COVID-19 has become one of the most widely talked about topics on social media. This research characterizes risk communication patterns by analyzing the public discourse on the novel coronavirus from four Asian countries: South Korea, Iran, Vietnam, and India, which suffered the outbreak to different degrees. The temporal analysis shows that the official epidemic phases issued by governments do not match well with the online attention on COVID-19. This finding calls for a need to analyze the public discourse by new measures, such as topical dynamics. Here, we propose an automatic method to detect topical phase transitions and compare similarities in major topics across these countries over time. We examine the time lag difference between social media attention and confirmed patient counts. For dynamics, we find an inverse relationship between the tweet count and topical diversity.