I believe you're focusing on the days aspect of that equation without considering what a lower T_double/p_detect could mean relative to hospitalizations.

If the real number of cases is doubling significantly faster than those detected -- say due to not testing them because their symptoms are mild enough to be missed entirely or to not seek treatment -- the calculations around things like mortality rate and severity of illness may yet change. As I understand it, scientists have already identified mutations of the virus that may also impact its ability to spread or injure infected hosts. I should note that these changes can swing both good and bad, I'm simply pointing out that there are more variables and that these variables changing may represent something not obvious from the equation.

I'm not an expert; I have fun reading about medical statistics. Most papers I've read take the time to examine these various confounding factors and what unexpected variations in their proposed equations might mean.

If the number of real cases double every N days, the number of people needing hospitalization is also likely to double every N days. If not, it probably means that the infection rate spread at different rates in high risk vs low risk part of the population, but unless those populations are strongly segregated from each other, that is unlikely.

This could go in both directions. In both cases, though it would require that the high-risk subpopulation is primarily infected by each other (or staff) instead of being infected from the general population. If high-risk individuals are primarily infected by low-risk individuals, the doubling time for each group should be the same.

Note that such an effect could go in both directions. The high risk subpopulation could get it at a higher rate the rest if we let the infection spread like wildfire in hospitals and elderly housing centers, or it could spread at a lower rate if the risk patients are kept isolated.

The point is that the mortality rate estimates are going to be totally out of whack without accurate measurements. As well as estimates of the currently infected.

This matters a lot. The first number is obviously important. The second matters so Italy can plan for whether things have peaked yet, or how many weeks out until it does.

The mortality does matter a lot if we plan to let everyone become exposed. However, if we do that, we must expect 15% of the people that are currently having enough symptoms to be diagnosed in Italy to perish. Even if only 20% of the infected are diagnosed there now, that is up 14 million people in Italy (if everyone gets it), out of which 2-3 million would die.

Or, if applied to the US, 10 million.

Alternatively, we can shut down the virus in the same manners that China, South Korea or Singapore have been doing, and keep it supressed until we develop efficient treatment.

Unfortunately, that means shutting down all affected regions really hard, and maintain a high level of supression for maybe 2 years.

However, if we are going to do so anyway, both the human and economic costs will be lower the earlier we begin.

If we allow exponential growth to continue with a doubling time of only a few days (even if we can move it from 2-3 days to 6 days), the time until this move is forced is very limited.

That assumes you are sampling uniformly, and not in already saturated clusters. The doubling time is also not a known constant, as it depends on actions taken etc, and uncertainty in it will broaden the distribution.

Obviously, all this is basic statistics and should be known to epiodemiologists, who hopefully have some input to policies.

For large N_obs, this will maybe be less important as you're going to find the severe cases anyway, but the uncertainty is significant in the beginning stages, and it is unfortunately these stages where actions will have most impact.

As long as growth remains linear in an exponential plot, you are nowhere near the saturation point. When about half the population is already infected, the number of new cases should grow roughly linearly. In Italy, it seems to still grow exponentially.

Itally has already passed the number of simultanous cases that their healthcare system can handle. I don't believe they really want to find out at what level the saturation point can be found. That could mean hundreds of thousands, potentially millions, dead.