Only time will tell. But usually, vaccines will induce longer immunity than the real diseases themselves because they have been optimized for immunogenicity.

There will be no eradication of the virus, certainly not world-wide, short of multiple waves of vaccinations. Not even with one type of vaccine, but more likely more than one.

Some of the developing world might get vaccinated with a cheaper, more traditional vaccine, but later. But there is no good vreason why you can't have huge vaccination campaigns with cooling trucks at least in their bigger cities.