Jérémy Vanhelst

Camille Ternynck

Hervé Ovigneur

Thibault Deschamps

Normative health-related fitness values for French children: The Diagnoform Programme Running head: Physical fitness reference

Keywords: Physical fitness; Children, Health, Percentiles

The primary objective of this study was to establish sex-and age-specific physical fitness percentiles for French children. The secondary aim was to assess sex, weight status and age differences for physical fitness levels in French children. A sample of 31,484 children (16,023 boys, 15,461 girls) aged 6-11 years participated in the Diagnoform programme. Cardiorespiratory fitness, muscular endurance, speed, flexibility and agility were assessed in this national programme. Percentile values were estimated as a function of age stratified by sex using a parametric method providing smooth centile curves and explicit formulae for the centile estimates. Values from the 10 th to the 90 th percentile are reported. The influence of body weight according to sex on the physical fitness level was also examined using an analysis of covariance adjusted for age. Physical fitness levels were slightly better in boys, except for agility and flexibility, in which girls performed better (Cohen's coefficient, 0.20-0.45; p < 0.001). All physical fitness tests were significantly associated with age (p < 0.0001).

In general, overweight and obese children had a significantly poorer physical fitness level compared with their normal-weight counterparts (p < 0.05). No difference was found between thin and normal-weight boys and girls, except for agility (p < 0.05). Reference values provide normative data for French children, and these data should be useful for identifying special needs for appropriate intervention programmes.

INTRODUCTION

Physical fitness is now widely recognized as an important determinant of health in children and adolescents. [START_REF] Ortega | Physical fitness in childhood and adolescence: A powerful marker of health[END_REF] Health-related physical fitness includes muscular strength, speed/agility and cardio-respiratory fitness. [START_REF] Ortega | Physical fitness in childhood and adolescence: A powerful marker of health[END_REF] Many studies have shown that each physical fitness component has a positive effect on the health status in young people. [START_REF] Ortega | Physical fitness in childhood and adolescence: A powerful marker of health[END_REF][START_REF] García-Hermoso | Is muscular fitness associated with future health benefits in children and adolescents? A systematic review and meta-analysis of longitudinal studies[END_REF][3] It has been suggested that cardiorespiratory fitness levels are associated with total and abdominal adiposity, and both cardiorespiratory and muscular fitness have been shown to be associated with established and emerging cardio-vascular disease risk factors. Moreover, improvements in muscular fitness and speed/agility have a positive impact on bone health, and both cardio-respiratory and muscular fitness enhancements have been shown to improve the quality of life in paediatric cancer patients, as well as having positive effects on mental health, self-esteem and academic performance. [START_REF] Ortega | Physical fitness in childhood and adolescence: A powerful marker of health[END_REF][START_REF] García-Hermoso | Is muscular fitness associated with future health benefits in children and adolescents? A systematic review and meta-analysis of longitudinal studies[END_REF][3] In two cohort studies performed in Swedish male adolescents, it was noted that low muscular strength and cardio-respiratory fitness were strong risk factors for major causes of death in young adulthood (suicide and cardio-vascular diseases), equivalent to other risk factors such as elevated body mass index (BMI) or blood pressure. [4][START_REF] Högström | Aerobic fitness in late adolescence and the risk of early death: a prospective cohort study of 1.3 million Swedish men[END_REF] For all these reasons, there is a real need to develop surveillance monitoring of physical fitness in children and adolescents.

Sex-and age-specific normative values are necessary to assess and interpret fitness status in children and adolescents. Moreover, establishing percentiles in youth is essential for screening children and adolescents according to their growth in comparison with those of the same age and sex. Fitness reference values in children and adolescents have been reported in many countries in Europe, the Americas, Oceania and Asia. [START_REF] Catley | Normative health-related fitness values for children: Analysis of 85347 test results on 9-17-year-old Australians since 1985[END_REF][START_REF] Castro-Pinero | Percentile values for muscular strength field tests in children aged 6 to 17 years: Influence of weight status[END_REF][START_REF] Castro-Pineiro | Percentile values for aerobic performance running/walking field tests in children aged 6 to 17 years: Influence of weight status[END_REF][START_REF] Eisenmann | Aerobic fitness percentiles for U.S. adolescents[END_REF][START_REF] Lee | Physical Fitness Levels in Korean Adolescents: The National Fitness Award Project[END_REF][START_REF] Ramos-Sepúlveda | Physical fitness and anthropometric normative values among Colombian-Indian schoolchildren[END_REF][START_REF] Hobold | Reference standards to assess physical fitness of children and adolescents of Brazil: an approach to the students of the Lake Itaipú region-Brazil[END_REF][START_REF] Secchi | Physical fitness and future cardiovascular risk in argentine children and adolescents: an introduction to the ALPHA test battery[END_REF][START_REF] Tambalis | Physical fitness normative values for 6-18-year-old Greek boys and girls, using the empirical distribution and the lambda, mu, and sigma statistical method[END_REF][START_REF] Miguel-Etayo | Physical fitness reference standards in European children: the IDEFICS study[END_REF][START_REF] Saint-Maurice | Establishing normative reference values for standing Broad Jump among Hungarian youth[END_REF][START_REF] Saint-Maurice | Establishing normative reference values for Handgrip among Hungarian youth[END_REF][START_REF] Sandercock | Centile curves and normative values for the twenty metre shuttle-run test in English schoolchildren[END_REF][START_REF] Santos | Physical fitness percentiles for Portuguese children and adolescents aged 10-18 years[END_REF][START_REF] Vanhelst | Physical Fitness Reference Standards in French Youth: The BOUGE Program[END_REF] Recently, Tomkinson established European normative values for several physical fitness components in youth aged 9-17 years. [START_REF] Tomkinson | European normative values for physical fitness in children and adolescents aged 9-17 years: results from 2 779 165 Eurofit performances representing 30 countries[END_REF] However, data on physical fitness levels in France are scarce, and only one study addressing the reference data for a French sample has been reported, and that study was restricted to adolescents. [START_REF] Vanhelst | Physical Fitness Reference Standards in French Youth: The BOUGE Program[END_REF] The primary objective of this study was to establish sex-and age-specific physical fitness percentiles for French children aged 6-11 years. The secondary aim was to assess sex, weight status and age differences for the physical fitness levels in French children. For the present study, data analysis was performed only with the DiagnoKid battery. Data were collected in several regions of France between 2010 and 2018 (see Supplementary file 1). All schools in France were invited to participate in the study, with each school director deciding whether to participate or not. If the invitation was accepted, the children were invited to participate. All data obtained from the organizer (Institut des Rencontres de la Forme) of the event were anonymized, and declared and approved by the Commission Nationale de l'Informatique et des Libertés (National Commission on Informatics and Liberty). The study objectives were explained carefully to both children and their parents, after which the children or their parents could accept or decline to participate in the study and to record anonymously their information technology data. Data were recorded by the organizer in an electronic data system. An audit of the complete dataset was performed, and aberrant data were excluded.

METHODS

This

Measurements

Participant characteristics

Body weight was measured with the participant wearing light clothes and without shoes to the nearest 0.1 kg using an electronic scale. Height was measured without shoes to the nearest 0.1 cm using a standard physician's scale. BMI was calculated as weight/height 2 (kg/m 2 ). The nutritional status was assessed using the International Obesity Task Force (IOTF) scale. [START_REF] Cole | Establishing a standard definition for child overweight and obesity worldwide: international survey[END_REF]

Physical fitness

The health-related physical fitness components were assessed using the Diagnoform battery tests, incorporating some Eurofit battery tests. [START_REF]Testing Physical Fitness EUROFIT Experimental Battery: Provisional Handbook[END_REF] This battery of tests assesses cardiorespiratory fitness, muscular strength (lower limbs), speed, flexibility and agility (see Supplementary file 2). All tests were performed once. Good reliability and validity in young people have been reported for all tests used in the present study. [START_REF] Ortega | Reliability of health-related physical fitness tests in European adolescents. The HELENA study[END_REF][START_REF] Mouraby | Reliability and validity of Diagnoform fitness test procedure[END_REF] Measurements were performed by the organizer and by other investigators who undertook special training to ensure that tests were administered correctly.

Cardio-respiratory fitness

The cardio-respiratory test was measured by a 20-m shuttle run-walk test during 6 min. [START_REF] Mouraby | Reliability and validity of Diagnoform fitness test procedure[END_REF] This test has been validated against the 20-m shuttle run reference test from Leger et al (1988) (r = 0.78; p = 0.001). [START_REF] Mouraby | Reliability and validity of Diagnoform fitness test procedure[END_REF] Children were instructed to run and walk as far as possible between two lines located 20 m apart during 6 min. The children ran as quickly as possible from the starting line to the other line and returned to the starting line at a fast walking pace, crossing each line with at least one foot throughout the complete test. The test began on the whistle and was concluded after 6 min. The distance covered by the child was recorded and is expressed in m.

Muscular strength

Muscular strength for lower limbs was assessed by the standing broad jump test. From a starting position immediately behind a line, standing with the feet approximately shoulder width apart, children jumped as far as possible with their feet together. The result was recorded in cm. A nonslip hard surface, chalk and a tape measure were used to perform the test.

Speed

Speed was assessed by running as fast as possible for 5 s. The children stood still in a comfortable position, feet behind the starting line, with no rocking movement. The test began on the whistle and was concluded when the countdown reached zero. The distance covered by the runner was recorded by a mark on the ground and is expressed in m.

Flexibility

Flexibility was assessed by a test measuring leniency and the capability to reach down as far as possible. From a standing position and with both legs straight, the children flexed their trunk and reached down as far as possible with their hands. Children maintained the position for 3 s. Results of this test were indexed: a score of 5 for placing the hands flat on the ground; 4 for fingers touching the ground; 3 for fingers reaching the ankle; 2 for fingers reaching the tibia; and 1 for fingers/hands reaching the knees.

Agility

Agility was assessed using the hopscotch test. Children stood still in a comfortable position, with their feet behind the starting line, and with no rocking movements. They then performed the hopscotch test as quickly as possible. The stopwatch was stopped when the child crossed the end line with both feet. The time taken to complete the test was recorded to the nearest tenth of a second.

Statistical analysis

Subject characteristics are described as mean ± SD, median and interquartile interval, or number (percentage). The influence of body weight on the physical fitness level was examined in boys and girls separately using an analysis of covariance model (or an ordinal logisitic regression) adjusted for age. In this model, covariates were age, sex, weight status (using the IOTF definition), and sex*weight status. Post hoc pairwise comparisons using normal weight as reference were performed using linear contrasts after applying Bonferroni correction.

To examine the association between age and each measurement of interest, and to investigate the need to provide age-specific reference intervals by sex, we used a non-parametric smoothing technique (loess regression). Age-specific reference intervals for each measurement of interest were constructed from the parametric method proposed by Royston and Wright (1998) providing smooth centile curves and explicit formulae for the centile estimates. [START_REF] Royston | A method for estimating age-specific reference intervals ('normal ranges') based on fractional polynomials and exponential transformation[END_REF] Further details about this method are available in Supplementary file 3. All analyses were computed using R software (version 3.5.2) and SAS software (version 9.4, SAS Institute Inc, Cary, NC, USA).

RESULTS

A total of 32,132 volunteers aged 4-17 years participated in the Diagnoform battery tests.

Because of the small sample size of participants aged 4, 5 and 12-17 years compared with other age-sex classes, we restricted the analysis to 31,484 participants aged 6-11 years.

Physical fitness measurements for the study sample are reported by age and sex in Table 1.

Overall, physical fitness levels were better in boys, except for agility and flexibility, in which girls performed better (p < 0.05). Differences for each component of physical fitness between boys and girls were small (Cohen's effect size, 0.2-0.5) (Table 1). All physical fitness tests were significantly associated with age (p < 0.0001). Scores obtained in physical fitness tests by boys and girls increased with increasing age.

Physical fitness according to weight status in boys and girls is presented in Figures 1 and2 and Supplementary file 4. In general, overweight and obese boys had significantly poorer physical fitness (cardio-respiratory, muscular strength, agility and speed) compared with normal-weight boys (p < 0.05; Figure 1). Similar results were found for overweight and obese girls (Figure 1). For flexibility, obese boys and girls had significantly lower scores than normal-weight boys and girls. Differences found between overweight and normal-weight boys and girls were small for all physical fitness components (Cohen's effect size, 0.2-0.5) (Supplementary file 4). Between obese and normal-weight girls, differences found were moderate for cardio-respiratory fitness, muscular strength and speed, and small for agility (Supplementary file 4). Between obese and normal-weight boys, differences found were large for cardio-respiratory fitness and muscular strength, moderate for speed, and small for agility (Supplementary file 4). No difference was found between thin and normal-weight boys and girls, except for agility, in which thin children performed better (p < 0.05; Figure 1). However, this difference was negligible (Cohen's effect size, < 0.2) (Supplementary file 4).

Sex-and age-specific percentile values (10 th , 20 th , 30 th , 40 th , 50 th , 60 th , 70 th , 80 th and 90 th percentiles) for the different physical fitness measurements are reported in Tables 2345, and the derived smoothed centile curves are shown in Supplementary file 5.

DISCUSSION

Establishing percentile values for physical fitness tests may help to identify the target population for primary prevention and may be useful for health promotion policies. The French health programme Diagnoform ® provides an opportunity to establish, for the first time, normative values for fitness components (i.e., cardio-respiratory fitness, muscular strength, speed, flexibility and agility) in French children.

The first finding from our study is that physical fitness levels were better in boys compared with girls, except for agility and flexibility, which agrees with results previously reported in children and adolescents. [START_REF] Secchi | Physical fitness and future cardiovascular risk in argentine children and adolescents: an introduction to the ALPHA test battery[END_REF][START_REF] Tambalis | Physical fitness normative values for 6-18-year-old Greek boys and girls, using the empirical distribution and the lambda, mu, and sigma statistical method[END_REF][START_REF] Miguel-Etayo | Physical fitness reference standards in European children: the IDEFICS study[END_REF][START_REF] Saint-Maurice | Establishing normative reference values for standing Broad Jump among Hungarian youth[END_REF][START_REF] Vanhelst | Physical Fitness Reference Standards in French Youth: The BOUGE Program[END_REF][START_REF] Golle | Physical Fitness Percentiles of German Children Aged 9-12 Years: Findings from a Longitudinal Study[END_REF][START_REF] Cadenas-Sanchez | Physical fitness reference standards for preschool children: The PREFIT project[END_REF][START_REF] De Oliveira | Physical fitness percentile charts for children aged 6-10 from Portugal[END_REF] These differences may be attributed to the distinct development, growth and maturation of boys and girls. [START_REF] Malina | Growth, Maturation, and Physical Activity[END_REF] In addition, cultural, social and environmental factors may affect the development of motor performance and physical activity lifestyle. Studies have reported that boys show better performance than girls in cardiorespiratory fitness and muscular strength because they are more physically active and have a higher fat-free mass. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF][START_REF] Ortega | Health inequalities in urban adolescents: role of physical activity, diet, and genetics[END_REF] Another outcome from our study is that physical fitness levels are associated with age. For boys and girls, the performance in physical fitness tests increased with increasing age. While studies of adolescents have shown a stagnation or decrease in physical fitness with age, an improvement in physical fitness with increasing age is reported in children. [START_REF] Tambalis | Physical fitness normative values for 6-18-year-old Greek boys and girls, using the empirical distribution and the lambda, mu, and sigma statistical method[END_REF][START_REF] Cadenas-Sanchez | Physical fitness reference standards for preschool children: The PREFIT project[END_REF][START_REF] Gontarev | Physical fitness reference standards in Macedonian children and adolescents: the MAKFIT study[END_REF] Our results suggest that older children have better motor competency compared with young children.

Berk suggests that motivation, concentration, the degree of motor skills, physical activity and body composition could also be factors that explain this difference. [START_REF] Berk | Development through the lifespan[END_REF] Regarding weight status differences, overweight and obese children (both boys and girls) have significantly poorer physical fitness levels compared with their normal-weight counterparts.

The impact of overweight and obesity on physical fitness in children has been extensively studied. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF][START_REF] Chen | Obesity, fitness and health in Taiwanese children and adolescents[END_REF][START_REF] Kim | Relationship of physical fitness to prevalence and incidence of overweight among schoolchildren[END_REF][START_REF] Brunet | The association between low physical fitness and high body mass index or waist circumference is increasing with age in children: the 'Québec en Forme' Project[END_REF][START_REF] Deforche | Physical fitness and physical activity in obese and nonobese Flemish youth[END_REF] The results from our study of French children confirm those of previous research indicating a negative relationship between weight status and fitness in children and adolescents. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF][START_REF] Chen | Obesity, fitness and health in Taiwanese children and adolescents[END_REF][START_REF] Kim | Relationship of physical fitness to prevalence and incidence of overweight among schoolchildren[END_REF][START_REF] Brunet | The association between low physical fitness and high body mass index or waist circumference is increasing with age in children: the 'Québec en Forme' Project[END_REF][START_REF] Deforche | Physical fitness and physical activity in obese and nonobese Flemish youth[END_REF] Body composition, especially fat mass, could have an influence on results.

One study suggests that excess body fat determined a lower performance in all tests requiring propulsion or lifting of the body mass, because the association became non-significant or was transformed into the opposite association after adjusting for fat mass. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF] It was concluded that when fat mass differences are taken into account, overweight and obese adolescents do not show a lower performance in weight-bearing physical fitness tests. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF] In contrast, physical fitness in thin youths has been far less studied. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF][START_REF] Chen | Approaching healthy body mass index norms for children and adolescents from health related physical fitness[END_REF][START_REF] Prista | Anthropometric indicators of nutritional status: implications for fitness, activity, and health in schoolage children and adolescents from Maputo, Mozambique[END_REF][START_REF] Mak | Health-related physical fitness and weight status in Hong Kong adolescents[END_REF] In our study, no difference was found between thin and normal-weight participants in physical fitness tests, except for agility: thin boys and girls had a better score in the agility test compared with their normal-weight counterparts. Studies examining the relationship between weight status and health-related physical fitness in youth have often reported a decrease in fitness with increasing BMI. [START_REF] Artero | Health-related fitness in adolescents: , and not only overweight, as an influencing factor. The AVENA study[END_REF][START_REF] Chen | Approaching healthy body mass index norms for children and adolescents from health related physical fitness[END_REF][START_REF] Prista | Anthropometric indicators of nutritional status: implications for fitness, activity, and health in schoolage children and adolescents from Maputo, Mozambique[END_REF][START_REF] Mak | Health-related physical fitness and weight status in Hong Kong adolescents[END_REF] Methodological differences in the physical fitness tests chosen might explain the discrepancies between the present study and previous findings. In addition, the hopscotch test used in our study to assess agility might explain the better performance of thin children compared with those of normal weight. Although thin children may have lower muscular strength, their body weight is lower. Thus, the hopscotch test may be considered as weight dependent. Our results suggest that the higher performance of thin adolescents resulting from their lower load to be moved could be counteracted by their lower fat-free mass.

Given that data on reference standards of fitness levels in French children are lacking, various aspects of the results from our study showing sex-and age-specific normative values for physical fitness tests are relevant. A fitness level below the 10 th percentile may be considered as deleterious for health because low scores on cardio-respiratory fitness and muscular strength are associated with increasing cardio-vascular risks. [START_REF] Ortega | Physical fitness in childhood and adolescence: A powerful marker of health[END_REF][START_REF] García-Hermoso | Is muscular fitness associated with future health benefits in children and adolescents? A systematic review and meta-analysis of longitudinal studies[END_REF][3] Even if our data on reference standards in this study are not directly associated with biological markers of cardio-vascular risks, we recommend the monitoring of children with a fitness level below the 30 th percentile for health promotion and intervention programmes. Our reference values can also be used to assess children's performance by health care practitioners, schools, sport clubs and by the children themselves. We propose that performance level may decline along a specific scale according to the following reference values: very low (X < 10 th percentile); low (10 th percentile ≤ X < 30 th percentile); medium (30 th percentile ≤ X < 70 th percentile); high (70 th percentile ≤ X < 90 th percentile); and very high (X ≥ 90 th percentile).

The current study has both strengths and limitations. Major strengths include the large sample of children from across France with sex-specific information, and the harmonization and standardization of assessment of physical fitness. The main limitation of the study is the cross-sectional design. Because of the changes in individual growth and maturation in children and adolescents, physical fitness reference values should be obtained from a longitudinal study with repeated measurements. Moreover, even if our study allows for the accurate detection of individual improvements, further studies should be performed to more fully characterize and identify cut-off points related to health outcomes for cardio-respiratory fitness, muscular strength, speed and agility. Finally, although the present data derive from a large sample across France, this study did not use a stratified sample design. Therefore, it is not possible to assume that the studied cohort is fully representative of the populations of French children and adolescents.

PERSPECTIVES

In summary, for the first time, our study provides sex-and age-specific physical fitness percentiles for French children aged 6-11 years old. Our findings complement those of Vanhelst et al (2017) in adolescents aged 10-15 years. [START_REF] Vanhelst | Physical Fitness Reference Standards in French Youth: The BOUGE Program[END_REF] These reference standards are of particular interest to teachers and health care practitioners in the development of intervention programmes for children identified with low physical fitness status.

  cross-sectional study was based on data from the French health programme Diagnoform ® (https://irfo.fr/). The principal aim of this national programme is to assess the physical fitness of a large sample of the French population from the age of 5 years. The programme is divided in four categories: (i) DiagnoKid for children aged 5-10 years; (ii) DiagnoTonic for adolescents and young adults aged 10-25 years; (iii) DiagnoActif for adults aged 25-60 years; and (iv) DiagnoSanté for those aged >60 years. The programme is carried out in large settings such as school playgrounds or sports club gymnasiums throughout France.

Legends


Y W Frayer

T Libsekal

L Medin

D Rook

R Tesfay

Macchiarelli

Variation in parietal bone thickness and structural arrangement in Eastern African erectus-like Homo: comparative evidence from late Early

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Variation in parietal bone thickness and structural arrangement in Eastern African erectus-like Homo: comparative evidence from late Early Pleistocene Uadi Aalad and Mulhuli-Amo, Danakil depression of Eritrea

CLÉMENT ZANOLLI 1 , LUCA BONDIOLI 2 , FRANCESCA CANDILIO 3 , ALFREDO COPPA 3 , DAVID W. FRAYER 4 , YOSIEF LIBSEKAL 5 , TSEGAI MEDIN 5,6 , LORENZO ROOK 7 , DAWIT TESFAY 3,8 , and ROBERTO MACCHIARELLI 9,10 .

1 UMR 5288 AMIS CNRS-Université de Toulouse Paul Sabatier, France, 2 Sezione di Bioarcheologia, Museo Nazionale Preistorico Etnografico "Luigi Pigorini", Rome, Italy, 3 Dipartimento di Biologia Ambientale, Università di Roma "La Sapienza", Italy, 4 Department of Anthropology, University of Kansas, Lawrence, KS, 5 National Museum of Eritrea, Asmara, Eritrea, 6 Institut Catala de Paleoecologia Humana i Evolució Social, Universitat Rovira i Virgili, Tarragona, Spain, 7 Dipartimento di Scienze della Terra, Università di Firenze, Italy, 8 Northern Red Sea Regional Museum, Massawa, Eritrea, 9 UMR 7194 CNRS-Muséum national d'Histoire naturelle, Paris, France, 10 Département Géosciences, Université de Poitiers, France.

Cranial vault thickness behaves as a highly variable feature in extant and fossil humans. While traditionally used to define H. erectus s.l. because of its relatively thick-walled bones, recent estimates show that, along the mid-sagittal plane and at the frontal and parietal eminences, bone thickness does not markedly differ between H. erectus s.l. and anatomically modern humans, nor does it reliably distinguish African and Asian erectus-like representatives. However, compared to the parasagittal areas, paleoanthropological information on thickness variation along the lateral parietal wall is scarce.

By using direct, CT-and microCT-based observations, we quantified the tabular and diploic thickness in two circa 1 Ma parietal bones from the 4.7 km apart but stratigraphically correlated sites of Uadi Aalad and Mulhuli-Amo, Eritrean Danakil depression. The first belongs to the nearly complete UA 31 cranium, noted for its blend of erectus-like and derived morphoarchitectural features. The second parietal bone is part of an incomplete cranium preliminarily labelled MA 64-88-378, assembled from nine elements collected on the surface in 2011-2014, whose global shape closely resembles UA 31.

While our records do not systematically sample homologous landmarks but rather comparable regions in UA 31 and MA 64-88-378, their relative thickness at various sites is nearly invariably diploë>outer≥inner table, even if proportions vary site-specifically. In all cases, intra-individual thickness variation tends to exceed that commonly observed in extant humans. We found a similar pattern, but expressing even more contrasted values, in the chronologically slightly younger parietal specimen from Gombore, at Melka Kunture, Ethiopia.

Funding support provided by: Italian Ministry for University and Research, Univ. "La Sapienza" of Rome ("Grandi Scavi"), Italian Ministry for Foreign Affairs, French CNRS.

Preferred presentation medium: poster Appropriate session: Paleoanthropology


Samson Lasaulce

Dr Chao Zhang
email: zhang@telecom-paris.fr

Vineeth Varma

Irinel Constantin Morarescu

Analysis of the tradeoff between health and economic impacts of the Covid-19 epidemic

Keywords: Epidemic, Covid-19, SARS-CoV2, SEIR model, epidemic management strategy, behavior model

Various measures have been taken in different countries to mitigate the Covid-19 epidemic.

But, throughout the world, many citizens don't understand well how these measures are taken and even question the decisions taken by their government. Should the measures be more (or less) restrictive? Are they taken for a too long (or too short) period of time? To provide some quantitative elements of response to these questions, we consider the well-known SEIR model for the Covid-19 epidemic propagation and propose a pragmatic model of the government decisionmaking operation. Although simple and obviously improvable, the proposed model allows us to study the tradeoff between health and economic aspects in a pragmatic and insightful way.

Assuming a given number of phases for the epidemic (namely, 4 in this paper) and a desired tradeoff between health and economic aspects, it is then possible to determine the optimal duration of each phase and the optimal severity level (i.e., the target transmission rate) for each of them. The numerical analysis is performed for the case of France but the adopted approach can be applied to any country. One of the takeaway messages of this analysis is that being able to implement the optimal 4-phase epidemic management strategy in France would have led to 1.05 million of infected people and a GDP loss of 231 billions e instead of 6.88 millions of infected and a loss of 241 billions e. This indicates that, seen from the proposed model perspective, the effectively implemented epidemic management strategy is good economically, whereas substantial improvements might have been obtained in terms of health impact. Our analysis indicates that the lockdown/severe phase should have been more severe but shorter, and the adjustment phase occurred earlier. Due to the natural tendency of people to deviate from the official rules, updating measures every month over the whole epidemic episode seems to be more appropriate.

INTRODUCTION

One of the goals of this work is to provide a simple but exploitable model to measure the quality of the epidemic management strategy implemented by a government to mitigate the health and macro-economic impacts of the Covid-19 epidemic. The quality is measured in terms of the tradeoff between the total number of infected people over a given period of time and the Gross Domestic Product (GDP) loss, under a constraint of the total number of infected people requiring Intensive Care Units (ICU). To reach this objective, we propose a behavioral model for governmental decision-making operations. Although we assume a simple measure for the quality of the lockdown measures and a simple dynamical model (namely, a classical susceptible-exposed-infected-removed (SEIR) model), the proposed approach is seen to be sufficient to constitute a first step into capturing and quantifying the tradeoff under consideration. In contrast with most studies conducted on the Covid-19 epidemic analysis where the primary goal is to refine the SEIR model (see e.g., [START_REF] Peng | Epidemic analysis of COVID-19 in China by dynamical modeling[END_REF][START_REF] Pribylova | SEIAR model with asymptomatic cohort and consequences to efficiency of quarantine government measures in COVID-19 epidemic[END_REF][START_REF] Victor | Estimation of the probability of reinfection with COVID-19 coronavirus by the SEIRUS model[END_REF][START_REF] Giordano | A SIDARTHE model of COVID-19 epidemic in Italy[END_REF][START_REF] Roques | Impact of Lockdown on the Epidemic Dynamics of COVID-19 in France[END_REF]) or employ the SEIR model by accounting for local variations (by using a given SEIR model per geographical region -see e.g., [START_REF] Domenico | Expected impact of school closure and telework to mitigate COVID-19 epidemic in France. COVID-19 outbreak assessment[END_REF]) or for the impact of class type (by age, sex, risk level -see e.g., [START_REF] Khawaja | Associations with Covid-19 hospitalisation amongst 406,793 adults: the UK Biobank prospective cohort study[END_REF]) our approach is to use the standard SEIR model for an entire country and choose a simple economic model to focus on the study of the tradeoff between health and economic aspects.

Although there have been many several interesting studies on the economic impact of Covid-19 (see e.g., [START_REF] Atkeson | What will be the economic impact of covid-19 in the US? Rough estimates of disease scenarios[END_REF][START_REF] Eichenbaum | The macroeconomics of epidemics[END_REF][START_REF] Baldwin | Economics in the Time of COVID-19[END_REF][START_REF] Fernandes | Economic effects of coronavirus outbreak (COVID-19) on the world economy[END_REF][START_REF] Mckibbin | The global macroeconomic impacts of COVID-19: Seven scenarios[END_REF]),the pursued goal of these studies is not to model the behavior of the government. As a consequence, the proposed tradeoff has not been analyzed, at least formally. In fact, the closest contribution to this direction would be given by [START_REF] Zakary | On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach[END_REF] where generic discrete-time epidemics over multiple regions are considered, the particular 4-phase structure is not considered, the focus is not on Covid-19, and the key aspect of the tradeoff analysis is neither developed nor analyzed. Additionally, the numerous studies available on the problem of the transmission rate control generally concern the continuous-time control approach. In this work, the focus is on a multiple phase approach (namely, 4 phases). In the literature dedicated to epidemic control, one can for instance find that some recent studies on how the lockdown strategies and quarantine can be planned in an optimal fashion [START_REF] Alvarez | A simple planning problem for covid-19 lockdown[END_REF][START_REF] Boujakjian | Modeling the spread of Ebola with SEIR and optimal control[END_REF][START_REF] Casella | Can the COVID-19 epidemic be managed on the basis of daily data?[END_REF][START_REF] Rawson | How and when to end the COVID-19 lockdown: an optimization approach[END_REF]. A common feature to all these works on optimal control and lockdown planning is that the policies under consideration, vary over time in a continuous manner, i.e., the lockdown policy is continuously evolving based on the infected population or just on time. However, from the perspective of a government, implementing such policies is not practical since daily changes of the epidemic control measures are difficult to be implemented and to be followed by people.

Summarizing, compared to the existing literature on epidemics modelling and control of epidemics, the main contribution of our work is fourfold:

• a model for capturing the tradeoff between health and economic aspect and therefore for the government decision-making operation is proposed and studied;

• the focus is on multiple phase control policies and not on general continuous-time control policies (to be precise, 4 phases are assumed, see Figure 1);

• the problem of finding the optimal features of the optimal epidemic management policy (i.e., the target severity level for each phase and the switching time instants) is stated and solved exhaustively.

Additionally, to refine the analysis, we assume a simple model for the natural time drift in terms of behavior of people;

• the numerical analysis of the tradeoff is dedicated to the Covid-19 epidemic and a case study for France.

Pre-lockdown phase

Lockdown phase

Post-lockdown phase

Adjustment phase effective reproduction number (or transmission rate)

Figure 1. One of the goals of this work is to determine numerically, for a given tradeoff between health and economic costs, the best 4-phase epidemic management policy that is, the best values for τ 0 , τ 1 , τ 2 , R 1 , R 2 , R 3 (the epidemic time horizon T and the natural reproduction number R 0 being fixed).

METHODS

Epidemic model

To model the dynamics of the Covid-19 epidemic globally i.e., over an entire country, we assume a standard SEIR model. Let us respectively denote by s, e, i, and r the fractions of the population:

being susceptible to be infected by the SARS-Cov2 virus, having been exposed to it, being infected, and being removed (including recoveries and deceases). The epidemic is assumed to obey the following continuous-time dynamics:

                                       ds dt (t) = -β(t)i(t)s(t) de dt (t) = β(t)i(t)s(t) -γe(t) di dt (t) = γe(t) -δi(t) dr dt (t) = δi(t) s(t) + e(t) + i(t) + r(t) = 1 (1)
where:

• β(t), t ∈ R, represents the time-varying virus transmission rate;

• γ denotes the rate at which the exposed subject develops the disease (this includes people presenting symptoms and asymptomatics). The period 1 γ is called the incubation period;

• δ denotes the removal rate and 1 δ is called the average recovery period.

We assume that the the control action u(t) taken by the decision-maker (the government or possibly a more local decision-maker) has a linear effect on the transmission virus rate. Additionally, the effectiveness of this action is assumed to undergo a non-controllable drift or attenuation effect due to the observed fact that people tend to relax their effort over time [START_REF] Dagnall | Bridging the gap between UK Government strategic narratives and public opinion/behavior: Lessons from COVID-19[END_REF][19], hence the presence of the attenuation factor a(t) yields:

β(t) = R 0 δ -u(t)a(t) (2)
where R 0 is the natural reproduction number (namely, without any control or population awareness), u(t) ∈ [0, U ] is the control action or severity level of the lockdown measures. Note that U corresponds to the most drastic or severe control action (in theory it could reach the value R 0 δ and make the reproduction number vanishing). In this work, u(t) is a piecewise-constant function. For the numerical analysis, we will assume a(t) to be a linearly decreasing function of time (as detailed in the next section). Therefore, one can define the time-varying effective reproduction number:

R(t) = β(t) δ = R 0 - u(t)a(t) δ . (3)
As illustrated by Figure 1, we are solving an epidemic control problem in which determining the function u(t) or R(t) amounts to jointly determining the switching instants τ 0 , τ 1 ,τ 2 and the targeted reproduction numbers R 1 , R 2 , R 3 ; T is a given period of time for the epidemic analysis. In particular, we will determine the best duration of the lockdown phase τ 1 and the corresponding targeted reproduction number R 1 .

Figure 1 shows for instance that if the lockdown measures taken are such the reproduction number is R 1 at the beginning of the lockdown phase, then, because of the drift induced by the typical human behavior, the effective reproduction number increases over time.

Time drift or people behavior model

We propose here a model for the attenuation function a(t), which quantifies the degree to which people relax their effort to implement the government management measures. As the attenuation effect is negligible when a new policy is released, we consider that a(t) = 1 when t ∈ {τ 0 , τ 1 , τ 2 }. The attenuation factor is assumed to increase over time in each phase, and we assume the following piecewise linear behavior between phases:

a(t) =                        1 for t < τ 0 , 1 -a 1 (t -τ 0 ) for τ 0 ≤ t < τ 1 , 1 -a 2 (t -τ 1 ) for τ 1 ≤ t < τ 2 , 1 -a 3 (t -τ 2 ) for t ≥ τ 2 . ( 4
)
where a 1 , a 2 , and a 3 respectively represent the attenuation coefficients during the lockdown phase, after the lockdown phase, and during the adjustment phase.

Decision-maker behavior model

The proposed model for the behavior of the decision-maker is based on the fact that it wants to obtain a given tradeoff between economic and health aspects. For the cost related to the economics loss, we assume the simplest reasonable model. That is, we assume that economic cost is quadratic in the control action. For the health cost, we assume that it is given by the number of infected people over the given period of time.

Therefore, the proposed overall cost consists of a convex combination of these two costs. By minimizing the overall cost, one realizes the desired tradeoff between economic and health aspects. On top of this we impose the number of patients requiring intensive care to be under a given threshold N ICU max . Thus, the corresponding minimization is performed under a constraint on the number of people infected at any time

t ∈ [0, T ]: σN i(t) ≤ N ICU max ,
N being the population size, N ICU max the maximum number of ICU patients, and 0 ≤ σ ≤ 1 is the percentage of infected people requiring intensive care. In France, official records state that the maximum cumulated number of ICU patients has reached 7 148 (on April 8, 2020) but the capacity over the whole territory has been evaluated to be greater than 15 000. By denoting α ∈ [0, 1] the weight assigned to the macroeconomic impact of the epidemic and K e > 0, K h > 0, µ > 0 some constants (parameters) defined below, obtaining the desired tradeoff amounts to finding a solution of the following optimization problem (OP) while fixing α to a given value: minimize

u(t)        αK e τ 0 +τ 1 0 u 2 (t)dt + 1 µ 2 1 τ 0 +τ 1 +τ 2 τ 0 +τ 1 u 2 (t)dt + 1 µ 2 2 T τ 0 +τ 1 +τ 2 u 2 (t)dt + (1 -α)K h [s(0) -s(T )]        subject to ∀t ∈ [0, T ], σN i(t) ≤ N ICU max τ 1 ≥ T min Equations (1) and (2) (5)
where:

• K e > 0 and K h > 0 are constants that weight the economic and health cost functions (they also act as conversion factors allowing one to obtain appropriate units and orders of magnitude);

• τ 0 and τ 1 represent the lockdown starting time and duration, respectively. T min is the minimum lockdown duration to make the lockdown policies effective. The quantity τ 2 represents the duration of the post-lockdown phase;

• the parameters µ 1 , µ 2 ≥ 1 accounts for possible differences in terms of economic impact between the lockdown and post-lockdown phases;

• s(0) and s(T ) are respectively the fractions of the population susceptible at the beginning and the end of the analysis.

We would like to make additional comments concerning the parameter µ 1 , µ 2 . The motivation for introducing µ 1 , µ 2 is twofold. First, after lockdown, people are more aware and act more responsibly than before lockdown. This means that automatic and costless population distancing typically occurs [START_REF] Atkeson | What will be the economic impact of covid-19 in the US? Rough estimates of disease scenarios[END_REF][START_REF] Greenstone | Does social distancing matter?[END_REF][START_REF] Andersen | Early evidence on social distancing in response to COVID-19 in the United States[END_REF]. Taking µ 1 , µ 2 ≥ 1 precisely amounts to having a smaller reproduction number without any cost for the government.

Additionally, as people typically tend to relax their effort to implement the epidemic management measures as time passes, it makes sense to assume that µ 1 ≥ µ 2 in our model. It also allows one to account for the fact that, after lockdown, the economic activity grows after the lockdown and the effects of the pandemic starts vanishing. This means that, in some sense we ignore memory effects due to lockdown measures.

Further refinements of the proposed model might be considered to account for the lockdown memory effects. This is out of the scope of the present paper but we believe that, this would correspond to assuming µ i < 1.

4-Phase optimal control with piecewise constant control actions

Solving analytically the optimization problem given by ( 5) is not trivial. However, since we restrict our attention to a certain class of control policies, the problem turns out to be solvable through exhaustive search. Assuming the attenuation factor a(τ 0 ) = a(τ 1 ) = a(τ 2 ) = 1 (no attenuation at the beginning of each phase) and a constant control action in each phase, by using the relation

u(t) = δ[R 0 -R(t)],
the OP (5) can be rewritten under a more convenient form for numerical purposes: minimize

(τ 0 ,τ 1 ,τ 2 ,R 1 ,R 2 ,R 3 )          αK e δ 2 (R 0 -R 1 ) 2 τ 1 + αK e δ 2 (R 0 -R 2 ) 2 τ 2 µ 2 1 + αK e δ 2 (R 0 -R 3 ) 2 [T -(τ 0 + τ 1 + τ 2 )] µ 2 2 + (1 -α)K h [s(0) -s(T )]          subject to ∀t ∈ [0, T ], σN i(t) ≤ N ICU max τ 1 ≥ T min R 2 > R 1 + 0.2 Equations (1) and (2). (6)
where R i represents the desired or target reproduction number over Phase i ∈ {1, 2, 3} without considering the attenuation factor (also, it is the reproduction number at the start of i-th phase). The second constraint is introduced here as there is a gap between lockdown reproduction number and after lockdown reproduction number. Finally, the conversion factors K e and K h are chosen as follows. The rationale behind the choice of K e is that when choosing α = 1 the GDP loss should correspond to the best estimations made by economists. The GDP loss over the lockdown period for a given country is denoted by ∆GDP, the conversion factor K e is chosen as follows:

K e δ 2 (R 0 -R 1 ) 2 τ 1 = ∆GDP. (7)
For France for example, the GDP loss during the lockdown has been evaluated (on April 20) to be around 120 billions e according to the OFCE [START_REF]Évaluation au 20 avril 2020 de l'impact économique de la pandémie de COVID-19 et des mesures de confinement en France[END_REF]. At last, the constant K h is merely chosen as K h = N , that is, when α = 0 the cost function corresponds to the number of people infected over the considered period of time.

RESULTS

General simulation setup

To perform exhaustive search over the sextuple of variables 1 δ = 1 0.1857 = 5.4 days, 1 γ = 1 0.16 = 6.25 days; these choices are consistent with many works and in particular the studies performed for France [START_REF] Salje | Estimating the burden of SARS-CoV-2 in France[END_REF] and Italy [START_REF] Casella | Can the COVID-19 epidemic be managed on the basis of daily data?[END_REF]. The population size is set to N = 66.10 6 , the maximum number of patients requiring intensive care is set to N ICU max = 15.10 3 , and σ = 1.5% [START_REF]Open Stats Coronavirus. Covid-19 statistiques/France[END_REF][25][26].

(τ 0 , τ 1 , τ 2 , R 1 , R 2 , R 3
Notice that this number is only reached for very small values of α (for which the total number of people infected over the analysis duration would be around 9 millions). The exposed population on March 1 2020 is initialized to N e(0) = 1.33.10 5 . This number is obtained from analyzing the data provided in [START_REF]Open Stats Coronavirus. Covid-19 statistiques/France[END_REF]. We consider the number of reported deaths at a given time to be a more reliable way of tracking the evolution of the pandemic rather than the reported number of infected people. Indeed, as soon as one examines absolute values, they typically become irrelevant. For example, during lockdown, because of the lack of tests and measurements, the real number of infected people was much higher than the official number. Now, even when tests were performed intensively, because of false positives, the absolute number of infected were again completely unreliable. For a prevalence of 1/1000 and a test reliability of 5% of false positives we see that the number of infected for 1000 people is declared to be about 50 whereas the actual number of infected is only 1. Motivated by these critical issues, we have considered figures which are much more reliable such as the number of deceases due to Covid-19 in France [23][24]. From the number of deaths and the global average rate (worldwide) of the mortality rate (in the range 0.3% -0.5% when averaged over classes of ages and countries), the reconstructed number of infected people turns out to be more accurate.

Therefore, by fixing the mortality rate to a given value (in [START_REF] Salje | Estimating the burden of SARS-CoV-2 in France[END_REF] for instance, the mortality rate averaged over all the classes of infected people is evaluated to be around 0.53% for France), one can estimate the exposed and infected population size 3 to 4 weeks before the measured number of deaths due to Covid-19.

For our computation, the initial conditions of the ordinary differential equations (ODE) equations are chosen as s(0) = 1 -e(0), i(0) = r(0) = 0, and the ODE is solved by using the Matlab ode45 solver.

Concerning the economic cost for France related to Covid-19, the GDP loss over the lockdown period is estimated by the OFCE [START_REF]Évaluation au 20 avril 2020 de l'impact économique de la pandémie de COVID-19 et des mesures de confinement en France[END_REF] to be 120 billions e and we have, as reliable figures, that τ France = 0.6. We therefore take K e = 7.379.10 9 e/day. Values of the reproduction number for the first two phases come from past and quite accurate evaluations (see e.g., [START_REF] Delfraissy | Sortie progressive de confinement: Prérequis et mesures phares[END_REF]).

The value R France 2 = 0.9 is less accurate and corresponds to the assumption that the government has been aiming at giving as much as freedom to the population while avoiding a second wave. Based on available statistics on Covid-19 in France [26], the attenuation coefficients of the drift model have been chosen as follows: a 1 = 0.1%, a 2 = 0.2%, a 3 = 0.2%. Unless stated otherwise, the economic impact parameters are chosen as µ 1 = 1.41 (i.e., µ 2 1 ∼ 2) and µ 2 = 1.3. Also, when α is assumed to be fixed, it is set to 10 -4 .

To justify the choice of the attenuation parameters, that is, a 1 = 0.1%, a 2 = 0.2%, a 3 = 0.2%, we apply the French policy into our model. By comparing the active cases obtained from our model and the statistics, it can be illustrated in Fig. 2 that our model matches well with the statistics, especially in the second-half of the plot, where the number of tests conducted are sufficiently large. This validate the choice of our model and parameters. The mismatch on March and April are mainly due to the lacking number of tests that were taken during the early outbreak of the pandemic, leading to a much lower reported number of active cases.

Statistics

Figure 2. Comparison between our model (blue curve) and the reported statistics (black curve) from March 1st to September 30th. When the number of tests are sufficiently large, our model matches well with the reported statistics. The reproduction number is linearly increasing during each phase, and is discontinuous during the transition between phases.

Optimal tradeoff between economic and health impacts

With the proposed government decision-making model, implementing a desired tradeoff between the health cost and economic cost merely amounts to choosing a given value for the parameter α. Figure 3 depicts for various values of α in the interval [10 -7 , 10 -4 ] the total GDP loss and number of infected people that is obtained after choosing the (quantized version of the) sextuple (τ 0 , τ 1 , τ 2 , R 1 , R 2 , R 3 ) that minimizes the combined cost given by Equation [START_REF] Domenico | Expected impact of school closure and telework to mitigate COVID-19 epidemic in France. COVID-19 outbreak assessment[END_REF]. At one extreme, when α is relatively large (α = 10 -4 ) (that is, when the government aims at minimizing the economic cost in the first place -always under the France Gross Domestic Product loss (in billions euros) The plots represent the possible tradeoffs between health cost (measured in terms of the total number of infected people) and economic cost (measured in terms of GDP loss) that can be obtained (by choosing the best epidemic management policy). In particular, with the assumed model, it is seen, in retrospect, that it would have been possible to divide the number of infected people by about 6 while saving about 10 billions e in terms of GDP with the optimal 4-phase strategy.

ICU
billions e. Which type of epidemic management strategy should be used to have such an outcome? The next sections provide a detailed analysis of the features of the optimal strategy.

Optimal features of the optimal epidemic management strategy

One of the important features for controlling the Covid-19 epidemic which has been well commented in newspapers in various countries is the lockdown starting time. To minimize the health cost, the answer is ready: the lockdown phase should always start as soon as possible. But when one wants to realize a tradeoff between health and economic costs, the answer is less immediate. For different values for µ 1 and µ 2 , Figure 4 provides the best day to start locking down the population, for one hundred values of α ranging from 10 -4 to 10 -6 . The main message of this figure is that, even for (relatively) large values for α (that is, when the economic cost dominates the health cost), the optimal lockdown starting day should be before March 4th (i.e., τ 0 ≤ 4). This clearly shows that, once an epidemic has been declared, invoking economic damages to delay the lockdown phase is not acceptable. Note that this conclusion holds when economic losses are assumed to be uniform over time (µ 1 = µ 2 = 1). When the economic cost associated with a

given intensity or severity level is lower after lockdown than during it (here µ 1 = 1.41 and µ 2 = 1.3), it is always optimal to start locking down as soon as possible. Note that our model does not capture the possible fact that population needs to be psychologically prepared to follow the lockdown measures. In France, by March 17, there were official figures about the epidemic which were sufficiently critical to make the 0 0.2 0.4 0.6 0.8 1

Weight assigned to the economical aspect ( ) 10 -4

Optimal lockdown starting day 1 = 2 =1, i.e., when the economical cost associated with a given severity level is the same during and after lockdown 1 =1.41, 2 =1.3, i.e., when the economical cost associated with a given severity level is lower after lockdown than during and after lockdown Figure 4. When economic losses are assumed to be uniform over time (µ = 1), there is some economic incentive to delay the lockdown, but this delay is seen to be at maximum 4 days. When economic losses are lower after lockdown than during it, it is beneficial to start the lockdown faster, up to 3 days delay.

population accept the measures whereas, starting on March 4 (the optimal starting date for µ 1 = µ 2 = 1) the situation might have not been critical enough to create full adhesion to government measures.

A second key feature of the Covid-19 epidemic control strategy was the lockdown phase duration namely, the value of τ 1 . To better explore the relationship between τ 1 and α (the tradeoff), we relax the lockdown duration constraint here and set T min = 1. For (µ 1 , µ 2 ) = (1, 1) and (µ 1 , µ 2 ) = (1.41, 1.3),

Figure 5 provides the optimal lockdown duration (in days) for values of α ranging from 10 -4 to 10 -6 . For (µ 1 , µ 2 ) = (1, 1) (i.e., when economic losses are uniform over time), the optimal duration ranges from 53 days to 83 days for a large fraction of the considered interval for α. Interestingly, we see that these values are relatively close to the lockdown duration effectively imposed in France namely 55 days. For larger values of µ 1 and µ 2 , the optimal lockdown duration is seen to be much smaller. Therefore, if economic losses are uniform over time, the French government policy seems to be very coherent. On the other hand, if the economic impact is smaller after lockdown, our study suggests shorter lockdown durations. In fact, our results show the existence of a critical value for the tradeoff parameter α above which the second phase of the management of the epidemic should not be present. This means that the optimal control consists of three phases instead of four.

To conclude this section, let us consider Figure 6. For the by default scenario studied in this paper ((µ 1 , µ 2 ) = (1.41, 1.3), α = 10 -4 ), the figure represents the evolution of number of infected people, that is, N i(t), and the transmission rate when the optimal policy is adopted. First, it is seen that for the health-economic tradeoff corresponding to α = 10 -4 , there is no interest in delaying the lockdown phase. When economic losses are assumed to be uniform over time (µ = 1), it is optimal to have a long lockdown (typically between 60 -80 days) whatever the tradeoff desired. But, if the economic losses of the third phase are less than during lockdown, the lockdown phase should be shorter.

By "lockdown" phase, the authors mean that it might be any type of phase for which the reproduction number is as low as 0.4 (versus the estimated 0.6 in France); very efficient digital tracing and intensive use of face masks is also an option which has been successfully adopted in countries such as South Korea (see e.g., [START_REF] Park | Contact tracing during coronavirus disease outbreak, South Korea, 2020[END_REF]). The optimal lockdown phase duration is seen to be about 1 month (instead of 2 for France). We see that the existence of an adjustment phase is part of the optimal policy. For this point, we see that the adjustment phase should have occurred much before in France (end of June versus end of September). The next section is precisely dedicated to the impact of the adjustment phase.

Lockdown policies with different R 0

Since the natural reproduction number R 0 depends on temperatures, population density, may vary over time due to mutation effects, and in any case is not known perfectly, it is of interest to study the impact of R 0 on the obtained characteristics for the optimal epidemic management policy. This is what Fig. 7 represents. It is seen that large variations on R 0 do not involve large variations on the starting day. For instance, moving from R 0 = 2 to R 0 = 3.5 only changes the optimal date by one day (Day 2 instead of Day 3), which confirms the need to act fastly even when the transmission is more limited (e.g., thanks to higher temperatures or lower population density). Note that this holds even if the economic impact is accounted for. It is also good for economical aspects to react fastly to an epidemics. For the optimal Figure 6. This figure represents the evolution of number of infected people (N i(t)) and the transmission rate when the optimal policy is adopted. reproduction number during lockdown (namely, R 1 ) it is also seen that moving a scenario in which R 0 = 2 to R 0 = 3.5 does not change very significantly the results: the target severity (or freedom) level would correspond to R 1 = 0.4 instead R 1 = 0.6, which shows that the severity should be high during lockdown even in countries or regions where transmission is more limited.

Mar

Alternatively, the impact of R 0 uncertainty can be evaluated by adding a perturbation on R 0 . It is assumed that what is known to determine the optimal parameters is R 0 = R 0 + ∆, ∆ being a Gaussian noise ∆ ∼ N (0, σ 2 ). The reproduction number have to stay in a given interval of physical relevance of the form [R min , R max ]. Thus the noise is imposed to stay in the interval

[-R 0 + R min , -R 0 + R max ]. With R 0 = 3.5, R min = 1 and R max = 4,
E ∆ [| R 1 -R 1 |] and E ∆ [| τ 0 -τ 0 |],
where R 1 and τ 0 are obtained with the noisy reproduction number R 0 . Remarkably, the impact of the corresponding noise on the results is seen to be very reasonable and does not affect the main conclusions drawn in the first version of the paper. This indicates that the conducted analysis is robust against some forms of uncertainties. But of course, as mentioned previously, a deeper analysis would be required to state more general conclusions.

Impact of the adjustment phase

Always for the typical scenario presented in the general setting part, Figure 9 depicts the evolution of the number of infected people (in France) for the policy effectively implemented over the period [March 1, September 30]. Here, only the adjustment phase is assumed to be optimizable. The figure allows one to quantify the impact of the severity level of the adjustment phase. Without the adjustment phase, the fraction of infected people is such that the number of people requiring intensive care exceeds the double of the maximum ICU capacity of France. However, by implementing measures such that R(t) < 1.2 over the adjustment phase, the constraint on the ICU capacity is not violated by the end of 2020. Furthermore, to avoid the overwhelming health service for a longer time, it is better to implement measures such that R(t) < 1 over the adjustment phase since the fraction of infected people will be non-increasing with R(t) < 1.

DISCUSSION

In this work, we propose to model the behavior of a government as far as the epidemic control is concerned.

The proposed model, despite its simplicity, has the merit of being able to capture the fundamental tradeoff between economic and health aspects. Obviously, to capture other effects such as the psychological effects of measures on people, a more general model should be considered. The proposed model allows one to provide quantitative answers to issues which have been well commented in the media. For example, even when a government chooses to assign a high value to the economic aspect, it is seen that the best strategy is almost always to implement a severe phase as soon as possible. This severe phase involves locking the population down, as most countries did, or to make intensive use of digital tracing and face masks as South Korea did. In the latter case, a loss in terms of privacy is the price to be paid for having more freedom.

When inspecting the obtained numerical results performed for France, it is seen that the optimal features for the lockdown/severe phase require targeting a reproduction number smaller than the one achieved in No adjustment phase Adjustment phase starts with R 3 =1.2

Adjustment phase starts with R 3 =0.9

Adjustment phase starts with R 3 =0.6

Starting day of the adjustment phase

  figure presented in Sec 3.4, for computational convenience, we take T = 210 days corresponding to 7 months from March 1st to September 30th) is assumed. The sets for the possible lockdown starting days, the lockdown duration (in days), post-lockdown phase duration, and the reproduction numbers are as follows: τ 0 ∈ {1, 2, ..., 30}, τ 1 ∈ {T min , T min + 1, ..., 90}, τ 2 ∈ {1, 2, ..., 120}, R 1 ∈ {0.4, 0.2, ..., 1.5}, R 2 ∈ {0.4, 0.2, ..., 1.5}, R 3 ∈ {0.4, 0.2, ..., 1.5}. Excluding Figure5, due to the physical characteristics of the epidemics in France, we set T min = 30 because the lockdown duration is at least 4 weeks or 1 month to make the lockdown effective in real systems[24][25][26]. The SEIR model parameters are as follows:

1 = 55 days with R France 0 = 3 . 5

 155035 and R France 1

Figure 3 .

 3 Figure3. The plots represent the possible tradeoffs between health cost (measured in terms of the total number of infected people) and economic cost (measured in terms of GDP loss) that can be obtained (by choosing the best epidemic management policy). In particular, with the assumed model, it is seen, in retrospect, that it would have been possible to divide the number of infected people by about 6 while saving about 10 billions e in terms of GDP with the optimal 4-phase strategy.

Figure 5 .

 5 Figure 5. When economic losses are assumed to be uniform over time (µ = 1), it is optimal to have a long lockdown (typically between 60 -80 days) whatever the tradeoff desired. But, if the economic losses of the third phase are less than during lockdown, the lockdown phase should be shorter.

  at 26/06 with R3=1.5 Post-lockdown phase starts at 03/04 with R2=0.7 Lockdown starts at 02/03 with R1=0.4

Fig. 8

 8 depicts the average bias for τ 0 and R 1 induced by uncertainty on R 0 . The average biases for τ 0 and R 1 are defined by

Reproduction number in lockdown phase R 1 Figure 7 .

 17 Figure 7. Comparison of lockdown policies with different R 0 . When the situation is worse, a sooner and more strict lockdown strategy should be applied.

France ( 0

 0 .4 vs 0.6), while having a shorter duration for the severe phase (1 month instead of 2). Note Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure 9 .

 9 Figure 9. The figure shows the impact of the adjustment phase. Only by imposing R(t) < 1.2 over this phase, the number of people admitted in ICU does not exceed the capacity of France by the end of 2020.

  capacity constraint) we see that the best epidemic management strategy leads to a GDP loss over the entire study period [March 1, September 30] is about 206 billions e with 7.16 millions infected, and 15 000 patients requiring intensive care. At the other extreme, when α is relatively small (α = 10 -7 ), the GDP loss reaches values as high as 295 billion e with a total number of newly infected people over the period [March 1, September 30] as low as 23 162. To evaluate the efficiency of the epidemic management strategy of the French government policy, we have represented the point corresponding to the estimated number of infected people and GDP loss by September 30; with our model, the GDP loss over the period of time of interest is 241 billion e and the total number of infected people is about 6.88 million. What the best

	200	220	240	260	280	300

tradeoff curve indicates is that there were management policies that would allow the French government to have a better "performance" both in terms of GDP loss and the number of infected people. For instance, we indicate a point for which it would have been possible to have about 1.05 million people infected (that is, about 6 times less than what is estimated with the current policy) while ensuring a total GDP loss of 231

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ACKNOWLEDGMENTS

The authors acknowledge the support of the French National Research Agency ANR and the French National Research Center CNRS. Part of this manuscript has been released as a pre-print at medRxiv [31].

FUNDING

This research work has been conducted within the ANR NICETWEET Project.

DATA AVAILABILITY STATEMENT

Publicly available datasets were analyzed in this study. This data can be found in the following website: www.santepubliquefrance.fr.

that some countries adopted measures which were more severe than France. For instance, China has been imposing the use of a given food supply system which was very efficient in terms of epidemic mitigation [START_REF] Wu | Characteristics of and important lessons from the coronavirus disease 2019 (COVID-19) outbreak in China: summary of a report of 72 314 cases from the Chinese Center for Disease Control and Prevention[END_REF][START_REF] Lau | The positive impact of lockdown in Wuhan on containing the COVID-19 outbreak in China[END_REF][START_REF] Fang | Human mobility restrictions and the spread of the novel coronavirus (2019-ncov) in china[END_REF]. Then, by planning an adjustment phase at the right time, we have seen that as a final result, the number of infected people can be reduced by a factor of 6 when compared to the current French policy (1.05 million of infected instead of 6.88 millions for the current policy) while having similar GDP losses (231 billions e instead of 241 billions e). We have seen that by considering a simple model as we have studied, the need for an adjustment phase could have been anticipated. Such a phase is necessary to avoid the number of patients under intensive care exceeding the capacity of the ICUs. Also, because of the natural tendency of humans to deviate from rules over time, it appears that measures should be updated about every month and not less frequently.

AUTHOR CONTRIBUTIONS

SL, CZ, VV, and CM designed the proposed model. SL and CZ conducted the review of the related literature. SL collected the data to calibrate the assumed model. CZ and SL performed the simulations. All authors contributed to the writing of the manuscript.