactured by most brands. In the end, we carried out topic modelling by using the LDA model. We found residents in the two countries are willing to share their views and feelings concerning the vaccine. Several death cases have occurred after vaccination. Due to these negative events, US residents are more worried about the side effects and safety of the vaccine.