Virginia had its first coronavirus death on Friday, March 14. Working backwards according to some numbers and calculations in a big article I highly recommend (Coronavirus: Why You Must Act Now), this means that, statistically, we probably had 692 to 1384 cases in the state that day, rather than the 41 confirmed. Obviously the real number of infections could be more or less, since we're talking exponential growth around 30% per day, which will amplify little differences. The range 692-1384 comes from 1% versus 0.5% mortality when medical services are good. If we assume that healthcare around here is great (0.5% mortality), that actually means 1 death implies 1000+ cases now, and 200 when the person got infected.
Well, hold on.
The big problem with the calculation is that it assumes the first 200 people caught the disease on the same day. That isn't totally impossible, given that COVID and SARS have super-spreading (and people flow across borders), but it isn't likely.
I'll try to fix that "big problem" in a minute. Let's look at the basic reasoning first, though.
The focus is on number of deaths because it's the only reliable count. That will be true regardless of whether people are quarantined, locked down, going to concerts, etc. Asymptomatic passengers get on planes and go to restaurants... corpses don't, really, and it's hard to miss the fact someone died. COVID-19 was discovered when someone died from an unknown virus in China. (It was seen in bats earlier, back in 2015, but never mind.)
For the first critical cases, medical staff can work their miracles. Later on, with overloaded hospitals, mortality goes up to 4 and 5% (it'll look higher thanks to an illusion of record-keeping as partial data pours in), but that isn't relevant early on. Thankfully 0.5% is the best estimate when everything's working (thanks go to Tomas Pueyo for explaining this so well in his article). Around 1 in 200 people wouldn't make it. That's on average, but the virus has not been mutating fast for a virus, and no one has a cure. It should be fairly consistent.
Let's try to address the inaccuracy I mentioned. It changes the emphasis more than the numbers, because of the wonders of exponents.
First we'll assume "low" mortality at 0.5%.
If the very 1st person who caught the virus in a region was the one who died, and we assume steady growth from 1 case, then there could be as few as 7 cases (on average) when that person died 17.3 days later (about how long it takes).
On the other hand, if the 100th sick person was the one who died, there would be 692 cases at that point, on average.
If the last of the 200 people was the one who died, or if they all contracted it on the same day, then there would be 1384 cases at that point.
The article estimated ~800, which is the exact same calculation as the one with "probable" next to it below, only rounding the exponent off to 3 for simplicity. Anyway, here are the calculations if you're curious (doubling happens every 6.2 days according to lots of data now):
1*2^(17.3/6.2) = 6.9
100*2^(17.3/6.2) = 691.8 ← probable
200*2^(17.3/6.2) = 1383.6
And if healthcare isn't quite as great and mortality is 1%, the expected total number of sick people on Friday would be more like:
1*2^(17.3/6.2) = 6.9
50*2^(17.3/6.2) = 345.9 ← probable
100*2^(17.3/6.2) = 691.8
Notice two of the numbers get repeated above (6.9 and 691.8). We're just highlighting different parts of the same curve. Neither mortality rate nor the patient's "position in line" for catching the virus alters that fundamental curve.
Looking at the reporting for Saturday, March 15, I see there are 6 states with 1 death each. Colorado and Georgia have about 100 cases each, but the others have a lot less: 45 for Virginia, 36 for Oregon, and Kansas and South Dakota with 8 and 9. Those numbers are systematically a lot less than 692, which should be a typical number of cases with 1 death reported so far. (Many would be silent or still incubating, so the discrepancy wouldn't be surprising even in a fantasy land of excessive testing.)
That's an average of 50 cases when we'd expect 692. So... for lack of a better approach, maybe multiply cases reported in the US by 14?
Unless I've missed something else major—other than that people get better and this changes the dynamics (but not much early in an outbreak).
Also, I must say calculus would give a slightly better estimate than 692... wait, scratch that! Exponential curves are neat. Integrate x*2^(17.3/6.2) from 1 to 200 and divide by 200. The mean number of cases by the time of the first death (whatever that person's position was in the first 200, averaging all the possible infection counts when they die, ie from 7 to 1384) is precisely the same, 692.
