Conceivably have you once suffered through a common cold? Perhaps did you get rid of the infection? Certainly you survived it! You won't cure a cold itself, yet your immune system generally develops resistance to a particular cold virus within a few days. We call that cured by your own immune response!

Human papilloma virus is identical in that HPV virus is just another virus. And you will generally create immunity to HPV. Nevertheless, HPV is better at evading your body's immune system than are the cold viruses. Moreover you have to persist with more diligence to develop immunity to HPV virus.
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