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

Streuspanne – Our Blog on Statistics and Jurojin / April 03, 2020

On 2 April, Germany reported 73,000 corona infections and 867 deaths. Italy counts 100,000 infections and 12,000 deaths. Based on these figures, the case mortality rate in Italy is more than 10 times higher than in Germany. Experts are clear: this difference is less an expression of a different health care system or a different population structure, but rather an expression of unknown dark figures.

The estimated number of unrecorded cases is not only needed to explain different case mortality rates (which experts assume to be approximately the same), but also to assess how the situation will develop. If a decision hast to be made in the near future about opening kindergartens, schools, parks and restaurants, it makes a large difference if 73,000 or 730,000 are infected.

Update on This Blog Post

The approach has met with great interest from many readers and colleagues, so we have decided to develop the approach further so that it can be used to determine the number of unreported cases even more effectively.

Of course, we will keep the suspense about the current number of undetected cases high until the end of the article. But please read the update for yourself and contact us for further questions.

Update on the Dark Figure Blog Entry 27.04.2020

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Model of Socially-Weighted Prevalence

Although we do not usually deal with epidemiology in the field of „Mathematics for Vehicle Development“, we have a lot of statistical expertise in dealing with incomplete data sets („missing data“) and non-representative samples. With the help of these statistical tools we have developed (led by Jochen Fiedler) the model of socially-weighted prevalence, which determines the number of unreported cases from the following considerations:

The virus has spread to all age groups. In younger age segments it should have spread more strongly due to higher mobility and number of contacts.
In detailed data sets with reported infections, this uniform mixing is not apparent. The virus seems to have spread more in the higher age segments.
This is due to the way testing is carried out in the countries. For example, in some countries only people with serious symptoms are tested, sometimes because capacities do not allow for more. These are mostly older people.
This allows us to draw conclusions about the number of unreported cases, especially in younger age segments.

Based on the data of April 2nd 2020, our model has determined that there must be at least 22,107 undetected infections in Germany in addition to the 73,268 reported infections. For Italy, 476,297 undiscovered infections are obtained (in addition to the 105,630 reported).

These estimated numbers of undetected infections alone reduce the difference of the case mortality rates to a factor 2 (instead of over 10).

In the following, we explain our approach in more detail:

Step 1: Derive Infection Rates per Age Segment From Detailed Data

The following table shows the deaths and reported infections (confirmed cases) in Italy by age segment:

Deaths and reported infections (confirmed cases) in Italy by age segment (as of 02.04.2020). — © Fraunhofer ITWM
Deaths and reported infections (confirmed cases) in Italy by age segment (as of April 2nd 2020).

In the last column, case mortality is given as the simple quotient of deaths and confirmed cases. Defining mortality in this way („open cases“) is controversial, because (besides the problem of the unknown number of unreported cases) it is additionally ignored that some currently infected persons will die.

However, in the next step we do not need the mortality rate and only look at the numbers of confirmed cases.

Step 2: We Determine the Prevalence per Age Segment

We know from population figures that, for example, 9.45 million of the 60.48 million Italians are between 50 and 59 years old. Of these 9.45 million, 20,975 have now been identified as infected. This percentage of infected people in the population is called prevalence.

The following table shows the prevalence by age segment and gender:

Prevalences in Italy by age segment and gender (as of 02.04.2020). — © Fraunhofer ITWM
Prevalences in Italy by age segment and gender (as of April 2nd 2020).

For example, 0.518 percent of those over the age of 90 are identified as infected, but only 0.221 percent of those aged 50-59. As explained at the beginning, the virus should have arrived in all age groups and spread most in those with the highest mobility and number of contacts. We attribute the different prevalences, with tendency towards older segments, to the fact that only few tests have been carried out for people with only mild symptoms who tend to be younger.

Step 3: Adjusting the Prevalence Yields an Estimate of the Lower Bound of the Number of Unreported Cases

The highest prevalence is in the age segment 90+ with 0.518 percent. This figure may be too low, because also older people might be asymptomatic; however, all other prevalences are certainly too low under our assumptions.

Therefore, all lower prevalences are now extrapolated to 0.518 percent. Consider the prevalence of 0.221 percent for the 50-59 year-olds as an example:

If there were 48,954 infections in the 50-59 age group, this segment would have the same prevalence as the over-90s. This yields 27,979 undetected infections in addition to the 20,975 reported.

If this calculation is repeated for all age segments, we obtain 257,000 unknown cases in Italy. An analogous calculation for Germany yields 20,400 unknown cases.

Step 4: Taking Into Account the Number of Contacts by Age Segment Further Increases the Estimated Number of Unreported Cases

We have seen that the high age segments – based on the reported infections – have the highest prevalence. In the step above, we have raised the segments with lower prevalence to reflect an even mix.

Based on a study by (Mossong, et al., 2008) we know the following numbers of social contacts:

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

The average number of contacts is, as expected, higher for the 0-69 year olds than for the group 70+. While the authors normalize the numbers in the last column to the 0-4 year olds, we again refer to the over-90 year olds. Their number of social contacts is not explicitly shown in the table. We assume that the use of the figures for 70+ represents a downward estimation.

Thus, the group of 50-59 year-olds has 1.31/0.81 = 1.61 times as many social contacts as the over-90s. Similar correction factors can be derived for all other age groups. These correction factors are then additionally multiplied by the dark figures estimated from the equal prevalence assumption of step 3. For the segment 50-59 this yields an estimated number of unreported cases of: 49,954 * 1.61 = 76,816. Taking these dark figures into account yields adjusted mortality rates for the different age groups, as shown in the following figure.

Comparison of recorded and adjusted case mortality rates in Italy (status 02.04.2020).

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