Estimating the impact of public health strategies on the spread of SARS‐CoV‐2: Epidemiological modelling for Toulouse, France

Summary The spread of severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2) and the resulting disease COVID‐19 has killed over 2 million people as of 22 January 2021. We have used a modified susceptible, infected, recovered epidemiological model to predict how the spread of the virus in France will vary depending on the public health strategies adopted, including anti‐COVID‐19 vaccination. Our prediction model indicates that the French authorities' adoption of a gradual release from lockdown could lead in March 2021 to a virus prevalence similar to that before lockdown. However, a massive vaccination campaign initiated in January 2021 and the continuation of public health measures over several months could curb the spread of virus and thus relieve the load on hospitals.

strategies on the resurgence of the epidemic, based on the strategy adopted by France. We also evaluated the potential impact of the vaccination campaign that was launched on 1 January 2021 on the spread of the virus.

| Statistical model
Earlier models for SARS-CoV-2 were based on published positive cases and do not take into account the patients' ages or any evolutive diffusion coefficient. 5,6 That is probably why the Johns Hopkins University predictive model underestimated the spread of the virus in Italy and overestimated its spread in France and the United Kingdom. Our model is a discretized version of a susceptible, infectious and recovered (SIR)-type model. 7 These compartmental models are well suited to studies of the spread of SARS-CoV-2 in different populations. 8,9 Our model includes a diffusion/transmission coefficient R 0 that varies with the likelihood of contagion, and two reduction coefficientsĉandq that describe the impact of public health measures on virus transmission. The model predicts how the SARS-CoV-2 virus would have evolved and projects the daily percentage of new positive cases. By cumulative effect, we therefore obtain a projection of the seroprevalence of SARS-CoV-2 in France.
We have used four variables ðS n ; P n ; Q n ; I n Þ, where S n is the number of healthy people on day n, and P i n is the number of undetected contagious carriers infected for i days ð1 ≤ i ≤ N T Þ. Similarly, Q i n is the number of detected contagious carriers infected for i days ð1 ≤ i ≤ N T Þ on day n; and I n is the number of people who were immunized. We assume that the risk of reinfection by SARS-CoV-2 after a first infection is negligible. N T is the number of days a person is contagious, and α is the percentage of the population tested each day. R 0 is the number of healthy people who a contagious person contacts and infects. We assume that R 0 varies over time and peaks when the virus load is maximal: 7 days after the start of infection. 10,11 We assume that the number of days a person is contagious is equal to the times of infection, that is, 20 days. 10,12 For N is the total population at the start of the epidemic phase,ĉ is the multiplier for the pace of the epidemic throughout public health restriction phases ð0 ≤ĉ ≤ 1Þ andq is the same multiplier during the quarantine period ð0 ≤q ≤ 1Þ.ĉ andq are set at 1 when there is no restriction or quarantine. The lower the values ofĉ orq, the greater the constraint which is applied to halt the spread of the virus. Some values ofĉ and the value ofq have been estimated in previous works by correcting the values predicted by the model using real data collected by the Toulouse Virology Laboratory. 3,13 ∀ n ∈ ⟦ d þ 1; þ∞⟧ĉ is defined asĉ ¼ argmin c jP n − P n ðcÞj n ∈ ⟦1; d⟧ And ∀ n ∈ ⟦ d 0 þ 1; þ∞⟧q is defined as :q ¼ argmin q jP n − P n ðqÞj n ∈ ⟦1; d 0 ⟧ We used data collected by the Toulouse Virology Laboratory from March 2020 to June 2020 to setq to 0.05. 13 The values ofĉ varied according to the public health restriction measures implemented in the Toulouse area. 3 N is given by On transition from day n to day n þ 1, we have According to Equation (1), the number of undetected contagious carriers on day n þ 1 is the number of untested, undetected carriers who were infected but not detected on day n.
According to Equation (2), the number of new undetected contagious carriers on day n þ 1 is the number of healthy people who were infected by undetected carriers at any stage of infection or by detected carriers at any stage of infection on day n.
According to Equation (3), the number of immunized people on day n þ 1 corresponds to the number of people immunized on day n plus the people who were on their last day of infection on day n, whether or not they were tested. Q 1 nþ1 ¼ 0 (no quarantine on day one, test results needed) According to Equation (4), the number of detected contagious carriers on day n + 1 is defined as the number of detected contagious carriers on day n plus the number of tested, but undetected contagious carriers on day n.
We set R 0 ¼ 2:2 at its peak, based on a national and regional French study, 14 and the international evaluations of the WHO. 15

| Study population
We estimated the initial model settings using data collected by the Toulouse Virology Laboratory ( Table 1). The total number of tests The percentage of new cases of SARS-CoV-2 per day was predicted using the initial parameters (Table 1). This estimation was based on the number of cases on the previous day and a contagion parameter ðR i 0 Þ that varied according to the day of infection, and on the administration parameters (quarantine, lockdown or restriction phases). We assumed that the COVID-19 vaccination campaign began on 1 January 2021. The theoretical efficacy of vaccination was set at 94% as stated by the Pfizer trial, 17 and we consider a subject to be immunized 7 days after the second dose (i.e., 28 days after the first injection).

| RESULTS
The match between the values predicted by the model and the values observed from 6 November 2020 to 1 January 2021 is given by

| Consequences of SARS-CoV-2 vaccination
As the SARS-CoV-2 epidemic will resume, despite any or all the measures adopted by the French authorities, mass vaccination must be rapidly introduced to limit virus spread and control the COVID-19 epidemic.
The release measures taken during the last few months are likely to result in 7% of the urban Toulouse population being seropositive at the end of January, rising to 7.8% at the end of

| Consequences of relaxing public health measures
As proactive vaccination strategies could keep the spread of SARS-CoV-2 under control, we assessed the date on which public health measures such as wearing masks could be relaxed without a resumption of the epidemic.
If an active vaccination campaign designed to control the spread of SARS-CoV-2 with 7500 people vaccinated per week (black curve, Figure 3) is adopted, it would be necessary to wait until 1 August 2021 before masks could be removed without a strong increase in the percentage of people testing positive (green curve, Figure 4a). If masks are removed on July 1 (blue curve, Figure 4a) or July 14 (red curve, Figure 4a), the rate of positive tests could become similar to those that led to the second lockdown on August 7 or early October.
An even more massive vaccination campaign (grey curve, Figure 3), with 8500 people vaccinated per week from the beginning of January, would enable wearing masks to be stopped on July 14 (red curve, Figure 4b). Hence, immunity, owing to vaccination or infection, would be about 32% which is insufficient to achieve herd immunity.
This indicates that strict hygiene rules and physical distancing will remain essential.

| DISCUSSION
We months. 31 We also assumed that the vaccinated people will not Our study has several limitations. The forecasts obtained with this SIR-type epidemiological model assume that its parameters remain stable over time. Like all mathematical models, there are potential biases associated with parameter estimation that can lead to biased projections. We have attempted to overcome this problem for the two parametersĉ andq that account for the impact of public health measures by correcting the predicted data using observed data. 3 For other parameters like R 0 , the herd immunity threshold is defined by 1 -1/R 0 , which implies that a higher R 0 requires a greater immune proportion of the population in order to block sustained transmission. 38 The estimated virus proliferation rates resulting from the application of various public health measures also assume that the population must continue to adhere to these measures stably over time. For example, we assume that mandatory mask wearing would continue when the second lockdown is released just as it was at the end of the summer in the  39 Although only six cases (around 1% among SARS-CoV-2 infected individuals) were detected in the Toulouse urban area between 1 January 2021 and 21 January 2021, subsequent studies are scheduled to assess its spread and the consequences for the model parameters.

ACKNOWLEDGEMENT
The English text was edited by Dr Owen Parkes.