Extension to the Question: How Can Statistics Be Used to Determine the Dark Figure of Corona Infections?

Streuspanne – Our Blog on Statistics and Jurojin / April 28, 2020

In the last post we presented a simple approach that can be used to statistically estimate the dark figure of corona infections (with regard to SARS-CoV-2). We originally developed the approach to explain at least a large part of the very different mortality rates between Italy and Germany (and less to estimate the number of unreported cases as precisely as possible). At the same time, the approach has aroused great interest among many readers and colleagues. We have therefore decided to develop the approach further so that it can estimate the dark figure even better.

After a brief repetition, we present further opportunities for improvement in this article. Of course we keep the expectations high. You will find the new figures at the end of the article. Before that, however, I would like to point out my co-authors. Firstly, my colleague Jochen Fiedler (you already know him from the last post) and secondly Christian Moritz (Faculty of Medicine at the University of Lyon), who supports us as an experienced scientist in correctly understanding the medical aspects.

Disclaimer: Every decision-maker who wants to incorporate the results of this post into their future decisions must absolutely note: The computations work with assumptions that are plausible but not strictly provable. Please contact us for any questions!

The Basic Idea Behind Socially-Weighted Prevalence

The starting point was the observation that, especially in Italy, older people were overrepresented in the number of reported infections. We did not attribute this to the spread of the disease, but to the test procedure: especially when the health system reaches its limits, mainly people showing symptoms are tested. At the same time, we kept in mind that the severity could be age-related.

In a first step, we determined how many undetected infections should be in the different age groups so that at least the same rate occurs in all groups (called prevalence). We took the highest prevalence of the 90+ age group as a basis and extrapolated all other groups to this value.

We then argued that not all groups have the same number of social contacts. Children attend kindergartens and schools; adults go to work; pensioners are mainly in smaller private social groups. Based on the data from April 2, 2020, we determined a dark figure of 22,107 unrecognized infections for Germany. At that time, 73,000 infections were officially reported, resulting in a total of approximately 95,000 infections. This would mean that around 80% of infections are actually recognized in Germany. As teased above, there are updated figures at the end of the article. In the disclaimer above, we strongly recommend to really read the sections on the way there and not just to scroll to the end.

Asymptomatic Cases Increase the Estimated Dark Figure

In our first adjustment step, we used the prevalence of the 90+ group for the Italian data, which should at least also occur in all other age groups in Italy. Of course, it happens even in that age group that not every infection is recognized. Keyword "asymptomatic cases". Based on data from the Diamond Princess (a cruise ship that was quarantined on February 5, 2020), the proportion of asymptomatic cases can be roughly estimated at 10 to 40 percent.

If now e.g. 25 percent of the infections are asymptomatic, then for every three detected infections there is one undetected - in the well-studied 90+ group. This results in a correction factor of 1.33, which applies equally to all age groups and the sum. For the unreported number of 22,107 infections (without asymptomatic cases) from April 2 this results in a corrected value of 29,000 undetected infections in Germany.

© Fraunhofer ITWM
If one takes into account the rate of asymptomatic cases, the dark figure increases in all age groups.

The Correct Date of the Dark Figure Estimation

Anyone being registered in the infection count today has already been tested a few days ago and has become infected a few days before the test. This time delay must be taken into account when handling the data. RKI is already correctly reflecting these delays in its reported figures. We currently think it would be appropriate to backdate our dark figure estimate by 9 days.

In order to bring the backdated dark figure estimate back to today's date, we have to consider the correct growth of the last 9 days. For this we can orientate ourselves on the current growth rates of new infections or the number of deaths. While deaths are recorded more reliably, their associated infections are further behind. The reported infections, on the other hand, have a significantly smaller delay, but are also more unreliable due to unreported asymptomatic cases and limited test capacities. To determine growth rates only, the data seems to be sufficient if averaged over a week (currently, the daily growth of reported new infections is fairly constant in the range of 1,800 - 2,800). We are actively investigating this as well.

The Effect of Measures and New Medical Knowledge

We only assumed that the infection was spread uniformly across all age groups in the first step. Immediately afterwards, we resolved the uniform prevalence and weighted it socially.

© Fraunhofer ITWM
Number of social contacts by Mossong, et al., 2008.

Four weeks ago, various measures were introduced to contain the spread of the virus. The typical numbers of social contacts have lost their validity. Non-digital social contacts have drastically reduced in number, and where they still take place, for example when shopping, they do so in larger distance radii. Can the contribution from school, workplace and (partially) friends be simply removed from previous statistics on social contacts? Here we keep an eye on new surveys of social scientists.

© Fraunhofer ITWM
The social interactions can be classified in different areas. Some of them are currently dropped (based on data such as from Fumanelli L, Ajelli M, Manfredi P, Vespignani A, Merler S (2012) Inferring the Structure of Social Contacts from Demographic Data in the Analysis of Infectious Diseases Spread. PLoS Comput Biol 8(9): e1002673. doi:10.1371/journal.pcbi.1002673)

Furthermore, it is discussed whether children are carriers of the virus to the same extent as adults. An interesting case from the French Alps leaves at least some doubts. The study by the French national health authority found that an infected 9-year-old boy with mild symptoms did not infect anyone, despite the high number of 172 people he came into contact with (including from three different ski schools). And this despite the fact that the infection rate of the entire group of sociable winter vacationers was on average three times higher than the national average. If this does not turn out to be an individual case, we will include corresponding study results in our model.

Our Current Estimate of the Dark Figure

As promised at the beginning, here is our current estimate of the dark figure for Germany.
Briefly summarized input:
• The currently reported data from April 26, 2020, with 155,177 infections detected
• backdating our estimate by 9 days
• the subsequent update to the reporting date with the growth rates of the infection numbers
• an assumed rate of asymptomatic cases between 0 and 40 percent. The 0 percent case defines our minimum estimate.

© Fraunhofer ITWM
Result of the estimation of actual infection numbers in Germany (status 26.04.2020)

Based on the data as of April 26th, 2020, we believe in at least 117,000 undetected infections, which should be added to the 147,177 detected infections. Assuming 40 percent asymptomatic cases, we speak of 298,000 undetected infections.

Thus, we estimate the total number of infections between 272,000 and 453,000.

Update to the Dark Figure Blog Post

Blog Entry