J Florescu

A Thariat

Rambeau

M Humbert

M De

La Losa

B Gery

C Florescu

J Thariat

Audrey Rambeau
email: a.rambeau@baclesse.unicancer.fr

Routine feasibility of postoperative chemoradiotherapy in head and neck squamous cell carcinoma at high risk of recurrence

Keywords: advanced head and neck cancer, postoperative chemoradiotherapy, squamous cell carcinoma, adjuvant treatment, treatment interruption

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

INTRODUCTION

Adjuvant postoperative radiation therapy has been standard practice in head and neck squamous cell carcinoma (HNSCC) at high risk of recurrence since the 1970s [START_REF] Fletcher | Radiotherapeutic Management of Surgical Recurrences and Postoperative Residuals in Tumors of the Head and Neck[END_REF]. Subsequently, several studies assessed adjuvant chemotherapy in grade III and IV SCC [START_REF] Stell | Adjuvant chemotherapy in head and neck cancer[END_REF][START_REF] Jassem | Chemotherapy in locally advanced head and neck cancer : a critical reappraisal[END_REF], and cisplatin associated to radiation therapy was shown to improve results [START_REF] Pignon | Meta-analysis of chemotherapy in head and neck cancer (MACH-NC): an update on 93 randomised trials and 17,346 patients[END_REF]. In the early 2000s, phase-III trials by the European Organization for Research and Treatment of Cancer (EORTC) [START_REF] Bernier | Postoperative irradiation with or without concomitant chemotherapy for locally advanced head and neck cancer[END_REF] and Radiation Therapy Oncology Group (RTOG) [START_REF] Cooper | Postoperative Concurrent Radiotherapy and Chemotherapy for High-Risk Squamous-Cell Carcinoma of the Head and Neck[END_REF] reported overall survival and locoregional control to be improved by 3 concomitant cisplatin courses (D1-D22-D43) at 100 mg/m² in HNSCC at high risk of recurrence [START_REF] Bernier | Defining risk levels in locally advanced head and neck cancers: A comparative analysis of concurrent postoperative radiation plus chemotherapy trials of the EORTC (#22931) and RTOG (#9501)[END_REF]. In 2010, The European Society of Medical Oncology (ESMO) recommended concomitant chemoradiotherapy after grade III-IV HNSCC surgery with microscopically positive margins (R1) and/or extracapsular rupture [START_REF] Grégoire | Squamous cell carcinoma of the head and neck: EHNS-ESMO-ESTRO Clinical Practice Guidelines for diagnosis, treatment and follow-up[END_REF]. Interruption of the radiation therapy is mainly due to toxicity [START_REF] Bernier | Postoperative irradiation with or without concomitant chemotherapy for locally advanced head and neck cancer[END_REF][START_REF] Cooper | Postoperative Concurrent Radiotherapy and Chemotherapy for High-Risk Squamous-Cell Carcinoma of the Head and Neck[END_REF], which is more frequent in case of associated chemotherapy and impairs prognosis, [START_REF] Bernier | Postoperative irradiation with or without concomitant chemotherapy for locally advanced head and neck cancer[END_REF][START_REF] Cooper | Postoperative Concurrent Radiotherapy and Chemotherapy for High-Risk Squamous-Cell Carcinoma of the Head and Neck[END_REF][START_REF] Bachaud | Combined postoperative radiotherapy and weekly cisplatin infusion for locally advanced Head and Neck carcinoma: Final report of a randomized trial[END_REF][START_REF] Zorat | Randomized Phase III Trial of Neoadjuvant Chemotherapy in Head and Neck Cancer: 10-Year Follow-Up[END_REF].

The main aim of the present study was to assess completion rates for postoperative chemoradiotherapy and the impact of various clinical factors. Secondary aims comprised assessment of the impact of treatment completion on progression-free (PFS) and overall survival (OS), and acute and late toxicity.

METHODS

A single-center retrospective study was performed in the Cancer Center of Caen (France), with approval from the CNIL data protection commission (n°2090357).

All patients receiving postoperative chemoradiotherapy for HNSCC at high risk of recurrence between 2007 and 2016 were included. Surgery was required to have been curative.

Patients with history of neck irradiation were excluded. Treatment was considered complete for 3 chemotherapy courses and full radiation dose without ≥3 days' interruption. Malnutrition was defined by body-mass index (BMI) <18 kg/m² or >5% body-weight loss. Toxicity was assessed according to the Common Terminology Criteria for Adverse Events (CTCAE, version 4.0) [START_REF]Common Terminology Criteria for Adverse Events (CTCAE) Version 4[END_REF]. Comorbidity was considered severe for scores ≥3 on any item of the Cumulative Illness Rating Scale (CIRS).

Quantitative variables were reported as median [range], and qualitative variables as number and percentage. The Fisher exact test was used to compare categorical variables and Kruskal-Wallis test for continuous variables continues. PFS and OS were estimated following Kaplan-Meier and compared on log-rank test [START_REF] Kaplan | Nonparametric Estimation from Incomplete Observations[END_REF]. The significance threshold was set at p < 0.05; p-values < 0.005 were considered suggestive [START_REF] Arends | ESPEN Guidelines on Enteral Nutrition: Non-surgical oncology[END_REF].

RESULTS

Population data

106 of the 163 patients receiving concomitant cisplatin chemoradiotherapy between 2007 and 2016 were retrospectively included (Figure 1).

There was male predominance: sex ratio 5.6. Median age was 58 years (range, 22-72 years).

Table 1 shows clinical characteristics. 27 patients (25.4%) showed malnutrition at diagnosis, 12 of whom received pre-treatment enteral feeding. 26 patients (24.5%) had severe comorbidity: cardiac pathology (n=7), lower-limb arteriopathy obliterans (n=7), high blood pressure (n=5), neurologic pathology (n=3), chronic lymphoid leukemia (n=3), or type-2 diabetes (n=3). 61 patients (67%) were active smokers and/or with active alcohol abuse.

Treatment

Surgery mostly consisted in: oropharyngectomy + neck dissection (n=18), exclusive neck dissection (n=18), interruptive pelvimandibulectomy + neck dissection (n=14), pharyngolaryngectomy + neck dissection (n=14), and partial glossectomy + neck dissection (n=9). Surgery included reconstruction in 34 cases (32%): free flap, n=16; local flap, n=15. Mean interval between surgery and chemoradiotherapy was 8.6 weeks (95% confidence interval (CI), [8.4-9.2]), and was >6 weeks in 96 cases (90.6%). Prognostic factors for chemoradiotherapy comprised: extracapsular rupture (n=80, 75.5%), ≥3 positive lymph nodes (n=57, 53.8%), R1 margins (n=29, 27.4%), perineural invasion (n=13, 12.3%), vascular embolism (n=8, 7.5%), or lymphatic embolism (n=5, 4.7%).

Scheduled radiation dose was 70 Gy (n=89, 84.8%), 66 Gy (n=11, 10.4%), 62 Gy (n=1), or 60 Gy (n=5, 4.8%). Radiation therapy was conformational in 57 cases (53.8%) and IMRT in 49 (46.2%)

Feasibility and acute tolerance of chemoradiotherapy

Adjuvant treatment was complete in 61 cases (57.5%). Low creatinine clearance was significantly associated with incomplete treatment (p=0,003); >5% body-weight loss during treatment showed suggestive association (p=0.026). (Table 2).

Radiation therapy was interrupted for more than 3 days in 16 patients: grade 3 cutaneomucosal toxicity (n=5), lung infection (n=3), severe weight-loss (n=5), general health deterioration (n=2).

Seven patients showed radiodermitis and 18 showed grade-3 radiomucositis. 72 patients (68.8%) received all 3 cisplatin courses, including 3 with dose reduction at the last course; 27 received 2 courses (25.5%), and 7 only 1. Reasons for interruption comprised: acute kidney failure (n=16), hepatotoxicity (n=17), general health deterioration (n=4), infection (n=3), severe mucositis (n=3), and ototoxicity (n=2). Five patients showed chemotherapy-induced grade 3-4 vomiting (4.7%). Two showed febrile neutropenia. Patients with initial malnutrition received significantly fewer courses (p=0.001), as did those with low glomerular filtration rate (p=0.001).

Enteral feeding during treatment showed suggestive association with incomplete chemotherapy (p=0.022). 37 patients had unscheduled hospital admission (34.9%): acute kidney failure (n=16), enteral feeding (n=5), sepsis (n=6), pain secondary to mucositis (n=5), vomiting (n=3), or other (n=2).

Survival

42 patients died during follow-up, including 23 from disease progression. Five-year OS was 58.9%. Median OS was 84.9 months (Figure 2). 32 patients showed recurrence (30.2%). Median PFS was 12.6 months [range, 1.7-89.4 months]. Five-year recurrence-free survival was 62.7%. Recurrence locations were locoregional only (n=8), metastatic only (n=16), or locoregional and metastatic (n=5). Five patients had a metachronous head and neck secondary location.

Complete chemoradiotherapy was not significantly associated with improved survival (p=0.81 for PFS and p=0.441 for OS). Likewise, non-interruption of radiation therapy was not associated with better PFS or OS (p=0.2872 and p=0.244, respectively), and nor was administration of all 3 chemotherapy courses (p=0.196 and p=0.707, respectively).

Late toxicity

Post-treatment kidney function data were available for 87 patients. Glomerular filtration rate was <60 mL/min in 13 patients, including 3 with <30 mL/min. Risk of chronic kidney failure showed a suggestive relation to age (p=0.019).

Nineteen of the 36 patients (52.8%) requiring enteral feeding during treatment recovered exclusively oral feeding within 3 months; 10 (9.4%) were never withdrawn from enteral feeding. There were 10 cases of osteoradionecrosis (9.4%). Primary sites comprised oral cavity (n=6), oropharynx (n=3) and oral cavity + oropharynx (n=1). Median post-treatment interval to onset was 14 months [range, 1.4-71.5 months]. Osteoradionecrosis was significantly associated with radiation dose (p=0.021) and suggestively associated with pre-treatment malnutrition (p=0.021).

DISCUSSION

The present study assessed the feasibility of postoperative chemoradiotherapy in locally advanced HNSCC displaying negative prognostic features. Treatment was performed as scheduled in 57.5% of cases. All 3 cisplatin courses were administered in 68%. The study population was older than the EORTC and RTOC cohorts (7): median age, 58 years versus respectively 53 and 54 years.

Primary sites were mostly oral cavity and oropharynx or, less often, larynx. Tumor stages and rates of R1 margins, capsule rupture and perineural invasion were comparable. Almost a quarter of patients had at least 1 severe comorbidity and would have been excluded from most therapeutic trials. 67% of patients were active smokers and/or alcohol abusers. Even so, treatment was complete in 57.5% of cases. Bernier [START_REF] Bernier | Postoperative irradiation with or without concomitant chemotherapy for locally advanced head and neck cancer[END_REF] reported a completion rate of 49%, and a study of exclusive chemoradiotherapy reported 58.9% [START_REF] Baine | Chemoradiotherapy for locally advanced squamous cell carcinoma of the oropharynx: Does completion of systemic therapy affect outcomes?[END_REF]. The only significant risk factor for treatment interruption was low pre-treatment glomerular filtration rate (p=0.003), although there was a suggestive relation to >5% weight-loss during treatment (p=0.026). Almost one-third of patients required unscheduled hospital admission, mainly for cisplatin nephrotoxicity or radiomucositis-related symptoms. Kidney function is critical for treatment. One-third of interruptions of chemotherapy directly implicated nephrotoxicity, which also accounted for half of the unscheduled admissions. Factors for cisplatin nephrotoxicity according to Motwani comprise age, dose, high blood pressure and hypoalbuminemia [START_REF] Motwani | Development and Validation of a Risk Prediction Model for Acute Kidney Injury After the First Course of Cisplatin[END_REF]. HNSCC requires high cisplatin doses of 100 mg/m² in patients who are often malnourished and thus at high risk of toxicity. In the present study, malnutrition was a factor for incomplete chemotherapy; missing retrospective data prevented analysis of blood pressure and hypoalbuminemia.

European [START_REF] Arends | ESPEN Guidelines on Enteral Nutrition: Non-surgical oncology[END_REF] and American [START_REF] Elliot | The clinical guide to oncology nutrition[END_REF] nutrition Societies recommend close dietary monitoring and oral supplementation to prevent interruption of radiotherapy in HNSCC. Enteral feeding is recommended in case of high mucositis risk, but intensive nutritional treatment is not necessary [START_REF] Roussel | Intensive nutritional care for patients treated with radiotherapy in head and neck cancer: a randomized study and meta-analysis[END_REF].

In the present study, severe weight-loss and enteral feeding were associated with interruption of chemotherapy. Despite a pre-treatment malnutrition rate of 25.4%, only half of these patients had enteral feeding. Preventing weight-loss during treatment is essential to avoid interruption. Patients with large pharyngeal, oral or supraglottic tumor show a malnutrition risk of 85%, compared to 7% for glottic primaries [START_REF] Anderson | Pretreatment risk stratification of feeding tube use in patients treated with intensity-modulated radiotherapy for head and neck cancer[END_REF]. Therefore, advanced T or N oral or oropharyngeal primary may indicate preventive gastrostomy to avoid malnutrition and dehydration in case of high toxicity risk.

The present retrospective design precluded reliable analysis of late toxicity such as ototoxicity or neurotoxicity. Analysis of causes of death was likewise unable to implicate disease progression.

Mortality due to cisplatin toxicity could not be assessed, which was a limitation in the study of late toxicity.

Parson et al. [START_REF] Parsons | An analysis of factors influencing the outcome of postoperative irradiation for squamous cell carcinoma of the oral cavity[END_REF] stressed the role of treatment duration: each week of interruption of radiation therapy is associated with 10-12% loss of local control [START_REF] Bentzen | Radiobiological considerations in the design of clinical trials[END_REF][START_REF] Withers | Transmutability of dose and time commentary on the first report of RTOG 90003[END_REF]. Interruption of radiation therapy plays a major role in tumor progression [START_REF] Rosenthal | A review of neoadjuvant chemotherapy for head and neck cancer: Partially shrunken tumors may be both leaner and meaner[END_REF][START_REF] Trotti | Postoperative accelerated radiotherapy in high-risk squamous cell carcinoma of the head and neck: long-term results of a prospective trial[END_REF][START_REF] Rosenthal | Importance of the treatment package time in surgery and postoperative radiation therapy for squamous carcinoma of the head and neck[END_REF]. Several studies showed that intervals of more than 6 weeks between surgery and initiation of radiation therapy are associated with poorer locoregional control [START_REF] Rosenthal | Importance of the treatment package time in surgery and postoperative radiation therapy for squamous carcinoma of the head and neck[END_REF][START_REF] Graboyes | Effect of time to initiation of postoperative radiation therapy on survival in surgically managed head and neck cancer[END_REF]. Baine et al. reported poorer survival in case of failure to complete concomitant chemotherapy [START_REF] Baine | Chemoradiotherapy for locally advanced squamous cell carcinoma of the oropharynx: Does completion of systemic therapy affect outcomes?[END_REF]. These prognostic factors did not emerge in the present study. The median interval between surgery and initiation of adjuvant therapy was 8.6 weeks, due to the high rates of reconstruction [START_REF] Cheng | Variables Associated With Feeding Tube Placement in Head and Neck Cancer[END_REF] (32%) and of severe comorbidity (24.5%) delaying cicatrization. Nevertheless, 5-year OS and PFS were 58.9% and 30% respectively, comparable to literature reports [START_REF] Bernier | Postoperative irradiation with or without concomitant chemotherapy for locally advanced head and neck cancer[END_REF][START_REF] Bernier | Defining risk levels in locally advanced head and neck cancers: A comparative analysis of concurrent postoperative radiation plus chemotherapy trials of the EORTC (#22931) and RTOG (#9501)[END_REF][START_REF] Kiyota | Adjuvant treatment for post-operative head and neck squamous cell carcinoma[END_REF].

CONCLUSION

The present rate of complete postoperative chemoradiotherapy was comparable to that in previous therapeutic trials. Concomitant cisplatin therapy is feasible as a routine attitude, despite comorbidities. Early nutritional support is a key to preventing weight loss and dehydration and unscheduled hospital admission, enabling chemoradiotherapy to be conducted under optimal conditions.

  Criteria for high recurrence risk comprised: lymph-node with capsule rupture, positive margins, ≥3 positive lymph nodes, perineural invasion and vascular or lymphatic embolism. Screening used the chemotherapy prescription software. 3D conformational radiation therapy used photons and electron beams between 2007 and 2013, or intensity modulation (IMRT) with integrated boost by 6MV photon beam as of 2011. Radiation was delivered in 5 fractions per week at 2 Gy per fraction. Concomitant chemotherapy used cisplatin at 100 mg/m² every 3 weeks (D1-D22-D43) associated to hyperhydration and prophylactic antiemetics.

Figure 1 :

 1 Figure 1: Inclusions flowchart.

Figure 2 :

 2 Figure 2: Progression-free (PFS) and overall survival (OS) according to Kaplan-Meier with 95%

Figure 1

 1 Figure 1: Inclusions flowchart

Figure 2 :

 2 Figure 2: Progression-free (PFS) and overall survival (OS) according to Kaplan-Meier with 95%

Table 1 :

 1 Patient data at diagnosis WHO: World Health Organization, NA: not available

	confidence intervals

Table 2 :

 2 Clinical factors and treatment completion IMRT: Intensity Modulated Radiation Therapy * during treatment


Bruno Vergès
email: bruno.verges@chu-dijon.fr

Cécile Dubois

Pauline Legris

Sous rubrique Mise au point Inhibiteurs PCSK9 en pratique clinique : données et interrogations PCSK9 inhibitors in clinical practice: Data and questions

Keywords:

Les inhibiteurs de la proprotein convertase subtilisin/kexin type 9 (PCSK9) sont de nouveaux traitements de l'hypercholestérolémie. Ils induisent une diminution marquée du taux de LDL-cholestérol plasmatique de 50 à 60 %. Parallèlement, ils réduisent significativement le risque d'évènements cardiovasculaires, comme cela a été démontré dans plusieurs études prospectives. Par ailleurs, les inhibiteurs PCSK9 diminuent le taux plasmatique de lipoprotéine (a) [Lp(a)] d'environ 25 %. Il semble que cette réduction du taux de Lp(a) participe au bénéfice cardiovasculaire, noté avec les inhibiteurs PCSK9, en particulier chez les patients ayant des valeurs de base de Lp(a) augmentées. Les mécanismes responsables de la baisse de Lp(a) sous anti-PCSK9 ne sont pas encore clarifiés. Le traitement par inhibiteurs PCSK9 est très efficace dans les hypercholestérolémies familiales sévères et permet de réduire, voire de supprimer, les traitements par LDL-aphérèses. Les données des études cliniques n'indiquent pas d'augmentation du risque de diabète sous inhibiteurs PCSK9. Cependant, la durée relativement courte de la plupart des études ne permet pas d'éliminer formellement un risque à moyen ou long terme.

Introduction

Les inhibiteurs de la proprotein convertase subtilisin/kexin type 9 (PCSK9) sont à notre disposition, depuis une période récente, pour le traitement de l'hypercholestérolémie. En France, leur prise en charge par l'Assurance Maladie est limitée aux formes sévères des hypercholestérolémies familiales. Dans le présent article, nous analyserons les effets des inhibiteurs PCSK9 sur les lipides et la prévention des accidents cardiovasculaires, puis nous aborderons les questions de leur efficacité chez les patients porteurs de formes sévères traitées par LDL-aphérèse, chez les sujets ayant une hypercholestérolémie familiale homozygote et chez les patients diabétiques, ainsi que de leurs effets secondaires potentiels.

Rappel sur PCSK9 et son histoire

En 2003, était rapporté, pour la première fois, une hypercholestérolémie familiale autosomique dominante liée à une mutation activatrice du gène de la proprotein convertase subtilisin/kexin type 9 (PCSK9) [START_REF] Abifadel | Mutations in PCSK9 cause autosomal dominant hypercholesterolemia[END_REF]. En 2005, plusieurs mutations « perte de fonction » de PCSK9 étaient décrites chez des sujets dont le taux de LDLcholestérol (LDL-C) était bas et le risque cardiovasculaire très faible [START_REF] Cohen | Low LDL cholesterol in individuals of African descent resulting from frequent nonsense mutations in PCSK9[END_REF][START_REF] Cohen | Sequence variations in PCSK9, low LDL, and protection against coronary heart disease[END_REF]. C'est à partir de ces observations que plusieurs travaux scientifiques ont pu déterminer la fonction de PCSK9 dans le métabolisme du LDL-C. PCSK9 est une pro-convertase qui est majoritairement exprimée et sécrétée par le foie [START_REF] Seidah | PCSK9: a key modulator of cardiovascular health[END_REF]. Une fois dans la circulation sanguine, PCSK9 se lie au domaine extracellulaire du récepteur au LDL (LDLR) présent à la surface des cellules. Après sa liaison au LDL-C, lorsque le LDLR est internalisé dans la cellule, sa liaison à PCSK9 le dirige vers la voie de dégradation lysosomale plutôt que vers son recyclage à la membrane plasmique [START_REF] Seidah | PCSK9: a key modulator of cardiovascular health[END_REF]. Ainsi, PCSK9 agit comme un inhibiteur endogène du LDLR. Les observations d'une accélération des lésions athérosclérotiques chez les animaux surexprimant PCSK9 et de leur réduction franche chez les animaux PCSK9knock-out (KO), a clairement montré l'intérêt potentiel d'une inhibition de PCSK9 afin de réduire le taux de LDL-C et le risque cardiovasculaire [START_REF] Seidah | PCSK9: a key modulator of cardiovascular health[END_REF]. C'est ainsi qu'une recherche a été menée afin de développer des agents inhibants PCSK9. Les premiers inhibiteurs de PCSK9 mis au point sont des anticorps monoclonaux humains dirigés contre la forme circulante de PCSK9. Actuellement, deux anticorps monoclonaux ont reçu une Autorisation de mise sur le marché (AMM) : l'alirocumab (Praluent ® , Sanofi/Regeneron) et l'évolocumab (Repatha ® , Amgen). Ces agents sont administrés par voie sous-cutanée.

Effets des inhibiteurs PCSK9 sur les lipides

Les anticorps anti-PCSK9 réduisent de façon très significative le taux plasmatique de LDL-C. Dans l'étude OSLER, réalisée chez des patients hypercholestérolémiques (70 % sous statines, 14 % sous ézétimibe), l'évolocumab, à la dose de 420 mg 1 fois par mois, ou 140 mg toutes les 2 semaines, a permis une réduction du taux plasmatique de LDL-C de 61 %, versus placebo [START_REF] Sabatine | Open Label Study of Long-Term Evaluation against LDL Cholesterol (OSLER) Investigators. Efficacy and safety of evolocumab in reducing lipids and cardiovascular events[END_REF]. Dans l'étude ODYSSEY, réalisée chez des patients hypercholestérolémiques (hypercholestérolémie familiale hétérozygote ou patients à très haut risque vasculaire) ayant un taux de LDL-C > 0,70 g/L sous dose maximale de statine tolérée, l'alirocumab à la dose de 150 mg toutes les 2 semaines a entraîné une réduction du taux de LDL-C plasmatique de 53 % par rapport au placebo [START_REF] Robinson | Efficacy and safety of alirocumab in reducing lipids and cardiovascular events[END_REF]. De façon non surprenante, les traitements pas inhibiteurs PCSK9 entrainent une réduction très significative du non-HDL cholestérol (en moyenne de 52 %) et du taux plasmatique d'apolipoprotéine B (apoB) (de 47 à 53 %) [START_REF] Sabatine | Open Label Study of Long-Term Evaluation against LDL Cholesterol (OSLER) Investigators. Efficacy and safety of evolocumab in reducing lipids and cardiovascular events[END_REF][START_REF] Robinson | Efficacy and safety of alirocumab in reducing lipids and cardiovascular events[END_REF].

Un autre effet lipidique important à signaler sous inhibiteurs PCSK9 est la réduction significative d'environ 25 % du taux de lipoprotéine(a) [Lp(a)] [START_REF] Sabatine | Open Label Study of Long-Term Evaluation against LDL Cholesterol (OSLER) Investigators. Efficacy and safety of evolocumab in reducing lipids and cardiovascular events[END_REF][START_REF] Robinson | Efficacy and safety of alirocumab in reducing lipids and cardiovascular events[END_REF]. Ce point sera développé ultérieurement dans cet article. Les effets des inhibiteurs PCSK9 sur les triglycérides (baisse de 12 à 17 %) et le HDL-cholestérol (augmentation de 4,6 à 7 %) sont modestes [START_REF] Sabatine | Open Label Study of Long-Term Evaluation against LDL Cholesterol (OSLER) Investigators. Efficacy and safety of evolocumab in reducing lipids and cardiovascular events[END_REF][START_REF] Robinson | Efficacy and safety of alirocumab in reducing lipids and cardiovascular events[END_REF].

Bénéfices cardiovasculaires des inhibiteurs PCSK9

Le bénéfice cardiovasculaire des inhibiteurs PCSK9 a clairement été démontré par plusieurs études prospectives. L'étude FOURIER a comparé l'effet de l'évolocumab (140 mg toutes les 2 semaines, ou 420 mg par mois) versus placebo, sur la survenue d'évènements cardiovasculaires, chez 27 564 patients avec des antécédents cardiovasculaire et un taux de LDL-C > 0,70 g/L sous statines, pendant un suivi médian de 2,2 années [START_REF] Sabatine | Evolocumab and clinical outcomes in patients with cardiovascular disease[END_REF]. Le traitement par évolocumab a permis une réduction significative de 15 % de l'objectif principal composé des décès cardiovasculaires, des infarctus du myocarde (IDM) et des accidents vasculaires cérébraux (AVC) non fatals, des hospitalisations pour angor instable et des revascularisations coronaires (p < 0,001) [START_REF] Sabatine | Evolocumab and clinical outcomes in patients with cardiovascular disease[END_REF]. Il était observé aussi, dans l'étude FOURIER, une réduction significative de l'objectif secondaire principal (décès cardiovasculaire, IDM et AVC non fatals combinés) de 20 % (p < 0,001), des IDM de 27 % (p < 0,001), des AVC de 21 % (p = 0,001), et des revascularisations coronaires de 22 % (p < 0,001) [START_REF] Sabatine | Evolocumab and clinical outcomes in patients with cardiovascular disease[END_REF]. L'étude ODYSSEY OUTCOMES a comparé l'alirocumab versus placebo, sur la survenue d'évènements cardiovasculaires, chez 18 924 patients après un syndrome coronaire aigu, pendant un suivi médian de 2,8 années [START_REF] Szarek | Alirocumab reduces total nonfatal cardiovascular and fatal events: The ODYSSEY OUTCOMES Trial[END_REF]. Il été observé, sous alirocumab, une réduction significative de 13 % des accidents cardiovasculaires non fatals (p < 0,0001) et des décès de 17 % (p = 0,02) [START_REF] Szarek | Alirocumab reduces total nonfatal cardiovascular and fatal events: The ODYSSEY OUTCOMES Trial[END_REF]. Une méta-analyse, à partir de 35 études randomisées contrôlées d'inhibiteurs PCSK9 regroupant 45 539 patients, confirment ces résultats en objectivant une réduction significative des IDM de 28 %, des AVC de 2 %, et des gestes de revascularisation coronaires de 22 % [START_REF] Karatasakis | Effect of PCSK9 inhibitors on clinical outcomes in patients with hypercholesterolemia: A meta-analysis of 35 randomized controlled trials[END_REF].

Effets des inhibiteurs PCSK9 chez les patients sous LDL-aphérèse

L'effet des inhibiteurs PCSK9 chez les patients atteints d'hypercholestérolémie familiale hétérozygote traités par LDL-aphérèse a été étudié dans l'étude ODYSSEY ESCAPE regroupant 62 patients traités par LDL-aphérèse toutes les semaines ou toutes les 2 semaines [START_REF] Moriarty | Alirocumab in patients with heterozygous familial hypercholesterolaemia undergoing lipoprotein apheresis: the ODYSSEY ESCAPE trial[END_REF]. Dans cette étude, le traitement par alirocumab a réduit de 75 % la fréquence des séances de LDL-aphérèse. Au terme de l'étude, en raison d'un contrôle satisfaisant du taux de LDL-C sous alirocumab, 63,4 % des patients n'étaient plus traités par LDL-aphérèse et la fréquence des LDL-aphérèses était diminuée de moitié chez 92,7 % des patients [START_REF] Moriarty | Alirocumab in patients with heterozygous familial hypercholesterolaemia undergoing lipoprotein apheresis: the ODYSSEY ESCAPE trial[END_REF].

Effets

des inhibiteurs PCSK9 chez les patients atteints d'hypercholestérolémie familiale homozygote Dans une étude réalisée chez 49 patients atteints d'hypercholestérolémie familiale homozygote sous hypolipidémiants, non traités par LDL-aphérèse, le traitement par évolocumab a permis, versus placebo, une diminution de 30,9 % du taux de LDL-C plasmatique [START_REF] Raal | Inhibition of PCSK9 with evolocumab in homozygous familial hypercholesterolaemia (TESLA Part B): a randomised, double-blind, placebo-controlled trial[END_REF]. Une réduction du LDL-C était observée chez tous les patients sous évolocumab, sauf un présentant une hypercholestérolémie homozygote par absence d'expression du récepteur au LDL. Chez huit patients atteints d'hypercholestérolémie familiale homozygote, l'évolocumab a permis une réduction très significative de 26,3 % du LDL-C chez les six patients présentant une mutation du gène du récepteur au LDL aboutissant à un LDL-récepteur défectueux, mais s'est montré inefficace chez les deux patients présentant une mutation du gène du récepteur au LDL engendrant une absence d'expression du récepteur [START_REF] Stein | Effect of the proprotein convertase subtilisin/kexin 9 monoclonal antibody, AMG 145, in homozygous familial hypercholesterolemia[END_REF]. Il a été montré in vitro, que l'alirocumab entraînait une augmentation très significative de l'expression des récepteurs au LDL sur les fibroblastes de sujets témoins, et que cet effet était seulement de 41 % sur les fibroblastes de patients atteints d'hypercholestérolémie familiale hétérozygote, de 14 % sur les fibroblastes de patients atteints d'hypercholestérolémie familiale homozygote avec récepteur au LDL défectueux, et nul sur des fibroblastes de patients atteints d'hypercholestérolémie familiale homozygote avec absence d'expression du récepteur au LDL [START_REF] Lambert | Normalization of low-density lipoprotein receptor expression in receptor defective homozygous familial hypercholesterolemia by inhibition of PCSK9 with alirocumab[END_REF]. Ainsi, il apparaît que les inhibiteurs PCSK9 ont une efficacité réelle chez les patients atteints d'hypercholestérolémie familiale homozygote, mais seulement chez ceux porteurs d'une mutation à l'origine d'un récepteur au LDL défectueux, mais pas chez les rares patients porteurs d'une mutation responsable d'une absence d'expression du récepteur au LDL.

Inhibiteurs PCSK9 et Lp(a)

Les inhibiteurs PCSK9 entrainent une réduction d'environ 25 % du taux plasmatique de Lp(a) [START_REF] Raal | Reduction in lipoprotein(a) with PCSK9 monoclonal antibody evolocumab (AMG 145): a pooled analysis of more than 1,300 patients in 4 phase II trials[END_REF][START_REF] Cao | A meta-analysis of the effect of PCSK9monoclonal antibodies on circulating lipoprotein (a) levels[END_REF]. Les mécanismes en cause ne sont pas encore complétement élucidés, mais plusieurs hypothèses sont avancées. Il faut savoir que PCSK9 est retrouvée présente sur la particule Lp(a) [START_REF] Tavori | PCSK9 association with lipoprotein(a)[END_REF]. Par ailleurs, il est noté, dans les différentes études cliniques, une certaine discordance entre la réduction du taux de LDL-C et de celle du taux de Lp(a). En effet, il a été noté, dans une étude clinique, une réduction moyenne du LDL-C de 52,8 %, alors que celle de la Lp(a) n'était que de 20,2 % [START_REF] Edmiston | Discordant response of low-density lipoprotein cholesterol and lipoprotein(a) levels to monoclonal antibodies targeting proprotein convertase subtilisin/kexin type 9[END_REF]. Par ailleurs, la réduction du LDL-C n'était pas statistiquement différente chez les patients dont la réduction de Lp(a) était inférieure à 10 % comparés à ceux dont la réduction de la Lp(a) était supérieure à 10 % (46 ± 8 % versus 57 ± 16 %, respectivement ; NS), ce qui indique qu'il existe très vraisemblablement des mécanismes spécifiques à la réduction de la Lp(a) sous inhibiteurs PCSK9. Ceci apparaît fortement suggéré par le fait que chez les patients atteints d'hypercholestérolémie familiale homozygote avec absence d'expression du récepteur au LDL, il est noté une réduction significative de Lp(a) alors que le taux de LDL-C demeure inchangé [START_REF] Stein | Effect of the proprotein convertase subtilisin/kexin 9 monoclonal antibody, AMG 145, in homozygous familial hypercholesterolemia[END_REF]. Une étude cinétique réalisée chez des primates a mis en évidence, sous alirocumab, une diminution significative de production de l'apo(a) sans effet sur son catabolisme [START_REF] Croyal | PCSK9 inhibition with alirocumab reduces lipoprotein(a) levels in nonhuman primates by lowering apolipoprotein(a) production rate[END_REF]. Chez des sujets sains, il a été observé une réduction significative de la production de Lp(a), sous alirocumab, sans modification de son catabolisme [START_REF] Watts | Controlled study of the effect of proprotein convertase subtilisin-kexin type 9 inhibition with evolocumab on lipoprotein(a) particle kinetics[END_REF]. In vitro, il a été montré que PCSK9 augmente la sécrétion d'apo(a) d'hépatocytes en culture, et que cet effet est aboli par les inhibiteurs PCSK9 [START_REF] Villard | PCSK9 modulates the secretion but not the cellular uptake of lipoprotein(a) ex vivo: an effect blunted by alirocumab[END_REF]. Si de nombreuses données suggèrent que les inhibiteurs PCSK9 sont susceptibles de réduire la production de Lp(a), les mécanismes moléculaires de cet effet demeurent inconnus. Par ailleurs, il faut noter qu'une accélération du catabolisme de la Lp(a) n'est pas formellement exclue. En effet, certaines données in vitro suggèrent que l'augmentation d'expression du récepteur au LDL à la surface des hépatocytes est susceptible d'accroître le catabolisme de la Lp(a) dans la mesure où l'élévation du nombre de récepteurs au LDL s'accompagne d'une augmentation de fixation de Lp(a) sur la membrane cellulaire [START_REF] Raal | PCSK9 inhibition-mediated reduction in Lp(a) with evolocumab: an analysis of 10 clinical trials and the LDL receptor's role[END_REF]. Si les mécanismes précis à l'origine de la diminution de la Lp(a) sous inhibiteurs PCSK9 demeurent mal compris, il n'en demeure pas moins que cette réduction de Lp(a) participe, en partie, au bénéfice cardiovasculaire observé sous inhibiteurs PCSK9. Ainsi, dans l'étude FOURIER, la réduction des décès cardiovasculaires, des IDM et des revascularisations coronaires urgentes est significativement plus importante chez les patients ayant une Lp(a) initiale > 120 nmol/L comparés aux patients ayant une Lp(a) ≤ 120 nmol/L (25 % versus 11 %, respectivement) [START_REF] O'donoghue | Lipoprotein(a), PCSK9 inhibition, and cardiovascular risk[END_REF]. Une analyse précise des données de l'étude FOURIER fait apparaître que chaque baisse de Lp(a) de 25 nmol/L sous évolocumab se traduit par une réduction de 15 % des évènements cardiovasculaires, et que l'évolocumab permet une réduction des évènements cardiovasculaires de 15 % si, soit le taux de LDL-C, sous traitement, soit celui de Lp(a), sous traitement, est inférieur à la médiane, et de 29 % si les deux valeurs de LDL-C et de Lp(a), sous traitement, sont inférieures à la médiane [START_REF] O'donoghue | Lipoprotein(a), PCSK9 inhibition, and cardiovascular risk[END_REF]. De la même façon, il est noté, dans l'étude ODYSSEY OUTCOMES, une réduction d'autant plus marquée des événements cardiovasculaires majeurs sous alirocumab que le taux initial de Lp(a) est élevé, et que la diminution du taux plasmatique de Lp(a) est un facteur propre de réduction des évènements cardiovasculaires indépendant de la baisse du LDL-C [START_REF] Bittner | Effect of alirocumab on lipoprotein(a) and cardiovascular risk after acute coronary syndrome[END_REF]. Ainsi, chez les patients de l'étude ayant un taux initial de Lp(a) au-dessus du 75 e percentile, la baisse de la Lp(a) sous alirocumab participe pour 27 % à la réduction des évènements cardiovasculaires [START_REF] Bittner | Effect of alirocumab on lipoprotein(a) and cardiovascular risk after acute coronary syndrome[END_REF].

Inhibiteurs PCSK9 et diabète

Dans l'étude ODYSSEY DM-DYSLIPIDEMIA, réalisée chez 413 patients diabétiques de type 2 (DT2) avec hyperlipidémie mixte (non-HDL-cholestérol > 1 g/L et triglycérides compris entre 1,5 et 5 g/L) sous statines, le traitement par alirocumab a permis une réduction du LDL-C de 43,3 %, de non-HDL cholestérol de 37,3 %, de la Lp(a) de 23,7 %, ainsi que du nombre des particules LDL petites et denses, comparé à un traitement hypolipidémiant autre (fénofibrate, oméga-3, niacine) ou une absence de traitement [START_REF] Ray | Alirocumab vs usual lipid-lowering care as add-on to statin therapy in individuals with type 2 diabetes and mixed dyslipidaemia: The ODYSSEY DM-DYSLIPIDEMIA randomized trial[END_REF]. Il n'était pas noté, dans cette étude, de modification de la glycémie à jeun, ni de l'hémoglobine glyquée (HbA1c) sous alirocumab. Les données de l'étude FOURIER, dans laquelle 40 % des patients étaient diabétiques, indiquent que la réduction du LDL-C plasmatique sous évolocumab est similaire chez les patients diabétiques et non-diabétiques [START_REF] Sabatine | Evolocumab and clinical outcomes in patients with cardiovascular disease[END_REF]. Parallèlement, le bénéfice cardiovasculaire sous évolocumab est identique chez les sujets diabétiques et nondiabétiques [START_REF] Sabatine | Evolocumab and clinical outcomes in patients with cardiovascular disease[END_REF]. En raison de l'augmentation du risque de diabète observée sous statines, il est logique de se poser la question d'un éventuel effet diabétogène des inhibiteurs PCSK9. Les données des différentes études cliniques ne permettent pas d'obtenir une réponse claire à la question. Une méta-analyse de 20 études randomisées de phase II ou III avec inhibiteurs PCSK9 versus placebo, regroupant 68 123 patients avec hypercholestérolémie sur un suivi médian de 78 semaines, a mis en évidence une augmentation significative de la glycémie à jeun de 1,88 mg/dL et de l'HbA1c de 0,032 % [START_REF] De Carvalho | Proprotein convertase subtilisin/kexin type 9 (PCSK9) inhibitors and incident type 2 diabetes: a systematic review and meta-analysis with over 96,000 patient-years[END_REF]. Cependant, aucune augmentation du nombre de DT2 n'était retrouvée dans cette méta-analyse. Une autre méta-analyse de 15 études contrôlées randomisées avec inhibiteurs PCSK9, regroupant 27 905 patients sur un suivi médian de 85,5 mois, n'objectivait pas d'augmentation du risque de DT2 [START_REF] Karatasakis | Effect of PCSK9 inhibitors on clinical outcomes in patients with hypercholesterolemia: A meta-analysis of 35 randomized controlled trials[END_REF]. Cependant, il faut noter que la durée moyenne des études est possiblement trop courte pour écarter formellement un risque de DT2 sous inhibiteurs PCSK9. En effet, plusieurs analyses de randomisation Mendéliennes indiquent un risque accru de diabète pour les génotypes PCSK9 associés à une réduction du LDL-C [START_REF] Ference | Variation in PCSK9 and HMGCR and risk of cardiovascular disease and diabetes[END_REF][START_REF] Lotta | Association between low-density lipoprotein cholesterol-lowering genetic variants and risk of type 2 diabetes: a meta-analysis[END_REF][START_REF] Schmidt | PCSK9 genetic variants and risk of type 2 diabetes: a mendelian randomisation study[END_REF]. Cependant, certains éléments sont à signaler, comme le fait que, dans certaines études de randomisation Mendélienne, le risque de diabète n'est observé que chez les sujets ayant une hyperglycémie à jeun non diabétique [START_REF] Ference | Variation in PCSK9 and HMGCR and risk of cardiovascular disease and diabetes[END_REF], ou que le risque de DT2 associé à certains génotypes de PCSK9 serait plus faible que celui associé à des génotypes de l'hydroxy-méthyl-glutaryl-coenzyme A (HMG-CoA) réductase ou de Niemann-Pick C1-like 1 (NPC1L1) donnant des réduction similaires du taux plasmatique de LDL-C [START_REF] Lotta | Association between low-density lipoprotein cholesterol-lowering genetic variants and risk of type 2 diabetes: a meta-analysis[END_REF]. Ainsi, les données de études génomiques et des études cliniques ne permettent pas d'écarter un risque de diabète sous inhibiteurs PCSK9, mais celui-ci pourrait être plus faible que sous statines.

Inhibiteurs PCSK9 et troubles neurocognitifs

Au cours de l'étude OSLER, il a été noté une augmentation des troubles neurocognitifs sous évolocumab versus placebo (0,9 % versus 0,3 %, respectivement). Cependant, cet effet secondaire n'a pas été observé dans les autres études cliniques avec inhibiteurs PCSK9. Dans une méta-analyse de 35 études contrôlées randomisées, il n'était pas noté d'augmentation significative du risque de troubles neurocognitifs sous inhibiteurs PCSK9 (odds ratio [OR] = 1,12 [intervalle de confiance à 95 % : 0,88-1,42]) [START_REF] Karatasakis | Effect of PCSK9 inhibitors on clinical outcomes in patients with hypercholesterolemia: A meta-analysis of 35 randomized controlled trials[END_REF].

Les points essentiels

• Les inhibiteurs PCSK9 sont des agents puissants induisant une diminution très significative du taux de LDL-cholestérol et une réduction des évènements cardiovasculaires majeurs. - ---------------------------------------------------------------------------------------------------------------

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•

  Par ailleurs, ils diminuent le taux plasmatique de lipoprotéine(a) [Lp(a)] d'environ 25 %. Cette diminution de Lp(a) apparaît participer au bénéfice cardiovasculaire observé sous inhibiteurs PCSK9, en particulier en cas de valeur initiale de Lp(a) élevée. • Les inhibiteurs PCSK9 sont très efficaces dans les hypercholestérolémies familiales sévères et permettent de réduire, voire de supprimer, les traitements par LDL-aphérèses. • Les données des études cliniques n'indiquent pas d'augmentation du risque de diabète sous inhibiteurs PCSK9. Cependant, la durée relativement courte de la plupart des études ne permet pas d'éliminer formellement un risque à moyen ou long terme. Déclaration de liens d'intérêt -Bruno Vergès déclare avoir reçu, au cours des 3 dernières années, des honoraires pour conférences et/ou expertise de Amgen, AstraZeneca/Bristol-Myers Squibb, Lilly, Merck Sharp & Dohme (MSD), Novo Nordisk. -Cécile Dubois et Pauline Legris déclarent ne pas avoir de liens d'intérêts.

-

  Glossaire des acronymes des études citées FOURIER : Further cardiovascular OUtcomes Research with PCSK9 Inhibition in subjects with Elevated Risk ODYSSEY : alirocumab, programme global d'essais cliniques de phase 3 ODYSSEY DM-DYSLIPIDEMIA : efficacy and safety of alirocumab versus usual care on top of maximally tolerated statin therapy in patients with type 2 diabetes and mixed dyslipidemia ODYSSEY ESCAPE : study of alirocumab in patients with heterozygous familial hypercholesterolemia (HeFH) undergoing low-density lipoprotein (LDL) apheresis therapy ODYSSEY OUTCOMES : evaluation of cardiovascular OUTCOMES after an acute coronary syndrome during treatment with alirocumab OSLER : Open label Study of evolocumab Long term Evaluation against LDL-C trial -

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Z Liu

P Magal

G Webb

Predicting the number of reported and unreported cases for the COVID-19 epidemics in China, South Korea, Italy, France, Germany and United Kingdom

Keywords: corona virus, reported and unreported cases, isolation, quarantine, public closings; epidemic mathematical model

Introduction

COVID-19 epidemics are currently a global crisis. Our goal in this paper is to develop a mathematical model to guide understanding of the dynamics of COVID-19 epidemics. In previous works [START_REF] Liu | Understanding unreported cases in the 2019-nCov epidemic outbreak in Wuhan, China, and the importance of major public health interventions[END_REF][START_REF] Liu | Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data[END_REF][START_REF] Liu | A COVID-19 epidemic model with latency period[END_REF], our team has developed differential equations models of COVID-19 epidemics. Our goal is to predict forward in time the future number of cases from early reported case data in regions throughout the world. Our model incorporate the following important elements of COVID-19 epidemics: [START_REF] Ahmed | Analysis of Caputo fractional-order model for COVID-19 with lockdown[END_REF] the number of asymptomatic infectious individuals (with no or very mild symptoms), [START_REF]COVID-19 Pandemic Planning Scenarios[END_REF] the number of symptomatic reported infectious individuals (with severe symptoms) and (3) the number of symptomatic unreported infectious individuals (with mild symptoms). With our model, we provide a prediction of the final size of the asymptomatic infectious cases, the reported cases (with severe symptoms) and unreported cases (with mild symptoms). We will apply the model in this paper to the epidemics in China, South Korea, Italy, France, Germany, and the United Kingdom.

The significance of transmission by asymptomatic and mildly symptomatic individuals is of major importance. It is well established that newly infected patients, who are asymptomatic or have only mild symptoms, can transmit the virus [START_REF] Guan | Clinical Characteristics of Coronavirus Disease 2019 in China[END_REF][START_REF] Mizumoto | Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan[END_REF][START_REF] Nishiura | Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19)[END_REF][START_REF] Qiu | Covert coronavirus infections could be seeding new outbreaks[END_REF][START_REF] Rothe | Transmission of 2019-nCoV infection from an asymptomatic contact in Germany[END_REF][START_REF] Wang | Evolving Epidemiology and Impact of Non-pharmaceutical Interventions on the Outbreak of Coronavirus Disease[END_REF][START_REF] Wölfel | Virological assessment of hospitalized patients with COVID-2019[END_REF]. In an early phase of the epidemic, the reported case data grows exponentially, which corresponds to a constant transmission rate. We assume that government measures and public awareness cause this early constant transmission rate to change to a time dependent exponentially decreasing rate. We identify this early constant transmission rate using a method developed in [START_REF] Liu | Understanding unreported cases in the 2019-nCov epidemic outbreak in Wuhan, China, and the importance of major public health interventions[END_REF]. We then identify the time dependent exponentially decreasing transmission rate from reported case data, and project forward the time-line of the epidemic course. Our model is applicable to COVID-19 epidemics in any region with reported case data, which can be updated to higher accuracy with on-going day-by-day reported case data. We refer to [START_REF] Ahmed | Analysis of Caputo fractional-order model for COVID-19 with lockdown[END_REF][START_REF] Chowell | Is it growing exponentially fast? ? Impact of assuming exponential growth for characterizing and forecasting epidemics with initial near-exponential growth dynamics[END_REF][START_REF] Jajarmi | A new fractional HRSV model and its optimal control: A non-singular operator approach[END_REF][START_REF] Kucharski | Early dynamics of transmission and control of COVID-19: a mathematical modelling study[END_REF][START_REF] Mustapha | Fractional modeling for the spread of Hookworm infection under Caputo operator[END_REF][START_REF] Roda | Why is it difficult to accurately predict the COVID-19 epidemic?[END_REF][START_REF] Verity | Estimates of the severity of COVID -19 disease[END_REF] for related results.

The organisation of this paper is as follows. In Section 2 we present a general differential equations model for current COVID-19 epidemics, applicable to any region for which reported case data is available. In Section 3 we provide a general method to parameterize and initialize the model. In Section 4 we apply the model to China, South Korea, Italy, France, Germany, and the United Kingdom. In Section 5 we discuss conclusions obtained from our analysis.

Model

The model consists of a mass action law epidemic model where the asymptomatic infectious and the unreported infectious can transmit the virus

         S (t) = -τ (t)S(t)[I(t) + U (t)], I (t) = τ (t)S(t)[I(t) + U (t)] -νI(t), R (t) = ν 1 I(t) -ηR(t), U (t) = ν 2 I(t) -ηU (t). (2.1)
This system is supplemented by initial data The transmission rate at time t is τ (t). Asymptomatic infectious individuals I(t) are infectious for an average period of 1/ν days. Reported symptomatic individuals R(t) are infectious for an average period of 1/η days, as are unreported symptomatic individuals U (t). We assume that reported symptomatic infectious individuals R(t) are reported and isolated immediately, and cause no further infections. The asymptomatic individuals I(t) can also be viewed as having a low-level symptomatic state. All infections are acquired from either I(t) or U (t) individuals. The fraction f of asymptomatic infectious become reported symptomatic infectious, and the fraction 1 -f become unreported symptomatic infectious. The rate asymptomatic infectious become reported symptomatic is ν 1 = f ν, the rate asymptomatic infectious become unreported symptomatic is ν 2 = (1 -f ) ν, where ν 1 + ν 2 = ν. The cumulative number of reported cases at time t is obtained by using the following equation

S(t 1 ) = S 1 > 0, I(t 1 ) = I 1 > 0, R(t 1 ) = R 1 ≥ 0 and U (t 1 ) = U 1 ≥ 0. ( 2
CR (t) = ν 1 I(t), (2.3)
and the cumulative number of unreported at time t is given by the formula

CU (t) = ν 2 I(t). (2.4)
The parameters and initial conditions of the model are given in Table 1 and a flow diagram of the model is given in Figure 1.

S I R U Symptomatic Asymptomatic τ S[I + U ] ν 1 I ν 2 I Removed η R η U

Method to estimate the parameters and initial conditions

We assume that 100 × f % of symptomatic infectious cases go unreported. The actual value of f is unknown and varies from country to country. We assume η = 1/7, which means that the average period of infectiousness of both unreported symptomatic infectious individuals and reported symptomatic infectious individuals is 7 days. We assume ν = 1/7, which means that the average period of infectiousness of asymptomatic infectious individuals is 7 days. These values can be modified as further epidemiological information becomes known.

At an early stage of the epidemic we assume that all the infected components of the system grow exponentially while the number of susceptible remains unchanged during a relatively short period of time t ∈ [t 1 , t 2 ], which corresponds to a constant transmission rate τ 1 . Therefore, we will assume that

I(t) = I 1 e χ2(t-t1) , R(t) = R 1 e χ2(t-t1) , U (t) = U 1 e χ2(t-t1) . (3.1)
We deduce that the cumulative number of reported satisfies

CR(t) = CR(t 1 ) + t t1 ν 1 I(θ)dθ (3.2)
and by replacing I(t) by the exponential formula in (3.1), we have

CR(t) = CR(t 1 ) + ν 1 I 1 χ 2 e χ2(t-t1) -1 . (3.3)
Hence it makes sense to assume that CR(t) -CR(t 1 ) has the following form

CR(t) -CR(t 1 ) = χ 1 e χ2(t-t1) -χ 3 . (3.4)
By identifying (3.3) and (3.4) we deduce that

χ 1 = χ 3 = ν 1 I 1 χ 2 . ( 3
I (t) = τ 1 S 1 [I(t) + U (t)] -νI(t), R (t) = ν 1 I(t) -ηR(t), U (t) = ν 2 I(t) -ηU (t).
By using (3.1) we obtain   

χ 2 I 1 = τ 1 S 1 [I 1 + U 1 ] -νI 1 , χ 2 R 1 = ν 1 I 1 -ηR 1 , χ 2 U 1 = ν 2 I 1 -ηU 1 .
Computing further, we get

             R 1 = ν 1 χ 2 + η I 1 , U 1 = ν 2 χ 2 + η I 1 , τ 1 = (χ 2 + ν)I 1 S 1 [I 1 + U 1 ] . (3.6)
By (3.5) we deduce that

I 1 = χ 2 χ 3 ν 1 . (3.7)
Finally we obtain

             R 1 = χ 2 χ 3 χ 2 + η , U 1 = ν 2 χ 2 χ 3 ν 1 (χ 2 + η) , τ 1 = (χ 2 + ν)(χ 2 + η) S 1 (ν 2 + η + χ 2 ) . (3.8)
The value of the basic reproductive number is

R 0 = τ 1 S 1 ν 1 + ν 2 1 + ν 2 η , (3.9)
which was derived in [START_REF] Liu | Understanding unreported cases in the 2019-nCov epidemic outbreak in Wuhan, China, and the importance of major public health interventions[END_REF].

During the exponential growth phase of the epidemic, τ (t) ≡ τ 1 is constant. When strong government measures such as isolation, quarantine, and public closings are implemented, we use an exponential decrease for a time-dependent decreasing transmission rate τ (t) to incorporate these effects since the actual effects of these measures are complex. The formula for τ (t) during this phase is

τ (t) = τ 1 , 0 ≤ t ≤ N, τ (t) = τ 1 exp (-µ (t -N )) , N < t.
(3.10)

The date N and the value µ are chosen so that the cumulative reported cases in the numerical simulation of the epidemic aligns with the cumulative reported case data after day N , when the public measures take effect. In this way we are able to project forward the time-path of the epidemic after the government imposed public restrictions take effect. The daily number of reported cases from the model can be obtained by computing the solution of the following equation:

DR (t) = ν 1 I(t) -DR(t), for t ≥ t 1 and DR(t 1 ) = DR 1 .
(3.11)

Predicting the cumulative number of cases

In this section we apply the method described in Section 3 to the data of confirmed cases in China, South Korea, Italy, France, Germany, and the United Kingdom. We predict the time evolution of a COVID-19 epidemic through its phases, we identify epidemic turning points by using the data for the cumulative number of reported cases and the daily number of reported cases, and we project the epidemic final sizes. As we can see in Figure 2, a major difficulty for the predictions from the reported case data, is to determine the date intervals for the exponential growth phase of the epidemic. For South Korea, the date intervals for the exponential growth phase of the epidemic is relatively clear from February 25 to March 1. This is because the measures imposed for contact tracing and social distancing were very strong. For Italy, the interval is more difficult to ascertain. We choose March 1 to March 10, but these values may change as more data of reported cases becomes available. For France, Germany, and the United Kingdom, the interval is very difficult to know with the present data. Again, future reported case data, will clarify these intervals. In general, the dates for the exponential growth interval and the date N , are key elements of our model, and they depend strongly on the implementation of social distancing measures. If these measures are implemented gradually, then the difficulty is increased.

(a) (b) (c)
Another difficulty in applying our model is how to fix the values of the parameters ν, f, η, µ, N . We take η = 1/7, which means that the average period of infectiousness of both unreported symptomatic infectious individuals and reported symptomatic infectious individuals is 7 days, ν = 1/7, which means that the average period of infectiousness of asymptomatic infectious individuals is 7 days. These values follow from the medical and epidemiological information and can be modified as further epidemiological information becomes known. N is around the time point of the implementation of the national prevention and control measures. The values of f is unknown, but information about the level of testing relates to the value of f . A decreased value of f corresponds to a greater final size of the epidemic. The increased testing can increase the value of f . Mortality can also be used as a reference to estimate the value of f . High mortality indicates high unreported ratio. In fact, from the values of f , N and µ, we can also obtain some information of the actual effects of these measures of testing, quarantining and isolation implemented by the governments in these countries.

The principle of our method is the following. By using an exponential best fit method we obtain a best fit of (3.4) to the data over a time [t 1 , t 2 ] and we derive the parameters χ 1 and χ 2 . The values of I 1 U 1 , R 1 and τ 1 are obtained by using (3.7)-(3.8). Next we fix N (first day of public intervention) to some value and we obtain µ by trying to get the best fit to the data.

In the method the uncertainty in our prediction is due to the fact that several sets of parameters (t 1 , t 2 , N, f ) may give a good fit to the data. As a consequence, at the early stage of the epidemics (in particular before the turning point) the outcome of our method can be very different from one set of parameters to another. We try to solve this uncertainty problem by using several choices of the period to fit an exponential growth of the data to determine χ 1 and χ 2 and several choices for the first day of intervention N . So in this section, we vary the time interval [t 1 , t 2 ], during which we use the data to obtain χ 1 and χ 2 by using an exponential fit. In the simulations below, the first day t 1 and the last day

t 2 vary such that Earliest day ≤ t 1 ≤ t 2 ≤ Latest day.
We also vary the first day of public intervention Earliest first day of intervention ≤ N ≤ Latest first day of intervention.

We vary f between 0.1 to 0.9. For each (t 1 , t 2 , ν, f, η, µ, N ) we evaluate µ to obtain the best fit of the model to the data. We use the mean absolute deviation as the distance to data to evaluate the best fit to the data. We obtain a large number of best fit depending on (t 1 , t 2 , ν, f, η, µ, N ) and we plot smallest mean absolute deviation MAD min . Then we plot all the best fit graphs with mean absolute deviation between MAD min and MAD min + 40.

Numerical simulations

The numerical simulations are presented in the chronological order of appearance for six countries (China, South Korea, Italy, France, Germany and United Kingdom). The data used are taken from [30, 31, 32, 33, 34, 35]. Before February 11, the data was based on confirmed testing. From February 11 to February 15, the data included cases that were not tested for the virus, but were clinically diagnosed. There were 17,409 such cases from February 11 to February 15 and there is a gap in the reported case data on February 11. The data from February 11 to February 15 specified both types of reported cases (i.e. the case tested for virus and the case clinically diagnosed). From February 16, the data did not separate the two types of reporting, but reported the sum of both types. We subtracted 17,409 cases (i.e. the new clinically diagnosed patient on Feb. 11) from the cumulative reported cases after February 15 to obtain the cumulative reported cases based only on confirmed testing after February 15.

Predicting the number of cases for mainland China

In Figure 3, the predictions are improving with time. We observe that if we apply our method too early as in Figure 3 (a) and (b) the predictions vary a lot. But once we pass the turning point as in Figure 3 (c) and (d) the predictions are becoming pretty good and do not vary too much until the end of the epidemic.

Predicting the number of cases for South Korea

The first confirmed case of the pandemic of COVID-19 in South Korea was announced on January 20, 2020. During the following four weeks (January 20 -February 17), South Korea monitored the potential spread of COVID-19 from existing confirmed patients, by using technological resources, such as tracking credit card use and checking CCTV footage of confirmed patients. On February 18, South Korea confirmed its 31 st case in Daegu. This 31 st patient had continued to go to gatherings of Shincheonji for days after showing symptoms. On February 19, the number of confirmed cases increased by 20 and on February 20 by 70, with a total of 346 confirmed cases by February 21, 2020, according to the Korean Centers for Disease Control (KCDC).

On February 20, the streets of Daegu were empty in reaction to the Shincheonji outbreak. On February 23, the government raised the coronavirus alert to 'highest level', and increased testing and contact tracing. Extensive measures were taken to screen the population for the virus, to isolate any infected people, and to quarantine those who were in contact with them. These rapid and extensive measures taken by the South Korean government have been judged successful in limiting the spread of the outbreak, without using the extreme measure of quarantining entire cities. In Figure 4, the predictions are improving with time. As in Figure 3, when we apply our method too early as in Figure 4 (a) and (b) the result is very uncertain.

Predicting the number of cases for Italy

On January 31, the first two cases of COVID-19 were confirmed in Rome. On February 22, the government announced a series of isolation measures, such as a new decree imposing the quarantine of more than 50,000 people from 11 different municipalities in Northern Italy. On March 1, nationwide measures began to be implemented. On March 8, Prime Minister Giuseppe Conte extended the quarantine lockdown to cover all the region of Lombardy and 14 other northern provinces. On March 10, Prime Minister Conte increased the quarantine lockdown to cover all of Italy, including travel restrictions, a ban on public gatherings, and a shut down of all commercial and retail businesses except those providing essential services, such as grocery shops and pharmacies. On March 20, the Italian Ministry of Health ordered tighter regulations on free movement. The epidemic in Italy can be divided into the exponential growth phase before February 22. It is reasonable to take N between February 22 and March 10, because the government gradually strengthened control measures to the whole country during the period February 22 to March 10.

Predicting the number of cases for France

On January 24, the first COVID-19 case was confirmed in Bordeaux, France. The epidemic in France can be divided into the exponential growth phase before March 14. It is reasonable to take N between March 14 and March 17. The reasons are as follows: (1) On March 14, many cultural institutions in the Paris region announced their closure, such as the Louvre, the Eiffel Tower, and institutions in other provinces such as the Chateau de Montsoreau Museum of Contemporary Art; (2) French President Emanuel Macron delivered a speech on March 16, announcing that France entered a "state of war", and closure measures would be imposed throughout all of France. On March 17, the "ban on foot" was officially launched.

Predicting the number of cases for Germany

A COVID-19 case was confirmed to have been transmitted to Germany on January 27, 2020. The epidemic in Germany can be divided into the exponential growth phase before March 13. It is reasonable to take N between March 13 and March 15, for the following reasons: (1) On March 13, 14 of the 16 German federal states closed their schools and nurseries; (2) On March 14, several federal states widened their measures to limit public activities. For example, Berlin, Schleswig-Holstein, and Saarland closed bars and other leisure venues. Cologne forbade all public events in the city center. In this work we use data for the cumulative number of reported cases and the daily number of reported cases for China, South Korea, Italy, France, Germany, and the United Kingdom. With this data, we project the future number of cases, both reported and unreported, in each country. For each country we observe that before the turning point the prediction is very uncertain and several of parameter values of f ν η may fit very well the data before the turning point. However, if we use enough reported case data, such as data up to the turning point, the prediction is pretty good and certain until the end of the epidemic.

A major difficulty for the predictions from the reported cases data, is to determine the date intervals of exponential growth phase. From the Figure 2 (a) this period is clear for mainland China (Jan 19-27). But it is much more difficult to decide for South-Korea (see Figure 2 (a)). By applying our method to the data for South Korea, we find the interval going from February 19 to February 24. This is because the measures imposed for contact tracing and social distancing were very strong. For Italy, the interval is difficult to ascertain (see Figure 2 (c)). We choose February 23 to March 10, but these values may change as more data of reported cases becomes available. For France, Germany, and the United Kingdom, the interval are also difficult to know with present data. Again, future reported case data, will clarify these intervals.

In the case of South Korea, the peak of the epidemic occurred approximately on February 29. In Figure 4, we see that our model agrees very well the data for South Korea. Accordingly to our model, the daily number of cases reaches a maximum of approximately 700 cases near the turning point February 29. Compared to South Korea, the public interventions in Italy, France, and Germany were relatively late. The peak of the epidemic occurs in Italy around March 14, and the peak of the maximum daily number of cases in our simulation is approximately 6 000, which agrees well with the daily reported cases data for Italy. For France, Germany, and the United Kingdom, the number of daily reported cases may still be rising. Our simulations captures these increasing values for exponential growth phase, but the advance to the next phase for both countries requires advanced data.

In general, the dates for the exponential growth interval, and the date N for the start of the new phase after the exponential growth interval, are key elements of our model, and they depend strongly on the implementation of social distancing measures. If these measures are implemented gradually, then the difficulty is increased. In future work we will allow for a series of N and µ values to account for a more complex implementation of measures that use time-dependent transmission function τ (t) depending on these values.

Another difficulty in applying our model is the assumption of the fraction f of total cases reported. The value of f is unknown, but we vary f between 0.1 to 0.9. In future work, we will allow a timedependent value for f , because in many applications the fraction of reported cases changes over the course of the epidemic. Our model incorporates social distancing measures through the time dependent transmission rate τ (t). It is evident that these measures should start as early as possible, and should be as strong as possible. The consequences of late public interventions may have severe consequences for the epidemic outcome. In future work we will apply our methods to other countries and regions within countries in an effort to better understand the evolution of COVID-19 epidemics. In [START_REF] Cotta | Mathematical Parameters of the COVID-19 Epidemic in Brazil and Evaluation of the Impact of Different Public Health Measures[END_REF][START_REF] Roques | Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France[END_REF] Bayesian's approach has been used successfully. Such a method could be used here as well, and that is left for future work.

In Table 3 we summarized the parameters giving the best fit to the data. The values of 1/ν (mean duration in days of the asymptomatic phase before symptoms onset) and 1/η (mean duration in days of the symptomatic phase) are very uncertain. Reported values have a wide range of variability. In an early report, a value of the asymptomatic phase was given as 5 days in [START_REF] Gandhi | Asymptomatic transmission, the Achilles' heel of current strategies to control, COVID-19[END_REF]. WHO reported that the asymptomatic period is on average 5 -6 days, but can be up to 14 days [START_REF] Who | Coronavirus disease 2019 (COVID-19)[END_REF]. In recent reports, patients were reported asymptomatic 4 -17 days after admission to hospitals [START_REF] Long | Clinical and immunological assessment of asymptomatic SARS-CoV-2 infections[END_REF], and 9.5 days [START_REF] Hu | Clinical characteristics of 24 asymptomatic infections with COVID-19 screened among close contacts in Nanjing, China[END_REF]. The Centers for Disease Control reported an average value of 6 days for asymptomatic transmission [START_REF]COVID-19 Pandemic Planning Scenarios[END_REF]. The period of median viral shredding was reported as 19 days pre-symptomatic patients and 14 days for symptomatic patients [START_REF] Long | Clinical and immunological assessment of asymptomatic SARS-CoV-2 infections[END_REF], and 22.6 days in pre-symptomatic patients and 25.2 days in symptomatic patients [START_REF] Noh | Asymptomatic infection and atypical manifestations of COVID-19: Comparison of viral shedding duration[END_REF], although viral shedding does not necessarily correlate to infectiousness.

In this work many questions need future considerations. First the data are questionable and may induce a bias in the estimation of parameters. Probably the most significant effect on the data is induced by the daily fluctuations of the number of tests. Such a large fluctuation has been observed in China and other countries when the medical doctors were trying to adapt the number of tests to the number of symptomatic patients. Such a phenomenon may induce a large variation in the number of reported cases. In order to improve our understanding of the real situation from the cumulative reported cases, we can improve the model in several ways. For example we can introduce age groups (see [START_REF] Prem | The effect of control strategies to reduce social mixing onoutcomes of the COVID-19 epidemic in Wuhan, China: A modelling study[END_REF][START_REF] Griette | Unreported cases for Age Dependent COVID-19 Outbreak in Japan[END_REF]) or we may examine big cities and their interconnections (see [START_REF] Wu | Nowcasting and forecasting the potential domestic and international spread of the COVID-19 outbreak originating in Wuhan, China: a modelling study[END_REF]). The method used to fit the data in the present paper has been extended and used successfully for age groups in Japan [START_REF] Griette | Unreported cases for Age Dependent COVID-19 Outbreak in Japan[END_REF]. Such a method is general and may probably be applied to other situations (patch models etc . . . ). In future work, we can also incorporate the number of tests (see [START_REF] Roques | Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France[END_REF]) which may significantly improve our understanding of COVID-19 epidemics.

Figure 1 :

 1 Figure 1: Compartments and flow chart of the model.

Figure 2 :

 2 Figure 2: In this figure we plot the cumulative number cases (with log scale) at the early beginning of the epidemic for (a) mainland China; (b) South Korea; (c) Italy; (d) France; (e) Germany; (f ) UK.

Figure 3 :

 3 Figure 3: In this figure we plot the cumulative number of cases of the left hand side and the daily number of cases on the right hand side. In (a) and (b) we use the data until February 5. In (c) and (d) we use the data until February 12. In (e) and (f ) we use the data until March 11. The best is obtained for t 1 = January 20, t 2 = January 30, 1/ν = 5 days, 1/η = 6 days, f = 0.6, µ = 0.1 and N = January 27. The smallest mean absolute deviation MAD min is 475.

Figure 4 :

 4 Figure 4: In this figure we plot the cumulative number of cases of the left hand side and the daily number of cases on the right hand side. In (a) and (b) we use the data until March 4. In (c) and (d) we use the data until March 8. (e) and (f ) we use the data until March 24. The best is obtained for t 1 = February 19, t 2 = February 24, 1/ν = 4 days, 1/η = 16 days, f = 0.9, µ = 0.26 and N = February 27. The smallest mean absolute deviation MAD min is 144.

Figure 5 :

 5 Figure 5: In this figure we plot the cumulative number of cases of the left hand side and the daily number of cases on the right hand side. In (a) and (b) we use the data until March 3. In (c) and (d) we use the data until March 10. (e) and (f ) we use the data until March 24. The best is obtained for t 1 = February 23, t 2 = March 10, 1/ν = 23 days, 1/η = 14 days, f = 0.4, µ = 0.11 and N = March 10. The smallest mean absolute deviation MAD min is 1129.

Figure 6 : 2 =

 62 Figure 6: In this figure we plot the cumulative number of cases of the left hand side and the daily number of cases on the right hand side. In (a) and (b) we use the data until March 17. In (c) and (d) we use the data until April 1. (e) and (f ) we use the data until June 10. The best is obtained for t 1 = March 1, t 2 = March 20, 1/ν = 6 days, 1/η = 15 days, f = 0.4, µ = 0.05 and N = March 15. The smallest mean absolute deviation MAD min is 2347.

Table 1 :

 1 Parameters and initial conditions of the model.

	Symbol	Interpretation	Method
	t 1	Date from which the epidemic grows exponentially	fitted
	S 1	Number of susceptible at time t 1	fixed
	I 1	Number of asymptomatic infectious at time t 1	fitted
	R 1	Number of reported symptomatic infectious at time t 1	fitted
	U 1	Number of unreported symptomatic infectious at time t 1	fitted
	τ (t)	Transmission rate at time t	fitted
	N	Date at which public intervention measures became effective	fitted
	µ	Intensity of the public intervention measures	fitted
	1/ν	Average time during which asymptomatic infectious are asymptomatic	fixed
	f	Fraction of asymptomatic infectious that become reported symptomatic infectious	fixed
	ν 1 = f ν	Rate at which asymptomatic infectious become reported symptomatic	fitted
	ν 2 = (1 -f ) ν	Rate at which asymptomatic infectious become unreported symptomatic	fitted
	1/η	Average time during which symptomatic infectious have symptoms	fixed

10 2 10 3 10 4 10 5 Cumulative reported cases (log scale) Mar 01 Mar 08 Mar 15 Mar 22 Mar 29 Apr 05 Apr 12 2020 10 2 10 3 10 4 Cumulative reported cases (log scale)

	Cumulative reported cases (log scale)	10 3 10 4						Cumulative reported cases (log scale)	10 2 10 3							Cumulative reported cases (log scale)	10 2 10 3 10 4 10 5
		Jan 19	Jan 26	Feb 02	Feb 09	Feb 16	Feb 23	Mar 01	Feb 19	Feb 26	Mar 04	Mar 11	Mar 18		Feb 21	Feb 28	Mar 06	Mar 13	Mar 20	Mar 27
								2020								2020			2020
				(d)							(e)						(f )
	Cumulative reported cases (log scale)	10 2 10 3 10 4
		Feb 25	Mar 03	Mar 10		Mar 17	Mar 24	Mar 31	Feb 24	Mar 02	Mar 09	Mar 16	Mar 23	Mar 30	Apr 06	Apr 13
								2020								2020

Table 2 :

 2 The parameters χ 1 , χ 2 , χ 3 are estimated by using the data to fit χ 1 exp(χ 2 t) to the cumulative reported cases data CR(t) + χ 3 (χ 3 is chosen to minimize the error of the fit). For each country the best fit of the data over the all period is obtained by choosing: (1) t 1 = January 27 and t 2 = January 30 for mainland China; (2) t 1 = February 19 and t 2 = February 24 for South Korea; (3) t 1 = February 23 and t 2 = March 10 for Italy; (4) t 1 = March 1 and t 2 = March 20 for France; (5) t 1 = February 24 and t 2 = March 11 for Germany; (6) t 1 = March 4 and t 2 = March 26 for United Kingdom. The parameters ν = 1/7 and η = 1/7. The values of I 1 , R 1 , U 1 , τ 1 are obtained by using (3.7), (3.8), and (3.4).

Table 3 :

 3 In this table, we summarize the values for 1/ν (mean duration in days of the asymptomatic phase before symptoms onset), 1/η (mean duration in days of the symptomatic phase), and f (the fraction of reported) obtained for the best fits to the cumulative reported data.

Funding: This research was funded by the National Natural Science Foundation of China (grant number: 11871007 (ZL)), NSFC and CNRS (Grant number: 11811530272 (ZL, PM)) and the Fundamental Research Funds for the Central Universities (ZL). This research was funded by the Agence Nationale de la Recherche in France (Project name : MPCUII (PM)).

Predicting the number of cases for United Kingdom

The United Kingdom was slow to implement public distancing measures. On 19 March, the government introduced the Coronavirus Act 2020, which gave discretionary emergency powers in restricting social care facilities, schools, police, the Border Force, local councils, and other public functions. The act went into effect on 25 March 2020. Closures to pubs, restaurants and indoor sports and leisure facilities were imposed by the Health Protection Regulations on March 21.

Conclusions

We have applied a method developed in [START_REF] Liu | Understanding unreported cases in the 2019-nCov epidemic outbreak in Wuhan, China, and the importance of major public health interventions[END_REF] and [START_REF] Liu | Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data[END_REF] to predict the evolution of a COVID-19 epidemic based on reported case data in that region. Our method uses early data, when the epidemic is in its exponential growth phase, corresponding to a constant transmission rate. In [START_REF] Liu | Understanding unreported cases in the 2019-nCov epidemic outbreak in Wuhan, China, and the importance of major public health interventions[END_REF] we demonstrated a method to identify this constant transmission rate. When public measures are begun in order to slow down the epidemic, a new phase begins, which we model with a time-dependent exponentially decreasing transmission rate in [START_REF] Liu | Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data[END_REF]. In [START_REF] Liu | Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data[END_REF] we applied this method to mainland China, and demonstrated the ability of our model to predict the forward time-line of the epidemic. In Figure 4 in [START_REF] Liu | Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data[END_REF], we showed how the prediction unfolded week by week, with increasing agreement with reported case data, in mainland China.

In Table 2 we summarized the parameters giving the best fit to the data.

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