Maybe have you once endured a runny nose? Probably did you eliminate the infection? Certainly you recovered! You won't cure a cold itself, but your system generally creates resistance to any cold infection within several days. That is called healed by your own immune system!

HPV is the same because human papilloma virus is just an additional infection. And you can normally develop immunity to HPV virus. Nonetheless, HPV virus is better at evading your body's immune system compared to the common cold viruses. Moreover you must persist a little harder to acquire immunity to HPV virus.
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