# Modelling the Evolution of COVID-19 in High-Incidence European Countries and Regions: Estimated Number of Infections and Impact of Past and Future Intervention Measures

## Abstract

**:**

_{t}) has been found in all studied countries and territories, as already suggested by the drop in the number of deaths over time. Interestingly, the impact of such major intervention measures seems to be the same in most of these countries. The model has also provided realistic estimates of the total number of infections, active cases and future outcomes. While the predictive capabilities of the model are much more uncertain before the peak of the outbreak, we could still reliably predict the evolution of the disease after a major intervention by assuming the subsequent reproduction number from the current study. A greater challenge is to foresee the long-term impact of softer intervention measures, but this model can estimate the outcome of different scenarios and help to plan changes for the implementation of control measures in a given country or region.

## 1. Introduction

## 2. Methods

#### 2.1. Data Collection

#### 2.2. Estimating New Infections and Deaths over Time

_{i}in a given day i is a function of the number of infections I

_{j}occurring in the previous j = 1…i−1 days, according to a previously calculated infection-to-death (ITD) probability distribution and an estimated infection fatality ratio (IFR) for each country (Equation (1)).

_{j}occurring in a given day j is a function of the number of infections I

_{k}in the previous k = 1…j−1 days, according to a serial interval (SI) distribution, and the reproduction number (R

_{t}) (Equation (2)).

_{t}) had an initial constant value R

_{0}(a parameter that was left to optimize during the fitting of the model independently for each country), and was assumed to change after an intervention to another value which was kept constant over time until a new intervention (the new R

_{t}value after a given intervention is also a parameter to optimize during the model fitting). For simplicity, the entire population was assumed to be susceptible to infection during model fitting, but for long-term predictions based on the resulting model parameters (see Section 3.5), the infected cases were assumed to be protected from further infection.

#### 2.3. Model Fitting

_{10}), given that the early stages of the epidemic in a given country might be dominated by infections that are not local. An exception was made in the case of small regions or countries with a low number of deaths, such as Iceland and La Rioja, in which observed deaths were included from the day of the first death (d

_{1}). Similarly, according to the original procedure [4], the initial infections of the model were assumed to be 30 days before this d

_{10}(or d

_{1}) day, starting with 6 consecutive days with the same number of infections, which was left as a parameter to be optimized in the fitting procedure. The initial infections for the first 6 days and the R

_{0}and R

_{t}values after each intervention were defined as parameters to optimize in the model. Fitting was done in the probabilistic programming language Stan, using an adaptive Hamiltonian Monte Carlo (HMC) sampler. Eight chains for 4000 iterations, with 2000 iterations of warmup and a thinning factor 4 were run. Running of 200 sampling iterations with 100 warmup iterations yielded very similar results in most of the cases, suggesting that convergence was achieved early in the fitting process. See more details in the original description of the model [4]. The original code is available at https://github.com/ImperialCollegeLondon/covid19model/releases/tag/v1.0.

_{t}is not necessarily the same in all areas. To simplify the model, here the number of interventions were reduced, and the possibility of having an end date for any intervention was added. The inclusion of only one intervention was found to be sufficient to explain the data, while the addition of further intervention steps did not significantly improve the fitting.

#### 2.4. Predictive Model from a Given Set of Parameters

#### 2.5. Estimating Active Cases from Model Predictions

_{i}in a given day i is a function of the number of the estimated new infections I

_{j}occurring in the previous j = 1…i−1 days, according to a previously calculated infection-to-recovery (ITR) probability distribution, after discounting the percentage of new infected cases with outcome of death from infection fatality ratio (IFR) (Equation (3)).

## 3. Results

#### 3.1. Model Suggests a Significant Impact of Intervention Measures on Disease Transmission

_{t}after major intervention is similar in all analyzed countries (except in Iceland and in La Rioja when elderly residences with reported cases were excluded), with mean values ranging from 0.57 (La Rioja region) to 0.71 (Germany), and an averaged value of 0.625. Interestingly, a recent calculation on the reproduction number (R) in Germany, estimated from a nowcasting approach on reported COVID-19 cases with illness onset up to three days before data closure, provides a current estimate of R = 0.71 (95% prediction interval: 0.59–0.82) [11], which is virtually the same as the one calculated here with the disease transmission model based on the reported deaths. Regarding the relative values of R

_{t}after intervention (in percentage relative to R

_{0}before intervention), they ranged from 12.0% (Spain) to 20.7% (Italy). These values seem to depend not only on the effectiveness of the intervention measures, but also on the evolution of the disease prior to intervention, described by R

_{0}, which seems to be different in each country (Table 3). As a warning note, this effect in relative terms was the one assumed to be constant for the same type of intervention in the different countries in the original application of the model [4], an assumption that does not seem to be valid.

_{0}values or to a delay in the initial number of infections at the early stage of the epidemic outbreak, but this is beyond this study.

#### 3.2. Estimated Number of Total Infections and Active Cases

#### 3.3. Predicting Discrete Distributions of Infections and Deaths

#### 3.4. The Reliability of the Predictions Depends on the Stage of the Epidemic Outbreak

_{t}values obtained after removing larger sets of dates increasingly deviate from the ones obtained with the entire set of dates. Similarly, the number of deaths predicted for the last week of data increasingly deviate from the reported ones when larger sets of dates are removed. However, the response of the model to the removal of data is different in the two cases analyzed here. In the case of Spain, the mean value for the predicted deaths in the last week does not dramatically deviate from the real one even when three weeks of data are removed, although the uncertainty increases (the 95% CI range increases with respect to the ones obtained with the entire set of dates). We can safely say that three weeks ago the model would have been able to reasonably predict today’s situation in Spain (as of 5 May 2020). In the case of La Rioja, the predictions get worse much faster upon removal of data, with larger deviations of R

_{t}and a worse prediction of deaths for the last week of reported data. Perhaps this different behavior of the model with shorter data is related to specific features of the disease evolution in each territory, or maybe the reasons can be found in the differences in sample size between Spain and La Rioja (i.e., removing dates from regions with an already small number of reported cases can introduce a larger uncertainty).

_{0}values are not dramatically different, the resulting R

_{t}values are completely wrong, which indicates that the model cannot estimate the impact of the intervention on the reproduction number before achieving the peak of the outbreak. This was actually the situation of the original study on 11 countries on 28 March [4], when the peak of the outbreak was still >1 week away in the majority of analyzed countries. Consistent with the validation test here, in those conditions (>1 week before the peak) the model underestimated the impact of the different intervention measures and predicted a much higher number of infections and death than the reported ones in the following days.

_{t}(after a major intervention) seem to be quite consistent across all countries (except Iceland), with mean values between 0.57 and 0.71, and an average value of 0.625. Considering this, the model was fitted again to the reduced set of data (up to one week before the peak), but this time assuming a locked value of R

_{t}= 0.625. Remarkably, in these conditions, the predictions are actually quite good (Figure 6F,L,R) and are indeed comparable to those obtained with the entire set of data. This validation test suggests that in cases in which the epidemic outbreak has not yet clearly passed its peak, especially when the available data is noisy (e.g., small sample size), it could be a better option to apply the model using a guess value of R

_{t}= 0.625 rather than trying to predict such an R

_{t}value by fitting.

#### 3.5. Modelling Long-Term Disease Progression in Different Scenarios

_{t}(very unlikely); (ii) slight increment to R

_{t}= 0.71 similar to the current situation in Germany (also quite unlikely given the activities that are planned to be allowed); (iii) further increment to R

_{t}= 1.0, which implies a doubling of the number of patients actively transmitting the disease (this could be a likely scenario for the period between 11 May and 22 June); or (iv) a much larger increment to R

_{t}= 1.8, similar to the situation in Iceland at the beginning of the epidemic, e.g., normal activities allowed but with extensive testing and isolation of detected infections (this is a likely scenario after 22 June). All these options have been considered for Spain and La Rioja, and the results are shown in Supplementary Materials Figures S2 (Spain), S3 (La Rioja) and S4 (La Rioja, excluding data from elderly residences). There is a further possible scenario that we can evaluate, which is a possible full return to the pre-pandemic period in 1 September 2020, with open schools, usual sports and cultural shows, etc., so this variable has also been considered in Supplementary Materials Figures S5 (Spain), S6 (La Rioja) and S7 (La Rioja, excluding data from elderly residences). We can see that in some of these scenarios, the possibility of a new epidemic outbreak is very clear. Figure 7 shows in more detail the predicted evolution for Spain in La Rioja (with and without elderly retirement home data) in a likely scenario, with R

_{t}= 1.0 between 11 May and 22 June and R

_{t}= 1.8 for the period afterwards. Assuming these reproduction numbers, we can foresee an outbreak in September/October. Obviously, the model does not consider possible intervention measures to be taken to limit the impact of this hypothetical outbreak, which might depend on how early the potential new infections could be detected.

_{t}in Equation (1)). However, in a situation such as the one in La Rioja where the cases inside the elderly retirement homes are reasonably isolated from the rest of the population and outside elderly residences the virtual totality of active cases seem to be detected and is thus unlikely to induce new infections, the effective R

_{t}values on the analyzed dates could be much lower than those in the hypothetical scenarios discussed here, which would suggest a more optimistic situation in the upcoming months. In any case, after 22 June, with travels between provinces in Spain allowed, it will be essential to be able to detect any new focus of infection that might induce a sudden outbreak. For comparison, assuming the same scenario in Iceland (R

_{t}= 1.0 between 11 May and 22 June 2020 and R

_{t}= 1.8 afterwards), no outbreak is predicted in the studied period, even when assuming a full return to pre-pandemic conditions on 1 September 2020 (data not shown). The model can thus be useful for evaluating the long-term impact of the implementation and/or removal of intervention measures on the disease evolution in a given country or region.

## 4. Discussion

_{0}is smaller than in other countries, probably because of a better control of the first detected cases thanks to a higher detection rate. Inclusion of a more dynamic R

_{t}may lead to significant improvements of the model, but also to increased noise in the fitting process unless more data can be considered.

## 5. Conclusions

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Probability distributions used in this work for (

**a**) serial interval, (

**b**) infection-to-death and (

**c**) infection-to-recovery.

**Figure 2.**Examples of random discrete distributions of samples of n = 1, 10, 100, 1000 expected deaths, according to the infection-to-death (ITD) probability distribution used in this work. For samples of a small size, the distribution can be very different from the probability curve shown in Figure 1b, while for samples of a larger size (e.g., 100 or 1000 deaths), the discrete distributions get closer to the probability distribution curve.

**Figure 3.**Estimates of infections and deaths over time after independent application of the model to each country and region. For each country, the left plot shows the predicted daily infections as compared with the reported ones, and the right plot shows the expected daily deaths as compared with the observed ones. Expected values are shown as blue bands: light blue (95% CI), dark blue (50% CI) and line (median). Observed values are shown as brown bars. The estimates of R

_{t}before and after intervention resulting from the model are the ones in Table 3. In all cases, after intervention, R

_{t}is significantly reduced to a value well below 1 and the number of new infections decreases. In the La Rioja second panel, daily observed deaths do not include those from elderly retirement homes, but daily reported cases are the total ones, since the daily reported cases outside elderly residences was not available for this work.

**Figure 4.**Estimated cumulative infections and active cases from an independent application of the model to each country and region. For each country, the left plot shows the estimated cumulative infections for each day as compared with the reported ones, and the right plot shows the expected active cases for each day (in some countries, they are compared with the reported ones). Expected values are shown as blue bands: light blue (95% CI), dark blue (50% CI) and line (median). Observed values are shown as brown bars. The estimates of R

_{t}before and after intervention resulting from the model are the ones in Table 3. In all cases, after intervention, R

_{t}is significantly reduced to a value well below 1 and the number of new infections decreases. In the La Rioja second panel, estimates are derived from observed deaths after excluding those from elderly retirement homes, but accumulated infections and active cases over time are the total ones, since the daily cases excluding elderly retirement homes was not available for this work.

**Figure 5.**Instances of discrete distributions of the estimated number of daily infections and deaths randomly sampled according to the serial interval and infection-to-death probability distributions, based on randomly selected sets of reproduction numbers and initial infections among the ones obtained by the fitted model (black lines), as compared to the reported data (brown bars). Each row represents a country, with two instances of discrete distributions. When the number of daily infections and deaths is large (e.g., Spain), the predicted discrete distributions are closer to the continuous ones (Figure 3), but when these numbers are smaller (e.g., La Rioja and Iceland), the discrete distributions resemble better the rough distribution of reported data over time. In the La Rioja second panel, observed deaths do not include those from elderly retirement homes, but daily reported cases are the total ones, since daily cases excluding elderly retirement homes was not available for this work.

**Figure 6.**Estimated deaths for Spain (upper row) and La Rioja (total data: middle row; excluding elderly residences: bottom row) derived from the model (

**A**,

**G**,

**M**) in comparison to the predictions obtained after removing the last week (

**B**,

**H**,

**N**), last 2 weeks (

**C**,

**I**,

**O**), last 3 weeks of data (

**D**,

**J**,

**P**) or keeping data just up to 1 week to the peak (

**E**,

**K**,

**Q**), the latter also after keeping constant R

_{t}= 0.625 (

**F**,

**L**,

**R**).

**Figure 7.**Forecast for daily deaths in Spain and La Rioja over the upcoming months (total data, after excluding data from elderly residences) assuming R

_{t}= 1.0 between 11 May and 22 June 2020 and R

_{t}= 1.8 afterwards.

Country/Region | Total Detected Cases | Cases per 100 k People | Total Reported Deaths |
---|---|---|---|

Spain | 219,329 | 469 | 25,613 |

Italy | 211,938 | 351 | 29,079 |

UK | 190,584 | 287 | 28,734 |

Germany | 163,860 | 198 | 6831 |

France | 131,863 | 197 | 25,201 |

La Rioja (Spain) | 3969 | 1256 | 336 |

La Rioja (Spain) ^{1} | 2988 | 952 | 143 |

Iceland | 1799 | 509 | 10 |

^{1}Excluding data from elderly retirement homes with reported cases. Reported deaths (excluding those in elderly retirement homes) correspond to 5 May 2020, but detected cases outside elderly retirement homes have been extrapolated from data on 15 May 2020.

Country/Region | Time of Intervention | Days from First 100 Cases to Intervention Time | IFR |
---|---|---|---|

Spain | 14 March | 11 | 0.926% |

Italy | 11 March | 16 | 1.090% |

UK | 24 March | 18 | 0.919% |

Germany | 22 March | 20 | 1.093% |

France | 17 March | 15 | 1.153% |

La Rioja | 14 March | 5 | 0.926% |

Iceland | 24 March | 8 | 0.556% |

**Table 3.**Estimated reproduction numbers (before and after intervention) and initial number of infections resulting from the model (shown mean values, with 95% credible interval).

Country/Region | R_{0} | R_{t} after Intervention | Estimated Infections in First 6 Days |
---|---|---|---|

Spain | 4.82 (4.18–5.51) | 0.58 (0.52–0.65) | 396 (153–819) |

Italy | 3.14 (2.93–3.38) | 0.65 (0.60–0.70) | 623 (370–964) |

UK | 3.60 (3.26–3.95) | 0.60 (0.50–0.70) | 749 (396–1276) |

Germany | 3.68 (2.91–4.57) | 0.71 (0.54–0.89) | 314 (70–876) |

France | 4.47 (3.93–5.06) | 0.64 (0.55–0.74) | 113 (43–242) |

La Rioja | 3.29 (2.41–4.48) | 0.57 (0.45–0.70) | 72 (8–239) |

La Rioja ^{1} | 2.55 (2.06–3.34) | 0.41 (0.21–0.59) | 123 (27–279) |

Iceland | 1.84 (1.37–2.38) | 0.26 (0.01–0.69) | 75 (24–162) |

^{1}Excluding data from elderly retirement homes with reported cases.

**Table 4.**Estimated cumulative infections as of 5 May 2020 (shown median values with 95% credible interval).

Country/Region | Estimated Total Infections | Detection Rate | % Population Infected | Estimated Active Cases |
---|---|---|---|---|

Spain | 2990K (2742K–3269K) | 7.3% (6.7–8.0%) | 6.4% (5.9–7.0%) | 415K (329K–538K) |

Italy | 3094K (2868K–3358K) | 6.9% (6.3–7.4%) | 5.1% (4.7–5.6%) | 445K (365K–551K) |

UK | 3800K (3406K–4292K) | 5.0% (4.4–5.6%) | 5.6% (5.0–6.3%) | 998K (739K–1369K) |

Germany | 793K (641K–1024K) | 20.6% (16.0–25.6%) | 0.9% (0.8–1.2%) | 245K (142K–451K) |

France | 2351K (2089K–2693K) | 5.6% (4.9–6.3%) | 3.6% (3.2–4.1%) | 482K (341K–715K) |

La Rioja | 38,505 (32,850–45,155) | 10.3% (8.8–12.1%) | 12.2% (10.4–14.3%) | 4746 (2821–8088) |

La Rioja ^{1} | 16,205 (13,095–19,972) | 18.4% (15.0–22.8%) | 5.2% (4.2–6.4%) | 945 (464–2059) |

Iceland | 2029 (1669–2647) | 88.7% (68.0–100%) | 0.6% (0.5–0.7%) | 141 (78–605) |

^{1}Excluding data from elderly retirement homes with reported cases.

**Table 5.**Reproduction number and expected deaths during the last week of reported data (29 April–5 May) after removing different sets of dates from the model.

Country/Region | Forecast Start Time | R_{0} | R_{t} | Deaths 29 April–5 May |
---|---|---|---|---|

Spain | 1791 (real) | |||

5 May (last day, original model) | 4.82 (4.18–5.51) | 0.58 (0.52–0.65) | 1653 (1386–1980) | |

28 April (1 week to last) | 4.87 (4.20–5.63) | 0.57 (0.48–0.65) | 1562 (1178–2048) | |

21 April (2 weeks to last) | 4.88 (4.13–5.69) | 0.54 (0.40–0.70) | 1470 (853–2466) | |

14 April (3 weeks to last) | 4.90 (4.03–5.84) | 0.50 (0.23–0.78) | 1344 (430–3435) | |

27 March (1 week to peak) | 4.08 (3.28–5.06) | 2.99 (0.78–4.37) | off-limits (2972–off) | |

27 March (1 week to peak) locked R_{t} | 4.56 (3.57–5.66) | 0.625 | 1864 (1223–2666) | |

La Rioja | 10 (real) | |||

5 May (last day, original model) | 3.29 (2.41–4.48) | 0.57 (0.45–0.70) | 19 (13–28]) | |

28 April (1 week to last) | 3.00 (2.33–4.17) | 0.71 (0.55-0.88) | 31 (18–50) | |

21 April (2 weeks to last) | 2.81 (2.24–3.83) | 0.84 (0.61–1.10) | 52 (22–111) | |

14 April (3 weeks to last) | 2.72 (2.17–3.74) | 0.98 (0.54–1.44) | 108 (16–358) | |

28 March (1 week to peak) | 2.76 (2.22–3.69) | 2.29 (0.80–3.35) | off-limits (37-off) | |

28 March (1 week to peak) locked R_{t} | 2.86 (2.22–4.00) | 0.625 | 19 (11–29) | |

La Rioja (excluding elderly residences) | 2 (real) | |||

5 May (last day, original model) | 2.55 (2.06–3.34) | 0.41 (0.21–0.59) | 5 (2–8) | |

28 April (1 week to last) | 2.46 (1.97–3.15) | 0.51 (0.29–0.72) | 7 (3–13) | |

21 April (2 weeks to last) | 2.45 (1.93–3.14) | 0.58 (0.28–0.87) | 10 (3–24) | |

14 April (3 weeks to last) | 2.44 (1.87–3.12) | 0.58 (0.14–1.10) | 13 (2–57) | |

28 March (1 week to peak) | 2.53 (2.04–3.34) | 1.88 (0.32–2.88) | off-limits (3–off) | |

28 March (1 week to peak) locked R_{t} | 2.61 (2.06–3.59) | 0.625 | 13 (8–20) |

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**MDPI and ACS Style**

Fernández-Recio, J.
Modelling the Evolution of COVID-19 in High-Incidence European Countries and Regions: Estimated Number of Infections and Impact of Past and Future Intervention Measures. *J. Clin. Med.* **2020**, *9*, 1825.
https://doi.org/10.3390/jcm9061825

**AMA Style**

Fernández-Recio J.
Modelling the Evolution of COVID-19 in High-Incidence European Countries and Regions: Estimated Number of Infections and Impact of Past and Future Intervention Measures. *Journal of Clinical Medicine*. 2020; 9(6):1825.
https://doi.org/10.3390/jcm9061825

**Chicago/Turabian Style**

Fernández-Recio, Juan.
2020. "Modelling the Evolution of COVID-19 in High-Incidence European Countries and Regions: Estimated Number of Infections and Impact of Past and Future Intervention Measures" *Journal of Clinical Medicine* 9, no. 6: 1825.
https://doi.org/10.3390/jcm9061825