In the beginning, it was doubling every 2 days. Then, IIRC, hovered around doubling every 3-4 days for some time.
The acceleration rate appears to have been plateauing (I wish I could imitate Cuomo's pronunciation lol) and on a slight decline.
However, IMO, there are a few problems with assessing the rate of infection.
The UK was only testing serious/critical cases in hospital, then they expanded to other patients, and now onto hospital staff, etc.
Nearly all of those serious cases will have contaminated others, but who haven't been tested. A proportion may have been asymptomatic, but still carriers; others would have had few symptoms (not worth calling a doctor about); others a moderate level (sub-emergency), and a further proportion needing hospitalisation themselves, thereby adding another case to the count.
Where there has been effective contact tracing and testing, case rates will appear higher. In other places, until there is an adequate antibody test, there's no way of knowing what the case rate has actually been.
Even where lockdowns are in place, essential workers can still get infected and contaminate others; as can those in lockdown but who still need to go out for shopping or others reasons; plus a percentage of people who haven't followed the lockdown measures.
Then there is the issue of, e.g., a cashier getting infected. A member of a lockdown family gets infected and goes back home. Few people living together, whether in a family setting or in a flat share can, or would want to, self-isolate (quarantine is bad enough).
In the (few) countries that have imposed a more severe lockdown (in some cases literally boarded up), the case rate peaked, then dropped dramatically - but such extreme measures wouldn't be tolerated in non-authoritarian regimes.
In other places, where there has been an effective tracing, testing and isolating / quarantining strategy, the case rate will also show a rapid decline.
As PPE and testing increase, and testing result times speed up, rates will fall and it will be safer to ease off and get the economy up and running again and to have some semblance of a more normal life (albeit a "new normal" one).
