Abstract
In response to the COVID-19 pandemic caused by SARS-CoV-2, governments have adopted a wide range of non-pharmaceutical interventions (NPI). These include stringent measures such as strict lockdowns, closing schools, bars and restaurants, curfews, and barrier gestures such as mask-wearing and social distancing. Deciphering the effectiveness of each NPI is critical to responding to future waves and outbreaks. To this end, we first develop a dynamic model of the French COVID-19 epidemics over a one-year period. We rely on a global extended Susceptible-Infectious-Recovered (SIR) mechanistic model of infection that includes a dynamic transmission rate over time. Multilevel data across French regions are integrated using random effects on the parameters of the mechanistic model, boosting statistical power by multiplying integrated observation series. We estimate the parameters using a new population-based statistical approach based on a Kalman filter, used for the first time in analysing real-world data. We then fit the estimated time-varying transmission rate using a regression model that depends on the NPIs while accounting for vaccination coverage, the occurrence of variants of concern (VoC), and seasonal weather conditions. We show that all NPIs considered have an independent significant association with transmission rates. In addition, we show a strong association between weather conditions that reduces transmission in summer, and we also estimate increased transmissibility of VoC.
Funding source: PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d’Aquitaine (see https://www.plafrim.fr ).
Acknowledgement
The authors thank the opencovid-19 initiative for their contribution to the opening of the data used in this article. This work is supported in part by the Inria Mission COVID19, project GESTEPID. The authors sincerely thank Jane Heffernan for scientific discussions and thorough proofreading of the article. We also thank Linda Wittkop, Jane Heffernan, Quentin Clairon, Thomas Ferté, and Maria Pietro for constructive discussions about this work. Experiments presented in this paper were in part carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d’Aquitaine (see https://www.plafrim.fr).
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Author contribution: AC, BPH, PM, MP and RT designed the study. LL and CV analyzed the data. AC, PM and MP implemented the software code. AC, BPH and MP interpreted the results. AC, BPH, LL, PM and MP wrote the manuscript.
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Research funding: Part of the experiments presented in this paper were carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d’Aquitaine (see https://www.plafrim.fr).
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Appendix A: Supplementary figures for hospitalization, NPIs, variants of concern and vaccination data
Figure 8 represents the elsewhere published (SI-VIC database) and publicly available data on prevalence and incidence of hospitalization for COVID-19 in 12 non-insular French regions. Representation of NPIs in all regions is available in Figure 9. Finally, representations of the VoC proportion and the vaccination coverage ramping up (1st dose) in the population in each region over time are given in Figure 10.
Top: total number of hospital bed occupied per 100,000 inhabitants (100, 000/N × Y
H
). Bottom: daily number of new hospitalizations per 100,000 inhabitants
Implementation of NPIs in all regions. Differences are only impacting school closure and curfew by few days.
Top: Percentage of either UK, SA or BR VoC over time. Bottom: Percentage of people who have received a first dose of vaccine.
Appendix B: Estimation of the hospitalization period using the correlation between the total number of individual hospitalized daily and the daily incident number of hospitalization
The relation between the total number of individual hospitalized daily denoted by H and the daily incident number of hospitalization H in is governed by
with D H the hospitalization period.
Using the data of daily incident number of hospitalization
Estimation of D H for the 12 regions: Île-de-France (IDF); Centre-Val de Loire (CVL); Bourgone-Franche-Comté (BFC); Normandie (Nor); Hauts-de-France (HDF); Grand Est (GE); Pays de la Loire (PL); Bretagne (Bret); Nouvelle-Aquitaine (NA); Occitanie (Occ); Auvergne-Rhône-Alpes (AURA); Provence-Alpes-Côte d’Azur (PACA).
| IDF | CVL | BFC | Norm. | HDF | GE | PL | Bret. | NA | Occ. | AURA | PACA | Nat. avg. | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| D H (days) | 18.3 | 19.5 | 18.0 | 20.6 | 18.6 | 17.7 | 16.7 | 19.6 | 18.1 | 17.4 | 17.0 | 18.1 | 18.3 |
Appendix C: Computation of the effective reproductive ratio
To compute the reproductive ratio R eff of our SEIRAH model
we apply the Next Generation Matrix approach [76]. The principle consists in focusing on three categories: (i) latent E, (ii) ascertained infectious I and (iii) unascertained infectious A with the following dynamics
Then, we build two matrices corresponding to: (1) V following the arrivals and departures from one other category and (2) F following the arrivals from another compartment exterior to the three categories. We have
It is then well known – see for instance Perasso et al. [77] for a proof – that
where ρ(FV−1) is the spectral radius of the Next Generation Matrix FV−1. Here, we have
with
We therefore obtain
Appendix D: Initial transmission rate and attack rate estimated using our population-based Kalman filter
Table 4 shows the estimation of the initial values for the transmission rate at the regional level.
Estimation of b init for the 12 regions: Île-de-France (IDF); Centre-Val de Loire (CVL); Bourgone-Franche-Comté (BFC); Normandie (Nor); Hauts-de-France (HDF); Grand Est (GE); Pays de la Loire (PL); Bretagne (Bret); Nouvelle-Aquitaine (NA); Occitanie (Occ); Auvergne-Rhône-Alpes (AURA); Provence-Alpes-Côte d’Azur (PACA).
| IDF | CVL | BFC | Norm. | HDF | GE | PL | Bret. | NA | Occ. | AURA | PACA | Nat. avg. | |
| b init | 0.789 | 0.767 | 0.784 | 0.773 | 0.781 | 0.809 | 0.761 | 0.765 | 0.768 | 0.789 | 0.786 | 0.778 | 0.779 |
Appendix E: Comparison of obtained attack rates with other studies
Our attack rates are compared to (i) those obtained by Hoze et al. [63] (see Table 5), and to (ii) 3 seroprevalence studies [64–66] (see Table 6).
Comparison of the estimated attack rates obtained in Hoze et al. [63] (first line) with our estimated attack rates (second line) at 3 dates for the metropolitan France.
| May 11, 2020 | October 31, 2020 | January 15, 2021 | |
| Hoze et al. [63] | 5.7% [5.1%; 6.4%] | 11% [9.7%; 12.4%] | 14.9% [13.2%; 16.9%] |
| Proposed estimates | 5.69% [5.61%; 5.77%] | 12.78% [11.98%; 13.66%] | 18.92% [16.76%; 21.43%] |
Comparison of the estimated attack rates obtained in 3 seroprevalence studies.
| May 2 – June 2, 2020 | May 4 – June 23, 2020 | Oct. 5 – Oct. 11, 2020 | |
| Auvergne-Rhône-Alpes* | 4.8% | – | – |
| Auvergne-Rhône-Alpes** | 4.48% | 4.54% | 7.15% |
| Bourgone-Franche-Comté* | 1.5% | – | 9.3% |
| Bourgone-Franche-Comté** | 6.04% | 6.14% | 9.33% |
| Bretagne* | 3.1% | – | – |
| Bretagne** | 1.75% | 1.79% | 3.51% |
| Centre-Val de Loire* | 2.1% | – | – |
| Centre-Val de Loire** | 5.13% | 5.26% | 8.15% |
| Grand Est* | 6.7% | 9% | 11.6% |
| Grand Est** | 10.85% | 11.36% | 12.72% |
| Hauts-de-France* | 2.9% | – | – |
| Hauts-de-France** | 5.42% | 5.64% | 8.64% |
| Île-de-France* | 9.2% | 10% | 14.8% |
| Île-de-France** | 12.57% | 13.06% | 17.97% |
| Nouvelle-Aquitaine* | 2.% | 3.1% | – |
| Nouvelle-Aquitaine** | 1.51% | 1.52% | 3.06% |
| Normandie* | 1.9% | – | – |
| Normandie** | 2.28% | 2.31% | 5.25% |
| Occitanie* | 1.9% | – | – |
| Occitanie** | 1.71% | 1.74% | 5.17% |
| Provence-Alpes-Côte d’Azur* | 5.2% | – | – |
| Provence-Alpes-Côte d’Azur** | 4.49% | 4.63% | 8.55% |
| Pays de la Loire* | 3.4% | – | – |
| Pays de la Loire** | 2.62% | 2.69% | 4.01% |
Appendix F: Regression model
The model writes as follow for each region i ∈ 1, …, 12:
Regression residuals, fixed and random effects are for the selected model given in Eq. (7) are given in Table 7. Covariance matrix is given in Table 8.
Scaled residuals, random and fixed effects of regression model (7).
| Scaled residuals | |||||
|---|---|---|---|---|---|
| Min | 1Q | Median | 3Q | Max | |
| −3.9394 | −0.4817 | −0.0033 | 0.5609 | 10.5139 |
| Random effects | |||||
|---|---|---|---|---|---|
| Variance | Std. dev. | Corr | |||
| σ region | 0.005159 | 0.07183 | |||
| σ Lock1 | 0.106598 | 0.32649 | −0.43 | ||
| σ Lock2 | 0.018070 | 0.13443 | 0.35 | −0.52 | |
| σ Curf-6PM | 0.002827 | 0.05317 | −0.29 | 0.12 | −0.39 |
| Fixed effects | |||||
|---|---|---|---|---|---|
| Estimate | Std. error | Df | t value | Pr
|
|
| A | −0.40919 | 0.02595 | 25.34463 | −15.766 | 1.30e−14 |
| Lock1 | −1.52327 | 0.09631 | 11.76340 | −15.817 | 2.75e−09 |
| Post-Lock1.1 | −0.77153 | 0.02037 | 4620.46129 | −37.869 |
|
| Post-Lock1.2 | −0.65810 | 0.01427 | 4634.29686 | −46.120 |
|
| Lock2 | −0.76626 | 0.04312 | 13.69861 | −17.771 | 7.47e−11 |
| Lock2.1 | −0.71299 | 0.02169 | 4644.40638 | −32.876 |
|
| Closed schools | −0.07150 | 0.00866 | 4639.83981 | −8.256 |
|
| Closed bars & rest. | −0.10746 | 0.01492 | 4623.96873 | −7.201 | 6.93e−13 |
| Barrier gestures | −0.61300 | 0.01963 | 4652.90602 | −31.229 |
|
| Curf. 6 PM | −0.35386 | 0.02629 | 59.55020 | −13.461 |
|
| Curf. 8 PM | −0.32590 | 0.01951 | 4404.57885 | −16.707 |
|
| Variants | 0.19505 | 0.02759 | 4605.07217 | 7.071 | 1.77e−12 |
| Weather | −1.03117 | 0.04013 | 4654.84448 | −25.696 |
|
| Bar & rest.: weather | 0.11877 | 0.05706 | 4501.30235 | 2.082 | 0.0374 |
Covariance matrix of regression model (7).
| α | Lock1 | Post-Lock1 – 1 | Post-Lock1 – 2 | Lock2 | Lock2 – red. | Closed schools | |
| α | 6.7 × 10−4 | −9.9 × 10−4 | 8.6 × 10−5 | −9.0 × 10−6 | 3.7 × 10−4 | 1.1 × 10−4 | −4.2 × 10−5 |
| Lock1 | −9.9 × 10−4 | 9.3 × 10−3 | 1.2 × 10−4 | −1.6 × 10−6 | −1.9 × 10−3 | −1.9 × 10−5 | −1.5 × 10−5 |
| Post-Lock1.1 | 8.6 × 10−5 | 1.2 × 10−4 | 4.2 × 10−4 | 6.0 × 10−6 | 1.7 × 10−4 | 1.2 × 10−4 | −1.9 × 10−5 |
| Post-Lock1.2 | −9.0 × 10−6 | −1.6 × 10−6 | 6.0 × 10−6 | 2.0 × 10−4 | 2.0 × 10−6 | 2.1 × 10−6 | −4.2 × 10−6 |
| Lock2 | 3.7 × 10−4 | −1.9 × 10−3 | 1.7 × 10−4 | 2.0 × 10−6 | 1.9 × 10−3 | 2.1 × 10−4 | 3.0 × 10−5 |
| Lock2.1 | 1.1 × 10−4 | −1.9 × 10−5 | 1.2 × 10−4 | 2.1 × 10−6 | 2.1 × 10−4 | 4.7 × 10−4 | 2.6 × 10−5 |
| Closed schools | −4.2 × 10−5 | −1.5 × 10−5 | −1.9 × 10−5 | −4.2 × 10−6 | 3.0 × 10−5 | 2.6 × 10−5 | 7.5 × 10−5 |
| Closed bars & rest. | −5.6 × 10−5 | −1.5 × 10−4 | −1.9 × 10−4 | 1.5 × 10−5 | −1.5 × 10−4 | −1.2 × 10−4 | −1.7 × 10−5 |
| Barrier gestures | −2.7 × 10−4 | 2.3 × 10−4 | −1.2 × 10−4 | −8.1 × 10−6 | −1.4 × 10−4 | −1.8 × 10−4 | 2.9 × 10−5 |
| Curf. 6 PM | 1.1 × 10−5 | 2.1 × 10−4 | 1.7 × 10−4 | 2.7 × 10−6 | −1.7 × 10−5 | 2.4 × 10−4 | 2.1 × 10−5 |
| Curf. 8 PM | 1.3 × 10−4 | −4.2 × 10−7 | 1.4 × 10−4 | 5.0 × 10−6 | 1.9 × 10−4 | 2.5 × 10−4 | −1.7 × 10−5 |
| VoC | 3.7 × 10−5 | −5.7 × 10−5 | −5.3 × 10−5 | 1.1 × 10−6 | 1.0 × 10−5 | 6.7 × 10−5 | −1.9 × 10−5 |
| Weather | 2.5 × 10−4 | −2.7 × 10−4 | 1.8 × 10−4 | −2.6 × 10−5 | 1.4 × 10−4 | 1.9 × 10−4 | −1.4 × 10−4 |
| Bar & rest.: weather | −8.5 × 10−5 | −1.1 × 10−4 | −5.5 × 10−4 | 2.7 × 10−5 | −6.3 × 10−5 | 2.3 × 10−4 | 1.2 × 10−4 |
| Closed bars & rest. | Barrier gestures | Curf. 6 PM | Curf. 8 PM | Variants | Weather | Bar & rest.: weather | |
|---|---|---|---|---|---|---|---|
| A | −5.6 × 10−5 | −2.7 × 10−4 | 1.1 × 10−5 | 1.3 × 10−4 | 3.7 × 10−5 | 2.5 × 10−4 | −8.5 × 10−5 |
| Lock1 | −1.5 × 10−4 | 2.3 × 10−4 | 2.1 × 10−4 | −4.2 × 10−7 | −5.7 × 10−5 | −2.7 × 10−4 | −1.1 × 10−4 |
| Post-Lock1.1 | −1.9 × 10−4 | −1.2 × 10−4 | 1.7 × 10−4 | 1.4 × 10−4 | −5.3 × 10−5 | 1.8 × 10−4 | −5.5 × 10−4 |
| Post-Lock1.2 | 1.5 × 10−5 | −8.1 × 10−6 | 2.7 × 10−6 | 5.0 × 10−6 | 1.1 × 10−6 | −2.6 × 10−5 | 2.7 × 10−5 |
| Lock2 | −1.5 × 10−4 | −1.4 × 10−4 | −1.7 × 10−5 | 1.9 × 10−4 | 1.0 × 10−5 | 1.4 × 10−4 | −6.3 × 10−5 |
| Lock2.1 | −1.2 × 10−4 | −1.8 × 10−4 | 2.4 × 10−4 | 2.5 × 10−4 | 6.7 × 10−5 | 1.9 × 10−4 | 2.3 × 10−4 |
| Closed schools | −1.7 × 10−5 | 2.9 × 10−5 | 2.1 × 10−5 | −1.7 × 10−5 | −1.9 × 10−5 | −1.4 × 10−4 | 1.2 × 10−4 |
| Closed bars & rest. | 2.2 × 10−4 | 1.6 × 10−5 | −1.5 × 10−4 | −1.2 × 10−4 | 3.3 × 10−5 | 1.5 × 10−4 | 5.1 × 10−5 |
| Barrier gestures | 1.6 × 10−5 | 3.9 × 10−4 | −1.7 × 10−4 | −2.0 × 10−4 | −4.7 × 10−5 | −5.2 × 10−4 | 2.9 × 10−4 |
| Curf. 6 PM | −1.5 × 10−4 | −1.7 × 10−4 | 6.9 × 10−4 | 2.4 × 10−4 | −3.2 × 10−4 | 1.8 × 10−4 | −8.1 × 10−6 |
| Curf. 8 PM | −1.2 × 10−4 | −2.0 × 10−4 | 2.4 × 10−4 | 3.8 × 10−4 | 6.4 × 10−5 | 2.8 × 10−4 | 1.1 × 10−4 |
| VoC | 3.3 × 10−5 | −4.7 × 10−5 | −3.2 × 10−4 | 6.4 × 10−5 | 7.6 × 10−4 | 9.1 × 10−5 | 2.9 × 10−4 |
| Weather | 1.5 × 10−4 | −5.2 × 10−4 | 1.8 × 10−4 | 2.8 × 10−4 | 9.1 × 10−5 | 1.6 × 10−3 | −1.3 × 10−3 |
| Bar & rest.: weather | 5.1 × 10−5 | 2.9 × 10−4 | −8.1 × 10−6 | 1.1 × 10−4 | 2.9 × 10−4 | −1.3 × 10−3 | 3.3 × 10−3 |
Appendix G: Comparison with other regression models
In this part, we compare our regression model to other regression models. We start by considering a simple model neglecting the weather (Model 1) and then we consider a model integrating the weather variable but neglecting the interaction with the bars and restaurants (Model 2) and the selected model (Model 3). Model 4 corresponds to Model 3 but without considering the delay of 7 days after the lockdowns. Table 9 summarizes the results. Figure 11 shows the fits.
Estimation of the associations between the transmission rates and the weather, the VoCs and the NPIs. Negative (resp. positive) values correspond to a decrease (resp. an increase) of the transmission rate.
| NPI/variants/weather | Model 1 | Model 2 | Model 3 | Model 4 |
|---|---|---|---|---|
| Lockdown 1 (delay of 7 days) | −83% [−86%; −80%] | −78% [−82%; −74%] | −78% [−82%; −74%] | −65% [−71%; −58%] |
| Post lockdown 1 – phase 1 | −54% [−56%; −53%] | −53% [−54%; −51%] | −54% [−56%; −52%] | −45% [−49%; −40%] |
| Post lockdown 1 – phase 2 | −49% [−51%; −47%] | −48% [−50%; −47%] | −48% [−50%; −47%] | −48% [−50%; −46%] |
| Lockdown 2 (delay of 7 days) | −49% [−53%; −44%] | −53% [−57%; −49%] | −54% [−57%; −49%] | −41% [−46%; −35%] |
| Lockdown 2 with open. | −38% [−41%; −35%] | −51% [−53%; −49%] | −51% [−53%; −49%] | −54% [−57%; −51%] |
| shops | ||||
| Closing schools | −15% [−16%; −13%] | −7% [−9%; −6%] | −7% [−8%; −5%] | −3% [−6%; −1%] |
| Closing bars & restaurants | 4% [1%; 7%] | −10% [−13%; −8%] | −10% [−13%; −8%] | −24% [−29%; −19%] |
| Barrier gestures | −63% [−64%; −61%] | −46% [−48%; −44%] | −46% [−48%; −44%] | −36% [−40%; −31%] |
| Curfew at 6 PM | −18% [−22%; −13%] | −30% [−33%; −26%] | −30% [−33%; −26%] | −28% [−34%; −23%] |
| Curfew at 8 PM | −5% [−8%; −2%] | −28% [−31%; −25%] | −28% [−31%; −25%] | −33% [−37%; −28%] |
| Proportion of variants | 44% [36%; 52%] | 20% [14%; 27%] | 22% [15%; 28%] | 6% [−2%; 15%] |
| Seasonal weather conditions | – | −63% [−65%; −60%] | −64% [−67%; −61%] | −73% [−76%; −69%] |
| Weather cond./closing | – | – | 13% [1%; 26%] | −32% [−42%; −19%] |
| bars & rest. | ||||
| AIC | −707 | −1486 | −1485 | 2224 |
Results of regression models of the transmission rate obtained with the population based Kalman filter using Model 3 given in Eq. (7). Top-Left: Mean fit with linear model without considering weather effect (Model 1). Top-Right: Mean fit with linear model considering linear weather effect (Model 2). Bottom-Left: Mean fit with linear model with linear weather effect and with closed bar and restaurants and weather interaction (Model 3). Bottom-Right: Mean fit with linear model with linear weather effect and with closed bar and restaurants and weather interaction (Model 4) but without taking into account the delay of 7 days after the lockdowns. Random effects are considered on the intercept, the lockdowns and the curfew at 6 PM.
Using the first model (AIC = −707), we obtain a negative association between the closure of bars and restaurants and the transmission rates which is not realistic. This is due to the fact that the bars and restaurants were open during summer when the transmission rate was very low with a effective reproductive number inferior to 1 in all regions. That is why in a second model, we add the weather variable. The AIC of this model is larger superior to the first ones (AIC = −1486). The third model assumes that there is an interaction between the closures of bar and restaurants and the weather to take into account the use of terraces (which have been expanded in many places since the beginning of the pandemic). The AIC is similar to the second model (AIC = −1485). The AIC of Model 4 is very large (AIC = 2224) compared to other ones validating the delay of 7 days.
We found that the three VoCs (alpha, beta, and gamma) are
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