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Covid-19 total deaths and fitted curves for different countries

 

Weekly summary (24 January 2021)

  • The increased in the daily death rates in the UK, Germany and the USA have slowed and for the UK and Germany there are signs they may be near their peaks.

  • In the UK, the total deaths in the pandemic, expressed as a percentage of the population are now the highest of all the countries shown here.

  • For Spain, France and Italy the declines in deaths following second wave peaks have now stopped or have started to increase again.  

  • It continues to be surprising how well the actual deaths in most of the countries shown here fit the modelled curves, which are based on simple mathematical formulae with only three adjustable parameters for each discrete wave of the pandemic.

Introduction

For the current Covid-19 pandemic, death statistics by country are widely available, with the Worldometers website being one of the most convenient sources.

There have also been many plots given in the media, including in the Guardian, the FT on-line, and elsewhere, of death rates from day of first death compared between countries (accredited to the John Hopkins University ).

I thought I'd see if I could reproduce these for myself, using a different selection of countries. This was easy enough to do using Excel and the Worldometers data.

Deaths per head of population

As politicians are keen to highlight, it is difficult to make valid comparisons between countries, but an approximate way of doing this is to compare the death rates per head of population, which I show in the graph below. These are the figures for total deaths to date divided by the population for each country.

The curves above show that even now the totals are comparatively small as a percentage of each country's population, with the highest being over 0.14% (which corresponds to about 1 in 700).  To put these figures into perspective, in the UK (and I presume elsewhere) the percentage of the population dying each year is just under 1% or 620,000 (2018 figure) in its population of 67 million.

As forecast last week, as a percentage of its population, the total deaths in the UK are now the highest of the countries shown here, an unenviable position to be in. The rate of increase for the UK is still the highest of all the countries shown here. 

For further comments, see the sections below on each country in turn.

Modelling

In the early stages of the pandemic, many sources were showing plots of log(total deaths) vs linear time as the initial expected exponential growth phase then appears linear. However although we have seen many of these plots in the media, forward projections of these figures are much rarer.

To project forward, an assumption is needed about how the death rates will change with time. There is a whole science of pandemic modelling about which I know very little. What I have seen involves a complex approach based on a large number of parameters and multiple differential equations. As I have no idea what assumptions are used in these models and how they work in practice, I've looked at a much simpler approach based entirely on the available data to date for death rates. I have ignored all the information on number of cases on the basis that these are entirely dependent on the amount of testing done, which at present is very limited in the UK at least.

Basis of approach

My approach is as follows. It seems to me that a plausible formula for the total deaths to a particular date is as follows:
                                                                                                                                                                              (1)

where t is the time since first death in days. For small t this curve is linear, which is equivalent to the initial expected exponential increase. For large t, this curve eventually saturates at a value = D, where the total number of deaths in the pandemic = exp(D).  The unknown parameters D and are found by least squares fitting to the reported total deaths to date.

A slight generalisation of this formula involves three unknown parameters instead of two:

                                                                                                                                                                      (2)

where t0 allows an adjustment to the (uncertain) date of first death. The constant t0 also allows a second wave to be added to the deaths from the first wave, and displaced in time relative to it.

To fit the above curves to the available data for a country, I compare the modelled and actual reported new deaths for each day, and sum  the square of the differences between them.   The Solver in Excel then allows the sum of the squares of the differences to be minimised by changing the unknown parameters D, , t0 in the above formulae.

In the early stages of the pandemic in each country, the numbers of reported deaths tend to rise exponentially with time (which appears as a straight line on a logarithmic plot). In this case, it is impossible to derive with any confidence a value for the "curvature" parameter, a. Without that, any projections of the numbers into the future are meaningless. Even as the epidemic progresses, small changes in the value of the parameter a can have a huge effect on the modelled number for the total number of people that will die in the epidemic.

It is important to note that I am not claiming any accuracy for these predictions. This approach is just one way of estimating future trends based on available data.

However, in China it appears the present phase of the pandemic is over. At the bottom of this page there is a study which shows that this approach was giving estimates of total deaths within a factor two of the final total, even at the early stage of the pandemic. 

Second Waves
With second waves now unfortunately becoming apparent in several countries, mainly in Europe, I have sought to model these by using the addition of two models each with one of the curves described above. For the second wave curve, I've used formula (2) above, which allows for the start time of the second wave to be adjusted relative to that of the first wave. To date, in all the countries a second wave is occurring in, the initial increase in deaths is much slower than in the first wave, and the total numbers look like being lower as well, although it is too early to be confident of this in most cases.

Third Wave
In the USA, there is now evidence of a third wave which I have now modelled in the same way as for the second waves seen elsewhere, but with another term for the third wave. Hence the overall total number of deaths is made up of the sum of three curves, model 1 + model 2 + model 3.

UK predictions

I first give here the results for the UK, followed by those for other selected countries. In all cases I have used the Worldometer data for total deaths to date, as a function of time.

For the UK, these are the daily figures from the DHSC that appear widely in the media. In these figures, for a death to be attributed to Covid-19, there must now have been a positive Covid-19 test within the last 28 days. These figures hence do not include deaths from Covid-19 where a test had not been performed, which may have occurred most often in care homes and the community. Conversely, any death, from whatever cause, within 28 days of a positive Covid-19 test is now included, including presumably obviously unrelated deaths such as car accidents.

The plot below shows the Worldometer data for the UK in terms of deaths to date (left logarithmic axis) and daily deaths (right linear axis). It also shows the results of fitting the second equation given above to the daily death values. As the daily figures show considerable fluctuations, with lower values at weekends, I now am now following many others in showing values averaged over the previous week (i.e. a rolling average).

On the plot below, also shown are the separate weekly registered death figures reported by the ONS (England) added to the numbers from the corresponding separate bodies for Scotland and Northern Ireland. These are all deaths where Covid-19 is mentioned on the death certificate, and are counted by date of registration of the death. These numbers are higher than those reported by the DHSC, which only counts those with positive Covid-19 tests. The registered death figures are also accompanied by information on all deaths and how they compare with the long-term average. These show a significant number of unaccounted for excess deaths where Covid-19 is not on the death certificate. These could be from other causes, e.g. heart attacks, cancers and strokes, which may be have increased due to the reallocation of NHS resources to handling the Covid-19, to the detriment of other forms of care.  Alternatively, or additionally, there may well have been some deaths caused by Covid-19 but not mentioned on the death certificate.  On 14 June, there had been about 65,000 excess deaths in the UK since the start of March 2020 - substantially higher than the 42,000 Covid-19 deaths reported by DHSC on that date. Since then the total number of deaths has been slightly lower than the 5-year average, so the excess deaths are declining slightly. This all goes to highlight, even today, the difficulty in being sure of how many people have died during a pandemic.

Note added 9 August: I have now stopped updating the ONS figures on the plot below, as they take a considerable time to work out. This is because the information, all in different formats, needs to be downloaded separately for England, Scotland and Northern Ireland and then added together!

In the UK, the first wave was met by a national lockdown that appeared to be sufficient to bring down cases and death rates after about 2 months. These measures were then eased during the summer, only for a second wave to emerge in the autumn. Initially this was tackled by a piece-meal series of local measures, few of which had the intended effect of driving cases down. The 4-week second national lockdown followed which was partially successful in bringing deaths back under control and overall cases were reduced slightly. However the dip in deaths was slight due to the more recent upsurge caused by the new variant and also by the easing of the second national lockdown restrictions.

In the last two weeks the daily deaths have increased dramatically and are now higher than in the first wave. The rate of increase has however slowed in the last week and it is possible the third peak will soon be reached. This follows the trend shown by new cases which are declining.

For the modelling, I have now added a third wave to fit to the recent upsurge in deaths. This week model 3 is consistent with a peak being reached in the few weeks. The total number of deaths in the third wave is now about 85,000 which is more plausible than the figure of 200,000 for last week. However only once the peak has been reached and a decline sets is the fitted curve likely to give a more reliable prediction for total deaths.   

In the plot below I show the results from model 1 (for the first wave), model 2 (for the second wave) and now a third model (model 3) for the recent upsurge. The total is the sum of all three waves and is a remarkably close fit to the actual values. 

Spain 

The plot below shows the daily deaths for Spain together with the fitted curves for both the first and second waves of the pandemic (Model 1 & Model 2), and the total (Model 1 + Model 2). After a strong first wave, Spain is now suffering from a smaller more gradual second wave. Until recently, the sum of the two models fitted the actual daily deaths quite closely but there is now clear evidence for a third wave occurring as in the UK and elsewhere.

Italy

The plot below shows the daily deaths for Italy together with the fitted curve. The curve is a pretty good fit to the actual deaths. Back in March, the pandemic gripped Italy a few weeks before other countries, and the deaths rose very sharply before falling back quite quickly, possibly due to the strict lockdown measures imposed. Deaths then stayed low, until about day 220 when an increase was first apparent.

The daily deaths in this second wave reached a very similar peak to the maximum in the first wave and a decline then began. The model 2 fitted to these values gives a total number of deaths at about 50,000 which is about 40% higher than the first wave total. In the last three weeks there has been a halt in the second wave decline with a possible third wave emerging, but weaker than in the UK or Spain.

France

The modelling now has two curves for the different waves. As in the UK and Spain, the increases in deaths in the second wave were slower than in the first wave. After a peak at about day 260, there was a slow decline underway. However, as with Italy, this appears to have halted - and a possible third wave may be starting, although like Italy this appears weaker than in the UK.

USA

As can be seen below, the USA numbers now show three distinct waves of the pandemic. After the first wave, and having been in decline for some time, the daily figures started increasing again around day 120. A second wave then occurred which appeared to peak around day 170. A slow second decline then took place before the latest upturn in deaths from about day 240.

I have modelled the USA figures using the sum of three different curves fitted to the daily death figures. The first curve is intended to match (approximately) the initial stages of the epidemic while the second curve is to account for the rise in deaths after around day 120.

I've now incorporated a third wave into the modelling to account for the latest increase in deaths. These are currently showing fluctuations up and down, but the fitted curve is no longer showing an unconstrained near exponential rise. The finite but implausible prediction for the total number of deaths in now just under 1 million.  The daily deaths in this third wave now significantly exceed those in the first two waves.

Sweden

Unlike all the other countries featured on this page, Sweden has not had a major lockdown. Hence a comparison with other countries that have much higher levels of restriction in interesting. Sweden has a significantly smaller population (about 10 million) and a lower population density than any of the other countries given here.  Sweden has experienced the familiar pattern of a first wave starting in March, followed by a decline and then an autumn second wave. 

The modelling of this second wave is now shown below within the total modelled deaths curve (I have not shown the separate model 1 and model 2 components). It is possible that the second wave is nearing its peak. The fitted curve suggests that the total second wave deaths will be about 20,000. 

 

Germany

In Germany, the approach has been completely different from Sweden with a huge amount of testing and follow-up of those infected, combined with a major lockdown. At present, the fitted curve has a final total death toll of about 9,000, in stark contrast to the UK, Italy, Spain and France. The fitted curve shows the peak in the daily numbers was reached on about day 35 and then tailing off well, despite some easing of lockdown rules. There is however now a strong second wave. 

The daily deaths are now dwarfing those in the first wave, although of course the first wave peak in Germany was comparatively very low. After a possible leveling off, probably caused by under-reporting over the Christmas/New Year holiday period, the daily numbers are again rising sharply and the predicted total number of deaths in the second wave is now a not impossible 150,000. There are some similarities between the current rapid increases in Germany and the UK, although the rise has been more gradual in Germany.

Brazil

For Brazil, unlike most of the countries looked at here, it is clear the actual deaths do not fit closely to the modelled curve. Following the initial rise, there was an unusual plateau, with almost constant values from day 80 to about day 160.  There was then a progressive decrease until around day 240. In the days since then, a second wave has developed despite it not being autumn in this southern hemisphere country. In the last week, the second wave deaths have risen even more steeply than last week. Under-reporting over the Christmas/New Year period may account for the small decline in numbers around days 280 - 290.

The modelling is now based on two separate curves for the first and second waves respectively. 

There is now international concern over a new variant of the virus in Brazil, and this, combined with an absence of social distancing, may well account for the current steep rise in deaths.

China

For China the present phase of the pandemic is apparently over.  It can therefore be used as useful test case for the modelling approach described above. The plot below the available data for China up to 50 days since the first death. Note immediately how closely the 2-parameter equation given above fits to the measurements, over almost the entire course of the pandemic. In this country at least, the equation seems to be more than plausible - it provides a good fit to the entire pandemic.

 

I have then tested how accurately the modelling approach predicts the final death toll, based on different amounts of the available data. The results are shown below. This graph shows how the predicted total deaths vary depending on the date at which the modelling is performed. All predictions were within a factor of two of the final value. This is unlike my experience with any of the other countries modelled to date. In all these cases, the predictions for the final death toll have been varying by large amounts, indicating the approach is not providing meaningful values.

It is however surely a coincidence that the prediction, performed on only the first six days of data (I excluded the values below 4 days from first death), is almost spot on the final figure.

 

 

 
All plots copyright Stephen Burch

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