Most predictions have a second peak.
The problem with that mitigation is fairly simple.
X number of people will get the disease. That number *can* change, but not much.
If you have a graph with that nice curve on it, think of the number of infected people as being the *area* under the line. That means the area under that line is going to be about the same.
A high peak with that area will lead to a quick drop off to zero. Area's all there, toute suite.
If you keep the line low early on, there are several options: (1) The line stays low but goes far, far to the right before it drops close to zero. It lasts for years and is an annoyance as it kills, but hospitals are barely inconvenienced while we're on lock down until there's a vaccine *and* the vaccine has been given to most of the population of a country. (2) You keep it fairly low so hospitals can manage, but it's more than an annoyance and we're still locked down. Risk is it can get out of control, this is an unstable scenario. (3) You manage to tamp it down, but as soon as you release restrictions you get another peak.
That last option is what most flattened-curve predictions give.
Even if you do actually eradicate the virus in an area, since different parts of the world have different programs going, it'll be really hard to prevent the virus from returning. That would require extreme measures to restrict and control travel. This would not just be airports, but might well require absolute closure of borders, both land and sea borders near international borders. If Florida's cleared and there's an outbreak in the Bahamas ...
We're still not actually sure of R0 and mortality for this bug.