Sure, except that's it's already happened. The US currently has something like twice as many vaccinated as recorded infections, with vaccination growing at something like 5x the rate of infections.

If viral infections actually followed am exponential growth curve then we would be close to 100% infections by now. Obviously that hasn't happened. In reality infections generally follow a logistic curve, in which only the very early part approximates an exponential curve.

Where did you get that virus is n^2 ?

If I'd have to guess, I'd say it is O(R^n)

it's tempting to think that, but no, in the simple non-mutating case, the exponent is fixed. the growth rate is exponential but the exponent does not grow with spread. they say the exponent ranges from 2-5 for covid, and drops with mitigation methods. that seems to clearly put it in n^2 to me.

that said, the mitigations do cause the exponent to fluctuate... and in countries where people are willing to cooperate it has dropped below 1 and the epidemic has been mitigated.

it is possible that this is actually a case calling where asymptotic analysis is inappropriate and in fact you have to resort to the raw data and measurement.

now that i think of it though, if you allow for mutations that are more infectious, then the exponent is not fixed and increases with spread... yikes.

I'm somehow still not convinced by your explanation. What exactly is N and what exactly is this measuring. I thought that N was a time period, and that the equation is something like

    Infected count = O(R^t)

But apparently you mean that N is the number of infection? And what are you measuring?

    ??? = O(n^R)

Where the ??? is also linear with the amount of people vaccinated?