Target practice

 

Personally, I think both the setting of targets during this epidemic, and debates about whether targets are genuinely met or have been fudged, are unhelpful distractions from the real issue of tackling the crisis. But just for the record, since Statistics is being dragged into this argument…

Also:

Mr Gove said 76,496 daily tests had been undertaken in the 24 hours up to 9am on May 3. This compares to the 122,347 tests carried out in the 24 hours to 9am on May 1 — the relevant period for when Matt Hancock, the health secretary, had set a deadline of undertaking 100,000.


Source and further details here


Update: again, just for the record… This is the UK government’s own graph on tests as of 5th May:

So, even disregarding arguments about what actually constitutes a test, current levels are significantly below the 100,000 tests per day target whichever definition is used.

Moving the goalposts

As you probably know, data on the state of the Coronavirus epidemic in the UK are presented and discussed at a daily press briefing. This includes a set of slides prepared by Public Health England (PHE) that give a statistical summary of the current status quo. There was slight complication in the presentation yesterday, however, since it was decided to include data on deaths that had occurred in care homes. This meant that there would be an additional increase of 5000 or so deaths. Not because these deaths had suddenly occurred, but because they had occurred at some point over the last couple of months and were now to be included in the count.

Here’s the slide on cumulative COVID-19 fatalities produced by PHE  from the day before the additional data were included.  

Now imagine increasing the UK value by around 5000 and the trajectory in the UK would be closer to that of the US than to that of any other European country.

But the equivalent PHE slide shown at yesterday’s press conference was this:

The UK value is now around 25k rather than 20k, due to the inclusion of the extra data, but the UK trajectory looks pretty much like that of several other European countries. And definitely much closer to those countries than to the US.

But how is this possible? The additional data can only make the UK trajectory worse in terms of numbers of fatalities, not better.

To answer this, you have to look at the fine print of the graphs. And actually there are 2 differences:

  1. In the first graph, countries are aligned so that day 0 for each country is the first day on which more than 0 cumulative deaths had been reported. In the second graph, this has been changed to 2000 cumulative deaths. This results in some realignment of the trajectories across countries.
  2. In the first graph, only data from day 15 are shown; in the second graph data are shown all the way from day zero. This has the effect of stretching the graph sideways in the second plot, which in turn has the effect of squashing the differences in the vertical direction.

So, both of these stylistic changes to the presentation have the effect of making the UK’s trajectory of COVID-19 fatalities look less extreme relative to those of European countries.

Now, there may be some perfectly innocent explanation as to why these changes were made and it may just be a coincidence that they show the UK numbers in a better light. Or, things were chosen this way by design. But in either case, there’s an important lesson here in how simple choices in presentation can lead to completely different interpretations of the same data.

Following the science

There’s been a lot of discussion lately about the efficacy and efficiency of the UK government response to the Coronavirus epidemic. There are many strands to this, but one concerns the speed with which policies of social restriction were introduced. And a lot of the debate has focused on two sporting events that were held shortly after lockdowns were introduced in many European countries, but before they were introduced in the UK: the Cheltenham Festival and the second leg of the Champions League tie between Liverpool and Atletico Madrid.

The Liverpool game was especially controversial because it had been known for some time that Madrid was already a focus for the Coronavirus outbreak in Spain. And while most other Champions League fixtures that were held that week were held behind closed doors, the decision was made to hold the Liverpool game with spectators, including 3000 travellers from Madrid.

The picture from both Cheltenham and Liverpool after each event is concerning, since both locations appear to have a higher rate of infection than would be expected (see here and here). But it will take careful analysis of the data to establish the extent to which these apparent effects can be properly attributed to the associated sporting events, and an even fuller analysis to determine whether the decisions to hold the events were anyway reasonable or not.

One argument, for example, that’s been presented to justify not holding matches behind closed doors is that there may be more transmission if people watch a match in many pubs rather than in a stadium. And in any case, it’s perfectly valid to argue that a higher rate of infection due to holding a sporting event has to be offset against the economic and other social costs of not holding it. So, even if it turns out that the 2 events in question are genuinely likely to have increased infection rates, this doesn’t in itself imply that the decisions to hold both events were wrong.

But here’s the thing… as with all aspects of planning for and responding to events connected with the epidemic, Science – and Statistics – provides a framework for decision making. In particular, it will give predictions about what is most likely to occur if different actions are taken and, in the case of statistical models, most likely also attach probabilities to different possible outcomes, again dependent on the course of actions taken.

Crucially, though, Science will not tell you what to do. It won’t tell you how to balance costs in terms of lives against that in terms of money. Or jobs. Or something else. That’s a political decision. Moreover, ‘Science’ isn’t a fixed static object that unveils itself in uniform and unchallenged forms. There are different sciences, all of which are constantly evolving, and any combination of which might lead to conflicting conclusions. Even different statistical models might not be in complete agreement. Science will help you understand the costs and benefits of actions that are available to you; but you must take responsibility for the choices you make on the basis of that information.

However, I’ve lost count of how many times politicians – especially in the UK – defend their actions by arguing ‘we followed the science’.  Here’s Health Secretary Matt Hancock in defence of the decision to hold the Cheltenham festival:

We followed the scientific advice and were guided by that science.

And here in defence of holding the Champions League cup tie:

This is of course a question for the scientists and what matters now is that people in Liverpool and across the North West get the treatment that they need and get the curve under control.

Neither comment is likely to be completely untrue – it would obviously be outrageous for any government in any situation to completely ignore scientific evidence – but both seem to be distractions from the fact that decision-taking is a political process which balances the various risks and costs involved.

The most Science can do is to provide an assessment of what those risks and costs are.


Here’s Brian Cox’s take on the same argument:

When you hear politicians saying ‘we’re following the science’ then what that means is they don’t really understand what science is. There isn’t such a thing as ‘the’ science. Science is a mindset, it’s about trying to understand nature.

And here’s the full section with video:

 

Moving on

The UK, in common with all other countries that have managed to suppress the worst effects of the Coronavirus pandemic though strong social controls, must now decide how and when to move out of the current status. Inevitably, tensions pull in different directions. On the one hand, a strict lockdown that eliminates most social interactions is safest in reducing transmissions of the virus; on the other hand, a complete relaxation of the restrictions is bound to lead to the transmission rate returning to its previous levels and the epidemic growing again at an exponential rate. If the objective were simply to minimise COVID-19 deaths in the anticipation of a vaccine being found, or simply to slow down the infection rate so that health services are not overwhelmed, then maintaining the current lockdown would be the best strategy.

But as is becoming increasingly clear, this strategy is unsustainable in the long-term for economic, social and indeed other health reasons. Inevitably, then, the question turns to which restrictions can be lifted without the epidemic again spiralling out of control. Or more bluntly: is it even possible to keep the epidemic under control with anything less than a total lockdown?

Recall from an earlier post, that the main statistic in determining the trajectory of an epidemic is the transmission rate, R, which is the average number of people an infected person will subsequently infect. Recall also that the value of R=1 is critical to understand the epidemic’s path; greater than that and the epidemic will grow exponentially; lower, and it will begin to fade. So, can R be reduced below 1 without having a total lockdown?

This graph, taken from ‘A Sustainable Exit Strategy: managing uncertainty, minimising harm‘, by the Institute for Global Change, paints a somewhat negative picture. It suggests, based on experience in different countries, that the only situations where the value of R was kept below the value of 1 was through strict lockdown.

The report goes on to make suggestions as to how a ‘traffic light system’, where different strengths of measures – red/amber/green – might be enacted depending on the current numbers of new cases, perhaps on a regional basis. But there are obvious logistic difficulties to stopping and starting commercial and business activities, so the question arises as to whether a uniformly applied set of measures, that stop short of a total lockdown, can reduce R to levels below 1.

Since other European countries are now starting to relax restrictions, evidence on this will become available in the coming weeks. But a detailed statistical study from Hong Kong already provides some information. This study has been published in the Lancet, as linked to in the following tweet:

The full report also can be found here.

Basically, the authors have mapped how the transmission rate of the disease has changed through time in Hong Kong, and related such changes to the social restrictions that were imposed at that time.

The following graph, taken from the report, shows the number of cases through to the end of March and the timing of restrictions that were imposed. At no point was a complete lockdown introduced in Hong Kong, though limitations on travel into the region and so-called community measures were introduced progressively in order to suppress the transmission rate.

Now compare this against a graph of the estimated transmission rate, R.

As you can see, the initial rate in Hong Kong was well above 2. The initial restrictions – mainly on travel and school restrictions – seemed to be sufficient to reduce the rate below 1 by the middle of February, but by the end of February there was evidence that the rate was creeping back above 1. Consequently, stricter restrictions were introduced, which seem to have stabilised the value of R to around 1. Admittedly, the best estimate of the current value of R is still slightly greater than 1, but the effect of the most recent restrictions are likely to take some additional days to filter through. Moreover, a value of R that is only very slightly greater than 1 does imply exponential growth, but at an initial rate that is likely to be manageable.

So, the Hong Kong experience – as evidenced by detailed statistical analysis – is that travel and community restrictions that fall substantially short of a total lockdown, can be sufficient in reducing Coronavirus transmission rates to levels that avoid the epidemic growing out of control. That said, there are obviously many other differences between the UK and Hong Kong, including demographic and social differences, as well as other aspects of epidemic control, so what works for Hong Kong may not be reproducible exactly in the UK. Nonetheless, it’s encouraging that the epidemic can be controlled with measures that are less draconian than those currently in place.

 

 

Testing times

As referred to in an earliest post, it’s been understood from the start of the epidemic that testing for Coronavirus is crucial for limiting its initial spread and as a way of mitigating against a second wave once the first wave has been brought under control. It seems reasonable therefore, at least in part, to judge the efficacy of a government’s response to the crisis by the extent of their testing regime. So how does the UK compare against other countries in this respect? Though it’s not the whole picture we can look at some statistics across countries.

This is a graph of the total number of tests per 1,000 people carried out through time for a number of countries:

Some of the main points are:

  • Even allowing for the fact that South Korea had a number of cases before most other countries, it carried out a large number of tests early on in the crisis. This is almost certainly one reason that it has been relatively successful in containing the epidemic, and has not needed to massively extend its testing capacity in recent weeks.
  • Italy’s outbreak started later than that of South Korea, but even allowing for that, it was slower in developing its testing capabilities. More recently though it has developed an extensive testing framework.
  • The US was extremely slow in ramping up testing facilities, though in recent weeks it has started to do much better.
  • The UK and Greece continue to have relatively limited numbers of tests. However, Greece has been commended for tackling the epidemic through other fast and decisive measures; the UK less so.

Another way of looking at the numbers is as a single snapshot of the test numbers – this time per 1 million capita – as of a couple of days ago.

It’s maybe no surprise that the UK is unable to match a country with a far smaller population like Iceland, but less clear why it can’t match Ireland, Italy or Germany.

But what about the relationship between testing and infection rates? The following graph compares per capita test rates against per capita case numbers:

Interpretation is not completely straightforward. One might hope that more tests per capita lead to a better control of the epidemic and therefore fewer cases per capita. But equally, countries that have so far had relatively few cases are likely not to have built up an extensive testing network yet. Also, a country that carries out more tests is likely to identify more positive cases.

Nonetheless, there are a number of takeaway points:

  • Though Iceland does the most tests per capita, it also has a large number of cases. But, as explained above, one reason it has more confirmed cases is likely to be simply because it’s done more tests. Moreover, Iceland has been praised for tackling the epidemic through a rigid contact tracing and isolation regime, and has been successful in keeping the number of deaths very low (currently just 9).
  • Italy, similarly, has a large number of cases despite carrying out many tests. Again, this will be partly because more tests are likely to lead to more cases, but is likely to be more strongly influenced by the fact that Italy was the first European country to be badly affected by the epidemic. It therefore had less evidence with which to decide how best to respond to the outbreak and was therefore relatively slow in ramping up testing. Furthermore, Italy is further ahead in its trajectory than other countries, so in future versions of this graph, other countries are likely to move further to the right than Italy does.
  • Though South Korea has kept its number of cases fairly low – especially for a country that was affected early on in the pandemic – its actual testing capacity is no greater than that of the US. However, as we saw in the earlier graph, its initial testing response was very much quicker. So, it’s not just the number of tests that’s important, it’s the speed with which the testing framework is put in place.
  • For countries with a large number of cases, the UK is weakest in terms of the number of tests.  This won’t be the only reason it has a large number of cases per capita, but it’s likely to be one of them.

In summary, based purely on the statistical evidence, the UK comes out relatively poorly compared to many other countries in terms of its response to the Coronavirus epidemic through building a platform for population testing.


One caveat. The fine print in each of these graphs makes clear that comparison across countries is not completely fair. Data are collected and recorded in different ways in different countries. For example, in the UK numbers are on people tested; in Italy they are on the number of tests carried out. This is likely to make some difference, as people who test positive are likely to be subsequently tested again to ensure they are negative. This would apparently count as 2 tests in the Italy data, but only 1 in the UK data. However, most test results come back negative. In Italy, for example, the rate of positive tests is around 13%. So, although the difference in data protocols might explain some of the difference in the UK and Italy numbers, it doesn’t explain most of it.

 

Prior infection

In an earlier post I referred to an Austrian study in which a random sample of the inhabitants of a region had been tested for the Coronavirus. In that article I stressed the importance of random sampling in order to be able to extrapolate the conclusions from the sample to a wider population. But a limitation of that study was the fact that tests were ‘standard’ COVID-19 tests, providing information only on whether an individual currently carried the disease or not. So, although the infection rate was only 0.33% in the sample, it was impossible to say what proportion of individuals in that sample had ever had the disease, perhaps having only slight or no symptoms.

This week a similar study has been reported based on a study in the German municipality of Gangelt, close to the Netherlands border. But in this case, the 500 individuals in the sample were tested for Coronavirus antibodies, a test which provides information on whether an individual has a prior COVID-19 infection, regardless of whether they currently have the disease.

Now, there has been something of a mis-handling of such tests in the UK, with millions of bought tests of this variety which were not properly evaluated prior to purchase and shown subsequently to be worthless in terms of their results. But as far as I can tell, this isn’t an issue for the tests used in the German study.

The results of the study are interesting, and shed both a positive and negative light on the effects of the epidemic.

First, some context. On February 15th, a couple of weeks after Germany reported it’s first case of Coronavirus, Gangelt held its annual carnival. Subsequently, several attendants at the carnival tested positive for Coronavirus, and shortly after the town became a hotspot for the infection within Germany. It’s widely assumed that the carnival was focal in causing the local hotspot and for attendants then assisting the spread of the disease countrywide. So, it’s thought that the infection rate in Gangelt is likely to be relatively high, which is one of the reasons it was chosen for this study.

However, the random sampling antibody study found that ‘just’ 14% of the sample are likely to have a prior infection. Still, 14% is considerably higher than the proportion that have tested positive in the region (around 3.5%), implying that the true death rate due to COVID-19 there is around 0.37% as opposed to the 2% or so that’s stated for Germany as a whole based on standard data. That’s obviously a welcome piece of news.

On the negative side, as discussed in earlier posts – here for example – one solution to the pandemic will occur naturally when a sufficient proportion of the population have a prior infection and are hopefully immune – though this is not guaranteed – from subsequent infection. But this is thought to be as much as 80% for COVID-19. So, if in a community that’s thought to have been a bit of a hotspot the previously infected rate is only 14%, it’s likely to be a long way short of the required threshold in the country as a whole. In other words, on the basis of this study, Germany – and presumably other countries too – are likely to be very far off from being able to relax social restrictions and rely on herd immunity to get them out of the pandemic.

For the UK, the Harvard epidemiologist William Hanage shows – using some back-of-the-envelope calculations – that to acquire herd immunity in the UK, a minimum of 600,000 COVID-19 deaths would occur. Though the UK government’s initial strategy seemed to be aimed at achieving herd immunity without a vaccine, their realisation of the scale of this number of fatalities and the impact it would have on both communities and the health service is almost certainly what led to a change of heart. But until a vaccine is found, the alternative is maintaining social restrictions to a sufficient extent to keep the transmission rate of the disease sufficiently low. As Hanage says:

This crisis is not close to over, quite the reverse. The pandemic is only just getting started.


A couple of comments:

1. Hanage’s calculation goes like this… At the time of writing there were around 100,000 COVID-19 confirmed cases in the UK. But the British Medical Journal suggests that only around 20% of infected people are actually tested positive. (Based on the Gangelt study the estimate would be 3.5/14 =25%). This implies that the actual number of people in the UK who have been infected is around 500,000. But if the UK is currently at the peak of its epidemic, and a similar number of people will be infected as the epidemic declines, that implies one million infected people at the end of the epidemic. But that means around 65 million people will remain uninfected and without immunity. Similarly, there have been 10,000 confirmed deaths due to COVID-19 in the UK. Assuming fatalities have also now peaked, there will be a total of 20,000 deaths once the epidemic has faded. Now, a very conservative estimate of the proportion of people in the population who need to be infected for herd immunity to kick in is 50%, which implies around 30 million people. But so far, as we’ve seen, it’s likely that only one million have been infected. So, we need 30 times as many people to be infected, and this will imply 30 times as many deaths; i.e. around 600,000.

2. Hanage’s comments are not meant to be fatalistic. His point is simply that there are many positive signs that show social distancing is working in terms of controlling the spread of the epidemic in most countries.  But, these measures are doing only that: controlling the spread of the epidemic. Though numbers of infections are growing in the community, they are still a long way from the sorts of numbers that will inhibit further spread of the epidemic via herd immunity. So until vaccines and therapies are available, it’s inevitable that some form of ongoing social controls will be required to stop the epidemic growing exponentially once more.

A/B test

In an earlier post I showed how a comparison of two different, though similar, provinces of Italy which had adopted different approaches to containing the Coronavirus outbreak could serve as a kind of statistical trial on the relative effectiveness of the strategies. A similar argument has now been made in respect of Ireland and the UK by Elaine Doyle. This is the first tweet in her thread, showing that Ireland and the UK started off with similar resources to fight the pandemic:

But, at the time of writing, the UK had far more fatalities than Ireland. Elaine writes:

As of today, there have been 320 deaths from the coronavirus in Ireland, and 9,875 deaths in the UK.

Moreover:

As of Saturday 11 April, there have been 6.5 deaths per 100,000 people in Ireland. There have been 14.81 deaths per 100,000 people in the UK. Guys, people have been dying at more than *twice the rate* in the UK

Elaine argues that since the countries are comparable in most respects, it’s reasonable to assume that the difference in outcomes is attributable to the difference in approaches.

The complete thread of her argument is available here. She writes:

You have a real-time A/B test happening *right in front of you

In other words, you can think of Ireland and the UK as two equivalent units. To one unit (Ireland) you’ve applied ‘treatment A’, a fast and decisive lockdown. To the other (UK) you’ve applied a slower, more gradual, lockdown. And the difference in outcomes is due to the difference in treatments.

This argument isn’t universally agreed on. Other experts argue that there are other fundamental differences between Ireland and the UK – for example, Ireland has a much greater proportion of its population living in rural areas – and these are just as likely to impact on the numbers of COVID-19 cases as the differences in the strategies applied. This is no doubt true. And a true A/B test  would have comprised random allocation of treatments to more than two countries. The lack of randomness in the allocation is a serious hinderance to interpretation, though a comparison of just 2 countries is dangerous in any case.

In time, it might be possible to compare many different countries according to the strategies adopted, accounting for different geographical and demographic factors, and decide which strategies are most effective. This type of analysis was discussed in an earlier post for the 1918-19 influenza epidemic, though I’m guessing we won’t want to wait 100 years to carry out such an analysis for the current pandemic.


Update: since writing this post, Elaine Doyle has written an article for the Guardian setting out her arguments.

Coronavirus statistics

I was thinking of writing a post giving a kind of glossary of various statistical terms and tools that are relevant for interpreting reports about the current pandemic, but Sylvia Richardson and David Spiegelhalter have already done a better job of it than I could have done. So, just follow the link here.


Sylvia Richardson has been elected to be the next president of the Royal Statistical Society, as of January 2021. David Spiegelhalter himself held this position from 2017-18.

Relative risk

One issue that’s constantly pushed by the media about the effects of COVID-19 is that it’s much more deadly for older people than for younger people. The following graph, for example, shows the age distribution for COVID-19 deaths in Italy as of 11, April:

Bear in mind, also, that there are fewer people in the older cohorts, so the differences in the rate of deaths per cohort are even greater.

But does this suggest that COVID-19 is affecting younger and older people to a different extent? In one sense clearly yes – more older people are dying than younger people. But, in another sense, no. Or at least, possibly no.

In this article David Spiegelhalter – whose podcast I mentioned in an earlier post –  considers the hypothesis that everyone’s risk of dying in the short term is affected by the current epidemic in exactly the same way, regardless of age.

Look, for example, at this diagram taken from David’s article:

These are deaths by age, per 100,000 of the population in England for a week in March, shown separately for males (left) and females (right). The solid dots correspond to COVID-19 deaths, the hollow dots are all other deaths.There are several points to notice:

  • The scale of the graph is logarithmic, so since the death counts – both COVID-19 and not – are approximately linear, the raw numbers will be increasing exponentially with age;
  • This is true for both males and females;
  • The patterns for COVID-19 deaths and others are roughly parallel on this logarithmic scale. This means that on a linear scale, one will be a multiple of the other.

Consider, for example, males in the 45-64 cohort. The death rates for COVID-19 and other causes are around .05 and 10 per 100,000 respectively. This means that the total risk of death including COVID-19 is 10.5:10 or 105% real-time to what it would have been with COVID-19 excluded. Loosely speaking, COVID-19 has raised the death risk for someone in that cohort by around 5%. But if you look at any of the other cohorts, the change is around the same. For example, in the 65-74 cohort, the respective death rates are around 2 and 40 per 100,000, leading to a ratio of 42:40 which is again 105%.

So, looked at this way, COVID-19 is affecting people of all ages in exactly the same way: it’s increasing the risk of death by around 5%. It’s simply that the death risk is so low for young people that an increase of 5% doesn’t lead to many additional deaths in absolute terms, whereas for older people a 5% increase in death rate leads to substantial increases in the observed numbers of deaths. Nonetheless, this provides an interpretation by which COVID-19 is non-discriminatory in terms of age.


A few caveats:

  1. This analysis is based on relatively few data. Though David’s article includes other analyses to support the hypothesis, he also concludes that further data will be required to verify it.
  2. Most of the deaths observed in the data summarised above will have been for people infected prior to lockdown measures having been introduced. It’s likely to be the case that lockdown measures will offer more protection to certain age cohorts than others, in which case the effect on death rates will be disproportionate across age cohorts.
  3. David also mentions in his article that health workers appear to be disproportionately affected in terms of COVID-19 – i.e. their additional death risk due to COVID-19 is greater than 5%.

Random sampling

rs

A recurrent theme in the COVID-19 posts to this blog is the difficulty in interpreting data and analyses due to the way data are collected. Different countries have completely different protocols for testing for the disease, and these protocols have also changed through time. Even the reported death counts are unreliable as a measure of the disease effects.

In one earlier post I mentioned two case studies where entire – though limited – populations had been tested: Vò, in northern Italy and the Diamond Princess cruise ship. Since these entire populations were studied, the data are 100% complete, but they are special cases, since they are closed populations where an outbreak of the epidemic is known to have occurred. But what about entire countries? It’s obviously impractical – at least in present circumstances – to test an entire population.

A statistically valid alternative in this case is random sampling – testing individuals randomly selected from the entire population. The proportion testing positive in the sample provides an estimate of the proportion in the entire population, and the bigger the sample, the better the estimate. Obviously there are logistical difficulties in testing genuinely randomly selected individuals, so various practical modifications are often implemented which have to be correct for in the analysis. But the principle is the same: to use information from a randomly selected sample of individuals to estimate the population level.

A study of this type has now been carried out for Austria. Full details of the analysis can be found here. In summary:

  • 1,544 individuals were included in the study which was carried out in the first week of April;
  • These individuals were identified by a stratification procedure: 249 Austrian districts were randomly selected; households in those districts were randomly selected; individuals within those households were randomly selected;
  • Such individuals were invited to participate in the study – the acceptance rate was 77%;
  • Hospitalised individuals were excluded from the study;
  • Final results were adjusted to correct for various factors including household size, gender and age.

The conclusion, after the correction for age, gender and other effects, is that the COVID-19 infection rate in the sample was 0.33%.

Now, bearing in mind that one solution to the epidemic is that a large number of the population acquire the disease, so building a ‘herd immunity‘, the figure of 0.33% is disappointingly small. Even allowing for sampling error, the true value in the population is predicted in the study to be at most 0.76%, whereas it’s thought that herd immunity will require around 60-80% of the population to have been infected.

However, this figure of 0.33% is just a snapshot in time of people who currently have the virus; it doesn’t say anything about the proportion of people who have had the virus – perhaps asymptomatically – and recovered. That figure, which is the figure of interest when discussing herd immunity, is bound to be bigger. But it’s impossible from this study to say by how much.

Moreover, extrapolating the 0.33% to the entire population of Austria would imply around 28,500 positive cases. By contrast, the number of active cases in Austria (as of today, 11 April) is recorded as 6,608:

austria.png

So, even as low an estimate as 0.33% for the countrywide infection rate implies a roughly four-fold increase in the number of cases above and beyond the official numbers.