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Covid-19 total deaths and fitted curves for different countries

 

Monthly summary (24 October 2021)

  • Until recently, the latest wave in the UK, fuelled by the the Delta variant appeared to have reached a peak with a small decline. However there is now preliminary indications of a further upturn in deaths, following the recent rise in cases and hospitalisations. It seems likely that as the autumn progress into winter, deaths will rise further unless additional restrictions as per the "Plan B" are introduced soon.

  • The largest Delta variant fuelled upturn in deaths in the countries featured here is occurring in the USA, but that now appears to be in delcine/

  • In almost all the other Western European countries covered here daily deaths showed varying levels of upturns no doubt due to the Delta variant, but most are now in decline.  These increases are modestly modest and I have not added a further model to fit to them. In some countries, it seems that the death rates are already declining slightly, unlike the UK and USA.

  • For the countries where I have modelled every separate wave of deaths, it continues to be surprising how well the actual deaths fit the modelled curves, which are based on simple mathematical formulae with only three adjustable parameters.

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 were 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 still comparatively small as a percentage of each country's population, with the highest (Brazil) now exceeding 0.28% (which corresponds to about 1 in 350).  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 a percentage of its population, the UK has now been "overtaken" by Italy, Brazil, and most recently the USA. 

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

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 an upsurge caused by the new variant and also by the easing of the second national lockdown restrictions.

The modelling is based on three separate waves.  The total number of deaths predicted in the third wave is now about 40,000 (unchanged from last week). This is very similar to the first wave. Note that the third and second wave models are overlapping and that the total number of deaths predicted in both is now about 90,000.  Hence the overall total for the three waves is predicted to be close to 130,000.  

In the plot below I show the results from model 1 (for the first wave), model 2 (for the second wave) and a third model (model 3) for the latest wave. The total is the sum of all three waves and is a remarkably close fit to the actual values.  It seems likely that the successful vaccination programme is speeding the current decline in the daily deaths, which continue to be below or "inside" the fitted curve, whereas after the first wave the decline was slower than the fitted curve ("fat" tail effect). 

The recent increases in cases, due to the new Delta variant have now been followed, since about day 480, by a modest rise in daily deaths. I have added a fourth model to account for this new wave.  The fitted curve now shows a peak number of daily deaths of around 130. The total predicted in this fourth wave is now about 20,000 but since about day 570 there are signs that the daily deaths are edging up with the rest of autumn and winter still to come. A further model, for a fifth wave, would need to be introduced to fit account for this.

Spain 

The plot below shows the daily deaths for Spain. After a strong first wave, Spain then experienced a smaller more gradual second wave. However from about day 300 the deaths started to rise again forming a third wave, as has happened in the UK. The peak of the third wave has been reached with a decline now underway.

I have now added a third wave model to these figures. The plot below shows all three models and their sum. The total number of deaths in the 3 curves in now about 60,000. The decline in the third wave deaths was briefly reversed to form a short lived fourth wave (not modelled) which has now subsided. Following a slow decline, there was a small upturn from about day 500 no doubt due to the Delta variant (not modelled). It now seems that this small wave has subsided again, to leave daily deaths at a low level.

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. Around day 305 the decrease in the deaths abruptly halted and a small further increase occurred which has now peaked and started dropping off again. To account for this, I have now added a third wave into the modelling. For Italy this third wave is clearly evident but is weaker than in other countries such as France and the UK. The sum of all three modelled waves gives an estimate of 100,000 for the total deaths.

A fourth wave started around day 370 which I have now included in the modelling.  The peak on this latest wave has been reached, and the predicted total number of deaths will be about 30,000 in this wave (an overall total of 130,000 for all four waves). A decline then took place and deaths reached a low level at around day 510. 

However, as with Spain, there was then a small upturn, starting at about day 520, no doubt due to the Delta variant. After declining from about day 570, there are now signs that daily deaths are edging up again, as in the UK.

France

The modelling now has four curves which give the best fit to the daily deaths, as for Italy above.  The sum of all modelled waves gives an estimate of about 110,000 for the total deaths.

In France, the last 3 waves have followed one another in quick succession, with the peak of each less than the one before. With the decline after the fourth wave, daily deaths reached a low level, before the latest, Delta variant fuelled rise, as elsewhere, started from about day 510. This latest peak was modest, with a decline in daily numbers to the present low level at day  600. I have not introduced an additional curve into the modelling to account for this most recent wave.

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. The third wave modelling is to account for the large increase in deaths from about day 240 onwards.

For the third wave, deaths have clearly peaked and the recent deaths are showing a somewhat erratic decline. The prediction for the total number of third-wave deaths in now about 400,000. This is clearly much higher than in the first two waves. The combined total for all three waves from the curves is about 615,000 - by far the largest national total in the world. Note that the deaths in the latest wave declined faster than the modelled curve, probably due to the vaccination programme, as in the UK. This rapid decline then stalled briefly and become more gradual - before returning to follow the fitted curve closely.

Since about day 500, the daily deaths which were at a low level started to increase again rapidly, following a large upturn in case numbers presumably due to the Delta variant. Of the countries featured here, this is largest increase due to the Delta variant. Is being attributed mostly to infections in those who have not been vaccinated. I've recently added a fourth model to fit to the latest upturn in deaths. It appears that the daily deaths peaked ay about 2000/day and are now in decline.  The total deaths in this wave is now modelled to be about 200,000.

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).  The second wave has peaked, and deaths were in rapid decline. However this decline then slowed and a small increase occurred before a further slow decline started. The fitted curve suggests that the total second wave deaths will be about 8,500. From about days 480 to 550 there were very few reported deaths in Sweden. However from about day 550 there appeared to be a very small increase, possibly due to the Delta variant, which is still continuing. 

 

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.

The second wave deaths have dwarfed those in the first wave, although of course the first wave peak in Germany was comparatively very low. The second wave deaths peaked and a strong decline ("inside" the modelled curve) followed. This then slowed and abruptly rose somewhat. This mini third wave (not modelled) has now also subsided and deaths were at a low level until about day 520. There was then a small upturn presumably caused by the Delta variant, which may now be declining.

Brazil

For Brazil, it is clear the the modelled curves do not fit the actual deaths as well as in most of the other countries shown here. In the first wave,  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.

The modelling is now based on two separate curves for the first and second waves respectively.  The actual second wave deaths increased strongly for many weeks but then abruptly  declining. The data for this country show large fluctuations and it is unclear how reliable the figures are.

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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