key: cord-315587-jelh8o3v
authors: Dye, C.; Cheng, R. C. H.; Dagpunar, J. S.; Williams, B. G.
title: The scale and dynamics of COVID-19 epidemics across Europe
date: 2020-06-29
journal: nan
DOI: 10.1101/2020.06.26.20131144
sha: 
doc_id: 315587
cord_uid: jelh8o3v

The frequently stated goal of COVID-19 control is to 'flatten the curve'; that is, to slow the epidemic by limiting contacts between infectious and susceptible individuals, and to reduce and delay peak numbers of cases and deaths. Here we investigate how the scale and dynamics of COVID-19 epidemics differ among 26 European countries in which the numbers of reported deaths varied more than 100-fold. Under lockdown, countries reporting fewer deaths in total also had lower peak death rates, as expected, but shorter rather than longer periods of epidemic growth. This empirical analysis highlights the benefits of intervening early to curtail COVID-19 epidemics: the cumulative number of deaths reported in each country by 20 June was 60-200 times the number reported by the date of lockdown.

The frequently stated goal of COVID-19 control is to "flatten the curve"; that is, to slow the epidemic by limiting contacts between infectious and susceptible individuals, and to reduce and delay peak numbers of cases and deaths. Here we investigate how the scale and dynamics of COVID-19 epidemics differ among 26 European countries in which the numbers of reported deaths varied more than 100-fold. Under lockdown, countries reporting fewer deaths in total also had lower peak death rates, as expected, but shorter rather than longer periods of epidemic growth. This empirical analysis highlights the benefits of intervening early to curtail COVID-19 epidemics: the cumulative number of deaths reported in each country by 20 June was 60-200 times the number reported by the date of lockdown.

The COVID-19 death toll has varied more than 100-fold across European countries; the total number of deaths reported in each country was 60-200 times the number reported by the date of lockdown.

Physical distancing flattens an epidemic curve by reducing the contact rate between infectious and susceptible individuals. A lower contact rate lowers the basic case reproduction number, R0, which reduces peak incidence and death rates and distributes the burden of illness over a longer time period (1-4). Flattening the curve protects health and health services.

For illustration, Figure 1 shows, with the aid of a dynamic SEIR epidemiological model (Supplementary Materials), how the COVID-19 epidemic in Switzerland might have been contained by flattening the curve. With an estimated basic case reproduction number of R0 = 2.33, the number of deaths reached a maximum of approximately 420 in week 15 (week beginning 6 April), and a total of 1800 deaths had occurred by week 25. If the basic case reproduction number had been higher at the outset (R0 = 3.50 in Figure 1 ) the epidemic would have been larger (2020 deaths) and narrower, peaking sooner and lasting for a shorter period of time. A lower basic case reproduction number (R0 = 1.75) would have reduced the size of the epidemic, and the 1300 deaths would have occurred over a longer time period (flatter epidemic curve). These calculations are consistent with epidemic theory, which shows how the final size of an epidemic (proportion of people infected, p) is a monotonically increasing function of R0, through the relationship = 1 − exp (− ) (5) .

Does this theory explain the differences in COVID-19 epidemics across European countries? Did countries with more effective COVID-19 control programmes ─ those that reported fewer cases and deaths ─ have flatter epidemic curves? How has the scale and shape of European epidemics been changed by interventions, especially physical distancing under lockdown? Figure 2 approaches these questions with respect to the number and distribution of deaths through time for 24 European countries. Rather than using a dynamic epidemiological model to describe these epidemics, such as an SEIR model which is constrained by assumptions of, for example homogeneous mixing and exponentially distributed time periods (Supplementary Materials), we used an empirical model (skew-logistic) to measure the rate of epidemic growth, the maximum number of deaths per week, the period of growth (time from one death to the maximum number of weekly deaths), and the rate of decline. The empirical model gives an excellent description of COVID-19 epidemics across Europe; it is also more discriminating between epidemics than estimates of the time-varying case reproduction number (Supplementary Materials, Figure S6 ). The total number of reported deaths was 5.6 (95%CI 0.29) times the peak number of weekly deaths across the 24 countries (R 2 = 0.98, Figure 3a ). The principal determinant of variation in the number of deaths among countries was neither the rate of epidemic growth (R 2 = 0.01, Figure 3b ) nor the rate of decline (R 2 = 0.03, Figure 3c ), but rather the duration of growth ─ the period over which the epidemic was allowed to expand (t = 3.91, p < 0.001, R 2 = 0.41, Figure  3d ).

Longer periods of epidemic growth were associated with lower growth rates (R 2 = 0.37, Figure  3e ), the expected consequence of physical distancing ( Figure 1 ). Against expectation, however, slower growth rates were associated with more deaths rather than fewer deaths overall. The reason is that, in comparisons between countries, slower growth was more than offset by longer periods of growth (the red line lies above the black line of equivalence in Figure 1e ).

The product of the rate (g) and duration of growth (d) defines net epidemic growth ─ how much an outbreak expands in size during the growth period. Net growth, measured by = , accounted for 88% of the inter-country variation in the number deaths (t = 12.4, p < 0.001, R 2 = 0.88; Figure 3f ). Although such a relationship is expected in principle, it is surprising that theory is upheld so faithfully in data collected in different ways across 24 diverse European countries.

Countries with larger populations generally reported more deaths (t = 6.83, p < 0.001, R 2 = 0.68); however, in multiple regression analysis, population size did not explain significantly more of the inter-country variation in deaths (t = 1.75, p > 0.05) than net growth, G (t = 6.39, p < 0.001). In other words, the effect of population size is accommodated within the measure of net growth. There is a hint in these data that the size of a COVID-19 epidemic increased more than linearly with national population size (N, millions), such that the number of deaths, = 51.1 . . The implication is that population size is more than an epidemic scaling factor, with effects that are not fully captured as deaths per capita. However, the exponent 1.32 is not significantly greater than unity (t = 1.64, p > 0.05). In any event, epidemics expanded more, and caused more deaths, in countries with larger populations.

Because the duration of growth ( Figure 3d ) and net growth (Figure 3f) can account for much of the inter-country variation in numbers of deaths, we expect, in the data as well as in theory, the timing of interventions to be a determinant of epidemic size. One summary measure of the implementation of control measures is the Government Response Stringency Index, a composite metric of lockdown policies based on nine indicators including school closures, workplace closures, and travel bans, with a score for each country varying between 0 and 100% (6) . Figure 4 shows how the total number of deaths (D) reported from each country up to 20 June increased in relation to the cumulative number that had been reported when the Stringency Index reached or exceeded 70% (DL). The form of the power relationship is ≈ 200 . . The geometric mean ratio of D to DL is 128 in both model and data, but with greater variation in the data (95%CI 155) than in the fitted model (95%CI 19) . Based on the model relationship, the expected number of deaths varied from 60 to 200 times the number at lockdown, as the latter ranged from its highest (more than 300 in Sweden) to lowest values (zero in e.g. Croatia, Romania, Slovenia).

To illustrate the differences between countries, Luxembourg achieved a Stringency score of 72% on 16 March having recorded only 1 death; by 20 June they had recorded 110 deaths. Spain achieved the same Stringency score of 72% a day later on 17 March, having recorded 309 deaths; by 20 June, Spain had recorded 27,000 deaths.

The outlying countries in Figure 4 raise questions for further investigation: Belgium locked down (73% Stringency) on 18 March with 5 deaths, and yet the epidemic grew faster than average (0.34/day) and for longer than average (34 days), generating a relatively high number of deaths. Unusually, Belgium has reported more COVID-19 deaths than all excess deaths, but only around 15% more (Supplementary Materials), whereas observed and expected COVID-19 deaths differ by a factor of 10 in Figure 4 . Sweden never achieved a Stringency score as high as 70% (maximum 46%) and reported a relatively large number of deaths (4900) among Scandinavian countries (fewer than 600 in each of Denmark and Norway). However, the general relationship in Figure 4 predicts a far higher death toll. The reasons why Belgium and Sweden do not conform to the general pattern in Figure 4 remain to be determined.

In conclusion, this analysis shows that European countries reporting fewer COVID-19 deaths had flatter epidemic curves ( Figure 2 ), but not in the way anticipated by simple epidemic theory ( Figure 1 ). Countries with fewer deaths overall did have smaller peak death rates, as expected, but they suffered fewer COVID-19 deaths due to shorter rather than longer periods of epidemic growth. In these comparisons between countries, longer epidemics were generally larger epidemics.

The cross-country comparison in Figure 4 shows that the overall death toll was lower in countries that locked down earlier, when fewer deaths had been reported. But the inference that a country with high COVID-19 mortality could have been like a country with a low mortality, had it acted sooner, comes with a caution: controlling COVID-19 epidemics through early intervention appears to be harder in countries with larger populations, for reasons that are not explained by this study.

Notice, in addition, that epidemic expansion was greater in more populous countries, not because they imported more infections (a scaling factor), though that might also be true (7, 8) , but rather because they had epidemics that grew for longer (a multiplication factor).

Taken together, these data expose the perils of delayed action during an epidemic with an average doubling time as short as 3 days. At this rate of growth, the daily death toll would have increased by a factor of ten within 9 days, which is shorter than the average time delay from one death to the time of lockdown, 11 days (ranging from -6 days in the Czech Republic to +43 days in Sweden). In effect, a COVID-19 epidemic in the average European country would have expanded more than ten-fold within the time it took to impose lockdown.

The need for speed is expected to apply to resurgent epidemics too. Behind this analysis lies the supposition, supported by antibody surveys (9-11), that only a small fraction of Europe's population as been exposed to infection (of the order of 5-10%), albeit a fraction that varies among countries. Like others (1, 3, 12), we assume that new COVID-19 outbreaks will threaten large numbers of susceptible people across Europe, a view reinforced by renewed outbreaks in several countries during June 2020. Facing the risk of further outbreaks of COVID-19, this analysis points to the benefits of taking action without delay. European Centre for Disease Prevention and Control, COVID-19. 2020.

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The copyright holder for this preprint this version posted June 29, 2020. . https://doi.org/10.1101/2020.06.26.20131144 doi: medRxiv preprint All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. .

The scale and dynamics of COVID-19 epidemics across Europe

This supplement to the main text provides data sources, methods of data analysis, the mathematical model of COVID-19 dynamics and additional data and Figs.

This analysis is based on COVID-19 deaths reported from 26 European countries, for which the abbreviations are tabulated below (1). We used reported deaths because they are likely to be more accurate than reported cases, both in absolute magnitude (scale of the epidemic) and in the distribution of deaths through time (shape of the epidemic). However, a comparison of COVID-19 deaths and all excess deaths in Switzerland (Fig 1, Fig S3 below) questions the completeness of death reports at the peak of the epidemic. Nevertheless, the large differences in numbers of reported deaths between countries, and in number of deaths per capita, are very unlikely to be due solely to differences in diagnostic and reporting methods (2) . Indeed, the clear patterns of association in Figs 3 and 4 of the main text suggest that both the data and the empirical model portray real effects, rather than artefacts, of COVID-19 epidemics across the 26 countries studied in this investigation.

Fig 1 was constructed with a compartmental model of SARS CoV-2 transmission, framed in ordinary differential equations, and representing a homogeneously mixing population divided among susceptible, exposed, infectious and recovered or died (SEIR), as follows: (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. . https://doi.org/10.1101/2020.06.26.20131144 doi: medRxiv preprint Fig S1. The SEIR model. The compartments denote those in the population that are Susceptible, Exposed, Infected and Recovered.

The variables S, E, I and R satisfy the ordinary differential equations:

A convenient recent reference is Ma (3), though we have adjusted the notation here, reserving certain symbols for use in other standard ways.

As this discussion focuses on the number that die due to the epidemic we adjust the SEIR model by splitting R, those that recover, to distinguish between Z, those that die, from those that recover well. This arrangement is depicted in Fig S2, and is a simplified version of that used by Dagpunar (4) , which discusses the outcomes of hospitalization. Fig S2. Adjustment of the SEIR model where R is divided into two compartments, R \ Z, those that recover and Z, those that die; where is the proportion that recover.

We allow this adjusted SEIR model to depend on certain parameters with the understanding that once these parameters are known the behaviour of the SEIR model is completely specified. There are nine parameters in the model: (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. . https://doi.org/10.1101/2020.06.26.20131144 doi: medRxiv preprint number of days from start of epidemic before observations began 5.351 initial number of exposed The last parameter is used to modify the last differential equation (4) to:

We denote by = ( 1, 2, … , 9) , the vector of parameters. With given, the four differential equations (1), (2), (3) and (5) 

where t is the day and N is the number of days of interest. However, the parameters are assumed unknown. We therefore used the standard method of Maximum Likelihood (ML) as given for example in Cheng (5) to estimate the parameter values.

Here we outline the approach (called fitting the model) used to estimate the parameters from a sample of observed daily deaths, this being used to prepare Fig 1 in the main text. Let the sample of observed number of daily deaths be denoted by

where is the number of deaths on day t and is the number of days observed. If the observations were made without error and if, with the right parameter values are correct for , then the death trajectory { ( , ) = 1,2, … , } would match the observed deaths Z in (7). So the model would be clearly successful in explaining deaths.

To include statistical uncertainty in the model we assume instead

where ( ) is random error. For simplicity the ( ) are assumed to be normally and independently distributed (NID) with standard deviation , i.e. 

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The copyright holder for this preprint this version posted June 29, 2020. . https://doi.org/10.1101/2020.06.26.20131144 doi: medRxiv preprint where is the random argument, and the parameters are fixed. In ML estimation, this is turned on its head so that Z is simply the known sample of observations now regarded as fixed and we write L as L( | ) = L( | )) calling it the (log)likelihood to indicate that it is now treated as a function of . The ML estimator is simply the value of at which L( | ) is maximized. i.e.

Nelder-Mead numerical search for the maximum was used. This goes through different i=1, 2, 3,… comparing the different L( , | ) to find , the best .

To simplify description of the estimation process, only fitting to deaths data, Z as in (7) has been described, but the method extends straightforwardly to include other data samples. For example

where is the number of active cases on day t. Fitting simultaneously to both Y and Z can be carried out by adding to the right-hand side of (10) a corresponding set of terms for Y. Numerical solution of the differential equations requires initial values for S, E, I, R. These are essentially scale invariant with ( + + + ) constant and independent of t. So the numerical integration can conveniently be done using S(0, ) = 1, E(0, ) some small quantity subsequently adjustable as its initial value is a parameter; with I and R also initially zero. The population size, also a parameter, is only needed to provide scaled values S, E, I, R at each step for comparison with the data Y and Z.

Epidemic shape and scale (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. . https://doi.org/10.1101/2020.06.26.20131144 doi: medRxiv preprint Model epidemics for Switzerland in Fig 1 differ in shape as well as size; they are not scale invariant. Fig S3 adjusts the peak sizes of epidemics in Fig 1 to show more clearly the differences in shape, i.e. the different distribution of deaths through time when an epidemic is "flattened" by reducing the case reproduction number.

In Switzerland, the relatively flat peak in reported COVID-19 deaths (Fig 1 of the main  text) differs from the estimated number of excess deaths (week 14), suggesting that COVID-19 deaths might have been under-reported in that week (Fig S4) . The number of excess deaths reported from European countries was approximately 29% greater than the number of COVID-19 deaths (in 8 countries for which estimates have been published; Fig S5) (7) . This is a small difference within each country, in comparison with the 100-fold difference COVID-19 deaths reported between countries. Week of 2020

Deaths/week Excess deaths/week All rights reserved. No reuse allowed without permission.

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Although the SEIR model is based on epidemiological principles, it does not accurately describe all of the European COVID-19 epidemics. In particular, the SEIR model does not easily replicate the slow rates of epidemic decline. For this reason we used a skew-logistic model to describe European epidemics:

where f(t) is the number of deaths per unit time. The growth rate of the epidemic is b, and f(t) converges to zero at a rate of decline d = b  2c when t  . Parameter τ positions the epidemic along the time axis. The model can be fitted to data ─ here deaths in each time interval ─ either by least squares (LSQ) or maximum likelihood methods (MLE). We fitted the model to data from 24 countries for which there were sufficient reported deaths (Figs 2 and 3 of the main paper), but added Bulgaria and Slovenia for the analysis in Fig 4. Examples of the estimates shown in Fig 2, (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. (8), implemented by ETH Zurich (9) . Re for each day is presented as the continuously varying (retrospective) average. While commonly used and reported, Re does not characterize and distinguish European epidemics as effectively as the empirical model ( Fig S6) .

Time trends in Re are similar for Germany and the United Kingdom (Fig S5b) , although they do show the lower rate of epidemic decline in the UK. In general, however, the skew-logistic model provides a more useful set of parameters to separate these two countries (and others), as in the shaded entries to the Table below. Besides the rate of decline, the two countries experienced differences in epidemic growth rate, the duration of epidemic growth, net growth, and the number of deaths at the epidemic peak, as shown in the following All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 29, 2020. . https://doi.org/10.1101/2020.06.26.20131144 doi: medRxiv preprint

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