Even without viral mutations there's lots that isn't yet known about the immune system reaction to the vaccines, including duration of protection.
How will that pan out? What is the mecahnism of the virus coming & going? Is it possible that the virus goes into hiding until the effects of the first vaccination wave wear off and then pops up again? Or are we hoping that the removal of a large number of people from the pool of available hosts means that the virus gives up and goes away? Or are we saying that, at this early stage, we just don't know?
(As usual, I'm not expecting a thorough answer - just point me in the direction of a good article...)
(My understanding) You are asking about "Herd Immunity". As more of the population have immunity from a disease so it gets harder for the virus to come across somebody it can infect e.g. If a person has the virus but never meets anybody who is not immune, that infection is stopped/closed off. The more of a population is immune, fewer infected people are able to pass their infection on to others.
As more and more people become immune so the virus spread is more and more contained/restricted as it is not being passed to people who can host an infection (and the virus needs to infect a new people, replicate and spread on to more people, etc.).
The result of this is the disease becomes contained or even eradicated. To eradicate the virus herd immunity has to be maintained for a fair time.
So the virus does not "give-up" and "go away", just infections are not being passed on.
The level of population immunity (through infection or vaccination) varies depending on the disease characteristics e.g. measles needs above 90% of the population to have immunity, Ebola 40%. For C-19 estimates of immunity levels needed for herd immunity are uncertain and changing but current thoughts seem around 60%-80%. Also I've not seen how the UK version Mutant ninja virus is going to impact the population immunity levels necessary to achieve herd immunity.
But is gets more complex as herd immunity also presents an "Evolutionary Pressure" on some virus in effect "encouraging" new strains to form.