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What % of infected are hospitalized
#1
The American Assn of Pediatricians gives weekly numbers for COVID in children.  Their numbers are extracted from the states' public data, and of course each state does it differently.  What counts as children may be ages 0-17, 0-18, 0-19, or 0-20.
Children and COVID-19: State-Level Data Report

Among their data was a state-by-state breakdown of what percent of children's cases resulted in hospitalization.  I thought this was a useful number to know, so I calculated it from the CDC tables by dividing
hospitalizations/100K for age group in last week  by
cases/100K for age group in last week

These numbers are likely to be low because cases are climbing and hospitalizations lag behind cases.  But it is a first crack at it.
From:
https://covid.cdc.gov/covid-data-tracker...csovertime
https://gis.cdc.gov/grasp/COVIDNet/COVID19_3.html

Week ending  7/31/2021
Age   Wk Cases/100K  Hosp/100K    Hosp/Cases
 0-4    59.8         1.2         2.0%
 5-11   84.8
         0.4         0.5%
12-15   96.4
         
12-17 ~104.5         0.7         0.7%
16-17  120.8
18-29  157.9         2.7         1.7% 
30-39  146.7         5.5         3.7% 
40-49  120.2         7.0         5.8%
50-64   84.1         8.3         9.9%
65-74   59.3         9.6        16.2%
75+     52.4       ~13.7        26.1%
75-84               12.7
85+                 16.3


Because the age ranges were different, I had to calculate some of the numbers from others, indicated by "~"

I find these numbers surprisingly high.  Ok, maybe 26% of 75+ have to be hospitalized.  That seems dramatic.  But one out of 6 cases in the age group 65-74 have to be hospitalized?  One out of 10 cases in 50-64?  One out of 200 K-6 student cases?  (Remember, they aren't vaccinated)

The average size of a primary school in the US is 442 (in 1999-2000).  If COVID burns through all the unvaccinated students, an average of 2 will need to be hospitalized.
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#2
POssibility of denominator being significantly larger than that derived from these statistics?
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#3
Certainly there is a far greater number of COVID infections than diagnosed infections.  But, diagnosed infections are one of very few statistics we have.

My surprise at the high percentage of hospitalized led me to think how few cases are actually being diagnosed.  I expect there are many out there that
  1. truly don't notice they are sick, 
  2. know they are sick but don't get a test because they don't want to be saddled with isolation (or those around them to be quarantined), 
  3. know they are sick but don't realize it could be COVID, or
  4. get a false negative (and believe it, or choose to believe it)
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#4
The August 4th IHME update on COVID for the US estimates (Fig 9) that 25% of the COVID cases in the US are detected.

The same document indicates excess deaths from COVID are 50% higher than reported deaths from COVID.
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