Increasing social distancing will naturally lower the exponential growth rate, but unless the growth becomes linear, there is no reason to believe that herd immunity is anywhere close.
In the asian countries, we more or less know that they have been able to contain the virus, instead of waiting for herd immunity. That meanst that they must have found most of the infected cases (or at least quaranteened them). This gives a good indication of how many deaths are caused by a given number of infected. The numbers are not different enough from Italy to think that more than (at most) a couple of hundred thousand in Italy have reached immunity. If so, it means that we have about 2 more orders of magnitude of growth before herd immunity can be reached. That would be hundreds of thousands of dead, if not millions, just in Italy.
However if the epidemic spreads from region to region, you could reach a local herd immunity in the first regions infected way before seing the country-wide curve becoming linear.
But thanks, trying to find a counter-argument to your post made me realize a single-curve model probably doesn't make a lot of sense to try and understand accurately the spread of an epidemic.
I would think that this can only give you the infected from a few clusters and you can be surprised by an unknown cluster of patients suddenly needing intensive care.
Edit: And due to the long incubation period, that cluster can be quite large.
Of course, after the disease starts propagating internally, testing becomes irrelevant, and it's time to reduce its spreading speed.
i absolutely don't understand how can someone give any ratio about this disease knowing how only the most severe cases got tested ( except in SK), and that the symptoms are almost undistinguisable from the flu.