Coronavirus in Europe: Doubling Time, Continuous Growth and Other Measures (Revised 18 May 2020)

This note discusses metrics that have been used frequently during the coronavirus pandemic: (i) the growth in the number of cases and deaths; (ii) the time they take to double; (iii) the re-infection ratio, generally known as the virus reproduction number R; and (iv) the excess mortality above the norm. The analysis here compares the UK with other countries in Europe, and is based on the daily counts disclosed by the relevant government agencies. Compound daily growth rates are extracted from weekly changes in order to smooth out short-term fluctuations; the visualisation of doubling is made clearer with a log base 2 plot; and comparability is improved by weighting for population, by standardising starting times, and by accounting for excess deaths with respect to the previous year. It is shown how growth curves behave as they converge to zero, the point at which there will have been no new cases for an entire week. It is also shown how the effects of the epidemic peaked when the doubling time lengthened to about two weeks, and the rate of growth in the total number of deaths fell to about 5% per day. A simple benchmark for R is constructed as the ratio of this week’s new cases to last week’s new cases, which reaches zero when the number of confirmed infections stabilises.

The World Health Organization’s daily COVID-2019 Situation Report provides information by country on (i) the number of laboratory-confirmed coronavirus cases and (ii) the number of attributed deaths. The European Centre for Disease Prevention and Control also collects this information, from health authorities worldwide. After validating the accuracy and reliability of the data, ECDC provides an updated data file each day, as used in this analysis. On 18 May 2020, the reported information for the five largest countries in Europe is as shown below, together with the aggregated position for all other countries in Europe:

It is generally accepted that the true number of individuals already infected is substantially higher than the numbers disclosed following laboratory testing. Also, there are known to be marked differences between countries in the basis for recording deaths, in the reporting delays that can arise, and in the scope and scale of screening tests. However, the available tally of confirmed cases and attributable deaths still represents an extensive accumulation of individual events, providing a sufficient census from which to make out similarities and differences between neighbouring European countries in the progress of the viral infection. In Figure 1 below, the conventional graph on the left shows the rapid growth in coronavirus cases to date, where the prolonged acceleration in the UK is particularly evident. On the right, the same data are plotted exponentially using a log base 2 scale on the vertical axis to display the doubling process.

Figure 1. Laboratory-confirmed coronavirus cases in the five largest European countries

Figure 1B shows how growth slows down progressively, with doubling time lengthening accordingly. The horizontal gridlines mark the doubling steps, which are enumerated on the vertical axis from a starting point of 10, doubling to 20, 40, 80 and so on. In addition, a number of guidelines have been drawn to illustrate doubling at different rates — from once per day to once every two weeks.

These growth curves are also evidenced by the calculations tabulated below. The cumulative counts, as collated by ECDC, are presented here on a weekly basis together with the number of days’ doubling time implied each week and the average growth rate per day for the week in question:

Table 1. Coronavirus cases. Each country’s population is given in red, as reported by the World Bank for 2018, along with the total count expressed per million as at 18 May 2020. In blue are the numbers of days over which total confirmed cases doubled, based on each week’s increase, and the third row gives the average daily growth rate each week.

Consistent with Figure 1, the quantitative summary in Table 1 verifies the systematic way in which the doubling of total confirmed cases slows down in each location. The calculations are based on weekly data, which helps not only to smooth out the effect of short-term fluctuations in daily counts but also to remove cyclical disturbances at weekends. More specifically, the cumulative total is divided by the comparative figure one week earlier to give (one plus) the rate of growth in total cases over the week, which is re-expressed here as the compound daily growth rate and updated each day as a moving average. The relationship between the rate of growth and doubling time is nonlinear: that is, one plus the daily rate of growth to the power of doubling time is equal to two.

The two metrics are used for the two vertical axes in Figure 2A below. This graph provides another way of visualising the epidemic curve, with its acceleration until an initial turning point is reached, followed by mainly declining growth that should fade eventually to zero, i.e. the point at which there will have been no new cases for an entire week. The early impact of the coronavirus in Italy (the blue trend line) and Spain (the red line) is particularly evident in Figure 2A, in contrast to the much lower rates of initial growth in France, Germany and the UK.

Figure 2. Daily rate of increase in the total number of cases. Growth is calculated as a daily rate over the past week and updated each day as a moving average. The right hand chart shows the number of days that elapsed between the first notification in each country and the date when positive test numbers began to multiply.

An enlargement of Figure 2A for the last four weeks is inset, again showing the rolling growth rate on the left hand axis and the implied doubling time for the total count on the right. The starting day has been adjusted so that zero on the horizontal axis marks the point at which case numbers began to rise, rather than the calendar date of the first recorded case. The relatively high growth rate in the UK that persists throughout the cycle is conspicuously evident.

Although the relationship is nonlinear, it is evident that the same information is provided by (i) the rate of growth in confirmed coronavirus cases, and (ii) the number of days that it takes for the total to double. The first of these metrics, the overall growth rate, has the advantage that it will reach the readily understandable level of zero when there have been no new cases for an entire week, as mentioned previously. Doubling time has the advantage that it lends itself to communication in plain English (e.g., “doubling every three and a half days”, or “twice per week”), drawing on basic notions about the spread of a contagious infection, such as the time taken for an infected individual to pass the virus on to one more person.

Figure 3. South Korea

It is worth comparing Figure 2A above with the experience in the first country outside China to have had an outbreak (South Korea, where they were active in contact tracing from the outset, and isolating the identifiable source). The pattern of initial acceleration and subsequent deceleration closed out with near-to-zero growth after four to five weeks, the time taken to contain the coronavirus from the date when the number of confirmed cases began to multiply. Across Europe, this outcome has taken longer to achieve.

Coronavirus deaths

The cumulative number of deaths in Europe as a whole is displayed in the stacked frequency chart on the left, starting from the first recorded death (on 15 February in France). The cumulative mortality counts are given below on a weekly basis in Table 2. The doubling times implied by the increase in the number of deaths each week are indicated in blue, beneath which is the average growth rate per day. Deaths per million inhabitants at 18 May 2020 are given in the last column, in red.

Table 2. Coronavirus deaths. Doubling time is given in blue, followed by the average daily growth rate each week. Deaths per million at 18 May 2020 are given in red, based on World Bank 2018 population estimates.

Figure 5A below plots cumulative number of deaths per million inhabitants since the count began to multiply (roughly speaking, this was when the third death was reported). Again, gridlines are drawn across the graph to help visualise the doubling process in the counts, and benchmark guidelines are included to illustrate differing doubling times. The chart on the right (Figure 5B) shows the degree of convergence in the rolling growth rates as they decline asymptotically to zero, a similar pattern to that seen earlier with regard to confirmed cases.

Figure 5. Coronavirus deaths in the five European countries with the greatest population. The right hand chart shows the rate of change in the total number of deaths, which is calculated as a daily rate over the past week and updated each day as a moving average.

Figure 5B displays the growth rate and doubling time from the sixth week after deaths began to rise to more than twelve weeks in the case of Italy. This section is highlighted because newly reported deaths tended to peak in each location after about six weeks. The graph suggests that the average doubling time in week six was about once every fortnight, and the average daily growth was about 5%. More specifically, across the five largest European countries, doubling time and growth developed as follows in the weeks after deaths started to climb in each country, with the sixth week’s averages emphasized:

Table 3. Average mortality growth and average doubling time. Averages across five countries are available for only the first nine full weeks after deaths started to increase. The tenth week excludes Germany, the eleventh also excludes Spain and the UK, and the twelfth is just for Italy (as at 18 May 2020).

Figure 6. Excess deaths. These are calculated as the difference between (i) total deaths from all causes in March & April 2020 and (ii) and the comparable base year figure; the number of deaths attributed officially to coronavirus (Covid) is then deducted to give the proportion of deaths that is Non-Covid. Mortality time series are obtained as daily counts from DESTATIS (Germany), INSEE (France), INE (Spain) and ISTAT (Italy, scaled up from 87% and 58% samples in March and April 2020 respectively), and, as weekly counts from ONS (for England & Wales only, plus counts on the same basis from the devolved statistical offices in Scotland and Northern Ireland).

The briefest period over which deaths have been rising is in Germany (67 days to 18 May), giving the shortest trend line in Figure 5A above. After exactly the same 67-day duration in each country, the reported outcome was as shown in Figure 6 on the left. This bar chart shows not only the reported total of deaths per million attributed to coronavirus in those nine and a half weeks (in dark blue), but also the estimated number of deaths above the norm that have not been attributed to coronavirus (in pale blue). With this approach, the ranking is made comparable not only through time (by starting each series at the point when the number of deaths began to climb) and across countries (by weighting by total population), but also in scope (by showing the contribution of the reported coronavirus deaths to all excess deaths).

With regard to excess mortality, this too is based on publicly-available information, specifically the counts published in each country for deaths from all causes during March and April 2020, from which the comparable figure for 2019 has been deducted (at the time of writing, the series for 2020 are only available up to 15 April for Italy and 19 April for Germany). These additional data have been downloaded from national statistical agency websites and scaled per million to produce the following plots showing the scale of excess mortality:

Figure 7. Excess mortality

Figure 7 above demonstrates clearly the unusual death toll in Spain, France, Italy and the UK in 2020. Interestingly, the major feature of the plot for Germany is the high mortality in 2018, during an influenza epidemic that started in February of that year. The central panel in the top row compares total mortality in 2020 directly with 2019 for all five countries, as a ratio each day, showing that the UK peaked at more than twice the 2019 level.

Finally, the question posed most frequently in the UK over recent weeks again concerns the daily count of newly reported deaths, at first asking when it might peak, and now whether it will peak again. To address this question in the European context, newly reported coronavirus deaths each day are displayed below in Figure 8, as the daily mortality averaged over the week, updated daily as before. The peaks are evident in this graph, as is the remarkably low mortality attributed to coronavirus in Germany. For the UK, the graph suggests a trajectory that is heading for the highest mortality rate amongst these countries.

Figure 8. New coronavirus deaths per day. The figures plotted for France and the UK are the adjusted series that incorporate care home deaths throughout.

Source: WHO Situation Report 18 May 2020

An alternative presentation (inset in Figure 9) shows the daily totals of newly reported deaths for the European region as a whole, by summing the contributions of each of the largest countries plus the rest of Europe. This stacked frequency chart reveals a degree of consistency in the overall impact, given that mortality and timing differences across national boundaries are offset in part through aggregation. As mentioned, the series for each country comprises the daily average over one week, which eliminates the effect of weekend under-reporting. With this in mind, it is interesting to compare the smoothed frequency distribution in Figure 9 with WHO’s worldwide aggregation of the raw data (inset beneath, starting with the initial outbreak in China); the cyclical weekend effects are clearly visible later on in the daily case counts for the wider European region (brown) and the Americas (yellow).

Containing the pandemic

The practical approach taken above can be extended to throw a little light on one further metric, the reproduction ratio R, which measures the number of individuals likely to contract the virus from one infected person. A recent study has estimated the initial reproduction number — known as R(0) — using data from Wuhan before the lockdown there. It has been found that each infected person may have transmitted COVID-19 to five or six others, i.e., about twice the earlier R(0) estimates of between two and three new infections. In addition, the authors tell us that the unconstrained doubling time in Wuhan appears to have been more like two to three days, rather than the prior estimates of six to seven days, and that the virus incubation period can be inferred from all this as between four and five days. Since the initial outbreak, the main objective in containing the pandemic has been to reduce R to less than one, such that each existing infection will cause less than one new infection, whereupon the disease should decline.

In order to estimate the number of new cases that may arise from a single source during the infectious period, epidemiological researchers build mathematical models that are calibrated for incubation and transmission duration (the serial interval), together with other influencing factors such as contact rates, durations and survival risks. It is said, for instance, that scientists at Public Health England use an age-stratified model that is adjusted for the risk of dying together with the time from infection to death to estimate their measure of ongoing transmission. In essence, though, the measure relates the number of individuals passing on the virus to the number then infected, which will equal one when the flow of new cases stabilises. Here, in Figure 10 below, a benchmark indicator is constructed from the publicly-available data as the ratio of new cases disclosed over a week to those over the previous week, noting that the one week lag roughly approximates the serial interval.

Figure 10A displays the benchmark ratio for the five largest countries in Europe, based on the time series of newly reported cases in each country. The graph shows how this is mainly tracking below one in each country (I have yet to investigate whether the disturbances in France and Spain in the last two weeks are attributable to data collection). In Figure 10B, a similar and smoother trajectory (with a one week lag) is evident in the benchmark ratio of newly reported deaths.

Figure 10. A simplified R ratio. Figure 10A compares this week’s new cases to last week’s new cases, which is equal to one when each new case is followed by just one new case a week later. The same calculation is applied to deaths in Figure 10B.

Concluding remarks

It is important to end by stressing that more detailed inferences are not feasible at this stage based solely on the reported counts. For example, comparable time series are required in order to improve the estimation of excess mortality, as outlined at BMJ Opinion by Professor Michel Coleman’s research group (7 May). In addition, validated revisions to existing series are required to reflect any reporting methods that may have changed, and to re-assign counts to the day of occurrence. More generally, further information is needed on the actual levels of infection within the population through wider testing.

Nevertheless, careful analysis and presentation of the headline figures (the tallies that are reported each day) is both feasible and essential, not only to be fully accountable to the public alongside Covid-19 Surveillance Reports, but also to provide useful benchmarks by which to judge the results of scientific analysis obtained with more complex models. This note — which has been updated every week or so — was first motivated by the puzzling mention of 5-day doubling (the seeding assumption in the Imperial College study) at the initial Downing Street briefings by ministers and advisers in mid-March, including the Prime Minister’s briefing of 16 March 2020. This was puzzling because the counts announced over the week to 16 March suggested that the daily growth in positive cases was already 26% (doubling every 3.0 days), and that the growth in the week before was 34% (doubling every 2.4 days, i.e. three times per week), which even then reflected the publicly available evidence from China and Italy.

Following preliminary discussion in the media (12 April, 19 April, 21 April), concern over the presentation of statistical information in the Briefings has grown, especially following the observations of Professor David Spiegelhalter, and letters to the press by Professor Bridget Shield and others (30 April, 4 May, 6 May, 9 May, 10 May). In this regard, perhaps there are lessons to be drawn from the analysis presented in this note:

1. Comparisons between headline figures by country can easily overlook the considerable differences in population. Proportionate scaling per million inhabitants may be adopted in order to draw more meaningful conclusions. Even so, social research often shows that variability amongst individuals within countries often dominates variability between countries, so the results of national comparison should be interpreted in the context of disaggregated regional comparisons, together with an understanding of population structures.
2. Direct comparisons at today’s date can also be misleading. Adjusting the starting point to when the first occurrence is reported in each location is one solution. Here, the origin has been adjusted to the date at which cases or deaths began to multiply, and also to when the threshold of one per million was reached. Realignment in this way allows comparisons to be made across countries at a similar point in the cycle.
3. Explicit graphical presentations, such as doubling time gridlines, make it easier to visualise the evidence.
4. Differences in the scope of reported coronavirus statistics have received much attention in the media. A simple stopgap is to compare mortality in 2020 with that in 2019, a fairly normal year, in order to gauge the relative contribution of deaths officially attributed to coronavirus to the estimated number of deaths above the norm.
5. For more complex indicators, benchmarks should be used to help members of the public ground their understanding in the headline statistical releases that are publicly available, such as the weekly benchmark ratio of new cases suggested here as an indicator of R.

Note. As shown in Table 2, with 5865 deaths in total in the UK by 6 April, and 1669 one week earlier, the count at the beginning of that week doubled every 3.9 days. The calculation of doubling time may be expressed mathematically as 3.9 = 7(ln2/(ln5865-ln1669)). The compound daily growth rate during the week was 19.7%, where 0.197 = exp((ln5865-ln1669)/7)-1 (note that the limiting case of the difference in logs is the instantaneous growth rate, i.e., compounding in continuous time, which is a fundamental construct in financial economics). The simplified R is the rate of change in first differences plus one. For example, from Table 1, it can be seen that the increase in the number of confirmed cases was 243695-219183 = 24512 last week, and 219183-186599 = 32584 in the previous week, yielding the simplified version of R based on new cases for the week of 32584/24512 = 0.75.

Previously: 8 May 2020, 28 April 2020, 21 April 2020, 14 April 2020, 7 April 2020, 31 March 2020. Personal note: I am not an epidemiologist, but there is cross-disciplinary connection in that my usual research is concerned with the econometric estimation of metrics that are constructed directly from reported data (like the observable growth rates and one week doubling times used here), how much information they may contain and how they may be compared internationally. Contact: s.j.mcleay@sussex.ac.uk & s.j.mcleay@sydney.edu.au.