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Alex de Visscher Coronavirus Covid-19 Superspreader Worldometers

SARS-Cov-2 modelling situation report

Introduction

As we start September, the UK situation regarding Covid-19 cases and deaths has changed somewhat.

Since the UK Government re-assessed the way deaths data is collected and reported, the reported daily deaths resulting from Covid-19 infections have (thankfully) reduced to a very low level, as we see from the UK Government Covid-19 reporting website.

Cases, however, as we see from the Government chart on the right, have started to rise again, although for a number of reasons the impact on deaths has been less than before. Note that this chart plots people testing Covid-19 positive (daily and total to date) against time.

I have integrated this real-world UK reported data with my model data to assess what is happening.

Reporting changes for UK deaths

As I reported in my August 17th post, reported daily deaths in England had previously set no time limit between an individual’s positive test for Covid-19, and when that person died.

The three other home countries in the UK had already been applying a 28-day limit for this interval. It was felt that, for England, this lack of a limit on the time interval resulted in over-reporting of deaths from Covid-19. Even someone who had died in a road accident, say, would have been reported as a Covid-19 death if they had ever tested positive, and had then recovered from Covid-19, no matter how long before their death the positive test had occurred.

This adjustment to the reporting was applied retroactively in England for all reported daily deaths, which resulted in a cumulative reduction of c. 5,000 in the UK reported deaths to up to August 12th.

Case numbers and antibody testing

You can see from the following Chart 10 that the plateau for modelled cases is of the order of 3 million. This startling view is supported by a recent Imperial College antibody study reported by U.K. Government here.

I have applied a factor of 8.3 to the reported cases in Chart 10 to bring them into line with the modelled cases, owing to significant under-reporting of the number of UK cases (based on positive Covid-19 tests).

Modelled Cases & Deaths development since Feb 1st - Uninfected, Cumulative Deaths, Uninfected & Seriously Sick
Modelled Cases & Deaths development since Feb 1st – Uninfected, Cumulative Deaths, Uninfected & Seriously Sick

The reported cases (defined, as above, by UK Government as people who have had a positive Covid-19 test) are just 337,168 as at September 1st, as we see from the following chart 9.

Modelled vs Reported Compartment development - Uninfected, Cumulative Cases & Deaths. Modelled Uninfected, All Infected & Seriously Sick
Modelled vs Reported Compartment development – Uninfected, Cumulative Cases & Deaths. Modelled Uninfected, All Infected & Seriously Sick

Testing, antibodies and case counting

The four pillars of Covid-19 testing include a single pillar of antibody testing, although it isn’t clear exactly which class of antibody is being tested. Not all antibody tests are the same.

It is also the case that despite more than 16 million Covid-19 tests having been processed in the UK to date (September 1st), the great majority of people have never been tested.

The under-reporting of cases (defining cases as those who have ever had Covid-19) was, in effect, confirmed by the major antibody testing programme, led by Imperial College London, involving over 100,000 people, finding that just under 6% of England’s population – an estimated 3.4 million people – had antibodies to Covid-19, and were therefore likely previously to have had the virus, prior to the end of June.

Even my modelled cases are likely to be a little under-estimated, and some update to my model’s calculation of cases will be made shortly.

Quite apart from the definition and counting of cases, according to a recent report by The Times, referencing this article from the BMJ, results obtained from some antibody testing might well be under-estimated too.

Stephen Burgess, from the Medical Research Council Biostatistics Unit at Cambridge University, and one of the authors, said. “It’s possible that somebody could have antibodies present in their saliva but not in their blood and it’s possible that somebody could have one class of antibody but not another class of antibodies.”

In particular, most antibody tests do not look for a type of response called IgA antibodies, which are made in mucus — in the mouth, eyes and nose. “In certain respiratory diseases, it’s well-documented that it’s possible to beat the infection with an IgA response,” he said.

When scientists have tested for IgA as well as the standard IgG antibodies, they have on occasions found hugely different results. In Luxembourg, IgA were found in 11 per cent of people compared with 2 per cent who tested positive using more conventional tests.

Dr Burgess said that calibrating tests using people who had been more severely ill may mean that a lot of asymptomatic infections are being missed.

The full report is here.

The Times concludes that it’s possible that herd immunity is closer than we think, with regional variations.

Reported Cases and Deaths

The following slide presentation shows only reported data for the UK. With Tom Sutton’s help, I have managed to link his previously developed Worldometers scraping code, which interrogates the daily updated Worldometers site for the UK, to retrieve reported cases and deaths data, to populate my MatLab/Octave model for Coronavirus (originally developed by Prof Alex de Visscher at Concordia University, Montreal).

This allows me to plot both modelled forecast data and reported data on the same charts, plotted from from the Octave forecasting model.

  • Reported UK Deaths vs.Cases since Feb 15th 2020, log chart
  • Reported UK Deaths vs.Cases since Feb 15th 2020, linear chart
  • Reported UK Deaths since Feb 15th 2020, linear chart
  • Reported UK Cases and Deaths since Feb 15th 2020, dual axis, log deaths, linear cases
  • Reported UK Cases and Deaths since Feb 15th 2020, linear dual-axis chart
  • Reported UK Cases and Deaths since Feb 15th 2020, log chart

Chart 3 shows reported deaths plotted against cases, on a log chart, and shows the log curve for deaths flattening as cumulative cases (on the linear x-axis) increase over time, indicating that the ratio of deaths/cases is reducing. This can also be seen very clearly on the linear scaled Chart 4.

Chart 5 shows cumulative deaths over time on linear axes, exhibiting the typical S-curve for infectious diseases; as of September 1st, daily deaths in the UK are in single figures.

Chart 6 shows deaths on a log y-axis (left) and cases on a linear y-axis (right).

Chart 7 plots both deaths and cases on linear y-axes (left and right respectively) for more direct comparison, and again we see that recently, since about Day 110 ( June 1st), cases have increased proportionately much faster than deaths. This date is fairly close to the time that the UK started to ease its lockdown restrictions.

Finally, Chart 8, plotting both deaths and cases on the same log y-axis, shows the relative progression over nearly 200 days since the onset of the pandemic.

These different views clearly show the recent changes in the way the epidemic is playing out in the UK population. Bear in mind that reported cases need something like a factor of 10 applied to bring them to a realistic figure.

Evidence for the under-estimation of Cases

The Imperial College antibody study referenced above is also in line with the estimate made by Prof. Alex de Visscher, author of my original model code, that the number of cases is typically under-reported by a factor of 12.5 – i.e. that only c. 8% of cases are detected and reported, an estimate assessed in the early days for the Italian outbreak, at a time when “test and trace” wasn’t in place anywhere.

A further sanity check on my forecasted case numbers, relative to the forecasted number of deaths, would be the observed mortality from Covid-19, where this can be assessed.

A study by a London School of Hygiene & Tropical Medicine team carried out an analysis of the Covid-19 outbreak in the closed community of the Diamond Princess cruise ship in March 2020.

Adjusting for delay from confirmation-to-death, this paper estimated case and infection fatality ratios (CFR, IFR) for COVID-19 on the Diamond Princess ship as 2.3% (0.75%-5.3%) and 1.2% (0.38-2.7%) respectively. See the World Health Organisation (WHO) description of CFR & IFR here.

In broad terms, my model forecast of c. 42,000 deaths and up to 3 million cases would be a ratio of about 1.4%, and so the IFR relationship between the deaths and cases numbers in my charts seems reasonable.

(NB since we know that the risk of death from Covid-19 is higher in older people, and the age profile of cruise ship passengers is probably higher than average, the Diamond Princess percentages are at the high end of the spectrum.)

Reasons for the reducing deaths/cases ratio

Reported deaths per case are reducing significantly, because:

a) we are more aware of taking care of older people in Care Homes (and certainly not knowingly sending Covid-19 positive old folks to them), sadly lacking in the early days of the pandemic in many countries;

b) relatively more young people are being infected as compared with older people because they are the ones working and going out more, and they have lower mortality than older people;

c) we have some better experience and palliative treatments to help some people recover (eg Dexamethasone as described at https://www.sps.nhs.uk/articles/summary-of-covid-19-medicines-guidance-critical-care/); and

d) daily cases are increasing, rather than reducing, as deaths are.

This is covered in a very good article by Rowland Manthorpe, technology correspondent, and Isla Glaister, data editor of Sky News, whose reports I have read before. The article makes very clear the changes in the age-profile of cases from early March to the end of July.

UK weekly confirmed cases by age, published by Sky September 2nd 2020
UK weekly confirmed cases by age, published by Sky September 2nd 2020

Another view of this is from The Times on September 5th, data sourced from Public Health England (PHE);

UK weekly confirmed cases by age, published by The Times September 5th 2020
UK weekly confirmed cases by age, published by The Times September 5th 2020

and, more specifically, here is how the proportion of cases has shifted between under 40s and over 50s from March until September.

Changing age profile of Covid-19 cases, published by The Times September 6th
Changing age profile of Covid-19 cases, published by The Times September 6th

Issues for modelling presented by local spikes

Modelling the epidemic for the UK is now really difficult, as most cases having an impact on the UK national statistics are nearly all caused by local outbreaks, or spikes – what I call multiple super-spreader events. Although that isn’t quite the right description, these are being caused by behaviour such as lack of social distancing, and maybe erratic mask-wearing on flights returning to the UK with pre- and even post-diagnostic cases on board.

The super-spreader events in the early days in Italy (and in the UK) were caused by people, unknowingly and asymptomatically infected, returning to their home countries from overseas and infecting others.

The increasingly frequent recent events we are seeing are caused, it seems to me, by people who ought, nowadays, to have more awareness of the risks, and know better, compared to those in the early days.

What would be needed to model such events is good local data for each one, and some kind of model for how, when and how often, statistically, these events might occur (aircraft, pubs, clubs, demonstrations, illegal raves and all the rest). Possibly even religious gatherings and other such cultural (including sporting) gatherings have a role.

So modelling this bottom-up is difficult – but feasible, hopefully. In any case, what is needed at the moment is a time-dependent way of handling the infection risks, in the context of these events, the way that lockdown easing points have been introduced to the model.

Worldometers/IHME forecasts and charts

I might say that modelling only by curve fitting, top-down, is pretty incomplete in my view. Phenomenological methods forecast the future based on the past with no ability to model or reflect changes in intervention methods, public behaviour and responses; and I see no capability in the methodology to take super-spreader events into account.

This might be difficult for bottom-up mechanistic modelling, but it’s impossible for broad, country-based curve-fitting, as no link can be made from input changes in government measures, population responses and individual behaviour, to their influence on outcomes.

I covered the comparative phenomenological and mechanistic methods in my previous posts on July 14th and July 18th.

In the charts that follow, we see that forecasts are made for three scenarios: current projections; mandates easing; and universal masks.

To do this, as IHME (Institute for Health Metrics and Evaluation at the University of Washington, USA) say at the IHME FAQ (Frequently Asked Questions) page, Worldometers/IHME forecasts rely on both statistical and disease transmission models: “Our current model is not a disease transmission model. It is a hybrid model that combines both a statistical modeling approach and a disease transmission approach, leveraging the strengths of both types of models, and scaling the results of the disease transmission model to the results of the statistical model.

This enables them to calibrate outcomes based on three outbreak management scenarios.

Illustrating the point, I show the IHME forecast for the UK, followed by that for the USA . First the UK:

Worldometers forecast for the UK, with three scenarios and error bounds

It seems that IHME forecasts for the UK, linked to the Worldometers UK site, are based on a broader view of UK deaths, relating to those where Covid-19 is mentioned on the death certificate, as defined by the UK Office for National Statistics (ONS), but not necessarily cited as the cause of death.

This is even though the Worldometers current reporting charts themselves are consistent with UK Government reported data, which presents deaths in all settings (including hospitals, care homes and the community) but only when Covid-19 is cited as the cause of death.

These ONS and IHME numbers are higher than the UK Government (and Worldometers) statistic. The daily numbers I have been using, presented by the UK Government, continue to be based on the narrower definition – Covid-19 as the cause of death on the death certificate.

Nevertheless, my main point here isn’t about the absolute numbers, but about the forecasting scenarios. We can see that the IHME methodology allows for several forecasting scenarios – current projections based on the interventions currently in place; mandates easing; and universal mask-wearing.

The US IHME forecast is presented similarly:

Worldometers forecast for the USA, with three scenarios and error bounds

In the case of the USA, the numbers are far larger for a much bigger population, and at worst the numbers are staggering. The Covid-19 deaths, currently 187,770 on this chart, had already exceeded Michael Levitt’s well-publicised curve-fitting Twitter forecast made in mid-July, indicating that by August 25th the USA excess deaths will have reduced to a very low level, and that the USA experience of the pandemic would essentially be over, with 170,000 deaths. It seems he agrees that forecast, or at least the way he expressed it, was a mistake.

In the USA case, the numbers are far larger for a much bigger population, and at worst the numbers are staggering. The current 187,770 already exceeds Michael Levitt's well-known curve-fitting forecast made a month ago, indicating that by August 25th the USA excess deaths will have reduced to a very low level, and that the USA experience of the pandemic would essentially be over, with 170,000 deaths. It seems he agrees that forecast, or at least the way he expressed it, was a mistake.
Michael Levitt’s well-publicised curve-fitting Twitter forecast made in mid-July, indicating that by August 25th the USA excess deaths will have reduced to a very low level, and that the USA experience of the pandemic would essentially be over, with 170,000 deaths
Michael Levitt's statement that his estimate of 170,000 reported deaths made 11 July was 7K too low.
Michael Levitt’s statement that his estimate of 170,000 reported deaths made on 11th July was 7K too low.

See Michael’s new UnHerd interview with Freddie Sayers.

As for excess deaths, no measure is without its issues, and the problem there is that Covid-19 deaths will probably have replaced deaths from some other causes (people go out less, so there will be less road accident deaths, for example).

This means that excess deaths reducing to zero isn’t by any means a sufficient test that the SARS-Cov-2 pandemic is all over bar the shouting.

IHME can predict several scenarios, as for the UK, and at best they are predicting 288,381 deaths by the end of the year for the USA. At worst their number is over 600,000. I’m sure things wouldn’t be allowed to get to that.

But these kinds of scenarios for different potential interventions, in combinations, or when eased, just aren’t going to work with curve-fitting alone, where, given just 3 (or at best, 4) parameters to do a least-squares fit of a Cycloid, Gompertz or more general Richards / General Logistics curve to the reported data, any changes to Government interventions and/or public response (even nationally, let alone for local spikes) can’t be reflected. It’s a top-down view of reported data (however well-cleansed) not a bottom-up causation model with the ability to make variations to strategies for intervention.

Mechanistic modelling is hard to do, takes longer and is more expensive in computer time (especially when trying to cover many countries individually); that is where a broader helicopter top-down view from curve-fitting can help to get started. But curve-fitting is not an actionable model for deciding between intervention methods.

I covered these methods in my blog posts on July 14th and July 18th as I was sanity checking my own outlook on modelling methods as between mechanistic modelling (the broad type of the model I use) and phenomenological / statistical methods.

The Imperial College resources

As I have already reported in my blog post on July 18th, Imperial College (and others such as The London School of Hygiene and Tropical Medicine) use a variety of model types and data sources (as do IHME) spanning both mechanistic and statistical methods (which include phenomenological techniques) for forecasts at different levels of detail and over different periods. These are described at the Imperial College’s Medical Research Council MRC Global Infectious Disease Analysis website, where this chart is presented, describing their different methods:

COVID-19 planning tools
Epidemiological models use a combination of mechanistic and statistical approaches.

and they go on to describe the key characteristics of the approaches:

Mechanistic model: Explicitly accounts for the underlying mechanisms of diseases transmission and attempt to identify the drivers of transmissibility. Rely on more assumptions about the disease dynamics.

Statistical model: Do not explicitly model the mechanism of transmission. Infer trends in either transmissibility or deaths from patterns in the data. Rely on fewer assumptions about the disease dynamics.

Mechanistic models can provide nuanced insights into severity and transmission but require specification of parameters – all of which have underlying uncertainty. Statistical models typically have fewer parameters. Uncertainty is therefore easier to propagate in these models. However, they cannot then inform questions about underlying mechanisms of spread and severity.

The forecasts they have made, as you can see, just as the IHME forecasts do, rely on several methodologies.

The table I have shown before from the pivotal Imperial College modelling team March 16th paper:

PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4 months (a month more than other interventions)
PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4 months (a month more than other interventions)

shows the capability to model a range of Non Pharmaceutical Interventions (NPIs) alone or in different combinations to arrive at forecasts based on such strategies. I covered the NPI variations in some detail in my August 14th blog post, and the mechanistic, statistical and phenomenological approaches in my July 14th blog post and July 18th post.

Discussion

My UK model is tracking quite well after a small change in intervention effectiveness since March 23rd to reflect the retroactive August 12th Government changes in counting deaths, and a slight easing of lockdown on day 105 (May 17th). We see a lot happening here and in other countries, with travel restrictions and quarantining measures changing all the time. It is unlikely that countries will revert to large scale lockdowns.

This is partly because lockdown is seen by many to have done its job; partly because of its negative economic and social impacts; and partly because we know more about the effects of the individual interventions available. Mechanistic modelling methods help discriminate between the effects of the different interventions.

One of the key factors in the choice of interventions is on the basis of longer-term outcomes – the effect of actions taken today on future “herd” immunity of the population, which I covered in my July 31st blog post.

I mention again the influential March 16th Imperial College paper in this respect which, while published nearly 6 months ago, does give an insight into the complexity and capability of modelling methods and data sources and intervention discrimination available to Government advisers.

Modelling on an overall national basis will need some enhancement to cope with the large number of local “spikes” and other events that we have been seeing recently.

Concluding comments

There are reasons for concern – the possibility that current spikes in cases might lead to a major “wave” in the epidemic; that autumn isn’t too far away; and that influenza and other related diseases such as SARS-Cov-2 are more prevalent in the autumn/winter months.

The BBC have reported that the return of students to Universities in the UK is expected to lead to a high risk of increasing the rate of Covid-19 cases. We will see.

I leave it to the Sky News summary to express closing thoughts, and some optimism.

The fear among government scientists is that if the outbreak gets out of control among young people, it will eventually leak into the more vulnerable parts of the population. What might look like a divergence between cases and deaths is actually just a larger lag. To find the answer to that, the best places to look are France and Spain, where cases are rising fast, but deaths and hospitalisations are still low. But whatever happens, we should remember: this isn’t March all over again. We test so much more. We know so much more about treatment. And we all understand how to change our behaviour. That is cause for optimism as we face the next six months.

Categories
Coronavirus Covid-19 Herd Immunity Imperial College Michael Levitt Office for National Statistics ONS PHE Public Health England Superspreader Sweden

Model update following UK revision of Covid-19 deaths reporting

Introduction

On August 12th, the UK Government revised its counting methodology and reporting of deaths from Covid-19, bringing Public Health England’s reporting into line with that from the other home countries, Wales, Northern Ireland and Scotland. I have re-calibrated and re-forecast my model to adapt to this new basis.

Reasons for the change

Previously reported daily deaths in England had set no time limit between any individual’s positive test for Covid-19, and when that person died. 

The three other home countries in the UK applied a 28-day limit for this period. It was felt that, for England, this lack of a limit on the time duration resulted in over-reporting of deaths from Covid-19. Even someone who had died in a road accident, say, would have been reported as a Covid-19 death if they had ever tested positive, and had then recovered from Covid-19, no matter how long before their death this had happened.

This adjustment to the reporting was applied retroactively in England for all reported daily deaths, which resulted in a cumulative reduction of c. 5,000 in the UK reported deaths to up to August 12th.

The UK Government say that it is also to report on a 60-day basis (96% of Covid-19 deaths occur within 60 days and 88% within 28 days), and also on the original basis for comparisons, but these two sets of numbers are not yet available.

On the UK Government’s web page describing the data reporting for deaths, it says “Number of deaths of people who had had a positive test result for COVID-19 and died within 28 days of the first positive test. The actual cause of death may not be COVID-19 in all cases. People who died from COVID-19 but had not tested positive are not included and people who died from COVID-19 more than 28 days after their first positive test are not included. Data from the four nations are not directly comparable as methodologies and inclusion criteria vary.

As I have said before about the excess deaths measure, compared with counting deaths attributed to Covid-19, no measure is without its issues. The phrase in the Government definition above “People who died from COVID-19 but had not tested positive are not included…” highlights such a difficulty.

Model changes

I have adapted my model to this new basis, and present the related charts below.

  • Model forecast for the UK deaths as at August 14th, compared with reported for 84.3% lockdown effectiveness, on March 23rd, modified in 5 steps by -.3%, -0% -0% and -0% successively
  • Model forecast for the UK deaths as at August 14th, compared with reported for 84.3% lockdown effectiveness, modified in 5 steps by -.3%, -0% -0% and -0% successively
  • Model forecast for the UK deaths as at August 14th, compared with reported for 84.3% lockdown effectiveness, on March 23rd, modified in 5 steps by -.3%, -0% -0% and -0% successively
  • Model forecast for the UK deaths as at August 14th, compared with reported for 84.3% lockdown effectiveness, on March 23rd, modified in 5 steps by -.3%, -0% -0% and -0% successively
  • Model forecast for the UK deaths as at August 14th, compared with reported for 84.3% lockdown effectiveness, on March 23rd, modified in 5 steps by -.3%, -0% -0% and -0% successively
  • Chart 12 for the comparison of cumulative & daily reported & modelled deaths to 26th April 2021, adjusted by -.3% on May 13th

This changed reporting basis reduced the cumulative UK deaths to August 12th from 46,706 to 41,329, a reduction of 5,377.

The fit of my model was better for the new numbers, requiring only a small increase in the initial March 23rd lockdown intervention effectiveness from 83.5% to 84.3%, and a single easing reduction to 84% on May 13th, to bring the model into good calibration up to August 14th.

It does bring the model forecast for the long term plateau for deaths down to c. 41,600, and, as you can see from the charts above, this figure is reached by about September 30th 2020.

Discussion

The relationship to case numbers

You can see from the first model chart that the plateau for “Recovered” people is nearly 3 million, which implies that the number of cases is also of the order of 3 million. This startling view is supported by a recent antibody study reported by U.K. Government here.

This major antibody testing programme, led by Imperial College London, involving over 100,000 people, found that just under 6% of England’s population – an estimated 3.4 million people – had antibodies to COVID-19, and were likely to have previously had the virus prior to the end of June.

The reported numbers in the Imperial College study could seem quite surprising, therefore, given that 14 million tests have been carried out in the U.K., but with only 313,798 positive tests reported as at 12th August (and bearing in mind that some people are tested more than once).

But the study is also in line with the estimate made by Prof. Alex de Visscher, author of my original model code, that the number of cases is typically under-reported by a factor of 12.5 – i.e. that only c. 8% of cases are detected and reported, an estimate assessed in the early days for the Italian outbreak, at a time when “test and trace” wasn’t in place anywhere.

A further sanity check on my modelled case numbers, relative to the number of forecasted deaths, would be on the observed mortality from Covid-19 where this can be assessed.

A study by a London School of Hygiene & Tropical Medicine team carried out an analysis of the Covid-19 outbreak in the closed community of the Diamond Princess cruise ship in March 2020.

Adjusting for delay from confirmation-to-death, this paper estimated case and infection fatality ratios (CFR, IFR) for COVID-19 on the Diamond Princess ship as 2.3% (0.75%-5.3%) and 1.2% (0.38-2.7%) respectively.

In broad terms, my model forecast of 42,000 deaths and up to 3 million cases would be a ratio of about 1.4%, and so the relationship between the deaths and cases numbers in my charts doesn’t seem to be unreasonable.

Changing rates of infection

I am not sure whether the current forecast for a further decline in the death rate will remain, in the light of continuing lockdown easing measures, and the local outbreaks.

Both the Office for National Statistics (ONS) and Public Health England (PHE) reported in early July a drop in the rate of decline in Covid-19 cases per 100,000 people in England.

Figure 2: The latest exploratory modelling shows the downward trend in those testing positive for COVID-19 has now levelled off

This was at the same time as the ONS reported that excess deaths have reduced to a level at or below the average for the last five years.

The number of deaths involving COVID-19 decreased for the 10th consecutive week

PHE reports this week that the infection rate is now more pronounced for under-45s than for over-45s, a reversal of the situation earlier in the pandemic. Overall case rates, however, remain lower than before; and although the rate of decline in the case rate has slowed for-over-45s, and is nearly flat now, for under-45s the infection rate has started to increase slightly.

Covid-19 cases rate of decline slows more for under-45s

The impact on the death rate might well be lower than previously, owing to the lower fatality rates for younger people compared with older people.

Herd immunity

Closely related to the testing for Covid-19 antibodies is herd immunity, a topic I covered in some detail on my blog post on June 28th, when I discussed the relative positions of the USA and Europe with regard to the spike in case numbers the USA was experiencing from the middle of June, going on to talk about the Imperial College Coronavirus modelling, led by Prof. Neil Ferguson, and their pivotal March 16th paper.

This paper was much criticised by Prof Michael Levitt, and others, for the hundreds of thousands of deaths it mentioned if no action were taken, cited as scare-mongering, ignoring to some extent the rest of what I think was a much more nuanced paper than was appreciated, exploring, as it did, the various interventions that might be taken as part of what has become known as “lockdown”.

The intervention options were also quite nuanced, embracing as they did (with outcomes coded as they were in the chart below) PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4 months (a month more than other interventions).

PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4 months (a month more than other interventions)
PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4 months (a month more than other interventions)

I had asked the lead author of the paper why the effectiveness of the three measures “CI_HQ_SD” in combination (home isolation of cases, household quarantine & large-scale general population social distancing) taken together (orange and yellow colour coding), was LESS than the effectiveness of either CI_HQ or CI_SD taken as a pair of interventions (mainly yellow and green colour coding)?

The answer was in terms of any subsequent herd immunity that might or might not be conferred, given that any interventions as part of a lockdown strategy would be temporary. What would happen when they ceased?

The issue was that if the lockdown measures were too effective, then (assuming there were any immunity to be conferred for a usefully long period) the potential for any subsequent herd immunity would be reduced with too successful a lockdown. If there were no worthwhile period of immunity from catching Covid-19, then yes, a full lockdown would be no worse than any other partial strategy.

Sweden

I mention all this as background to a paper that was just published in the Journal of the Royal Society of Medicine as I started this blog post, on August 12th. It concerns the reasons why, as the paper authored by Eric Orlowski and David Goldsmith asserts, that four months into the COVID-19 pandemic, Sweden’s prized herd immunity is nowhere in sight.

This is a somewhat polemical paper, as Sweden is often held up as an example of how countries can succeed in combating the SARS-Cov-2 pandemic by emulating Sweden’s non-lockdown approach. I have been, and remain surprised by such claims, and now this paper helps calibrate and articulate the underlying reasons.

Although compared with the UK, Sweden had done little worse, if at all, despite resisting the lockdown approach (although its demographics and lifestyle characteristics are not necessarily comparable to the UK’s), compared with their more similar nearest neighbours, Norway, Denmark and Finland, Sweden has done far worse in terms of deaths and deaths per capita.

I think that either for political or for other related reasons, perhaps economic ones, even some otherwise sensible scientists are advocating the Swedish approach, somehow ignoring the more valid (and negative) comparisons between Sweden and the other Scandinavian countries, as opposed to more favourable comparisons with others further afield – the UK, for example.

I have tried to remain above the fray, notably on the Twittersphere, but, at least on my own blog, I want to present what I see as a balanced assessment of the evidence.

That balance, in this case, strikes me like this: if there were an argument for the Swedish approach, then a higher level of herd immunity would have been the payoff for experiencing more immediate deaths in favour of a better outcome later.

But that doesn’t seem to have happened, at least in terms of outcomes from testing for antibodies, as presented in this paper. As it says “it is clear that nowhere is the prevalence of IgG seropositivity high (the maximum being around 20%) or climbing convincingly over time. This is especially clear in Sweden, where the authorities publicly predicted 40% seroconversion in Stockholm by May 2020; the actual IgG seroprevalence was around 15%.

Concluding comments

As I said in my August 4th post, the outbreaks we are seeing in some UK localities (Leicester, Manchester, Aberdeen and many others) seem to be the outcome of individual and multiple local super-spreading events.

These are quite hard to model, requiring very fine-grained data regarding the types and extent of population interactions, and the different effects of a range of intervention measures available nationally and locally, as I mentioned above, applied in different places at different times.

The reproduction number, R (even nationally) can be increased noticeably by such localised events, because of the lower overall incidence of cases in the UK (something we have seen in some other countries too, at this phase of the pandemic).

While most people nationally aren’t directly affected by these localised outbreaks, I believe that caution – social distancing where possible, for example – is still necessary.

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Adam Kucharski Alex de Visscher Coronavirus Covid-19 David Spiegelhalter Superspreader Worldometers

Model updates for UK lockdown easing points

Introduction

As I reported in my previous post on 31st July, the model I use, originally authored by Prof. Alex de Visscher at Concordia University in Montreal, and described here, required an update to handle several phases of lockdown easing, and I’m glad to say that is now done.

Alex has been kind enough to send me an updated model code, adopting a method I had been considering myself, introducing an array of dates and intervention effectiveness parameters.

I have been able to add the recent UK Government relaxation dates, and the estimated effectiveness of each into the new model code. I ran some sensitivities which are also reported.

Updated interventions effectiveness and dates

Now that the model can reflect the timing and impact of various interventions and relaxations, I can use the epidemic history to date to calibrate the model against both the initial lockdown on March 23rd, and the relaxations we have seen so far.

Both the UK population (informally) and the Government (formally) are making adjustments to their actions in the light of the threats, actual and perceived, and so the intervention effects will vary.

Model adjustments

At present I am experimenting with different effectiveness percentages relating to four principal lockdown relaxation dates, as described at the Institute for Government website.

In the model, the variable k11 is the starting % infection rate per day per person for SARS-Cov-2, at 0.39, which corresponds to 1 infection every 1/0.39 days ~= 2.5 days per infection.

Successive resetting of the interv_success % model variable allows the lockdown, and its successive easings to be defined in the model. 83.5% (on March 23rd for the initial lockdown) corresponds to k11 x 16.5% as the new infection rate under the initial lockdown, for example.

In the table below, I have also added alternative lockdown easing adjustments in red to show, by comparison, the effect of the forecast, and hence how difficult it will be for a while to assess the impact of the lockdown easings, volatile and variable as they seem to be.

DateDay Steps and example measuresChanges to % effectiveness
23rd March52Lockdown starts+83.5%
13th May105Step 1 – Partial return to work
Those who can work from home should do so, but those who cannot should return to work with social distancing guidance in place. Some sports facilities open.
-1% = 82.5%

-4% = 79.5%
1st June122Step 2 – Some Reception, Year 1 and Year 6 returned to school. People can leave the house for any reason (not overnight). Outdoor markets and car showrooms opened.-5% = 77.5%

-8% = 71.5%
15th June136Step 2 additional – Secondary schools partially reopened for years 10 and 12. All other retail are permitted to re-open with social distancing measures in place.-10% = 67.5%

+10% = 81.5%
4th July155Step 3 – Food service providers, pubs and some leisure facilities are permitted to open, as long as they are able to enact social distancing.+20% = 87.5%

-6% = 75.5%
1st August186Step 3 additional – Shielding for 2m vulnerable people in the UK ceases0% = 87.5%
Dates, examples of measures, and % lockdown effectiveness changes on k11

After the first of these interventions, the 83.5% effectiveness for the original March 23rd lockdown, my model presented a good forecast up until lockdown easing began to happen (both informally and also though Government measures) on May 13th, when Step 1 started, as shown above and in more detail at the Institute for Government website.

Within each easing step, there were several intervention relaxations across different areas of people’s working and personal lives, and I have shown two of the Step 2 components on June 1st and June 15th above.

I applied a further easing % for June 15th (when more Step 2 adjustments were made), and, following Step 3 on July 4th, and the end (for the time being) of shielding for 2m vulnerable people on August 1st, I am expecting another change in mid-August.

I have managed to match the reported data so far with the settings above, noting that even though the July 4th Step 3 was a relaxation, the model matches reported data better when the overall lockdown effectiveness at that time is increased. I expect to adjust this soon to a more realistic assessment of what we are seeing across the UK.

With the % settings in red, the outlook is a little different, and I will show the charts for these a little later in the post

The settings have to reflect not only the various Step relaxation measures themselves, but also Government guidelines, the cumulative changes in public behaviour, and the virus response at any given time.

For example, the wearing of face coverings has become much more common, as mandated in some circumstances but done voluntarily elsewhere by many.

Comparative model charts

The following charts show the resulting changes for the initial easing settings. The first two show the new period of calibration I have used from early March to the present day, August 4th.

On chart 4, on the left, you can see the “uptick” beginning to start, and the model line isn’t far from the 7-day trend line of reported numbers at present (although as of early August possibly falling behind the reported trend a little).

On the linear axis chart 13, on the right, the reported and model curves are far closer than in the version in my most recent post on July 31st, when I showed the effects of lockdown easing on the previous forecasts, and I highlighted the difficulty of updating without a way of parametrising the lockdown easing steps (at that time).

Using the new model capabilities, I have now been able to calibrate the model to the present day, both achieving the good match I already had from March 23rd lockdown to mid-May, and then separately to adjust and calibrate the model behaviour since mid-May to the present day, by adjusting the lockdown effectiveness at May 15th, June 1st, June 15th and July 4th, as described earlier.

The orange dots (the daily deaths) on chart 4 tend to cluster in groups of 4 or 5 per week above the trend line (and also the model line), and 3 or 2 per week below. This is because of the poor accuracy of some reporting at weekends (consistently under-reporting at weekends and recovering by over-reporting early the following week).

The red 7-day trend line on chart 4 reflects the weekly average position.

Looking a little further ahead, to September 30th, this model, with the initial easing settings, predicts the following behaviour, prior to any further lockdown easing adjustments, expected in mid-August.

Chart 12 for the comparison of cumulative & daily reported & modelled deaths, on the basis of 83.5% effectiveness, modified in 4 steps by -1%, -5% -10% and +2% successively
Chart 12 for the comparison of cumulative & daily reported & modelled deaths

Finally, for comparison, the Worldometers UK site has a link to its own forecast site, which has several forecasts depending on assumptions made about mask-wearing, and/or continued mandated lockdown measures, with confidence limits. I have screenshot the forecast on October 1st, where it shows 48,268 deaths assuming current mandates continuing, and mask-wearing.

My own forecast shows 47,201 cumulative deaths at that date.

Worldometers forecasts on the basis of mask wearing vs. no mandated measures, with confidence limits
Worldometers forecasts on the basis of mask wearing vs. no mandated measures, with confidence limits

Alternative % settings in red

I now present a slideshow of the corresponding charts with the red % easing settings. The results here are for the same initial lockdown effectiveness, 83.5%, but with successive easings at -4%, -8%, +10% and -6%, where negative is relaxation, and positive is an increase in intervention effectiveness.

  • Chart 12 for the comparison of cumulative & daily reported & modelled deaths to 26th April 2021, on the basis of 83.5% effectiveness, modified in 4 steps by -4%, -8% +10% and -6%% successively
  • Chart 12 for the comparison of cumulative & daily reported & modelled deaths to 30th Sep 2020, , on the basis of 83.5% effectiveness, modified in 4 steps by -4%, -8% +10% and -6%% successively
  • Chart 4 for the comparison of cumulative & daily reported & modelled deaths, plus reported trend line, on the basis of 83.5% effectiveness
  • Model forecast for the UK deaths as at August 8th, compared with reported for 83.5% lockdown effectiveness
  • Model forecast (linear axes) for the UK deaths as at August 8th, compared with reported for 83.5% lockdown effectiveness, modified in 4 steps by -4%, -8% +10% and -6%% successively

The model forecast here for September 30th is for 49, 549 deaths, and the outlook for the longer term, April 2020, is for 52,544.

Thus the next crucial few months, as the UK adjusts its methods for interventions to be more local and immediate, will be vital in its impact on outcomes. The modelling of how this will work is far more difficult, therefore, with fine-grained data required on virus characteristics, population movement, the comparative effect of different intervention measures, and individual responses and behaviour.

Hotspots and local lockdowns

At present, because the UK reported case number trend has flattened out and isn’t decreasing as fast, and because of some local hotspots of Covid-19 cases, the UK Government has been forced to take some local measures, for example in Leicester a month ago, and more recently in Manchester; the scope and scale of any lockdown adjustments is, therefore, a moveable target.

I would expect this to be the pattern for the future, rather than national lockdowns. The work of Adam Kucharski, reported at the Wellcome Open Research website, highlighting the “k-number”, representing the variation, or dispersion in R, the reproduction number, as he says in his Twitter thread, will be an important area to understand.

The k-number might well be more indicative, at this local hotspot stage of the crisis, than just the R reproduction number alone; it has a relationship to the “superspreader” phenomenon discussed for SARS in this 2005 paper, that was also noticed very early on for SARS-Cov-2 in the 2020 pandemic, both in Italy and also in the UK. I will look at that in more detail in another posting.

Superspreading relates to individuals who are infected (probably asymptomatically or pre-symptomatically) who infect others in a closed social space (eg in a ski resort chalet as reported by the BBC on February 8th) without realising it.

The hotspots we are now seeing in many places might well be related to this type of dispersion. The modelling for this would be a further complication, potentially requiring a more detailed spatial model, which I briefly discussed in my blog post on modelling methods on July 14th.

Superspreading might also need to be understood in relation to the opening of schools, in August and September (across the four UK home countries). It might have been a factor in the Israel experience of return to schools, as covered by the Irish Times on August 4th.

The excess deaths measure

There has been quite a debate on excess deaths (often a seasonal comparison of age-related deaths statistics compared with the previous 5 years) as a measure of the overall position at any time. As I said in a previous post on June 2nd, this measure does mitigate any arguments as to what is a Covid-19 fatality, and what isn’t.

The excess deaths measure, however, has its own issues with regard to the epidemic’s indirect influence on the death rates from other causes, both upwards and downwards.

Since there is less travel (on the roads, for example, with fewer accidents), and many people are taking more care in other ways in their daily lives, deaths from some other causes might tend to reduce.

On the other hand, people are feeling other pressures on their daily lives, affecting their mood and health (for example the weight gain issues reported by the COVID Symptom Study), and some are not seeking medical help as readily as they might have done in other circumstances. Both factors tend to increase illness and potentially death rates.

Even as excess deaths reduce, then, it may well be that Covid-19 deaths increase as others reduce. Possibly a crossover with seasonal influenza deaths, later on, might be masked by the overall excess deaths measure.

As I also mentioned in my post on July 6th, deaths in later years from other causes might increase because of this lack of timely diagnosis and treatment for other “dread” diseases, as, for example, for cancer, as stated by Data-can.org.uk.

So no measure of the epidemic’s effects is without its issues. Prof. Sir David Spiegelhalter covered this aspect in a newspaper article this week.

Discussion

The statistical interpretation and modelling of data related to the pandemic is a matter of much debate. Some commentators and modellers are proponents of quite different methods of data recording, analysis and forecasting, and I covered phenomenological methods compared with mechanistic (SIR) modelling in previous posts on July 14th and July 18th.

The current reduced rate of decline in cases and deaths in some countries and regions, with concomitant local outbreaks being handled by local intervention measures, including, in effect, local lockdowns, has complicated the predictions of some who think (and have predicted) that the SARS-Cov-2 crisis will soon be over (some possibly for political reasons, some of them scientists).

Even when excess deaths reduce to zero, this doesn’t mean that Covid-19 is over, because, as I mentioned above, illness and deaths from other causes might have reduced, with Covid-19 filling the gap.

There are also concerns that recovery from Covid-19 as a death threat can be followed by longer-lasting illness and symptoms, and some studies (for example this NHLBI one) are gathering evidence, such as that covered by this report in the Thailand Medical News.

This Discharge advice from the UK NHS makes continuing care requirements for discharged Covid-19 patients in the UK very clear.

It is by no means certain, either, that recovery from Covid-19 confers immunity to SARS-Cov-2, and, if it does, for how long.

Concluding comments

I remain of the view that in the absence of a vaccine, or a very effective pharmaceutical treatment therapy, we will be living with SARS-Cov-2 for a long time, and that we do have to continue to be cautious, even (or, rather, especially) as the UK Government (and many others) move to easing national lockdown, at the same time as being forced to enhance some local intervention measures.

The virus remains with us, and Government interventions are changing very fast. Face coverings, post-travel quarantining, office/home working and social distancing decisions, guidance and responses are all moving quite quickly, not necessarily just nationally, but regionally and locally too.

I will continue to sharpen the focus of my own model; I suspect that there will be many revisions and that any forecasts are now being made (including by me) against a moving target in a changing context.

Any forecast, in any country, that it will be all over bar the shouting this summer is at best a hostage to fortune, and, at worst, irresponsible. My own model still requires tuning; in any case, however, I would not be axiomatic about its outputs.

This is an opinion informed by my studies of others’ work, my own modelling, and considerations made while writing my 30 posts on this topic since late March.