Death Rates and Testing Rates - Why Fewer Germans Die of COVID-19

Yesterday, a couple of friends approached me with the same questions: why does is look like COVID-19 is less deadly in Germany than in many other countries? So had I a closer look at the numbers at Worldometers. This graph illustrates the issue:

Deaths relative to reported COVID-19 cases on 3/25/2020. Note that the y-axis is logarithmic.

The graph shows the "raw case fatality rate" (CFR) for nine countries. The raw CFR is calculated by simply dividing the number of reported deaths by the number of reported cases; higher numbers indicate a more deadly epidemic. The nine countries in the graph fall into three categories:

• ~ 0.5% CFR (Australia, Germany, Austria) in green
• ~ 1.5% CFR (Switzerland, US, Denmark) in blue
• ~ 7-10% CFR (Spain, Iran, Italy) in red

It is well understood why the third group has a much higher rate. In all three countries, the large number of cases has overloaded the hospital system, and only a small fraction of patients that need a ventilator can get one; most of the other patients who cannot get a needed ventilator die, thereby increasing the death rate dramatically. But what is the reason for the differences between the first two groups?

Better Healthcare?

Theoretically, it could be that a better health care systems manage to keep more patients alive. But looking at the countries in the first two groups, this can be excluded: Switzerland and Denmark both have universal health care that is at least as good as health care in Germany, Austria, and and Australia.

More testing?

Another potential reason for the differences between the first two groups is that some countries perform more tests than others. Let's look at how many tests some of these countries have done; since the countries are vastly different in size, the numbers are normalized to tests per one million inhabitants:

This chart shows that the three countries with the lower CFR rates (Australia, Germany, and Austria) have done substantially more testing than two of the countries with the intermediate CFR rates (Switzerland and United States). This is a strong indicator that the observed higher CFR rates are caused by lower levels of testing. But let's look at some of the differences in more detail.

Limited testing in the USA and Switzerland

The US in particular had problems to get sufficient test capacity online; the COVID tracking project shows that fewer than 1,000 tests were performed in the US until March 4.  Testing capacity has increased in the last 10 days, which resulted in a dramatic rise in the confirmed case numbers. The US now has the highest number of active COVID-19 cases (72,702 on 3/26/2020), and the highest number of new cases per day. If the trend from the last 3 days continues, the US will have the highest number of total confirmed cases by tomorrow, passing both China and Italy.
Switzerland has chosen a very restrictive testing policy that excluded anyone with mild COVID-19 symptoms, unless they had additional risk factors like pre-existing medical conditions. The restrictions even excluded people with symptoms who had been traveling to high-risk regions like northern Italy.

Extensive testing in Germany, Austria, and Australia

Germany has done 120,000 tests per week in the last month. The testing guidelines require testing for anyone with symptoms who has been in contact with a confirmed COVID-19 case, works in health care, or has medical risk facors, but make testing for all other people with disease symptoms optional, depending on available capacity.
Austria reported that 3,138 tests had been completed by March 4. That's about four times more tests than the US, but Austria's population is 37 times smaller than the US population. Austria has announced an ambitious program to vastly increase testing capabilities.
Australia has performed a very high number of tests, both relative to the number of citizens and relative to the number of confirmed cases. The Australian testing guidelines are similar to those of other countries in that they require testing anyone with symptoms who has been in close contact with another COVID-19 infection; but while Germany limits this to confirmed cases, Australia also includes probable cases, even if not yet confirmed. Like Germany, Australia allows tests of symptomatic patients without additional factors if capacity is available.

The special case: Denmark

Denmark differs from the other five countries in groups 1 and 2 in that is has a raw CFR near 1.5% like the US and Denmark, but has done a similar number of tests (relative to population size) as Germany, Austria, and Australia. The reason for this is that Denmark has been able to slow down the exponential growth of the epidemic, as this graph of new confirmed cases indicates:

Daily new cases in Denmark show effectiveness of local restrictions

Denmark managed to stop the daily increase in new infections, reducing the number of new cases from 252 on March 11 to 132 two weeks later. For comparison, look at the daily new cases in the Netherlands:

On March 11, the Netherlands had 121 new cases. In the next two weeks, the number grew almost eight-fold, to 852 cases.
One effect of controlling the COVID-19 epidemic is that the raw case fatality rate (CFR) increases. This is due to the time lag between the diagnosis (the positive test) and the death. This can be a bit difficult to understand, so we'll look at it in the next section.

Time lag effect on fatality rates

To illustrate the time lag effect on estimated fatality rates, let us imagine a small epidemic that starts with 2 cases. Every week, the number of infections doubles, so that we have 4 cases in week 2, 8 cases in week 3, 16 cases in week 4, and 32 cases in week 5.
We'll further assume that every second person dies from the disease, but that it is a slow death that happens two weeks after diagnosis. So nobody will die in weeks 1 and 2; 1 person will die in week 3; and so on. For each week after the first death, we calculate the raw case fatality rate CFR by dividing the total number of deaths by the total number of cases. Here's a little table that shows the numbers for the first few weeks:

CFR calculation example 1:
Doubling every week, 50% fatality, 2 weeks between diagnosis and death

We can see right away that the calculated raw CFR is much lower than the actual fatality rate of 50%: the raw CFR is just 12.5%! This is typical for an epidemic in a exponential growth phase, as long as there is a significant time between diagnosis and death. The lower raw CFR reflects the growth of the epidemic between infection and death. In our example, the number of cases grows 4-fold in the 2 weeks between infection and death, so the raw CFR is 4-fold too low.
Another way of looking at this is to say that most of the infections are too young to die at any point. In week 4, for example, we have a total of 16 cases, but 8 of them are new, and 12 of them are from the previous week. Only 4 cases are at least 2 weeks old, and only this group has died by week 4: 50% of the 4 people.
We can use this knowledge to calculate a time-corrected case fatality rate (we'll call it tCFR). One way to do this is to look back at the earlier total case numbers, and use these for the CFR calculation. So to calculate the tCFR in week 4, we divide the total number of death in week 4 by the total number of cases 2 weeks earlier: 2 divided by 4 gives the correct fatality rate of 50%.

So what happens when an epidemic starts to slow down? Let's look at the extreme case, where we somehow stop all future infections after week 6:

CFR calculation example 2:
New infections drop to zero in week 7

We see that the raw CFR picks up in week 7, and increases to the expected 50% in week 8. At this point in time, all cases had enough time to die. If an epidemic is just slowed down rather than stopped completely, the effect will be somewhere between examples 1 and 2 above: closer to the the correct CFR, but still somewhat lower. Over time, as the number of new cases becomes smaller and smaller relative to the total cases, it will get closer and closer to the real CFR.


Now let us get back to COVID-19, and calculate some time-corrected CFRs. Studies have shown that the average time between onset of symptoms and death is about 19 days for COVID-19. If we assume that it takes 4 days between onset of symptoms and diagnosis, we can look at the confirmed case numbers 19-4 = 15 days in the past to get a time-corrected CFR. Here are the numbers that we get for six countries when we divide reported death on 3/25/2020 by the number of reported cases on 3/10/2020:

Raw CFR for 3/25/2020, and time corrected CFR using reported cases from 3/10/2020

The numbers for the time-corrected CFR are very large, between 9% and 215%. This reflects the very rapid growth of reported cases in the selected countries between 3/10 and 3/25, which ranged from 6.6-fold for Denmark to to 68-fold for the US. As discussed, a part of the observed increase is due to increased testing.

What does it mean that the time-corrected CFR rates for the US and Spain are above 100%? Well, the number says that the number of people who died is larger than the number of people who were diagnosed 2 weeks ago. The obvious cause for this is that the number of confirmed cases on 3/10 was substantially lower than the number of actual infections. This statement is actually true for all countries in the list: we know that the infections-fatality ratio is somewhere in the range of 0.5% to 1%, at least as long as the health care system is not overloaded.

If we compare the corrected CFR rates to the known IFR rate, what we get is the ratio of actual infections to confirmed cases. If we assume an IFR of 1%, then the reported number of cases in Australia reflects only 1 out of 9 actual infections. That sound incredibly high at first glance, but we must remember that this includes all cases that are "too young" to be diagnosed. Around March 10, the apparent doubling time in Australia was about 3 days. If we allow 6 days between infection and first symptoms, and another three days until the COVID-19 test was done and reported, we have 9 days. This is 3 doubling times, and therefore an 8-fold increase in infections! In other words, the number of actual infection in Australia on 3/10 was 9-fold higher than the reported case number primarily because most infections were pre-symptomatic ("too new").

This kind of analysis really just gives us "ballpark" estimates, since we do not have exact data about the delay between symptom onset, testing, and reporting. If all tests were done and reported within a day of symptom onset, then we'd look at just 2 doubling times, and more than 50% infections that escaped detection. Other uncertainties also remain, for example with respect to time between diagnosis and death, and for the "true" IFR, numbers which may be different from those seen in other countries. Indeed, the most credible IFR calculations resulted in a numbers closer to 0.5-0.6%.

But it appears likely that Australia did indeed test about a quarter to half of all infections, and possibly more. For Germany and Denmark, the number is slightly lower. Since Denmark has slowed the spread of the epidemic 2 weeks ago, the raw CFR of 2% can also be used to estimate test coverage; if we assume a true IRF of 0.5%, it indicates that about 25% of all infections were tested.

For the other countries, the corrected CFR numbers are much higher, indicating a larger number of infections that are not reflected in the official case numbers. For Switzerland, the overall factor is about 60, which includes both pre-symptomatic infections and cases that do not meet the stringent test requirements. For the United States and Spain, the actual number of infections on 3/10/2020 was probably more than 100-fold higher than the reported numbers (949 cases for the US; 1,695 cases for Spain).

Take home lesson

The COVID-19 testing practices in many countries are restrictive and have excluded a fraction number of infections.  When tests are limited to symptomatic patients only, the reported case numbers understate actual infections by a factor at at least 2, and possibly 8, excluding "young" infections, even if no further restrictions are in place. However, many countries have additional testing restrictions, for example by requiring severe symptoms or contact to a confirmed case. In countries with low test capacity and/or stringent test requirements, time correction analysis reveals under-reporting factors of 100 or higher.

Differences in testing rates can quickly be identified by comparing raw case fatality rates; low case fatality rates, like seen for Germany, Austria, and Australia, indicate higher testing rates. This analysis is supported by analyzing local testing restrictions, capacities, and number of tests performed.

Understanding that "confirmed case" numbers are likely to under-estimate the actual size of the epidemic is essential to choose sufficiently strict containment measures. Taking reported case numbers at face value is one of the reasons why so many countries all over the world have not enacted effective measures in time.
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I understand that many of my readers will find the claim of 100-fold under-reporting hard to believe, but some epidemiologists have issues similar statements. Empirical confirmation is rare, but can be found in Italy, where testing requirements differ between regions. One study where an entire town of 3,000 people was tested after the first confirmed case found 89 infections in the first round of tests, and 6 infections in a second round. This corresponded to an infection rate of 3%, which was 200 times higher than the reported infection rate for Italy at that time. By quarantining the infected individuals, the epidemic was stopped in this town without any additional infections.