There's an important #vaccine threshold you may not be aware of. This thread will give you a primer on "Critical Vaccine Coverage". Call it CVC.
CVC is the coverage needed to push R below 1 EVEN WITHOUT OTHER INFECTION CONTROL MEASURES.

If left unchecked, R is greater than one and infections grow exponentially. But if R is below 1 that means infections will eventually die down, they can no longer spread exponentially. One infected person infects less than one other when R is below 1.

Let's go through an example of CVC.
Assume that without any control measures, Ro = 3. Assume vaccines reduce transmission by 85%.
Then CVC = (1-1/Ro)/.85
= (1-1/3)/.85 = .78
With these assumptions, we need to cover 78% of the random popn to reach "critical vaccine coverage".

Put another way, 78% coverage forces R to drop below 1 just with vaccines. There are other factors that can sway this number, like higher/lower uptake among the greatest spreaders or more transmissible variants.

Once this theoretical CVC is reached, infections do not immediately stop, they still have momentum for some time as infections decay. But as long as vaccines *maintain* the assumed effectiveness as far as transmission, the numbers will decline.

Important to note that first dose effectiveness is lower than the 85% I assumed and may decline faster over time. But hopefully the example shows how vaccines can reach a critical threshold and "force" decline.
Also important to note that we get there a lot faster and with fewer

ICU patients and lives lost if we combine a vaccine rollout with public health restrictions. That's what the UK did and today they reported zero deaths for the first time in over a year.
https://twitter.com/BNODesk/status/1391786009587822594

This article by Marc Lipsitch inspired the thread, in case you want to read a more nuanced and detailed description of what I tried to generally sketch out. https://ccdd.hsph.harvard.edu/2020/12/17/covid-19-vaccines-and-herd-immunity/