As in my twitter comment at you, the random variables "treatment" and "infection" are very much correlated, so you shouldn't just multiply them like you would for independent random variables.
If you're going to say "infection = 0.4" ... and we know that hospitals are overwhelmed unless ≲1% of population catches it per month ... then it follows that, if you get infected, then you are almost definitely in a future scenario in which hospitals are overwhelmed. (Plug in "treatment=0".)
More precisely, we can oversimplify to two scenarios:
Your assumption that infection = 0.4 means that you think probably >50% chance that the second bullet point is the one that will come to pass. (Why are you skeptical about social distancing? It's working in Hong Kong... When thousands are dying each day, there would be a lot of political will for drastic measures, right?)
Anyway, under those assumptions, you should be able to prove that P(treatment | infection) ≈ 0.
So I guess I'm suggesting the simpler formula, risk = base_rate_untreated_mortality * age * sex * npec * context * infection, or something like that. Unless you decide you're very optimistic about social distancing succeeding, and consequently setting infection << 10%, in which case the "infection & treatment" scenario now becomes a noticeable contribution.
You are correct, and that simpler model gives an even greater risk. I'm skeptical about social distancing because hospitals become overcrowded once 1/1000 of the population gets infected, and they need one month to process the hospitalized. With that pace, the quarantine would need to last 83 years. Even if this estimate is wrong by 10x that implies quarantine duration of 8 years. So much about flattening the curve. The best hope is a vaccine, so the quarantine lasts for approx 1 year, but maybe much shorter if more resources are invested and barriers (such as rigorous testing requirements, China could be of help here) somehow avoided.
Wei Dai already talked about it here: https://www.lesswrong.com/posts/RukXjEvMfqDKRJaup/what-will-be-the-big-picture-implications-of-the-coronavirus?commentId=p6xZhhJLMBRdfXhe5
Lombardy has a population of 10M, at 5K confirmed infections they got overcrowded, 1/2000 of the population got confirmed infected, let's say the true number was 1/1000. I didn't check the Wei Dai's math but his number is similar "0.1% of Hubei's population have a confirmed infection, and its hospitals are already at the breaking point".
"In Daegu, 2,300 people were waiting to be admitted to hospitals and temporary medical facilities, Vice Health Minister Kim Gang-lip said. A 100-bed military hospital that had been handling many of the most serious cases was due to have 200 additional beds available by Thursday, he added."
And they have 10x hospital beds compared to the US.
By the way, that's another Roko, I'm not that guy :)
Oops, sorry confusing my Roko's!!
I was figuring there's 1 open hospital bed per 1000 people where I live (USA), which lets ~0.5% get infected per month. Are you sure it wasn't 0.1% hospitalized, rather than 0.1% infected? It doesn't really matter for this conversation, but it's still something I'd like to know.
I think we're largely in agreement. The question is, over a year (antivirals could be faster, fingers crossed), will people get sick of the increasingly large piles of bodies and demand more social distancing, or will they get sick of social distancing and demand less of it? Or swing back and forth each month??? It's hard to say...
Why are you skeptical about social distancing? It's working in Hong Kong... When thousands are dying each day, there would be a lot of political will for drastic measures, right?
I think you're right about social distancing working, and if you live in a country that has the capacity to mount an effective response, I'd probably put p(treatment) = 0.7 or so. Remember we won't see any effect from what Italy has just done for at least a week because of incubation, and if it doesn't work they'll just keep escalating the isolation and quarantine, and we know that a high enough response works (China, South Korea).
Also, I think there's a better than even chance that p(young + no preexisting conditions) is much lower than either individually - since the absolute numbers of young people in a lot of those studies were low.
Also also, and maybe the OP took this into account, the corrections for delay to death and underreporting skew the death rates even more strongly towards older patients.
I wouldn't discount the possibility of a saving throw in the case of the virus approaching its natural attack rate - massive mobilization to provide at least basic medical care (oxygen) on a huge scale. The UK government has floated ideas that sound a lot like that (field hospitals outside cities), and there has already been a colossal expansion in the production of protective gear in China. So I would put p(Treatment | infection) at 0.2 or so if you live in the UK or somewhere similar.
Finally, and possibly for the above reasons, Rob Wiblin estimated a probability of the same 1/10th as high as the OP, here and again here.
I think Rob Wiblin is confused about the death rate with overwhelmed hospitals - link. He thinks it's 1.6%, but I think it's really 5-15%.
I really don't think P(treatment) should be an input to your analysis at all; it should be an intermediate result, if it's even worth mentioning at all. P(treatment) lumps together wildly different things. For example, compare P(treatment | 1% of the population is infected) vs P(treatment | 50% of the population is infected). The former requires 50x more treatment capacity. I'm not saying here that P(treatment | 50% of the population is infected) is definitely <5% or anything like that; I'm saying, at the meta-level, that the value of P(treatment | 50% of the population is infected) is an intuitively-understandable quantity whose value is worth debating directly, whereas P(treatment) is not.
Likewise, I think P(treatment | infection) lumps together very different scenarios, some where almost nobody gets infected but you personally get unlucky, and others where almost everyone gets infected.
Assuming that he read your comment and the comments of people on his FB saying similar things, I think Rob is confident that aggressive testing and social distancing measures will arrest the spread (as they already have in at least 3 countries!), along with expansion of capability (already happening w.r.t. masks!), will ensure that we get sort-of-adequate access to healthcare, even if things are somewhat overwhelmed, like in Wuhan, so doubling or 5x-ing their mortality rate is a better guide to what is likely to happen, rather than guesstimating based on no treatment.
Well, that's entirely possible.
But I also think it's possible that the US (where I live) would be worse-off (in terms of hospital overwhelmed-ness) than the heart of Wuhan. The logic is: Hubei is <5% of the population of China, and the central government could draw on the resources of the other >95% of the country to marshal a response. And the other 95% was unencumbered by the extreme social distancing that Hubei was undergoing.
By contrast, as far as I know, there could be exponentially-growing community transmission in every city in the USA right now. (After all, we know about the Seattle outbreak because there happened to be the Seattle flu study checking random people in Seattle, not because we were checking random people in every city and only found community transmission in Seattle. If I understand correctly.) If there's a crisis everywhere at once, then obviously marshaling a response is harder, not least because the people trying to marshal the response are hampered by the social distancing measures.
Not to mention various other differences between the US government and Chinese government and South Korean government etc. :)
Epistemic status: high uncertainty.
Even if I'm wrong this model could be useful to others, you can plug your own numbers. Please correct my assumptions in the comments if you think they're wrong or you have new/better data.
Chance of infection
We don't know what percentage of people will get infected. Some experts say 40-70%. Around 50% of cases are symptomatic, but for young people it's closer to 90%.[1] Asymptomatic cases seem not to be infectious. The rate of infection among young people is being underreported because the symptoms are milder and not detected as covid but are mistaken for common flu.[1] According to the stats, the chance of infection is very small for children and 4x lower for 20yo than 60yo. [2] This is wrong, but we can accept it and move on because having mild symptoms is not fatal and hence equivalent to not having covid at all. From now on, by "infection" I mean "noticable infection". The chance of being asymptomatic is already factored in. Note: say 100% of people get exposed and the risk of infection grows linearly from 0yo to 60yo, that would give us total of about 50-70% of an infected population. Best effort estimate of infection chance:
infection = 0.4
Lack of treatment
We don’t know what percentage of people who have coronavirus die. That’s because most of infected people so far had received hospital care. Once the pandemic spreads, most people will not receive hospital care because hospitals will be full. What would the mortality rate for untreated patients be?
"every demographic has approximately equal hospitalization rates, which other sources suggest are 15% to 20%."[2][3]
Those "demographics" are crude estimates, but still 0-49 had the same admittance rate as older. With regards to those 20% who require hospitalization:
"5% of people who are diagnosed with Covid require artificial respiration. Another 15% need to breathe in highly concentrated oxygen - and not just for a few days."[2]
Some studies show 6% of people in critical condition, while still 18% of people have a severe case.[4] It's not clear what exactly "severe" means. The best I can estimate with that is:
base_rate_untreated_mortality = 0.12
Adjusting for demographics
How does age affect that? The data on that is messy but seems like <50yo have around 3x smaller chance of a "critical condition".[4] The chance is 2% for 15-49, 2.5% for people of all ages with no preexisting condition, 6% overall. Note: those people received treatment, so this is a lower bound. What happens in completely untreated cases is still unknown.
Age (30yo) seems to lower the chance by 1.5x
age = 1/1.5
Having no preexisting conditions seems to lower the chance by 2x.
npec = 1/2
Note: age and having no preexisting conditions are correlated. In this model, together they offer a protection of 3x.
Being male increases the chance by 1.5x.[5]
sex = 1.5
Contextual adjustments
No data controlled both for age and for preexisting conditions. There is some probability that the quadrant of (young, no-preexisting condition) have very low mortality. No idea what that probability is so I'm going to say 50%.
quadrant = 0.5
There is a chance that the virus will mutate into low-mortality strain. High amount of infected in Germany and still very low deaths - indicates that this may already happened.
mutate = 0.1
The "cure" which disables transmission may be found, and manufactured in enough quantities. [6]
cure = 0.1
Climate may be an important factor in slowing the disease down.
climate = 0.3
Even if (young, no-preexisting condition) receive preferential treatment, the hospital systems seem to break down at 5K infected in an area of around 20M people (Italy[7], South Korea[8]). Inside that area, the total number of infected may be 10M, even with some containment at the peak of infection it may be 4M, implying 800K people needing hospitalization, implying 1.25% percent of getting treatment if infected at that time. Oxygen tanks could get depleted at some point. Also, we may run out of doctors. Due to high viral load, doctors are under larger chance of being infected:
"A high number of medical workers have been infected — 10% in the Lombardy region in the north, where the virus first appeared."[7]
Still, containment measures may work well, so the hospitals will not be overcrowded, which implies that treatment will be available.
treatment = 0.2
context = (1 - quadrant) * (1 - mutate) * (1 - cure) * (1 - climate)
age30_treated_mortality = 0.002
risk = (base_rate_untreated_mortality * age * sex * npec * (1 - treatment) + age30_treated_mortality * treatment) * context * infection
print(int(1/risk))
Update
As steve2153 says, infection chance and treatment chance both depend on which scenario we are in: population-wide infection vs small-size infection. Therefore the model should be simpler:
risk = (base_rate_untreated_mortality * age * sex * npec) * context * infection
References
[1] https://www.medrxiv.org/content/10.1101/2020.03.04.20031104v1.full.pdf
[2] https://www.reddit.com/r/China_Flu/comments/fbt49e/the_who_sent_25_international_experts_to_china/
[3] https://slatestarcodex.com/2020/03/02/coronavirus-links-speculation-open-thread/
[4] https://www.nejm.org/doi/full/10.1056/NEJMoa2002032
[5] https://www.worldometers.info/coronavirus/coronavirus-age-sex-demographics/
[6] https://www.reddit.com/r/COVID19/comments/fe2gwq/sarscov2_cell_entry_depends_on_ace2_and_tmprss2/
[7] https://nypost.com/2020/03/07/italy-calls-in-retired-doctors-to-help-combat-coronavirus-epidemic/
[8] https://www.scmp.com/week-asia/health-environment/article/3052885/south-korean-coronavirus-patient-dies-home-600-wait