Sorry – while I do remember the existence of logistic growth from school, my knowledge doesn't go much beyond that, either, so I can't be of any help there.
As for your original question – I've been wondering about the same thing.
My personal back-of-the-envelope calculation was more along the lines of "If the vaccine is 90 % effective (the numbers given for Germany during the Delta wave and before the start of the third round of vaccinations were closer to 90 than to 95 %) in preventing hospitalisation and 75 % (for example) of the population are vaccinated, it means that case numbers in hospitals are 75 % x 1 + 25 % x 10 = 3.25 times higher than they would be with full vaccinations.", though I suppose there are enough holes that could be poked into that calculation, too.
A three-fold reduction in case numbers is nothing to be sneezed at, but on the other hand as long as the spread is still behaving mostly exponentially instead of turning into logistic growth and slowing down again that's only one-and-a-half doubling periods, so not that much of a buffer, either. Of course the vaccines are also somewhat (if not as well as was originally hoped) preventing some spread, so with full vaccination growth rates themselves would also be lower and hospital cases be reduced by more than threefold, but whether that's then enough, who knows?
In the end I decided there's no point in worrying myself further here, because there are too many unknown factors in getting a reasonable result that I can't easily answer and ultimately I have no desire in becoming a full-time epidemiologist.
As for your original question – I've been wondering about the same thing.
My personal back-of-the-envelope calculation was more along the lines of "If the vaccine is 90 % effective (the numbers given for Germany during the Delta wave and before the start of the third round of vaccinations were closer to 90 than to 95 %) in preventing hospitalisation and 75 % (for example) of the population are vaccinated, it means that case numbers in hospitals are 75 % x 1 + 25 % x 10 = 3.25 times higher than they would be with full vaccinations.", though I suppose there are enough holes that could be poked into that calculation, too.
A three-fold reduction in case numbers is nothing to be sneezed at, but on the other hand as long as the spread is still behaving mostly exponentially instead of turning into logistic growth and slowing down again that's only one-and-a-half doubling periods, so not that much of a buffer, either. Of course the vaccines are also somewhat (if not as well as was originally hoped) preventing some spread, so with full vaccination growth rates themselves would also be lower and hospital cases be reduced by more than threefold, but whether that's then enough, who knows?
In the end I decided there's no point in worrying myself further here, because there are too many unknown factors in getting a reasonable result that I can't easily answer and ultimately I have no desire in becoming a full-time epidemiologist.