## Higher case fatality rates can signal smaller infection totals

Apr 17, 2020

No one knows exactly how many “actual” Covid-19 cases there are in any given country. We know the number of “total” or “confirmed” cases, but these are just the cases that have been confirmed with a test. We can begin to estimate how many actual cases there are using the number of deaths.

If we have an estimate of the death rate (calculated as total deaths divided by total cases), then we can estimate the actual number of cases. If the death rate (case fatality rate, or CFR) were 1%, then 1 death would equate to 100 cases.

However, Covid-19, as with most pandemics, grows at an exponential rate. One person infects several others, who in turn infect others. Exponential growth is marked by *doubling time*: how long it takes the total number of infections to double. So far, Covid-19 has exhibited the ability to double about every 4 days.

The overall average “run” of the disease is about 3 weeks: it takes 3 weeks to go from infection to death. During that time, if the doubling time is 4 days, the disease will double 5 time.

So, with a CFR of 1%, a single death represents 100 cases three weeks ago. 5 doublings of 100 cases are: 100 > 200 > 400 > 800 > 1,600 > 3,200.

On the other hand, the WHO and others have suggested the mortality rate is higher than that. Globally, the CFR stands at about 3.5%. At that rate, 1 death is 3.5% of 28 cases. If 28 cases were to double 5 times, the result would be 28 > 57 > 114 > 229 > 457 > 914.

So, 30,000 deaths at a 1% mortality rate in the United States represents the possibility of 96 million infections (or something like 1/3rd of the USA). On the other hand, 30,000 deaths at a 3.5% mortality rate suggests 27 million infections. 30,000 deaths at a 4.5% mortality rate (which is more inline with the USA’s specific reality) suggests 21 million infections.

The actual death rate is far from certain because of the dearth of testing. But the fact that many different nations in many different contexts are seeing roughly the same kinds of percentages gives some weight in my book to the idea that the order of magnitude for the CFR is right, even if the precision is not quite there yet.

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