Add to that a possibility of asymptomous infection. Not only that, but assuming this would be a parasitic or viral infection, them killing the host, especially before spreading, would not be beneficial for survival, so the infection would probably become nonleathal to majority, because the surviving strands would be the ones that stay hidden the longest.

In addition, if “the efficient erradication” missed a one zombie, what guaranties are there that it was JUST one zombie? Could you trust someone who has been in contact witha a zombie, but claims not being infected? Have you been in contact with a zombie recently, mayhaps? Are you sure you haven’t been?