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Yeah, I have serious doubts about the totals presented so far given the requirement for tests that were not used or available for so many weeks.
In a few years the epidemiologists and statisticians will go over the death rates for the period of this pandemic and will be able to make some estimates of the actual death toll. They will have numbers for deaths from all causes, which were at a fairly steady rate before, and will be able to make adjustments for such things as fewer auto accicdent fatalites due to reduced travel. I suspect the verified death toll from COVID-19 will be less than 2/3 of these estimates.
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I suspect that a few governors who agree (or pretend to agree) with Trump that the virus is less dangerous than its reputation -- that they are not being aggressive about finding and counting cases. So they pass the word along to not count deaths that are possibles but not demonstrated by a virus test etc.
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The Governor of Florida fired the head of his stats reporting agency because she wouldn't lie and phony up numbers her boss wanted phonied, is that what you describe as not being aggressive?
She's since started putting out what she considers correct numbers in an effort to both expose the lies and probably win a significant award somewhere down the road.
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Run the numbers the other way. The current use of "verified" by testing for death toll numbers keeps out those who died with symptoms consistent with COVID-19.
So the estimates made after the fact will include those untested individuals implicitly. Would have to look up actual figures to be more exact, but for example using made up figures maybe there were 1 million deaths in total on average a month before the pandemic. During the pandemic the figures show up as 60,000 a month higher on average using the high of about 2,000 a day dying at the peak and verified.
The statisticians would then look at the breakdown of the deaths by category. If they saw for instance an average of 1,000 fewer auto accident fatalities, another 1,000 a month fewer accidental deaths working at a job, and so on that could be linked to people doing those things less during the pandemic, that would reduce the number of expected deaths during that period below that average seen before the pandemic. Some portion of that difference can be inferred to be from COVID-19.
If the number of deaths from other causes averaged 950,000 a month during the pandemic, that leaves a difference of 50,000 from the expected rate. With data from epidemiological studies such as post pandemic antibody tests they may estimate that 30,000 +/- 5,000 of those deaths not directly attributed to COVID-19, but in that 50,000 difference were from the disease.
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Just the increased deaths in senior care homes that have not been verified because they don't have tests easily available could be an additional 10,000 in the last 3 months. Time and proper analysis will tell but the current administration is working to minimize numbers wherever possible.