No, this is only true if the R value is constant in time. Not all functions with a positive first derivative are the exponential function.
Because otherwise the growth is indeed exponential, yeah?
Of course, the epidemic curve described by that function would indeed bounded below by an exponential function on part of its range, but the same is true of any function with positive derivative, and calling for example f(x) = x^2 exponential for that reason makes the term meaningless.
This is presumably what the OP meant by “as long as R>1, the curve is exponential”. But this is literally equivalent to saying “as long as f’(x) > 0, f is exponential” which is just not a useful concept.
The initial spreading will be exponential in the beginning - as every actual curve illustrates - if the disease is left unchecked as OP suggested, until R_t values go down again due to immunity. That's all I meant to say.
[1] https://www.sciencedirect.com/science/article/pii/S002251932...
exponential in the beginning == this part of the function can be approximated by a function ae^xb where a>0 and b>1
vs.
first derivative bounded away from zero in the beginning == any function that increases, including linear functions with constant first derivative and polynomials with linear first derivative
Or do you think all increasing functions are the same..?
It is easy to show that for any function whose derivative is continuous and positive at 0, there is an exponential function (properly translated such that they agree at 0) that has similar properties.
You should be specific about what properties you're talking about. What you're saying is that any function increasing function grows faster than some exponential function on a finite interval.
Still, you can observe f(x) on x ∈ [0, 1] and see that it is growing linearly. And you can observe g(x) on ∈ [0, 1] and see that it is growing exponentially.
I do not see the value in discussing the rate of exponential growth of f. Where as for g, there is a parameter with value ln(2).
If data looks like f, don't try to fit an exponential function to it (whether it's a least-squares fit, or any other objective f > g.
Are you trying to say anything else?
In a nutshell, the German government and every other country on earth is doing what you propose, except that they are vaccinating at the same time and try to keep the curves flat (but your use of "slow" also suggests this, so it's not clear what else you're suggesting).