1 Introduction

>> There is more than enough reason for a new paper: while the numbers are rising more slowly, it is unclear whether the situation will improve sustainably, and winter is still on the horizon. We therefore consider it our scientific and professional duty to once again point out the need for a change in strategy. The approaches of the past few months must be examined impartially, instead of a kind of tunnel vision that continues and even increasingly relies on suboptimal strategies. We have already pointed out in previous documents1 the option of a (missed) alternative narrative, a narrative that, in the sense of a "civil society jolt," abandons the old Bergamo motif of the persistent, ever-increasing threat and shows the population a more optimistic and viable path that shapes the daily problems of isolation (e.g., when caring for the sick and those in need of care) and the restrictions on fundamental rights differently than in recent months. The constructive capacities of civil society must be mobilized, and the top-down strategies of the 19th century should be replaced by modern infection control methods that have long been the standard in other areas and are based on a contemporary understanding of organizational and social theory.
A first step would be the compilation and communication of reliable figures based on an adequate basic understanding of epidemiology, the central theme of this paper (6.1). A second step would be a contemporary concept of prevention, namely the supplementation of general preventive measures such as distancing and tracking with the protection of vulnerable population groups particularly affected by the consequences of the infection (so-called stable control, see Tp2, Chapter 3.22). The urgency of such a changed approach is growing daily, because if we avoid the discussion of prioritization criteria today with the repeatedly reiterated argument that "locking them away is impossible" (a striking association), this blockage will come back to haunt us at the latest when it comes to prioritizing vaccination. How can we specifically and purposefully allocate access to vaccination to the large group of "elderly" that our society encompasses if we cannot already protect the vulnerable today in a way that preserves dignity and humanity? Should we be worried about the mood in the queues at vaccination centers?
Inconsistent concepts and errors do not disappear if they are ignored. Based on the “Third Way” formulated in May of this year (Schrappe et al. 2020), we therefore first focus on the current epidemiological development and derive the need for specific prevention measures from the current dominance of the sporadic (homogeneous, “creeping”) type of spread. In the case of an asymptomatically transmitted infectious disease, a strategy that is based solely on contact tracing and no longer has the resources available to protect the groups of people most affected by the outcome is ineffective. Only the combination of both measures, namely tracing and protection, makes sense. However, the reasons for the lack of acceptance of this (actually obvious) approach in the political sphere lie deeper and include, for example: For example, it can be found in the assumption that the numbers (reporting rates) we look at for daily control represent a realistic reflection of reality, meaning that we have the epidemic at least quantitatively under control.
This is not the case. The real trend has long since shifted away from the random sampling we test daily and is largely taking place in the significantly larger portion of the population that is not currently being tested (the problem of the unreported number of cases). Therefore, we neither know where we stand nor whether the thresholds used are valid, because these are irrelevant and are clearly being eroded by the developments.
The group of authors recognizes the difficulty of political action in the current situation. In this spirit, we would like to recall the preamble to the first position paper on April 5, 2020, and emphasize once again that this is not criticism for its own sake, but rather a constructive proposal for redesigning the approach. This position paper therefore proposes two newly developed indices that—given the continued lack of cohort studies—promise improved recording and management. However, we continue to be concerned about the "high-pitched tone" that accompanies the discussion. This is not about being dogmatic, but about managing a crisis situation. The most important and also wisest thing in such a situation should be to listen and to engage openly with different viewpoints. If this ability were lost in this crisis, social damage would have occurred that would be almost irreparable. We hope that it will not come to that and that a reversal is possible, initially at the level of communication.

Thesis 1: A change of strategy is necessary. The asymptomatic route of infection and the current state of (sporadic) spread make controlling the infection through contact avoidance and tracing alone impossible. The most important prerequisite is valid, reliable, and achievable metrics that can be used to design specific prevention programs for vulnerable groups in the population. Two concrete, newly developed proposals (notification index NI and hospitalization index HI) are presented here. Against this background, clinging to the Bergamo-style threat narrative is incompatible with modern methods of risk communication and management.
2. On epidemiology
2.1. The epidemiological model
The epidemiology of infectious diseases differs from the epidemiology of other diseases (e.g., heart attacks) in one crucial respect: while they are being discussed (recorded, investigated, etc.), these events can spread to an often unimaginable extent. A heart attack patient, on the other hand, cannot "infect another patient with a heart attack"; two heart attacks are therefore (with a grain of salt) independent of each other. The situation is quite different with Covid-19, because two infections with this virus can certainly be related to each other, as the second infection can be caused by the first. Historically, this insight, which seems almost self-evident today, is not all that old. As late as the 19th century, there were views that infectious diseases were caused by third substances (e.g., the well-known miasmas, evaporations from the earth, or metaphysical constructs), which, after all, could be responsible not only for cholera but, in principle, also for heart attacks.
But the situation is actually more complicated than one might think, because – certainly counterintuitive in the current situation – Covid-19, as an infectious disease caused by SARS-CoV-2, is a relatively simple process: two people come close to each other, one is infected, droplets form, and now the infection has spread to the second. But why was cholera so difficult to understand back then? Because it wasn't primarily about human-to-human transmission, but rather about a pathogen reservoir, namely drinking water, be it a well, as during the cholera epidemic in London, or the Elbe River in Hamburg in midsummer. A pathogen reservoir can also be formed by animals, which is why the discussion about mink on animal farms is so significant – just imagine if the SARS-CoV-2 virus were to spread to native animal species and mutate in the process; this would dramatically change the situation. Terms have been coined for these infection patterns: human-to-human transmission is the prototype of an epidemic situation, the drinking water setting in cholera is referred to as an endemic situation (en-demie: persisting in the population as long as the well is not closed)3.
The epidemic and endemic types have one thing in common: with sufficiently meticulous observation, the focus or reservoir can be identified. This applies with two important exceptions: the pathogen must be identifiable in an endemic situation (this was the difficulty in the 19th century before Robert Koch: In the case of cholera, there was evidence of an infectious agent, but the pathogen had not yet been identified, thus lacking definitive proof). In an epidemic situation, one relies on the infected being identifiable as such, because otherwise the "foci" cannot be identified. This is precisely the problem with SARS-CoV-2/Covid-19: The majority of infected people are not ill and do not display any other symptoms, so the extent of spread in the population remains largely unknown.
Similar to HIV infection, these asymptomatically infected individuals render part of the infection process invisible, so that cases repeatedly emerge completely unexpectedly "out of nowhere," which in turn can lead to outbreaks and, of course, cause disease. The automatic reflex is then often to test "everyone," so to speak, i.e., the entire susceptible population. But this is not only very difficult in practice, but also ineffective, because there is always a way for an infectious agent to gain influence regardless - someone tested today can become infected tomorrow from someone who is only tested the day after tomorrow, and so on. In such a situation, one sees a pattern that partly runs through identified outbreaks, i.e., under an epidemic dynamic, and partly consists of individual, seemingly unrelated cases of infection. This third type of infection is referred to as "sporadic" spread.
With SARS-CoV-2/Covid-19, the development can be traced very clearly. At the beginning, from February to the end of April 2020, it was an epidemic pattern, with clusters that were known to a certain extent and perceived as very threatening, but were ultimately contained.4 However, due to asymptomatic transmission, 100 percent control was not possible, so that a "creeping" spread followed during the summer months, almost below the threshold of awareness. The focus of public perception also shifted, as the threat scenario of a local emergency was replaced by the fear of an uncontrolled spread in the background that "could no longer be controlled." Winter was approaching.
This development has been observed and described time and again in "epidemics." Descriptions of the plague are classic: a family buys fabric and other goods (including rat fleas) on an incoming ship; the plague quickly appears in their house, prompting the police to intervene; a lockdown is imposed, etc., and everything seems fine. One to two weeks later, a city-wide outbreak suddenly breaks out. What has happened? The pathogen has spread through the rat and rat flea population, leading to outbreaks in numerous locations simultaneously. While there are no rat fleas in the coronavirus, the creeping spread during an apparent dormant period also occurs here. How often have countries (e.g., Switzerland, Italy for a time, etc.) been praised as those who "did it right," only to then realize that it was only a brief respite (the same will happen in Asia).
Leaving the narrative tone and returning to the analytical approach, one can highlight two outstanding landmark facts in the case of SARS-CoV-2/Covid-19 that shape the entire development and make it understandable, but also clearly highlight the wrong decisions:
(1) the fact of asymptomatic transmission, which was already confirmed in February by the investigation of the outbreak on the Princess Diamond (Mizumoto et al. 2020) and
(2) the now leading importance of the sporadic spread type, which results from asymptomatic transmission and describes the infection dynamics from the area of the non-tested population (see Tp4, Chapter 1.1.1).

Point (1) plays a particularly important role in the choice of data collection methods for frequency distribution. As early as February, and at the latest by March, it was clear that representative cohort studies were needed to make and justify the most important decisions. Unfortunately, this was not done in Germany despite the clear evidence, so that today we ultimately only have indirect evidence of the extent of the spread of SARS-CoV-2/Covid-19. Chapter 2.3 discusses this in more detail, including an estimate of the so-called unreported number of cases (using different approaches) and the foreseeable deficits in management during the coming phase of population vaccination.
Point (2) describes the basis for the ongoing spread in the population and the predictable failure of the strategy based solely on tracking (see Tp5, Chapter 2). While it has a certain effectiveness at the beginning of the epidemic and in containing epidemic outbreaks, it remains ineffective in a sporadic pattern. Instead, concepts based on general prevention measures, including tracking, and specific, targeted approaches for vulnerable population groups are appropriate.
The following chapter will address the issue of frequency measures again. Due to the misperceptions regarding asymptomatic transmission and sporadic spread just described, the fiction is maintained that the infection numbers (notification rates) reported daily by health authorities reflect the true frequency in the population, coupled with the assumption that the threshold values calculated from them will have some kind of control effect. This view should not be accepted; it can easily be refuted.
Hypothesis 2: The SARS-CoV-2/COVID-19 epidemic is characterized by its transmission by asymptomatically infected individuals and the sporadic type of spread, which has largely replaced focal ("epidemic") spread. This characterization has direct consequences for quantitative assessment and the design of preventive measures.

2.2. Frequency measures: the misuse of the term
“Incidence” and its consequences
All frequency measures used in epidemiology have in common that they represent characteristics or events in relation to a known population (denominator). This can be done either in the form of a cross-sectional study (prevalence) or in the form of a cohort study (incidence). In infectious disease epidemiology, the special feature to note (see above) is that the characteristics are not stable but change rapidly due to infection.
Since official language always refers to “incidence”, “incidence rate” and “7-day incidence”, special attention must be paid to the characteristics and prerequisites of the concept of incidence (see textbooks of epidemiology, selected here: Fletcher, Fletcher and Wagner 1996, see p. 77 ff).
The following aspects are essential for the use of the term “incidence” in an infection event:
(1) the population under study must be known as a reference value (in order to be able to form a denominator),
[Example: Data on nosocomial infections must refer to the number of patients examined (incidence rate) or even to the number of days of observation (specific incidence rate).]
(2) the population must be fully studied (or a representative sample must be formed),
[A study of nosocomial postoperative wound infections cannot be limited to three randomly selected, adjacent rooms on Ward A, but must cover the entire hospital or refer to a justified, randomly selected sample of rooms.]
(3) the same population must be examined completely over a period of time in order to actually record and assess the new emergence of an infection,
[A study on the incidence of nosocomial postoperative wound infections cannot be conducted on Mondays in three rooms on Ward A, on Tuesdays in three rooms on Ward B, etc. (unless this is a representative sample), but all rooms and thus all patients must be examined every day throughout the observation period – and not just symptomatic patients, see Covid-19.]
(4) the population under investigation must consist of persons free of any characteristics at the time of entry into the study period,
[A study on the incidence of nosocomial postoperative wound infections always refers to patients who have not previously had such an infection; patients who, for example, already have an infection upon admission are not included in the incidence calculation (see present-on-admission issues in the context of quality assessment and surveillance of nosocomial infections).]
(5) For the same reason, the period of observation
The period should be long enough to really observe the re-emergence of the feature.
[If, when recording nosocomial wound infections, one only observes the three rooms for two days, wound infections will often not be detected if they occur later (the observation period should therefore be at least four weeks). Likewise, it makes no sense to record the new occurrence of SARS-CoV-2/COVID-19 during a one-week observation period if the PCR is positive for 14 days and the incubation period alone is five days.]

In the current situation, there is not a single characteristic that would be necessary for the use of the term “incidence” (see Figure 1):
• The population is unknown (point 1), because no representative samples are used; instead, tests are conducted on an ad hoc basis. Applied to the field of nosocomial infections, with its 700,000 cases annually, this would mean that surveys on this problem would be conducted according to the "we need to look here" principle.
• the population (or a sample) is not fully examined (see point 2), but different people are tested on Mondays than on Tuesdays (and so on),
• the population is not examined over a period of time (point 3), which is diametrically opposed to the concept of incidence in the sense of “new cases in a period of time”,
• the carriers of the trait at the beginning of the study period are not excluded (point 4), although their infection can no longer “reoccur”, and
• The investigation or reporting period of one day or seven days is inadequately short, because an infection with an incubation period of five days and PCR detectability of 14 days (approximate values) cannot be fully described (point 5). The European Center for Disease Control (ECDC) therefore assumes a reporting period of 14 days, which is still very short, but at least more reasonable.5

Ultimately, the situation is easily resolved: the daily reports from the Robert Koch Institute (RKI) do not represent an incidence or a "7-day incidence," but rather a combination of several "1-day incidences," which, upon closer inspection, are identical to the daily prevalence.6 We are therefore dealing with a compilation or addition of several, unsystematically generated point prevalences, which describe frequencies in daily changing samples, which are then summarized to form a "7-day value." This construct can best be described using the (somewhat outdated) term "period prevalence." The terms are presented graphically in Fig. 1.
The term "7-day incidence," for example, is therefore not used at the European level; the European Center for Disease Control (ECDC) correctly refers to notification rates.8 It would be advisable for German usage to return to an appropriate use of the basic terms here; one could, for example, speak of a "7-day notification rate." Chapter 2.5 proposes an alternative that is composed of several parameters (notification index).
The question remains: why is this point so important?
The problem is enormous and dominates the entire subsequent discussion, because the use of the term "incidence" suggests that one has knowledge of the number of new infections that actually occur within a given period (7 days), and that one can accurately gauge the development from this. This perspective is very powerful because it insinuates a basis for action on which far-reaching political and societal decisions are made. However, this basis for action is not sustainable, and on the other hand, it blocks the urgently needed path to using more sensible approaches.
As explained in the following chapter, the development has long been determined by the infection dynamics in the unstudied population. Another way to put it is: the number of unreported cases dominates the quantitative reality of the epidemic and the threshold values used. However, we find ourselves (through our own fault) in the difficult situation of only being able to approximate the number of unreported cases indirectly, since cohort studies, which we currently need nationally and regionally (e.g., in large cities), have been neglected against all expert advice. To reiterate: the effects of this deficit will become even more dramatic in the future, because we will urgently need reliable incidence data if, for example, we want to demonstrate the success of vaccination campaigns and possibly also therapeutic strategies.
Thesis 3: The term "incidence" is misused in reporting. The RKI's frequency data are unsystematically obtained, event-related prevalence values that are added over 7 days (best described as period prevalence). The impression created by the term "incidence" or "7-day incidence" that one knows about the status of the epidemic and the number of new cases actually occurring in a given period is misleading and undermines the credibility of political action.

2.3. On the dark figure: Estimation of the magnitude
Since cohort studies have not yet been conducted, (significantly less meaningful) indirect methods must be used to assess future developments. In particular, the number of unreported cases and thus the actual frequency of SARS-CoV-2/Covid-19 infections and their various manifestations cannot be estimated otherwise. The following methods are generally available for this purpose, all of which only allow a rough approximation:
• Considerations on plausibility,
• simple models under different assumptions,
• international comparisons,
• Seroprevalence studies based on antibody determinations.

Seroprevalence studies, such as those conducted to determine how high a proportion of schoolchildren in a city have already been infected with COVID-19, are based on the detection of specific antibodies (at a specific point in time) and take a cumulative view, essentially recording all infections in the past period (questions regarding antibody persistence are excluded here). In contrast, studies of active infection (PCR, antigen detection using rapid tests) focus on the specific infection status, thus determining how many people are infected at a specific point in time (see Figure 2).

2.3.1. Considerations on plausibility
To emphasize right from the start: Considerations of plausibility do not represent a methodologically rigorous derivation. However, such considerations can provide a guideline. One basis for this can be, for example, one of the most important epidemiological observations of recent months: the significant increase in the rate of positive PCR results (test positivity rate), evident both in Germany and internationally, coupled with a massive increase in testing frequency.
As can be seen in Figure 3, the rate of positive test results in Germany during the summer months was below 1% for a long time. This was not unusual at the time, as the number of tests conducted weekly rose to approximately 1 million per week. It was reasonable to assume that the increased testing volume would result in relatively fewer infections being detected. However, in recent months, it has been observed that the relative frequency of positive test results (test positivity rate) has increased significantly, despite significantly more testing. Recently, 9% of the tests conducted were positive; abroad, up to 20% of the tests conducted were positive (Spain, Switzerland, France).
In view of this situation, the analytical question naturally arises: if 9% of the 1.5 million people tested in Germany recently (in one week) have a positive PCR result, what would it look like if – hypothetically – another group of 1.5 million people were tested in the same week (laboratory resources, etc., are not taken into account in this consideration).
Is it plausible to assume that the test positivity rate in this "neighboring" group is zero? Such an assumption, that the positive test results would clearly only refer to the tested population, is of course not very plausible. Instead, one would expect a much higher test positivity rate than zero in the second test group, if not 9%, then at least 4%, or perhaps only 1%—which would correspond to 16,000 additional cases in the second group in the week under consideration.

Thesis 4: Given the steady increase in the test positivity rate observed in Germany and internationally, it is almost impossible that the number of newly occurring infections relates solely to the tested population (reported rate). Therefore, numerous infections must be expected to occur in the non-tested population.
occur (dark figure).

2.3.2. Simple models under different assumptions
Consequently, it seems not only obvious but imperative to assess the prevalence of SARS-CoV-2/COVID-19 infection in the untested population under different assumptions. As shown in Figure 4, the total number of newly reported infections per week has increased in recent weeks (since October 30th, to be precise) to over 100,000 cases and currently stands at over 130,000 cases. With 1.5 million people tested, this corresponds to a test positivity rate of 9%.
Furthermore, Figure 4 assumes a total population of 83 million in Germany. In seven consecutive, event-related cross-sectional samples (“Monday to Sunday”), a total of 130,000 infections were detected among 1.5 million people (see Figure 1). These 130,000 infections were extrapolated to the total population of 83 million people in Germany, resulting in a key figure (referred to as the “7-day incidence rate,” see Chapter 2.2) of approximately 150/100,000 inhabitants.
However, this approach practiced by the RKI is based on two
relevant requirements:
1. the figure of 130,000 infections is not only relevant for the 1.5 million people tested, but represents an estimate of the frequency in the entire population, and
2. The untested population of 81.5 million people has an infection rate of zero.

However, these assumptions are neither sound nor justified. As already explained in the previous chapter, before applying the estimates to the total population, one must instead make assumptions about how frequently the infection also occurs in the untested population. Since we do not have reliable data on this due to the lack of cohort studies, one must resort to estimations across various orders of magnitude, as summarized in Table 1.
The left column of Table 1 contains assumptions regarding the number of unreported infections; these refer to infections with a positive test result (usually PCR) occurring in the same week, not to seroprevalence data (antibody detection, see Fig. 2). Even assuming that 0.5% of tested individuals in the untested population have an active infection (compared to 9% in the tested population), an additional 407,500 infections would occur (see column 3). In relation to the total number of infections present this week, over 75% of the infections would be due to the number of unreported infections (column 5). If one bases oneself on the results of the testing of the entire population in Slovakia (SZ 17.11.2020), which have now been communicated in the press, which identified a total of 38,000 antigen-positive persons (approx. 1% of cases) who were previously unknown using an antigen test (“rapid test”) among 3.6 million tested persons, and if one transfers these results to Germany (line 3 in Table 1), then the 130,000 known infected persons, assuming a prevalence of 1% of cases, would be offset by the large number of 815,000 unknown infected persons; in total, the number of unreported cases would be responsible for 86.2% of the infections. In agreement with the data on seroprevalence (see Chapter 2.3.4.), this finding indicates that a significant number of unreported cases must be expected in Germany, possibly even with an unreported number of up to a factor of 6-7 (Table 1, 6th column).
Due to a lack of data, such a representation can only be hypothetical. It is very early to extrapolate the data from Slovakia, where attempts are being made to survey the entire population. In particular, the problem of false-positive results in low-prevalence populations must be expected (see Tp4, Chapter 1.1.2). Nevertheless, there is strong evidence that the spread in the German population has also progressed so far that the figures derived from the tested population are not relevant for assessing the overall situation. This will be discussed in the following chapters.

Hypothesis 5: A simple estimate shows the quantitative dominance of infections in the untested population ("dark figure" under various assumptions) over the number of known, newly reported infections. Based on the prevalence of IUI from the total population survey of Slovakia, one obtains an additional 815,000 infections in the untested population for Germany compared to 130,000 known reports in one week. Consequently, the threshold values based solely on the reported infections after testing (reporting rates) cannot be considered significant.

2.3.3 International comparisons
In all European countries, there was a significant increase in newly reported daily infections in late summer. The threshold of 100,000 newly reported cases in 7 days was reached:
- in Germany on 30.10.2020,
- in France on 10.10.2020 and
- in Spain on 22.10.2020

France even exceeded the 200,000 cases per week mark on October 24, and the 300,000 mark on November 4, 2020. The figures are easier to compare using daily figures relative to the population (see Figure 5), especially allowing for the inclusion of countries with smaller populations such as Sweden. The threshold of 20 cases (smoothed over 7 days) per 100,000 inhabitants was exceeded:
- in Germany on 4.11.2020,
- in France on 9.10.2020,
- in Spain already on 8.9.2020 and
- in Sweden on 31.10.2020

The four countries discussed here are only about 9 weeks apart.
The presented trends show a clear homogeneity, especially in recent months, regardless of the different national approaches and containment strategies. Even Sweden has not been exempt from the increase in cases this fall. It is reasonable to suspect that these are essentially the same dynamics, shifted by a few weeks. Similar differences can also be observed when comparing the federal states in Germany. In all states, the increase in the number of cases has been accompanied by an increase in the test positivity rate.
Of course, this observation does not fundamentally rule out the possibility that each European country has its own country-specific dynamics, which can be significantly influenced by national measures. However, another interpretation is also possible and quite likely, which states that we are seeing a temporally staggered, but essentially uniform, dynamic of virus spread in the different countries, characterized by a continuous increase and building on the sporadic spread during the summer months (whether the lockdown measures are now leading to a real change remains to be seen).
The power of the trend becomes particularly clear when choosing the cumulative representation, which was common at the beginning of the epidemic (see Figure 6). It is difficult to imagine that a trend reversal can be expected anytime soon in an epidemic characterized by moderate infectiousness and transmission through asymptomatic carriers, and exhibiting such dynamics.

Thesis 6: International comparisons also suggest the epidemic's rapid spread. These figures indirectly indicate that a complete recording of all cases is virtually impossible, that a high number of unreported cases must be assumed, and that a complete reversal of the trend through contact restrictions alone is extremely unlikely.

2.3.4 Seroprevalence studies
In the absence of cohort studies, IgG antibody levels can be determined in the population to determine the number of infections already recovered (so-called seroprevalence studies). However, it has not yet been conclusively clarified how long these antibodies remain detectable post-infection and whether they are present in every recovered case (e.g., even in asymptomatic cases) (Poland et al. 2020, Wajnberg et al. 2020, Gudbjartsson et al. 2020B). However, these uncertainties must be accepted because it is crucial to gain an impression of the "cumulative" infection rate in the population, as important decisions, for example, regarding school operations, the planning of containment measures, and the planning of the vaccination campaign, depend on this.
A further limitation must be emphasized: the authors' group is not in a position to conduct a systematic review on seroprevalence studies using their own resources (which, however, would be urgently needed and would need to be continuously updated). Therefore, only exemplary studies can be cited here. Given this assumption, three types of seroprevalence studies can be broadly distinguished:
• Investigations of outbreaks,
• Examinations coupled with a diagnostic offer and
• representative surveys (random selection).

The so-called Heinsberg study by Streeck et al. (2020) must be considered representative of the first form; they examined the transmission routes and the focus-related data on spread and prognosis. However, the seropositivity (15.5%, 5%, 3%, 1%, 5 ...
Only one country began cohort studies early enough, namely Iceland (Gudbjartsson et al. 2020A). The subsequent seroprevalence study showed that only half of the infections detected in the cohort (seroprevalence) were detected by PCR (Gudbjartsson et al. 2020B). These studies, published prominently, have been repeatedly criticized in Germany (including with regard to their statements on lethality), but the described finding is nevertheless remarkable: according to this study, the number of unreported cases is twice the cumulative PCR prevalence. The seroprevalence in the second study was 0.9% of the population; the study was conducted between April and early July 2020 (for the temporal relationships, see Fig. 7).
Another important study published recently was conducted after the so-called first wave in Spain (Pollán et al. 2020). This study, too, was not based on random testing but used a representative, random sample of subjects. Seropositivity was 5% (over 10% in Madrid), and only 19% of those with positive antibody results had a history of PCR testing, which corresponds to an underreported number of cases by a factor of five.
In Germany, the Munich KoCO19 study investigated an indirect relationship between cumulative PCR prevalence and seropositivity (Radon et al. 2020, Hoelscher et al. 2020A). In this early study, the number of unreported cases was a factor of four, and antibody testing identified four times more cases than were "officially" known through PCR testing. Seropositivity in the Munich population was 1.8%.
In a small study among Deutsche Bahn employees in early July, the seroprevalence was between 1.3 and 3.0%. One person was PCR-positive, but the number of study participants was too small to estimate the number of unreported cases (Anonymous 2020A). In an Austrian study (included here) among employees of a company, the seropositivity was 1.88% (Wiedermann et al. 2020). In a study of almost 16,000 children in Bavaria, a seropositivity of up to 1.13% was found over time, with an overall unreported figure of a factor of six (Hippich et al. 2020). While this representation is incomplete, two aspects should be noted:
(1) In such a dynamic infection process as the SARS-CoV-2/Covid-19 infection, the timing of the study plays a crucial role. As can be clearly seen from Figure 7, the evaluated and published studies date from the period during or after the so-called first wave. It is therefore completely open how the question of seropositivity should be assessed at present. If one disregards issues such as antibody persistence, etc., it is unlikely to be lower than in the studies compiled here, but rather – if one considers the
Considering the enormous number of cases currently being recorded (see Figure 6), the figure is likely to be significantly higher, because it is likely that not all cases can be recorded (overflow). Figure 7 shows the time course.
(2) The second point, of course, concerns the estimation of the infection rate of the population at this time. While fewer tests were conducted in the spring, the current overload of the facilities means that many cases are likely to go unreported—consequently, both cases are likely to cancel each other out. In any case, the number of unreported cases is between a factor of 2 and a factor of 6, so (soon)
• based on the cumulative number of 1 million infections in Germany, twice that number, 2 to 6 million, is to be expected (2.4% to 7.2% of the population),
• a seroprevalence of 10% is within reach in the cities (on November 15, 2020, 23,833 cases in Munich [rounded taking into account the number of unreported cases: 50,000 to 150,000 cases, corresponding to 3.3% to 9.9% of the population] and 47,697 cases in Berlin [rounded: 100,000 to 300,000 cases or 2.7% to 8.1% of the population]).

Recent press reports on seroprevalences of 53% in nursing homes in Madrid and of 37% among staff support this perspective (SZ of November 12, 2020).

Hypothesis 7: The available seroprevalence studies were conducted very early in the epidemic, mostly in direct connection with the so-called first wave. The cumulative perspective of antibody tests indicates an unreported number of cases between a factor of 2 and a factor of 6 compared to the cumulative findings from PCR diagnostics. Initial data from Madrid have been published that suggest partial immunization of the population.

2.4. Limits: Understandability, reliability, accessibility
There should be no doubt that successful crisis management relies on numbers and thresholds (see the ad hoc statement of the authors' group dated October 14, 2020). Both at the organizational level and at the level of the overall system, learning processes and behavioral change are only possible if the reason for and success of the measures can be communicated using numbers. However, there are important basic prerequisites for the successful use of thresholds: the numerical values must
• be understandable and transparent,
• they must be reliable and valid, and
• they must be accessible.

These requirements represent basics from the area of corporate and system management and have been continuously and extensively discussed in the healthcare sector over the last two decades, for example in the discussion on quality and patient safety (for example, the discussion on quality indicators should be remembered).
1. Transparency and comprehensibility: The importance of the limit values used has been repeatedly addressed in the thesis papers: lack of delimitation of the sample size, lack of representativeness of the sample, etc. In the present thesis paper 6.1.
2. Lack of reliability. The threshold and the numbers used for control must be valid and reliable: valid in the sense that they correctly identify a problem, reliable in the sense that they can be reliably collected. The scientific data on validity can be quickly summarized: there is no data that states that a threshold of x/100,000 inhabitants is associated with a positive course of the epidemic or a successful intervention. The argument that, beyond a certain threshold, tracking by public health authorities is no longer possible is (a) unsupported, and (b) implausible simply because, with the dominant type of sporadic spread, no one can assume that public health authorities, no matter how equipped, can even manage this (already Tp2, Chapter 2.1.3). The biggest problem, however, relates to the lack of reliability, i.e., the reliability of the measurement. If – as derived in the previous section – the calculation is dominated primarily by the dark figure, then such limit values cannot be reliably determined and are therefore of no use. In addition, one must bear in mind that a value that cannot be reliably determined can never be valid.
3. Accessibility: The resolution of the Chancellor/MP Conference (Anonymous 2020B No. 3) of May 6, 2020, still specified a threshold (reporting rate) of 50 newly reported infections per 100,000 residents per week for the introduction of restrictions. It was quite rightly pointed out that this threshold did not apply in the event of a cluster outbreak. The resolution of the Chancellor/MP Conference (Anonymous 2020C) of October 14, 2020, effectively tightened the limits to 35 per 100,000 residents, and no further mention was made of the distinction between clusters and sporadic spread. In fact, however, each of these limits is completely out of reach. The renewed adherence to the limit of 50/100,000 at the conference on November 16, 2020, lacks any realistic basis, given the current reporting rate of just under 150/100,000 for the entire Federal Republic (Anonymous 2020D). Research and practical experience on this topic provide a clear answer: if there is any means of rendering control through numerical values and limits ineffective, it is the feeling of citizens (and employees, too) that the limits are beyond their reach anyway and cannot be met despite all efforts (so-called crowding out).

These findings should guide future development. To describe the apparent blockage, the authors' group used the term "hermeneutic tunnel vision" in their ad hoc statement of October 14, 2020. This refers to the fact that further decisions are continually being made based on unsustainable figures, which, precisely because of the unreliability of the figures, are unsuccessful (or only erratically successful), so that further decisions appear necessary in exactly the same vein. This approach cannot be successful.
In the authors' view, establishing these figures by law (amendment to the Infection Protection Act of November 18, 2020) is not a remedy, as they are materially meaningless. However, since indicators and thresholds appear necessary for overcoming the epidemic, the next section attempts to develop more reliable quantitative measures that could remedy this situation, even in the absence of cohort data.

Thesis 8: Key figures and thresholds used for control must, according to the findings of modern organizational theory and system control, be reliable, valid, transparently developed, understandable, and implementable by the population (attainability). The currently used thresholds, which were also included in the amendment to the Infection Protection Act (e.g., "35 cases/100,000 inhabitants"), primarily lack measurement reliability because they cannot be distinguished from the unreported number of cases. However, unreliable thresholds may also be invalid, i.e., they cannot be meaningfully applied because they do not measure what they are intended to measure. Furthermore, the targets ("get back below 50/100,000") are unrealistic and therefore violate the aforementioned requirement of attainability.
2.5. Alternative I: notification index on the dynamics of the epidemic
The question of indicators that allow for control—as long as cohort studies are not available—and that possess acceptable validity and reliability should really be at the top of the agenda. Unfortunately, there is little evidence of this.
Therefore, the group of authors proposes a proposal for discussion under the term "notification index" and invites interested institutions to participate in this fundamentally important discussion. This index describes the dynamics of the epidemic and thus allows for far more sophisticated control than is possible through the notification rate alone. The higher the index, the more likely it is that the epidemic will develop its own dynamics and become uncontrollable.
The following points or parameters were included in the creation of this notification index:
• As shown above, the reporting rates used so far have low reliability and therefore lack validity insofar as one could derive a prognostic statement about the development of a (sub)population from them.
• Therefore, reporting rates are primarily determined by the test positivity rate.
to supplement this, as is the approach of the European Centre for Disease Control (ECDC)9 (see also FAZ of October 10, 2020). The test positivity rate represents the percentage of positive test results relative to the tests performed (not the number of people tested). The risk is higher when – as is currently the case in Germany – the reporting rate (number of newly reported infections per 100,000 inhabitants in a defined period) increases and at the same time an increase in the rate of positive test results (most recently to 9%) is observed. If the reporting rate increases and the test positivity rate decreases, then the situation is not as dramatic, as a kind of "dilution" is evidently taking place.
• However, the number of tests performed within a given time period (without considering the results) also plays a significant role, because otherwise, reducing the number of tests could lead to a decrease in reported cases and thus a reduction in risk. Testing frequency must be related to the population size, e.g., tests performed per 100,000 inhabitants. The lower this testing rate, the higher the risk that cases will go undetected due to underreporting.
• Furthermore, the heterogeneity of case occurrence must be considered. A clear approach is used here: the greater the proportion of cases occurring in clusters within a population, the lower the risk. This assessment is based on the fundamental epidemiological model of the SARS-CoV-2/COVID-19 epidemic (see Chapter 2.1), which assumes that epidemic outbreaks are easier to control than sporadic spread. It is proposed that clusters of 20 or more inhabitants be counted as such, and that heterogeneity be represented as the quotient of the percentage of infected individuals in these clusters and the percentage of sporadically infected individuals.

Therefore, the higher the heterogeneity (the greater the proportion of people infected via outbreaks), the lower the risk, and correspondingly the risk increases when the relative proportion of sporadically infected people increases.
By considering these four relationships together, an index can be created that is called the notification index NI (or notification index):

NI = -----------

M stands for the reporting rate (new cases/100,000 inhabitants per period (per week)), T+ for the test positivity rate (% of PCR-positive tests), Tn for the number n of tests carried out in this period and H for the heterogeneity described above.

The practicality and importance of this notification index can easily be illustrated with examples:
Case 1: A city of 1 million inhabitants has a weekly reporting rate M of 100/100,000 inhabitants. The test positivity rate T+ is 10%, the number Tn of tests in the same week is 1,000 per 100,000 inhabitants, and the heterogeneity H is 30% of infected individuals from clusters and 70% of sporadically infected individuals (30/70). The notification index is 100 x 10 / 1,000 x 0.43 or 2.33. Assessment: The high T+ of 10% and the significant role of sporadic spread (70%) indicate a high risk despite the low reporting rate.
Case 2: A city of 1 million inhabitants has a weekly reporting rate M of 120/100,000 inhabitants. The test positivity rate T+ is 4%, the number Tn of tests in the same week is 3,000 per 100,000 inhabitants; the heterogeneity H remains unchanged at 30/70. The notification index is 120 x 4 / 3000 x 0.43 or 0.37. Assessment: Many tests are being carried out, and the T+ is low, so despite the high percentage of sporadic spread, the overall risk is low.
Case 3: A region of 1 million inhabitants has a weekly reporting rate M of 100/100,000 inhabitants. The test positivity rate T+ is 10%, the number Tn of tests in the same week is 1,000 per 100,000 inhabitants; however, the heterogeneity H shows a pronounced effect from a cluster outbreak (outbreak in a meat processing plant), such that 80% of the infected individuals originate from this cluster and only 20% from sporadic spread. The notification index is 100 x 10 / 1,000 x 4 or 0.25. Assessment: Although the T+ is very high at 10%, since the infection mainly originates from the cluster (high value for H), the notification index is low, much lower than in case 1, where the dynamics are due to sporadic spread.

These 9: As an alternative to the common threshold of a "7-day reporting rate," the so-called notification index is presented here, which incorporates the reporting rate M, the test positivity rate T+, the number of tests performed Tn per 100,000 inhabitants, and a heterogeneity index H. The heterogeneity index H is based on the ratio of infected individuals from cluster outbreaks (epidemic pattern) to sporadic spread patterns. According to the basic epidemiological model used here, the sporadic spread type is assigned a higher risk than the more easily controlled epidemic type (cluster).

2.6. Alternative II: Hospitalization Index to Measure the Burden on the Healthcare System
The notification index presented in the previous chapter provides information on the dynamics of infection within the population, but it should not be used as a measure of disease severity, the risk to the population, or the strain on the healthcare system. For example, if a high notification index indicates that (frequent) illnesses are almost exclusively treatable in outpatient settings or even asymptomatic, the strain on the healthcare system is likely to be lower than if the same index value were to be met with a high proportion of hospitalized patients with severe illnesses.
One possibility, of course, would be to rely solely on the hospitalization rate of infected individuals (inpatient treatment relative to all infected individuals, see Chapter 3.2) to assess the strain on the healthcare system and the disease burden on the population. However, this rate provides no information about the quantitative strain, as the total number of patients is not taken into account. It therefore makes sense to combine the hospitalization rate (HR) with the notification index presented above. Since hospitalization occurs, or can occur, several days after the diagnosis of infection, the notification index (NI) should be calculated five days prior to hospitalization or the calculation of the hospitalization rate (HR).
This construct is referred to here as the Hospitalization Index (HI) and is calculated according to

NI = ----------- x HR or HI = NI x HR

Here, HI stands for Hospitalization Index and HR for Hospitalization Rate (% of hospitalized patients); for the other parameters, see the formula in Chapter 2.5. The notification index NI and the parameters it contains must be backdated by five days. This index and the associated procedure are largely uncharted territory; therefore, one should also consider omitting the heterogeneity factor (as a component of NI) when calculating the HI.
3. Data on the outcome of the
SARS-CoV-2/Covid-19 infection
3.1. General
It feels like being transported back to March of this year, when the cumulative presentation of case numbers dominated and heightened the prevailing threat scenario to an unbearable degree. While this approach to case numbers (reporting rates) has been partially abandoned – despite all methodological shortcomings – there is currently a new cumulatively described phenomenon whose incessant increase is announced daily: the number of cumulatively reported, continuously rising intensive care patients.
Of course, it cannot be denied that the number of hospitalizations, patients requiring intensive care and ventilation, and also the number of deaths are rising – the current epidemiological situation unfortunately suggests nothing else. The Bergamo narrative is being used again (behind the scenes), instead of addressing the situation.
Possible solutions were discussed. This does not mean that
Instead of perpetuating illusory contact restrictions for months on end, differentiated concepts should be developed that prioritize the protection of those individuals at particularly high risk of complications and ensure optimal use of existing resources. As stated in the ad-hoc statement of the author group dated October 14, 2020: this would represent a new narrative, a credible resetting of objectives, and a meaningful alternative to the numbing repetition of the same threat scenarios. The art of risk communication: the constant repetition of the same threat is not part of it.

3.2. Hospitalization rate
The hospitalization rate of known SARS-CoV-2/COVID-19 infections has declined significantly since spring and is currently around 6-71 TP3T (see Fig. b). Assuming an underreporting factor of 2 or 6 (see Chapter 2.3.2), considerably lower values are to be expected, as the hospitalization rate refers to the total number of infected individuals, not the number of reported infections (reporting rate). Surprisingly, the increase in the average age of hospitalized COVID-19 patients (unlike the mortality rate) has not yet been reflected in the hospitalization rate.
If one takes the staff and patient/resident/care recipient groups of facilities according to Sections 23 (hospitals), 36 (nursing homes), and 33 (care facilities) as a basis and presents the trend in the hospitalization rate (Fig. 9, excluding hospital patients and nursing home residents), a clear decrease in the hospitalization rate is also evident. In the interest of balanced risk communication, it would be appropriate at this point to emphasize the successes of the developments in prognosis and therapy, and thus the threat.
to lighten the scenario a bit.

These 10: The rate of hospitalizations has decreased significantly over time. For staff in hospitals, nursing homes, and care facilities, the decrease is currently continuing; for the total number of all Covid-19 patients (see Fig. 8), the rate has stabilized at a level of 7%.

3.3. Need for intensive care and ventilation
In recent weeks, there has been an increase in the number of intensive care patients with SARS-CoV-2/Covid-19 infection (see Fig. 10). Due to the different data sources, it is difficult to make a statement about the relative development of the need for intensive care (compared to the overall development of case numbers). However, the particularly important prognostic parameters of the need for ventilation and mortality are trending favorably and are decreasing significantly (see Fig. 10). This would also be an opportunity to communicate positive news, as some
Intensive care physicians suggest it (SZ 12.11.2020).
When assessing the increase in COVID-19 patients requiring intensive care, the question remains whether some of this increase is caused by the now standard practice of testing non-COVID-19 patients for COVID-19. Analogous to the common expression "mortality with or from COVID-19," the more appropriate term here would be "requiring intensive care with or because of COVID-19." Precise data on this distinction are not available.
Assuming a prevalence of 1% in the population (see Chapter 2.3.2), this would mean that out of approximately 20,000 intensive care patients, there would be 200 patients who do not require intensive care due to Covid-19 (but would then need to be cared for and treated as such).
The availability of intensive care capacity has caused considerable confusion. Figure 11 illustrates the data: It clearly shows that in recent weeks there has not only been an increase in intensive care occupancy by Covid-19 patients and a decrease in available capacity, but also a significant absolute decrease in the total available intensive care bed capacity. Staff shortages are frequently cited as the reason, but it is worth questioning whether these are not intensive care beds that are designated as such in the hospital plan, for which public funding has been allocated, and for which bonuses were offered during the first lockdown to keep them available.
These 11: The number of COVID-19 patients requiring intensive care has recently increased significantly, which explains part of the reduced available intensive care capacity. Over the entire period, there has been a decrease in the ventilation rate and, above all, in the mortality rate of COVID-19 patients requiring intensive care. The decrease in the total number of reported intensive care beds is puzzling, as this could also contribute to the reduction in available intensive care capacity.

3.4. Mortality – Update
As clearly shown in Figure 8, the case fatality rate of SARS-CoV-2/Covid-19 infection has been decreasing since the beginning of the epidemic and is below 1%. The RKI's case fatality rates based on reporting rates (a frequently cited point of criticism, see Tp2, Chapter 2.1.1 No. 4) are also slowly decreasing.
The epidemiological literature now contains good, population-based studies on this topic (e.g., Gudbjartsson et al. 2020B in Iceland), which suggest an overall fatality rate of less than 0.51 TPIs is likely. The age distribution and the capacity of the healthcare system play a crucial role: In a large meta-analysis, the infection fatality rate—that is, the mortality rate not based on reported cases but on the total number of infected individuals—increases from near zero to 0.41 TPIs at age 55, reaching 141 TPIs at age 85 (1.31 TPIs at age 65, 4.21 TPIs at age 75) (Levin et al. 2020). Studies of cluster outbreaks have limited relevance to this topic but can contribute important information under certain conditions.
The age-related factor is also evident in the RKI data when considering the mortality of hospitalized COVID-19 patients (see Fig. 8, turquoise-blue line). In September, the mortality rate of hospitalized COVID-19 patients was significantly below 101 TP3T, which is extremely favorable compared to the mortality rate of "normal" non-COVID-19 pneumonia (8-131 TP3T across all age groups, cf. Bundestag statement Schrappe 2020A). Currently, however, the value has risen above 101 TP3T, which is likely related to the increase in the average age – thus, there is no longer a significant difference when comparing hospitalized patients with COVID-19 and community-acquired pneumonia, e.g., caused by pneumococci. Furthermore, the long-term consequences of the increasingly traumatic effects of ventilation therapy are unlikely to differ significantly, based on future comparative studies. A calming of the situation by professional voices that do not downplay the risks but establish the connection is urgently needed.
Collectives without unreported cases and with complete data on mortality are the infected employees of facilities according to Section 23 (hospitals), Section 36 (nursing homes), and Section 33 (care facilities). As shown in Figure 12, a steady decline in mortality can be observed over the course of the epidemic for these three groups, if one disregards—and this is particularly important—the initial increase among nursing home staff (red line).

These 12: The mortality rate of patients infected with SARS-CoV-2/Covid-19 decreases significantly over time. The very clear age dependency appears to be confirmed. This also applies to the mortality rate of employees of facilities according to Sections 23, 36, and 33 of the Infection Protection Act (IfSG), which can be considered well-documented with regard to this endpoint.

4. Outlook
This position paper, 6.1, focuses on the epidemiological situation and criticizes in particular the inadequate methodological and substantive quality of the data currently used for information and control purposes, and the threshold values derived from them. Looking ahead, it must be pointed out that even the best data and methodologically sound threshold values can only fulfill their purpose if the underlying concept of prevention is well-designed and the potential for socio-political conflict is not exacerbated. For example, it remains to be seen whether the promise in the amended Infection Protection Act that social isolation of vulnerable individuals is not permissible will truly be implemented, or whether this is merely a legal fig leaf, meaning that in reality, violations of dignity and humanity will continue to occur without those affected having any recourse.
From an epidemiological perspective, two further points should be added as an outlook: firstly, the development of a sensible testing strategy, and secondly, the initiation of cohort studies. The testing strategy has been addressed in detail in previous position papers (Tp2, Chapter 2.1.3; Tp3, Chapter 1.3; Tp5, Chapter 2.2.1) and in the statement to the Bundestag on October 28, 2020 (Schrappe 2020B). The primary task is to first identify the crucial question and then develop appropriate procedures. An indiscriminate expansion, as was pursued during the summer months, must be rejected in any case. Rapid tests will, of course, play a prominent role in the near future; this will be addressed specifically in Position Paper 6.2 under the heading of prevention.
The second point concerns cohort studies, which remain urgently needed. The aim here is to address the problems with sample selection that have now been described in detail. Although these studies should have been initiated immediately from the outset according to scientific criteria, prospective cohort studies still need to be started today. They must include randomly selected population samples that are regularly (e.g., every 14 days) tested for new SARS-CoV-2/COVID-19 infections (longitudinal design, PCR and/or antigen tests, and secondarily, antibody testing). It is important to note that cross-sectional studies on seroprevalence (antibodies) are not considered cohort studies because they are retrospective. Cohort studies allow for key conclusions regarding the development of incidence, transmission routes, symptoms, prognosis, and risk groups. Furthermore, cohort studies are essential for planning and evaluating vaccination campaigns.
This brings us to the final point: Many aspects that have been overlooked and not adequately considered in the discussions of recent months will resurface during the planning of vaccination and/or vaccination campaigns. These primarily concern questions of prioritization, but also, of course, the assessment of effectiveness at the population level. As health services research already points out: the controlled study is one thing, the effectiveness (and adverse effects) at the population level is another.