Following the identification of the new coronavirus SARS-CoV-2 in Wuhan (China) in December 2019 and its global spread, large epidemics of COVID-19 have ensued in Europe. In response to the rising numbers of cases and deaths and to preserve health systems, European countries—as with those in Asia—have implemented measures to control their epidemics. These large-scale non-pharmaceutical interventions vary between countries, but include social distancing (such as banning large gatherings), border closures, school closures, measures to isolate symptomatic individuals and their contacts, and large-scale lockdowns of populations with all but essential internal travel banned. Understanding whether these interventions have had the desired effect of controlling the epidemic, and which interventions are necessary to maintain control, is critical given their large economic and social costs. The key aim of these interventions is to reduce Rt, a fundamental epidemiological quantity that represents the average number of infections generated at time t by each infected case over the course of their infection.

In China, strict movement restrictions and other measures (including case isolation and quarantine) began to be introduced from 23 January 2020, which achieved a downward trend in the number of confirmed new cases during February and resulted in zero new confirmed indigenous cases in Wuhan by 19 March 2020. Studies have estimated how the values of Rt changed during this time in different areas of China, from around 2–4 during the uncontrolled epidemic to below 1 (refs. 1,2).

Estimating Rt for SARS-CoV-2 presents challenges, owing to the high proportion of infections that are not detected by health systems1,3,4 and to the regular changes in testing policies, which resulted in different proportions of infections being detected over time and between countries. Initially, most countries had the capacity to test only a small proportion of suspected cases and reserved tests for severely ill patients or for high-risk groups (for example, the contacts of positively tested individuals).

An alternative way to estimate the course of the epidemic is to calculate backwards from observed deaths to the number of infections. We introduce a Bayesian mechanistic model linking the infection cycle to observed deaths, inferring the total population infected (attack rates) as well as Rt. We assess whether there is evidence that interventions have so far been successful at reducing Rt to values below 1. We simulate a hypothetical counterfactual scenario in which Rt remains at starting levels to estimate the deaths that would have occurred without interventions.

Reported deaths are likely to be far more reliable than case data—although reported death data still have limitations. First, early deaths attributable to COVID-19 may have been missed. Second, there is variation in the reporting of deaths by country and over time. Third, reporting delays are expected and can be both systematic and random in nature. We attempt to overcome these data limitations by using a consolidated data source, incorporating noise in our observational model, partially pooling of information between countries and performing a sensitivity analysis under scenarios of underreporting to test our conclusions (Supplementary Information).

Our model relies on fixed estimates of some epidemiological parameters, such as the onset-to-death distribution, the infection fatality rate and the generation distribution, that are based on previous work5,6; we perform a sensitivity analysis on these parameters. Our parametric form of Rt assumes that changes in Rt are an immediate response to interventions rather than gradual changes in behaviour, and it does not include importation or subnational variation. We assume that individual interventions have a similar effect in different countries, and that the efficacy of these interventions remains constant over time. Our framework infers Rt from mortality data, while accounting for time lags since infections occurred. As a result, even with perfect data and partial pooling, we cannot perfectly predict the current value of Rt. However, the credible intervals on Rt show the self-consistent behaviour that is a hallmark of a fully Bayesian analysis throughout the entire period we study, exhibiting appropriate shrinkage as more data become available (Supplementary Videos 13).

Italy was the first European country to begin major non-pharmaceutical interventions, and other countries followed soon afterwards (Extended Data Fig. 4). The onset of interventions ranged between 2 March and 29 March 2020. We analysed data on mortality from COVID-19 in 11 European countries until 4 May 2020, at which point lockdowns were relaxed in Italy and Spain. For each country, we model the number of infections, the number of deaths and Rt (Fig. 1). Rt is modelled as a piecewise constant function that changes only when an intervention occurs. Each country has its own individual starting Rt before interventions took place. For all countries, interventions are assumed to have the same relative effect on Rt and are informed by mortality data across all countries. The only exception is that we use partial pooling to introduce country-specific effects of the effectiveness of the last intervention introduced in the study period in a country (which is usually lockdown).

Fig. 1: Country-level estimates of infections, deaths and Rt for France, Italy, Spain and the UK.
figure 1

Left, daily number of infections. Brown bars are reported infections; blue bands are predicted infections; dark blue, 50% credible interval; light blue 95% credible interval. The number of daily infections estimated by our model drops immediately after an intervention, as we assume that all infected people become immediately less infectious through the intervention. Afterwards, if Rt is above 1, the number of infections will start growing again. Middle, daily number of deaths. Brown bars are reported deaths; blue bands are predicted deaths; credible intervals are as in the left plot. Right, Rt. Dark green, 50% credible interval; light green, 95% credible interval. Icons are interventions, shown at the time at which they occurred.

Estimated infections, R t and effect sizes

In all countries, we estimate there are orders-of-magnitude fewer infections detected (Fig. 1, Extended Data Figs. 1, 2) than true infections, most likely owing to mild and asymptomatic infections as well as limited testing capacities and changes in testing policy. In Italy, our results suggest that—cumulatively—2.8 (2.2–3.5) million people (all parenthetical ranges refer to 95% credible intervals) have been infected as of 4 May 2020, which gives an attack rate of 4.6% (3.6–5.8%) of the population (Table 1). In Spain (which has also experienced a large number of deaths), we estimate that 5.5% of the population (2.6 (2.1–3.3) million people) have been infected to date. Germany, the most populous country in our study, is estimated to have one of the lowest attack rates at 0.85% with 710,000 (550,000–930,000) people infected. Belgium has the highest estimated attack rates of 8%, followed by Spain with 5.5%. Although there have been few reliable national serological studies7, initial small-scale surveys in Austria8 and Denmark9 closely align with our estimates. A much larger study in Spain is very closely aligned with our estimates10. To some extent, these initial results validate our choice of infection fatality rate.

Table 1 Total population infected by country

Averaged across all countries, we estimate the initial Rt to be 3.8 (2.4–5.6), consistent with previous analyses1,11. These estimates are informed by our choice of generation-interval distribution and the initial growth rate of observed deaths. A shorter assumed generation time results in lower starting Rt (Supplementary Discussion 3). The initial values of Rt are also uncertain, owing to (a) importation (rather than local transmission) being the dominant source of new infections in the early period of the epidemic and (b) possible under-ascertainment in deaths, particularly before testing became widespread. We perform sensitivity analyses around these parameters (Supplementary Discussions 10, 11).

We estimate large reductions in Rt in response to the combined non-pharmaceutical interventions. Our results—which are driven more by countries with advanced epidemics and larger numbers of deaths—suggest that these interventions have together had a substantial effect on transmission, as measured by changes in the estimated Rt. At the time of this study, we find current estimates of Rt to range from a posterior mean of 0.44 (0.26–0.61) for Norway to a posterior mean of 0.82 (0.73–0.93) for Belgium, with an average of 0.66 across the 11 countries—an 82% reduction compared to the pre-intervention values. For all countries, we find that current interventions have been sufficient to drive Rt below 1 (probability Rt < 1.0 is greater than 99% across all countries we consider) and achieve control of the epidemic. These conclusions are corroborated by studies from individual countries—France12, Spain13, Germany14 and the UK15—over a similar period, which arrive at very similar estimates despite different methodologies and data. For example, a previous study12 estimates an Rt of 0.67 for France using hospitalization records (we estimate 0.68); for Germany, the Robert Koch Institute reports Rt of 0.76 using electronically notified cases14 (we estimate 0.71). The retrospective stability of our model (Supplementary Videos 13) is variable when the implementations of interventions are very dissimilar; an example of this is seen in Sweden, where interventions were dissimilar to other countries and led initially to large uncertainty. Our model uncertainty is also dependent on the magnitude of Rt; this occurs because infections are a nonlinear function of Rt and are sensitive to small increases. Uncertainty shrinks greatly when Rt is reduced. Examples of this effect are seen in all countries, but it is most pronounced in Belgium and France; these countries show large uncertainties in the number of infections in the early period of the epidemic. Our choice of parameterizing Rt using piecewise constant functions means that we cannot capture the fine-scale variation that could be achieved by using additional covariates.

Lockdown has an identifiable large effect on transmission (81% (75–87%) reduction) (Fig. 2). The close spacing of interventions in time (Extended Data Fig. 4) means that the individual effects of the other interventions are not identifiable (Fig. 2). Our partial pooling model requires only one country to provide a signal for the effect of a given intervention, and this effect is then shared across all countries. Although this sharing can potentially lead to initial over- or under-estimation of the effect of an intervention, it also means that a consistent signal for all countries can be estimated before that signal is present in data from an individual country16. Therefore, this sharing is potentially useful for generating early warnings, by leveraging what happened in countries with earlier epidemics to inform countries with more-recent epidemics.

Fig. 2: Effectiveness of interventions on Rt.
figure 2

Our model includes five covariates for governmental interventions, adjusting for whether the intervention was the first one undertaken by the government in response to COVID-19 (red) or was subsequent to other interventions (green). Mean relative percentage reduction in Rt is shown with 95% posterior credible intervals. If 100% reduction is achieved, Rt = 0 and there is no more transmission of COVID-19. Lockdown is significantly different from the other interventions; the other interventions are not significantly different from each other, probably owing to the fact that many interventions occurred on the same day or within days of each other (as shown in Extended Data Fig. 4). Results are derived from a model that represents 11 countries with a total population of 375 million and 128,928 reported COVID-19-related deaths up to 4 May 2020.

Estimated effect of interventions on deaths

Extended Data Table 1 shows total deaths forecast from the beginning of the epidemic up to and including 4 May 2020 under our fitted model and under the counterfactual model, which predicts what would have happened if no interventions were implemented (and Rt = R0; that is, the initial Rt estimated before interventions came into effect).

By comparing the deaths predicted under the model with no interventions to the deaths predicted in our intervention model, we calculated the total deaths averted in our study period. We find that across 11 countries 3.1 (2.8–3.5) million deaths have been averted owing to interventions since the beginning of the epidemic; Extended Data Fig. 5 compares the actual total deaths to the counterfactual total deaths. The counterfactual model without interventions is illustrative only, and reflects the assumptions of our model. We do not account for changes in behaviour; in reality, even in the absence of government interventions we would expect Rt to decrease and therefore would overestimate deaths in the no-intervention model. Conversely, we do not consider the effect on the infection fatality rate as a result of an overwhelmed health system in which patients may not be able to access critical care facilities, which would underestimate the number of counterfactual deaths. In the Supplementary Information, we show further counterfactual estimates under differing assumptions of the generation distribution and onset-to-death distribution and all scenarios broadly show the same trends. Given this agreement across differing scenarios, we believe our estimates for the counterfactual deaths averted to be plausible.


During the ongoing transmission of SARS-CoV-2 in Europe, we analyse trends in the numbers of deaths to assess the extent to which transmission has been reduced. Representing the infection process associated with COVID-19 using a semi-mechanistic, joint Bayesian hierarchical model, we can reproduce trends observed in the data relating to deaths and produce empirically driven predictions that are valid over short time horizons.

We estimate that there have been many more infections than are currently reported. The high level of under-ascertainment of infections that we estimate here is probably due to the focus on testing in hospital settings, which misses milder or asymptomatic cases in the community. Despite this, we estimate that only a relatively small minority of individuals in each country have been infected (Table 1). Our estimates imply that the populations in Europe are not close to herd immunity (about 70% if R0 is 3.8)17. Furthermore, with values of Rt below 1 in all countries, the rate of acquisition of herd immunity will slow down rapidly. Our estimates for attack rates during our study period are consistent with those reported from national serological studies7. Similarly, comparable studies estimating Rt all agree that the number as of 4 May 2020 is less than 1.

To our knowledge, our modelling approach is unique in pooling information from multiple countries at once. Using this approach means that we require a central consolidated data source (such as data from the European Centre of Disease Control (ECDC)), and also that trends in some countries will be affected by those countries with more data. We argue that this effect is beneficial, in that it helps to minimize idiosyncrasies in the data16, as well as to improve consistency of estimates over time. Although our qualitative conclusions surrounding the effect of interventions and the finding that Rt is less than 1 are robust to our choice of whether to incorporate pooling or not, the ability to use a greater extent of available data and share information across countries in a statistically principled manner markedly improves the consistency of model predictions across the study period (Supplementary Videos 13).

Most interventions were implemented in rapid succession in many countries, and as such it is difficult to disentangle the individual effect sizes of each intervention. In our analysis, we find that only the effect of lockdown is identifiable, and that it has a substantial effect (81% (75–87%) reduction in Rt). Taking into account country-specific effects, the effect size of lockdown remains large across all countries (Supplementary Fig. 29).

We acknowledge the limitations of existing mortality data relating to COVID-19—in particular, deaths outside hospitals may be underreported. However, by using the ECDC data, we rely on a comprehensive data source that is refined and updated each day in a systematic process. Our sensitivity analysis of underreporting and statistical-measurement noise suggests that we may slightly underestimate the attack rates in some countries, but this does not change our overall conclusions pertaining to Rt. However, even if the data were complete, our method cannot surmount the time lag between infections and deaths and can only fully identify trends in infections 2–3 weeks earlier. Extensions of our model could use case, hospitalization or intensive care data, but reconciling the different biases inherent in these sources while ensuring parsimony is challenging and would require additional assumptions.

The modern understanding of infectious disease, combined with a global publicized response, has meant that nationwide interventions could be implemented with widespread adherence and support. Given the observed infection fatality ratios and the epidemiology of COVID-19, major non-pharmaceutical interventions have had an effect in reducing transmission in all of the countries we have considered. In all countries in this study, we find that these interventions have reduced Rt below 1, and have contained their epidemics at the current time. When looking at simplistic counterfactual models over the whole epidemic, the number of potential deaths averted is substantial. We cannot say for certain that the current measures will continue to control the epidemic in Europe; however, if current trends continue there is reason for optimism.



Our model uses daily consolidated death data from the ECDC for 11 European countries currently experiencing the COVID-19 epidemic: Austria, Belgium, Denmark, France, Germany, Italy, Norway, Spain, Sweden, Switzerland and the UK. The ECDC provides information on confirmed cases and deaths attributable to COVID-19. For population counts, we use the United Nations Population Division age-stratified counts18.

We also catalogue data on the nature and type of major non-pharmaceutical interventions. We looked at the government webpages from each country as well as their official public health webpages to identify the latest advice or laws being issued by the government and public health authorities. We collected the following: school closure ordered; case-based measures; public events banned; social distancing encouraged; lockdown decreed; and the time of the first and last intervention. A full list of the timing of these interventions and the sources we have used is provided in the Supplementary Notes, Supplementary Table 2.

By using the ECDC data, we rely on a consolidated data source compiled by the ECDC, who include many sources of data each day, constantly refining and updating data using a comprehensive and systematic process. However, despite the rigorous protocols, countries may vary in the specifics of the data that they report to the ECDC. For example, there is variation in reporting (that is, community versus hospital) and time lags. Despite these issues, we use ECDC data to ensure as much consistency as possible across all countries.


A visual summary of our model is presented in Extended Data Fig. 3; details are provided in the Supplementary Methods.

We fit our model to observed deaths according to ECDC data from 11 European countries. The modelled deaths are informed by an infection-to-death distribution (Supplementary Fig. 1; derived from assumptions about the time from infection to the onset of symptoms and about the time from the onset of symptoms to death), and the population-averaged infection fatality ratio (adjusted for the age structure and contact patterns of each country, as discussed in the Supplementary Methods, Supplementary Table 3).

Given these distributions and ratios, modelled deaths are a function of the number of infections. The number of infections is modelled as the product of Rt with a discrete convolution of the previous infections. Individual components of this convolution sum are weighted by the generation time distribution (the average time from the infection of one person to the time at which they infect another; Supplementary Fig. 2). In our work, we approximate the generation time distribution using the serial interval distribution. Rt is a function of the initial Rt before interventions and the effect sizes from interventions, in which interventions are modelled as piecewise constant functions.

Following the Bayesian hierarchy from bottom to top gives us a full framework to see how interventions affect infections, which can result in deaths. A schematic of our model is shown in Extended Data Fig. 3. To maximize the ability to observe the effect of interventions on deaths, we fit our model jointly for all 11 European countries, and use partial pooling of information between countries with both individual and shared effects on Rt. Partial pooling operates on the last intervention, which is—in most cases—lockdown. The effect of partial pooling can be seen in Supplementary Discussion 12, Supplementary Fig. 29. We chose a balanced prior that encodes the prior belief that interventions have an equal chance of having an effect or not, and ensure a uniform prior on the joint effect of all interventions (Supplementary Fig. 3). We evaluate the effect of our Bayesian prior distribution choices and evaluate our Bayesian posterior calibration to ensure our results are statistically robust.

We perform extensive model validation and sensitivity analyses. We validate our model by cross-validation over a 14-day period (Supplementary Discussion 1, Supplementary Table 1) and we show the fits for holdout samples in Supplementary Figs. 515. We check the convergence of the Markov chain Monte Carlo sampler (Supplementary Fig. 4). We consider the sensitivity of our estimates of Rt to the mean of the generation distribution (Supplementary Discussion 3, Supplementary Figs. 16, 17). We further show that the choice of generation distribution does not change our counterfactual conclusions (Supplementary Fig. 18). Using univariate analyses and uninformative priors, we find (Supplementary Fig. 19) that all effects on their own serve to decrease Rt (Supplementary Discussion 4). We compare our model to a non-parametric Gaussian Process model (Supplementary Discussion 5). To assess the effect of individual countries on the results, we perform a ‘leave one country out’ sensitivity analysis (Supplementary Discussion 6, Supplementary Figs. 20, 21). To validate our starting values of Rt, we compare our model against an exponential-growth linear model (Supplementary Discussion 7, Supplementary Fig. 22). Instead of a joint analysis, we consider fits of our model to individual countries (Supplementary Discussion 8, Supplementary Figs. 2326). We perform a sensitivity analysis with respect to the onset-to-death distribution (Supplementary Discussion 9, Supplementary Fig. 27). We validate our probabilistic seeding scheme through an importance-sampling leave-one-out cross-validation (Supplementary Discussion 10). We consider a model extension with a constant, probabilistic under-reporting (Supplementary Discussion 11), finding that Rt does not change substantially (Supplementary Fig. 28).

Our model is different to other approaches (such as EpiEstim19) that use the discrete renewal equation. We use the renewal equation as a latent process to model infections and propose a generative mechanism to connect these infections to death data. Simply applying the renewal equation directly to death data requires positing a mechanism in which deaths in the past can cause future deaths (see, for example, ref. 20). In addition, for Rt, we are able to use a functional relationship in which non-pharmaceutical interventions can have a direct effect on Rt.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this paper.