R0 is partly socially determined.
The key points are these:
1) tests are a lagging indicator of where the infection was a week or so ago in the population
2) it's not perfect, because in order to conserve test kids, they are presuming infection at many medical centers for people presenting with related symptoms
3) you might argue "but that OVERSTATES cases!" -- and it does, but ONLY in people already seeking medical attention, not in the great unwashed like those who were at Spring Break.
4) The key is whether enough people who are unknown carriers, took this seriously enough, in the last couple of weeks, for the therefore-lowered transmission, to have pulled down the cases that might be about to happen, enough that the residual infections do not _still_ overflow the ERs.
(no good single verb tense for THAT one!)
IF we knew that those seeking medical care, were (as it were) draining the resevoir of potentially uninfected, I'd agree with your assessment of the parabolic shaped curve.
That is, the size/shape of the parabola will be fixed for each different value of social distancing. A simple parabolic model assumes the % of population distancing is constant over time.
Finally -- China is announcing an upsurge in new cases, they're trying to blame on foreigners. Some have predicted several waves since each wave makes people scatter back to social isolation, lowering immediate morbidity / mortality but preventing herd immunity. And then there's the issue of antibody-dependent enhancement, which the Chicoms sure as hell aren't on any soapbox about.
We were on an exponential curve. Still are, really but now we are at the top in the recovery phase. # new deaths flat. So as far as “new cases” we entered Recovery phase ten days ago. As a nation.