Let’s say we successfully flatten the curve, and we avoid the massive run on hospitals we would otherwise have in a couple weeks. Now what? Is the expectation that the disease will disappear for lack of transmission? Or are we expecting that a significant number of us still get it, and we just need that to happen at a more convenient (over a longer period of) time? If, as I suspect, the latter - don’t we actually need sufficient continued transmission to keep it going at a preferred rate?
Yes, the latter, as I understand it. Here’s a good eight-minute PBS video on the subject: What This Chart Actually Means for COVID-19 - YouTube
Flattening the curve is only meant to deal with slowing the rate of infection so that the health care system is not overburdened with a sudden, simultaneous onslaught of cases as is happening now. It has less to do with decreasing the number of infected and more with ensuring necessary resources for healthcare. The thinking is the more people being properly managed under professional care the less damage and fatality there will be. A tragic example as to the consequences of not flattening the curve is to see what’s currently occurring in Italy.
Instead of a curve, Cuomo used the analogy of a wave. A six foot wave washing over a town is bad news, but not nearly as bad news as a 20 foot wave.
Note that eventually there is a good chance that good treatments and vaccines will be available. So if the delay is long enough people may not suffer the bad effects at all.
Straightening the hill. 
Some day the law might get them but the mountain never will.
I’m glad I’m not the only one with that damn song stuck in my head.
^This.^ Squinting at the two curves on the chart and applying thumbnail calculus to it, the area under the two curves looks the same. That means the total number of infections will wind up the same no matter what we do, but keeping the peak under that dotted line means fewer will die, a win in my book.
Psst… other way around… 
Okay wait. Maybe it’s just a Friday afternoon and I need caffeine, but … slowing the infection rate would reduce the total number of people infected. Right? If 2 people each infect 2 people who each infect 2 people, then you’re up to 14 cases, if I’m doing my math right. If 1 person infects 1 person who infects 1 person, then you’re up to 3 cases. Even if they lifted the quarantine and the rate of infection went up, you’re still going to have a way lower total number of people infected if you reduce the infection rate.
ETA: I guess it depends on whether we get the infection rate threshold to less than 1.0 or not.
Okay, so sorry everyone has to read my whole thought process, but now that I thought it through, it’s clear to me that either of the situations described in the OP are possibilities. If we can slow the transmission rate from 3.0 to 2.0, for example, then eventually everyone would still get it, they’d just get it more slowly, because any number great than 1 multiplied by itself over and over again will eventually reach infinity. But if we slow the transmission rate from 3.0 to 0.5, then the disease would actually die out, because a number less than 1 multiplied by itself over and over again eventually reaches zero. Or if the transmission rate fluctuates between <1 and >1, then not everyone would get it but the disease wouldn’t die out, either.
This paper, first shared here by bump in this thread is very important to get some sense of what is out there relative to the concept and what might be required as the “now what?”.
First the limitations of the paper - they used just one set of possible assumptions, including that children were just as contagious as adults (and the best current evidence is that they surprisingly are much less so); and they assumed that the asymptomatically infected are 2/3s as contagious as those with symptoms, while more recent data suggests that it is more like half as contagious.
But still.
It is being cited as very sobering because by that model all of what they would call “mitigation”, slowing but not stopping the spread, results in overwhelming healthcare system capacity 8-fold.
They are concluding that the goal has to be suppression:
Problem though. Do that combination of all for just 5 months and you suppress cases numbers to below surge capacity but when you release it, well after 5 to 6 weeks you explode again, as bad as before. (Which happens to coincide with influenza season starting up again. :eek:)
Which was exactly the OP’s point.
Alternatively you can hold the suppressive measures in place for the 18 months that it takes to get a vaccine to market. OR try doing that with
The suppressive level interventions would be expected to be in place on for 2/3s of the 18 months period with “holidays” a third of the time.
For mitigation they found that cancelling mass gatherings did extremely little. Isolation at home of symptomatic cases for 7 days with voluntary quarantine of household contacts for 14 days (assuming 50% compliance), and social distancing of the over 70 crowd, was the least poorly effective approach, and did better than suppression with release after 5 months.
So under the assumptions of the model (using current case fatality rates, assuming children are as infective as adults, and that the asymptomatic are 2/3s as contagious as the symptomatic), the “now what?” is a very long very haul or having it with a surge delayed but not decreased.
Under those assumptions.
You got it. Technically, flattening the curve has no direct correlation to total number of infected. The number might go up or it might go down based on a number of unknown factors. But if you keep the infection rate down, the number of people who can be treated and get better goes up, and the possibility of people getting infected goes down.
And a series of three (and a 1/3) six-foot waves may have the same volume of water as a 20 foot wave, but will cause far less damage than the 20 foot wave would
Another useful analogy might be - if four planes land at the airport at the same time, the shuttle buses to the parking garage are going to be packed - leading to crowding, shoving, and general unpleasantness and injury (getting knocked over, or having your foot run over by a heavy bag) - but if the planes come in with a half hour gap between them, the shuttles are a much more pleasant situation, even though the same amount of passengers have used them that day.
That whooshing sound was the General Lee passing over your head. 
I like an action hero analogy -
If the hero can position himself in the hallway so he only has take on one minion at a time he can beat up a dozen of them. If they all attack him at once from all sides (which somehow they never do!) he’s dead.
We want to be the action hero taking on one of the dozen minions at a time, not having to fight 12 at once. It’s a long fight scene, like some of the fight scenes in Netflix’s Daredevil, but even without superpowers Matt could win like that. Exhausted by the end but still standing.
And therein lies the problem. The hero gets tired. He can fight 4 or 6 or 8, either together or sequentially, but after that he is tired and defeated. After being defeated, the remaining minion do the same damage as they would have before. The only difference is that it took a lot longer if he fights them serially.
If you let a minion or two through, slowly, they actually create more heroes while they are doing damage. By the time the last minion gets through, there is nothing but heroes left and the last minion does no damage at all.
And of course, the longer the fight lasts, the more time there is for another hero to arrive (analogous to giving us the time to actually increase the amount of resources available to care for sick people)
I interpreted the analogy wrong. Ignore my comment.
The new article from Thomas Pueyo (of Coronavirus: Why You Must Act Now-fame) spells out the case for suppression (the ‘Hammer’) followed by a staggered response of measures that are increased when infectiousness threatens to rise again (the ‘dance’) in (gory) detail, as well as giving the consequences of delaying the strategy, or following a route of mere ‘mitigation’ (hint: they’re not pretty).