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Edited by: Zisis Kozlakidis, International Agency For Research On Cancer (IARC), France

Reviewed by: Lin Wang, University of Cambridge, United Kingdom; Gui-Quan Sun, North University of China, China

This article was submitted to Infectious Diseases - Surveillance, Prevention and Treatment, a section of the journal Frontiers in Medicine

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The COVID-19 epidemic was reported in the Hubei province in China in December 2019 and then spread around the world reaching the pandemic stage at the beginning of March 2020. Since then, several countries went into lockdown. Using a mechanistic-statistical formalism, we estimate the effect of the lockdown in France on the contact rate and the effective reproduction number _{e} of the COVID-19. We obtain a reduction by a factor 7 (_{e} = 0.47, 95%-CI: 0.45–0.50), compared to the estimates carried out in France at the early stage of the epidemic. We also estimate the fraction of the population that would be infected by the beginning of May, at the official date at which the lockdown should be relaxed. We find a fraction of 3.7% (95%-CI: 3.0–4.8%) of the total French population, without taking into account the number of recovered individuals before April 1st, which is not known. This proportion is seemingly too low to reach herd immunity. Thus, even if the lockdown strongly mitigated the first epidemic wave, keeping a low value of _{e} is crucial to avoid an uncontrolled second wave (initiated with much more infectious cases than the first wave) and to hence avoid the saturation of hospital facilities.

COVID-19 epidemic was reported in the Hubei province in China in December 2019 and then spread around the world reaching the pandemic stage at the beginning of March 2020 (

The basic reproduction number _{0} corresponds to the expected number of new cases generated by a single infectious case in a fully susceptible population (_{0} associated with the COVID-19 epidemic, leading to values from 1.4 to 6.49, with an average of 3.28 (_{0} can be interpreted as the product of the contact rate and of the duration of the infectious period, and since the objective of the lockdown and associated restriction strategies are precisely to decrease the contact rate, an important effect on the number _{e} of secondary cases generated by an infectious individual is to be expected. This value _{e} is often referred to as “effective reproduction number,” and corresponds to the counterpart of _{0} in a population that is not fully susceptible (_{e} > 1, the number of infectious cases in the population follows an increasing trend, and the larger _{e}, the faster this trend. On the contrary, if _{e} < 1, the epidemic will gradually die out. The control measures in China have been shown to have a significant effect on the COVID-19 epidemic, with growth rates that shifted from positive to negative values (corresponding to _{e} < 1) within 2 weeks (_{e}. Fitting a SEIR epidemic model to time series of reported cases from 31 provinces in China, Tian et al. (_{0} = 3.15 before the implementation of the emergency response in China, a value that was divided by more than 20 once the control measures were fully effective. Using contact surveys data for Wuhan and Shanghai it was estimated in Zhang et al. (

Standard epidemiological models generally rely on SIR (Susceptible-Infected-Removed) systems of ordinary differential equations and their extensions [for examples of application to the COVID-19 epidemic, see (

In a previous study (_{0} in France, and we found a value of 3.2 (95%-CI: 3.1–3.3). Although the number of tests at that stage was low, an advantage of working with the data from the beginning of the epidemic was that the initial state of the epidemic was known.

Here, we develop a new mechanistic-statistical approach, based on a SIRD model (

estimating the effect of the lockdown in France on the contact rate and the effective reproduction number _{e};

estimating the number of infectious individuals and the fraction of the population that has been infected by the beginning of May (at the official date at which the lockdown should be relaxed).

We obtained the number of positive cases and deaths in France, day by day from Santé Publique France (

The mechanistic-statistical framework consists in the combination of a mechanistic model that describes the epidemiological process, a probabilistic observation model and an inference procedure.

The dynamics of the epidemic are described by the following SIRD compartmental model:

with

The model is started at a date _{0} corresponding to April 1st. The initial number of infectious _{0}) = _{0} is not known and will be estimated. The total number of recovered at time _{0} is also not known. However, as the compartment _{0}) = 0, thereby considering only the new recovered individuals, starting from the date _{0}. We fixed _{0}) = 3523, the number of deaths at hospital by March 31. The initial _{0}) is between 65 and 67 million cases. For our computation, we assumed that _{0}) (at least when

The ODE system (1) was solved thanks to a standard numerical algorithm, using Matlab^{®}

The number of cases tested positive on day _{t} carried out on day _{t} the probability of being tested positive in this sample:

The tested population consists of a fraction of the infectious cases and a fraction of the susceptibles: _{t} = τ_{1}(_{2}(

with κ_{t}: = τ_{2}(_{1}(

Each day, the number of new observed deaths (excluding nursing homes), denoted by

Note that the time

The unknown parameters are α, γ, κ, and _{0}. We used a Bayesian method (

The likelihood _{t} is known, we get:

with _{i} the date of the first observation and _{f} the date of the last observation. In this expression _{0} through _{t} and

The posterior distribution corresponds to the distribution of the parameters conditionally on the observations:

where π(α, γ, κ, _{0}) corresponds to the prior distribution of the parameters (detailed below) and

Regarding the contact rate α, the initial number of infectious cases _{0} and the probability κ, we used independent non-informative uniform prior distributions in the intervals α ∈ (0, 1), _{g}, was obtained in Roques et al. (_{g} is depicted in _{0} at the beginning of the epidemic was known (equal to 1), and did not need to be estimated. Thus, we estimated in Roques et al. (

The numerical computation of the posterior distribution is performed with a Metropolis-Hastings (MCMC) algorithm, using 5 independent chains, each of which with 10^{6} iterations, starting from the posterior mode. To find the posterior mode we used the BFGS constrained minimization algorithm, applied to ^{®} function ^{®} codes are available as

Denote by ^{*}(^{*}(^{*}(^{*}(

The observation model (3) implies that the expected cumulated number of deaths on day ^{*}(

To assess model fit, we compared these expectations and the observations, i.e., the cumulated number of cases tested positive, _{0} the number of cases tested positive by March 31 (_{0} = 52, 128) and the cumulated number of deaths _{0} the number of reported deaths (at hospital) by March 31 (_{0} = 3123). The results are presented in

Expected number of observed cases and deaths associated with the posterior mode vs. number of cases actually detected (total cases). The blue curve corresponds to the expected number of cases tested positive ^{*}(_{0} is the number of cases tested positive on March 31 (_{0} = 52 128).

The pairwise posterior distributions of the parameters (α, _{0}), (α, γ), (α, κ), (γ, _{0}), (γ, κ), (κ, _{0}) are depicted as

The effective reproduction number can be simply derived from the relation _{e} = α/(β + γ) when _{e} is therefore easily derived from the marginal posterior distribution of the contact rate α (since we assumed β = 1/10; see section 2.2). It is depicted in _{e} of 0.47 (95%-CI: 0.45–0.50).

Posterior distribution of the effective reproduction number _{e} in France.

The marginal posterior distribution of _{0} indicates that the number of infectious individuals at the beginning of the considered period (i.e., April 1st) is 1.4·10^{6} (95%-CI: 1.1·10^{6} − 1.8·10^{6}). The computation of the solution of (1) with the posterior distribution of the parameters leads to a number of infectious ^{5} infectious cases (95%-CI: 1.3·10^{5} − 2.1·10^{5}) and (^{6} infected cases including recovered (95%-CI: 2.0·10^{6} − 3.2·10^{6}). The dynamics of the distributions of

Distribution of the number of infectious cases

Many studies focused on the estimation of the basic reproductive number _{0} of the COVID-19 epidemic, based on data-driven methods and mathematical models [e.g., (_{0} was about 3.3. We focused here on an observation period that began after the lockdown was set in France.

We obtained an effective reproduction number that was divided by a factor 7, compared to the estimate of the _{0} carried out in France at the early stage of the epidemic, before the country went into lockdown [a value _{0} = 3.2 was obtained in (_{e} = 0.47 is significantly below the threshold value 1 were the epidemic starts dying out.

The decay in the number of infectious cases can also be observed from our simulations. It has to be noted that, although the number of infectious cases is a latent, or “unobserved” process, the mechanistic-statistical framework allowed us to estimate its value (_{0} = 3.2, the herd immunity threshold, corresponding to the minimum fraction of the population that must have immunity to stop the epidemic, would be 1 − 1/_{0}≈69% [a threshold of 80% was proposed in (_{e} would approach the initially estimated value of _{0}, and the second wave would start with about 1.6·10^{5} infectious individuals (in comparison with the few cases that initiated the first wave in France) and about 64·10^{6} susceptible individuals. Keeping a low value of _{e} is therefore crucial to avoid the saturation of hospital facilities.

We deliberately chose a parsimonious mechanistic model with a few parameters to avoid identifiability issues. Possible extensions include stage-structured models, where the infectious class _{e} and the number of susceptible cases. Although herd immunity is far from being reached at the country scale, it is likely that the fraction of immune individuals strongly varies over the territory, with possible local immunity effects [e.g., by April 4 the proportion of people with confirmed SARS-CoV-2 infection based on antibody detection was of 41% in a high-school located in Northern France (

Publicly available datasets were analyzed in this study. This data can be found here:

LR, EK, JP, AS, and SS conceived the model and designed the statistical analysis. LR and SS wrote the paper. LR carried out the numerical computations. All authors reviewed the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This manuscript has been released as a pre-print at medRxiv (

The Supplementary Material for this article can be found online at: