I highly doubt we'll have an exponential curve for manufacturing supply.
Even a 2x increase is incredibly difficult. Just look at n95 manufacturing, a much easier problem to solve. We never saw exponential curve in its manufacturing.
Not much reason to believe vaccination rollout will be exponential. Maybe quadratic.
That’s what they said about the human genome project, back when it looked like it was going to take decades because of how slow the progress had been. (They finished ahead of schedule.)