By chance have you a while ago been down with acute rhinitis? Likely did you eliminate the virus? Certainly you recovered! You can't heal a common cold directly, however your body normally develops immunity to any cold virus within a couple of weeks. We call that being cured by your personal immune system!

HPV is the same since HPV virus is simply another virus. Therefore you could generally establish immunity to HPV virus. However, HPV virus is more adept at hiding from your body's immune system than are the cold viruses. So you have to try a little harder to obtain immunity to HPV.
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