Although the online campaigns of anti-vaccine advocates, or anti-vaxxers, severely threaten efforts for herd immunity, their reply behavior--the form of directed messaging that can be sent beyond follow-follower relationships--remains poorly understood. Here, we examined the characteristics of anti-vaxxers' reply behavior on Twitter to attempt to comprehend their characteristics of spreading their beliefs in terms of interaction frequency, content, and targets. Among the results, anti-vaxxers more frequently conducted reply behavior with other clusters, especially neutral accounts. Anti-vaxxers' replies were significantly more toxic than those from neutral accounts and pro-vaxxers, and their toxicity, in particular, was higher with regard to the rollout of vaccines. Anti-vaxxers' replies were more persuasive than the others in terms of the emotional aspect, rather than linguistical styles. The targets of anti-vaxxers' replies tend to be accounts with larger numbers of followers and posts, including accounts that relate to health care or represent scientists, policy-makers, or media figures or outlets. We discussed how their reply behaviors are effective in spreading their beliefs, as well as possible countermeasures to restrain them. These findings should prove useful for pro-vaxxers and platformers to promote trusted information while reducing the effect of vaccine disinformation.