Given the sheer diversity of HPV strains, I'm doubtful that we'll ever eradicate it, as a number of them are either genuinely or largely benign, and an out and out viral eradication campaign is a long term, sustained and global effort that is invariably very expensive.

Actually getting the vaccines that cover the high risk strains into the developing world is a much more obtainable goal, and even that is extremely challenging for a number of reasons.