Youssef Gergesa
email: youssef.gerges@insa-lyon.fr.

Kerem Ege

Laurent Maxit

Nicolas Totaro

Jean-Louis Guyader

Vibrations, Shocks and Noise Mid Frequency Vibroacoustic Modeling of an Innovative Lightweight Cab -Floor/Cavity Interaction

to an innovative lightweight cab.

Introduction

The City Lightweight and Innovative Cab (CLIC) project is a scientific collaboration gathering different public and private organizations. The aim is to propose an innovative truck cab responding to new European legislations in security and CO2 emission. A very high strength steel will be used to lighten the structure. This could affects directly the acoustic environment of the cab. In order to control the noise requirements at the design stage, it is then necessary to be able to simulate the vibroacoustic behavior of a truck cab in the mid frequency range. In this context, a methodology is developed at INSA Lyon based on Statistical modal Energy distribution Analysis (SmEdA) [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF][START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF]. This method is considered as substructuring approach for vibroacoustic problems and post-process for finite element methods (FEM). It is based on the knowledge of the uncoupled subsystem modes that can be computed using the FEM [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF][START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF]. In this paper, one focuses on the fluid-structure coupling of the cab floor with the interior cavity when they are represented by non-coincident meshes.

Vibroacoustic modeling with non-coincident finite element mesh

FEM method remains the most common modeling method in industrial applications despite the huge number of degree of freedom (dof) (for information purposes, the structural FEM model contains 6Mdof). Moreover, in the mid frequency range, the vibroacoustic behavior of the global system is not so easy to analyze with the FEM results and SmEdA concept allows substructuring the global problem. To study the interaction between the floor and the interior cavity, one can perform modal FEM calculations on two independent uncoupled subsystems: the floor and the cavity. The energy equation of motions of the coupled subsystems can then be written and easily resolved. In these equations, the spatial couplings of the subsystem modes are expressed through the intermodal works which is deduced from the mode shapes. When dealing with non-coincident meshes of the two subsystems, the intermodal works cannot be calculated directly and an interpolation process is required as described below.

Interpolating the acoustic entities on the structural nodes or vice versa ensure the continuity of the problem. One presents here a projection method based on the barycentric coordinate system:

Let A, B, C and G be the geometrical points associated to four nodes in the coordinate system R(O,x,y,z). G is the barycenter of the triangle ABC if and only if one attributes real weights a, b and c to the nodes A, B and C respectively, verifying the equation:

aGA + bGB + cGC = 0
This equation verifies the eigenvalue problem associated to the matrix system below:

[ ] { } { }
a, b and c should be positive and normalized so that their sum is equal to 1:

a + b + c = 1 (a,b,c ) ≥ 0
Figure 1 illustrates the problem where the green mesh presents the fluid elements and the red mesh the structural elements. For each node Gi (of the red structural mesh) one aims to find the weights associated to A, B and C (of the green fluid mesh).

Finally the acoustic mode shapes on the node G i is obtained with:

Ψ Gi = aΨ A + bΨ B + cΨ C
The example chosen to illustrate the method is the floor of the cab coupled to the cavity. Figure [START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF] shows the shape of the first acoustic pressure mode of the cavity on the floor interface and figure [START_REF] Dong | A methodology for including the effect of a damping treatment in the midfrequency domain using SmEdA method[END_REF] shows the projection of this mode on the floor nodes. This result shows the efficiency of the proposed method and allows extending this work to the vibroacoustic interaction.

Vibroacoustic interaction in mid frequency

SmEdA is based on the linear equation of power exchanged between coupled subsystems. Energy equipartition among subsystems modes is not assumed and one deals with energy levels of individual modes of subsystems in a frequency range [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF][START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF]. The modal coupling loss factor between the mode p of the floor and the mode q of the cavity is given by:

β pq = β W* β ω
β W is the modal spatial coupling factor and β ω is the modal spectral coupling factor. Figure (4) shows the spatial and spectral modal coupling factor in a frequency band, figure (5) shows the modal coupling factor and the floor/cavity shapes of the strongest coupled modes. As we can observe, this couple of modes is strongly coupled due to frequency coincidence.

Conclusion

This work presents a vibroacoustic study of a truck cab in the mid frequency range. A projection method in noncoincident mesh case is illustrated and SmEdA method is used for the vibroacoustic study. The next step will be to introduce damping materials to improve the vibroacoustic comfort [START_REF] Dong | A methodology for including the effect of a damping treatment in the midfrequency domain using SmEdA method[END_REF].

Fig. 1 .

 1 Fig. 1. Non-coincident mesh Fig. 2. Acoustic mode shape at the floor interfaces Fig. 3. Acoustic mode shape projected on structural nodes

Fig. 4 .

 4 Fig. 4. Spatial (top) and spectral (bottom) coupling factors in a Fig. 5. Modal coupling loss factor and mode shapes of the frequency range strongest coupled modes

Acknowledgements

This work was co-funded by the French government (FUI 12 -Fonds Unique Interministériel) and European Union (FEDER -Fonds européen de développement régional). It was carried out in the framework of the LabEx CeLyA ("Centre Lyonnais d'Acoustique", ANR-10-LABX-60) and the research project CLIC ("City Lightweight Innovative Cab") labelled by LUTB cluster (Lyon Urban Truck and Bus), in partnership with Renault Trucks, Arcelor-Mittal, ACOEM, ALTRAN, FEMTO-ST (Univ. de Franche-Comté) and LVA (INSA de Lyon).


Jean-Bernard Fabre
email: jb.fabre@humanfab.com

Laurent Grelot

William Vanbiervielt

Julien Mazerie

Raphael Manca

Vincent Martin

Managing the combined consequences of COVID-19 infection and lock-down policies on athletes: narrative review and guidelines proposal for a safe return to sport

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

INTRODUCTION

The SARS-CoV-2 (COVID-19) pandemic is a global public health emergency. According to the data available, from the beginning of the epidemic to 23 August 2020, a total of 23 256 567 people were infected, and 805 733 died in 202 countries, as reported by the Coronavirus Resource Centre of the John Hopkins University of Medicine. 1 Many research teams proposed that COVID-19 and β-coronaviruses are

To cite: Fabre J-B, Grelot L, Vanbiervielt W, et al. Managing the combined consequences of COVID-19 infection and lock-down policies on athletes: narrative review and guidelines proposal for a safe return to sport. BMJ Open Sport & Exercise Medicine 2020;0: e000849. doi:10.1136/ bmjsem-2020-000849 ► COVID-19 is currently characterised by influenzalike symptoms and respiratory troubles; in some cases, multiorgan failure may occur and especially cardiac injury. The main difficulty is that symptoms of COVID-19 vary among individuals, and athletes may be affected by no apparent sign of the disease. This could be a real danger for athletes when returning to their usual training and play. ► Isolation could have a significant impact on the physical and mental state of the athletes leading to detraining. A reduction in aerobic capacities, muscular power and flexibility and a diminution of muscle mass are expected. ► Combined effect of detraining and COVID- 19 symptoms may influence the arousal of post-viral fatigue syndrome, thus influencing the capacities of athletes to perform exercise and increasing the risk of injury.

What are the findings ► COVID-19 may lead to multiorgan failure, and especially cardiac and vascular injuries. Allowing athletes to return to practice on the sole basis of cardiac assessment may underestimate risks. To have a more global assessment and insights about the severity of the disease, blood-and urine-specific biomarkers are recommended. from the same family 2 3 with a 79.0% nucleotide identity to SARS-CoV and 51.8% identity to MERS-coronavirus (MERS). [START_REF] Ren | Identification of a novel coronavirus causing severe pneumonia in human: a descriptive study[END_REF] Moreover, research have demonstrated that COVID-19 genome has 96% of similarity with the entire genome of bat coronavirus. [START_REF] Zhou | A pneumonia outbreak associated with a new coronavirus of probable bat origin[END_REF] The SARS-CoV-2 is predominantly spreading through the respiratory tract and uses the same receptor as SARS-CoV (ACE2). The principal source of contagion is most likely human-to-human aerosol transmission, which happens essentially through contaminated droplets, hands or surfaces. Virus particles present in secretions from an infected person's respiratory system infect others through direct contact with mucous membranes. [START_REF] Adhikari | Epidemiology, causes, clinical manifestation and diagnosis, prevention and control of coronavirus disease (COVID-19) during the early outbreak[END_REF] Also, the median incubation period is between 2 days and 12 days (median 5.1 days). Thus, based on the data from the previous coronavirus epidemics (SARS-CoV and MERS), COVID-19 has first been described as influenza viral infections and a pathogen that primarily targets the respiratory system (ie, type 2 pneumocytes) with pneumonia and acute respiratory distress syndrome (ARDS) for the most severe cases. [START_REF] Ksiazek | A novel coronavirus associated with severe acute respiratory syndrome[END_REF] After few months and a better understanding of the disease, it appears that COVID-19 also causes direct myocardial injury, 8 disseminated intravascular coagulation and multiorgan failure. [START_REF] Zang | The interaction of circRNAs and RNA binding proteins: an important part of circRNA maintenance and function[END_REF] One of the main difficulties is that symptoms of COVID-19 vary among individuals, and even young athletes can be infected with no apparent symptom of the disease. This could be a real danger for amateur or professional athletes when returning to their usual training and subsequently when they will return to intensive play. Indeed, as an emerging infectious disease, COVID-19 late effects and sequelae are still unknown for both the moderate and severe forms. The disease spread rapidly across the globe owing to the unique properties of the virus (ie, extraordinary genetic diversity, highly contagious, easy spreading, relative resistance to climatic variations), [START_REF] Mackenzie | COVID-19 a novel zoonotic disease caused by a coronavirus from China: what we know and what we don't[END_REF] causing an unprecedented pandemic forcing governments to impose an almost global quarantine. At the beginning of 2020 (January-March), the whole world, including the world of sports, entered an extreme and unknown situation, where, gradually, all sports competitions were postponed and any organised training or practice was banned. [START_REF] Corsini | Football cannot restart soon during the COVID-19 emergency. a critical perspective from the Italian experience and a call for action[END_REF] In the absence of a vaccination or an antiviral therapy, physical distancing emerged as the key step against COVID-19. Isolation, of course, did not allow athletes to follow their usual training and competitive routines. Regardless of duration, isolation could have a significant impact on the physical and mental state of an athlete. Staying in quarantine can have negative effects on physiological systems, especially aerobic capacities, muscular performance and body composition. This well-known phenomenon is called detraining. [START_REF] Mujika | Detraining: loss of training-induced physiological and performance adaptations. Part I[END_REF] After at least 4 weeks of quarantine, depending on country lock-down measures, the rates of new infection begin to plateau and even decline in some countries. This glimmer of hope has created an enthusiasm and a collective euphoria about the resumption of normal sports activity. To reduce injury risk and promote the return of performance, having a better understanding of the combined effect of detraining and COVID-19 infection seems to be relevant. Hence, there is an emergency for a global and purposeful approach to help the sports medicine community to ensure safety and wellness of all athletes before they return to sport. We propose here a global model of evaluation to (1) reduce any lethal risk of practice, especially myocarditis and sudden cardiac death (SCD); (2) evaluate the combined consequences of the disease and detraining on the physical abilities and biological profile of athletes; and (3) monitor postinfection fatigue symptoms.

COVID-19 SYMPTOMS AND COMPLICATIONS

Usual symptoms of the disease are fever, cough, myalgia, fatigue, slight dyspnoea, sore throat, headache and conjunctivitis. [START_REF] Chen | Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study[END_REF][START_REF] Chen | Analysis of clinical features of 29 patients with 2019 novel coronavirus pneumonia. Zhonghua Jie He He Hu Xi Za Zhi= Zhonghua Jiehe He Huxi[END_REF][START_REF] Yang | Clinical course and outcomes of critically ill patients with SARS-cov-2 pneumonia in Wuhan, China: a single-centered, retrospective, observational study[END_REF] Gastrointestinal involvement was also reported in a smaller percentage of cases, with diarrhoea, nausea and vomiting. [START_REF] Li | COVID-19 patients' clinical characteristics, discharge rate, and fatality rate of meta-analysis[END_REF] Only around 10% of cases become serious, with dyspnoea, hypoxaemia and an extensive (>50%) destruction of the lung parenchyma leading to fibrosis. [START_REF] Pascarella | COVID-19 diagnosis and management: a comprehensive review[END_REF] In around 5% of cases, the disease evolved in a critical condition, with respiratory failure, pneumonia, shock, multiorgan failure and, in the most serious cases, death, which is almost always caused by progression to ARDS and multiorgan failure. [START_REF] He | Coronavirus disease 2019: what we know?[END_REF][START_REF] Wu | Risk factors associated with acute respiratory distress syndrome and death in patients with coronavirus disease 2019 pneumonia in Wuhan, China[END_REF][START_REF] Xu | Pathological findings of COVID-19 associated with acute respiratory distress syndrome[END_REF] Acute hypoxaemia may cause obstinate dyspnoea with the need for oxygen therapy administration through high-flow nasal oxygen, or through the application of a noninvasive positive pressure (with oronasal or face masks, helmets). 21 22 Unfortunately, in the case of O 2 saturation worsening, orotracheal intubation and invasive mechanical ventilation are mandatory in intensive care unit. However, SARS-CoV-2 infection does not only affect lungs.

CARDIOVASCULAR COMPLICATION OF COVID-19

Despite COVID-19 clinical manifestations are essentially respiratory, major cardiac and vascular (ie, vasculitis) complications are being reported. One of the proposed mechanisms of myocardial injury includes a cytokine storm that develops during severe COVID-19 illness. This may lead to decrements in cardiac function, similar to those seen in other forms of sepsis, with features that overlap with classic forms of 'stress' or catecholamineinduced cardiomyopathy. [START_REF] Clerkin | Coronavirus disease 2019 (COVID-19) and cardiovascular disease[END_REF] To date, this appears to be a self-limiting phenomenon confined to the severe phases of the illness in individuals with coexisting medical conditions, such as hypertension, type 2 diabetes and cardiovascular disorders. 15 24 Alternatively, COVID-19 may directly infect myocardial cells, thereby leading to myocarditis with lymphocyte-rich inflammatory histology, acute impairment of cardiac muscle function and potentially residual chronic scars with increased vulnerability to malignant ventricular arrhythmias. Importantly, myocarditis has recently been reported in patients who did not present symptoms, such as fever or respiratory difficulties. [START_REF] Paul | Myocarditis revealing COVID-19 infection in a young patient[END_REF] One of the major difficulties is that symptoms of COVID-19 vary among individuals. [START_REF] He | Coronavirus disease 2019: what we know?[END_REF] Lavezzo and colleagues (unpublished data, 2020) reported in a cohort study conducted in the town of Vò Euganeo in Italy that around 50-75% of individuals with positive RT-PCR throat swab results remain asymptomatic, while others develop mild influenza-like symptoms. It is therefore difficult to differentiate COVID-19 from other respiratory diseases, 26 27 which may be a real threat. Interestingly, many professional team sport players in extremely good fitness condition have been infected (unpublished data, French Soccer Federation, 2020), demonstrating that athletes are not spared by the disease. Although definite evidence is lacking, athletes may beat higher risk of developing myocarditis than the general population. Sports activity can influence the susceptibility to infections, depending on the intensity and duration of the physical exercise. [START_REF] Nieman | Is infection risk linked to exercise workload?[END_REF] While moderate physical activity may improve immunological defences, 28 29 intense and prolonged training or competition lower the immunity by reducing salivary secretory IgA, lactoferrin and lysozyme altering the T-cell response. [START_REF] Brolinson | Exercise and the immune system[END_REF] All mechanisms may increase the vulnerability of athletes to viral infections. 29 30 The myocarditis associated with COVID-19 may be an additional threat since acute cardiac injury is observed in almost one-fifth of patients, with a 50% survival rate. [START_REF] Doyen | Myocarditis in a patient with COVID-19: a cause of raised troponin and ECG changes[END_REF] Myocarditis has been traditionally considered as the cause of life-threatening ventricular arrhythmias and SCD in athletes. The prevalence and acute and delayed clinical implications among infected people who experience mild illness or who remain asymptomatic remain completely unknown. Also unknown is the incidence of silent myocardial inflammation that lingers long after the resolution of typical COVID-19 symptoms, a form of disease that may uniquely affect athletes during resumption of training and competition. This could be a real danger for amateur or professional athletes. There is a need to prevent lethal risk, especially myocarditis and SCD risk, prior to any practice.

CARDIAC TESTING IN ATHLETES PREVIOUSLY INFECTED WITH COVID-19

From the perspective of heart health, the preparticipation evaluation (PPE) is traditionally considered as a tool to screen for occult cardiovascular diseases that predispose the athlete to SCD. [START_REF] Drezner | AMSSM position statement on cardiovascular preparticipation screening in athletes: current evidence, knowledge gaps, recommendations and future directions[END_REF] In the context of COVID-19, it would be prudent to adopt a broader approach of the cardiovascular PPE. In addition to using the PPE to search for rare genetic and congenital conditions, it is recommended to look for cardiovascular sequelae of COVID-19 to ensure a safe return to sport. Though imperfect, medical history and physical examination are valuable tools for identifying athletes with underlying myocardial inflammation and/or overt myocarditis. The use of 12-lead ECG is crucial in the initial evaluation of athletes. Specialist must seek patterns that reflect myocardial inflammation such as T-wave inversions and new ST segment changes. [START_REF] Drezner | International criteria for electrocardiographic interpretation in athletes: consensus statement[END_REF] Repolarisation abnormalities are the most common alterations in acute myocarditis, being detected in 40% of the patients. The most notable ECG feature of early repolarisation is ST segment elevation, which may vary in terms of morphology, location and degree. T-wave inversion can also be observed particularly in black athletes but confined to leads V1-V4. [START_REF] Calore | Electrocardiographic anterior T-wave inversion in athletes of different ethnicities: differential diagnosis between athlete's heart and cardiomyopathy[END_REF] For more details about acute myocarditis and ECG evaluation, readers should refer to the excellent review of Vio and colleagues. [START_REF] Vio | Myocarditis in the athlete: arrhythmogenic substrates, clinical manifestations, management, and eligibility decisions[END_REF] However, as failure to identify athletes with myocarditis by preparticipation screening ECG has been reported before, [START_REF] Corado | Trends in sudden cardiovascular death in young competitive athletes after implementation of a preparticipation screening program[END_REF] blood sampling as a complementary approach during the PPE is recommended. 37 38 Biomarkers indicative of myocardial injury are indeed elevated in myocarditis. In a recent meta-analysis review about the effects of COVID-19 on cardiac biomarkers, Li and colleagues [START_REF] Li | COVID-19 patients' clinical characteristics, discharge rate, and fatality rate of meta-analysis[END_REF] found troponin I significantly increased in patients with COVID-19 with severe disease compared with those with milder infection. Increases in troponin I, creatine kinase (CK-MB) and N-terminal pro b-type natriuretic peptide (NT-pro BNP) are indicators of possible cardiac damage during COVID-19 infection, and three case reports have found fulminant myocarditis 39 40 and cardiac tamponade [START_REF] Hua | Life-threatening cardiac tamponade complicating myo-pericarditis in COVID-19[END_REF] after COVID-19 infection. We then suggest that troponin I, CK-MB and NT-pro BNP must be routinely measured in infected athletes before they return to play.

This will help to identify those infected athletes that may require additional testing and medical care prior to return to play. However, allowing athletes to return to practice on the sole basis of cardiological assessment may underestimate risks since the COVID-19 can lead to multiorgan failure. There is thus a need for a broader monitoring prior to return to practice.

MULTIORGAN THREAT AND GLOBAL ASSESSMENT FOR INFECTED ATHLETES

In about 5% of the cases, SARS-CoV-2 infection does not only affect lungs and heart. As commonly observed in sepsis, liver and kidney alterations might occur. [START_REF] Zhang | Liver injury in COVID-19: Management and challenges[END_REF] Hepatocytes degenerate, liver sinusoids are hyperaemic with microthrombi and can lead to a metabolic disorder. So far, these complications are reported only in severe forms, but less is known about athletes infected with milder symptoms and the potential detrimental effect on liver and kidney of an early return to sport, which would result in misadapted physiological responses to exercise. We then suggest adding specific blood and urine biomarkers to the PPE to help understand the severity of the infection in athletes before a return to sport. Recent work reported that C reactive protein (CRP) levels are increased in patients with COVID-19, and it has been shown that median CRP values are strongly correlated with disease severity and prognosis. [START_REF] Ruan | Clinical predictors of mortality due to COVID-19 based on an analysis of data of 150 patients from Wuhan, China[END_REF] Based on a global review of laboratory markers, complete blood count, interleukin 6, alanine aminotransferase, aspartate aminotransferase, albumin, CRP, lactate dehydrogenase, procalcitonin, ferrintin, [START_REF] Mehta | COVID-19: consider cytokine storm syndromes and immunosuppression[END_REF] troponin I, CK-MB and NTpro BNP can be relevant to explore the level of severity of COVID-19 in patients. [START_REF] Li | COVID-19 patients' clinical characteristics, discharge rate, and fatality rate of meta-analysis[END_REF] 25 45-49 To prevent asymptomatic acute kidney injury that may result from COVID-19 infection, we recommend assessment of traditional urine biomarkers such as the glomerular filtration rate of the kidney,

Open access

Fabre J-B, et al. BMJ Open Sp Ex Med 2020;0:e000849. doi:10.1136/bmjsem-2020-000849 creatinine, albumin urea, cystatin C, beta-trace protein, beta-2 microglobulin and fatty acid-binding proteins. 50 51 For a better understanding, biomarkers are presented in table 1. During the first month of practice, tracking the evolution of kidney and liver biomarkers could be of great help to understand the consequences of the viral infection and prevent a putative worsening due to a too rapid and/or too high energy expenditure. In that perspective, oxygen consumption (VO 2 ) evaluation could be an additional valuable approach.

VO 2 EVALUATION FOR ATHLETES INFECTED WITH RESPIRATORY SYNDROME

The data obtained from various groups worldwide and the 31 provinces of China suggest that the clinical symptoms of COVID-19 are more or less similar to that of SARS-CoV infection. 52 53 SARS is associated with pulmonary complications in the form of pulmonary fibrosis and bronchiectasis. [START_REF] Peiris | Clinical progression and viral load in a community outbreak of coronavirus-associated SARS pneumonia: a prospective study[END_REF] A reduction in aerobic exercise capacity 3 months after hospital discharge has been reported in some adult survivors of the SARS-CoV. [START_REF] Ong | Air, surface environmental, and personal protective equipment contamination by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) from a symptomatic patient[END_REF] This reduction could last up to 12 months after the onset of the illness. [START_REF] Hui | Impact of severe acute respiratory syndrome (SARS) on pulmonary function, functional capacity and quality of life in a cohort of survivors[END_REF] Such impairment was consistently shown to be at variance with the lung function of these subjects, which was either normal or showed only mild abnormalities. Yu et al [START_REF] Yu | Longer term follow up of aerobic capacity in children affected by severe acute respiratory syndrome (SARS)[END_REF] reported that aerobic capacity, reflected by peak VO 2 , was also reduced in a cohort of asymptomatic children 6 and 15 months after the acute episode of SARS-CoV. Moreover, whatever the severity of symptoms experienced, because of lock-down, athletes will certainly have a reduction in maximal and submaximal aerobic performance. These losses in aerobic performance, due to detraining, due to sedentarism-induced detraining, [START_REF] Narici | Impact of sedentarism due to the COVID-19 home confinement on neuromuscular, cardiovascular and metabolic health: physiological and pathophysiological implications and recommendations for physical and nutritional countermeasures[END_REF] alter cardiovascular function and muscle metabolic potential. [START_REF] Mujika | Detraining: loss of training-induced physiological and performance adaptations. Part I[END_REF] VO 2 testing is an important clinical assessment tool since it provides a composite evaluation of the respiratory, cardiac and metabolic systems. Exercise testing is thus a more reliable assessment method of functional outcome than resting pulmonary function tests. [START_REF] Mehta | COVID-19: consider cytokine storm syndromes and immunosuppression[END_REF] In the case of athletes infected with COVID-19 with minor or no symptom and a normal cardiological assessment, a VO 2max test must be performed. In more severe cases, a follow-up of the peak VO 2 and ventilatory threshold as indicator of improving aerobic capacity could be useful, as recommended in patients infected with SARS-CoV. 54 55 COVID-19 could also have an impact on metabolic adaptation during exercise. [START_REF] Yu | Longer term follow up of aerobic capacity in children affected by severe acute respiratory syndrome (SARS)[END_REF] The evaluation of the respiratory exchange ratio could help to understand the effect of the disease on metabolic adaptations to exercise in athletes with moderate symptoms or asymptomatic. In the case of athletes who recovered from a severe infection, complementary measurement should be proposed. Lipid profile assessment before and after exercise and glycaemia measurements during the incremental VO 2 test are currently performed on patients suffering from metabolic disorder to exercise. [START_REF] Fittipaldi | Cardiorespiratory performance and acute effect of high-intensity exercise on lipid profile in hypertensive sedentary older adults with and without diabetes mellitus[END_REF] This approach could be proposed to get insights into the consequences of the viral infection on metabolic adaptations to exercise, together with an evaluation of other performance determinants.

ATHLETES INFECTED WITH COVID-19, LOCK-DOWN AND PHYSICAL PERFORMANCE DETERMINANT

Athletes are used to reduced activity periods throughout their sport careers, usually coinciding with the end of their competition period, illness, injury or other factors. Nevertheless, infected athletes are facing the cumulative effects of the illness and detraining on their physical performance. Because COVID-19 is a new type of coronavirus (SARS-CoV-2) that shares 96% of his genotype with the SARS-CoV of 2002, [START_REF] Zhou | A pneumonia outbreak associated with a new coronavirus of probable bat origin[END_REF] one can speculate that they share the same effect on muscles. So and colleagues [START_REF] So | Severe acute respiratory syndrome and sport[END_REF] reported loss of muscle mass in patients infected with mild and severe form of SARS-CoV. They concluded that it can affect force production and locomotion. Ong et al [START_REF] Ong | Pulmonary function and exercise capacity in survivors of severe acute respiratory syndrome[END_REF] suggested that the reduced exercise capacity was probably related to myalgia. Additionally, the reduction in physical activity due to lockdown policies may also have affected muscle mass, and more generally, body composition. We then suggest that the recovery of muscle mass with training may be longer in athletes previously infected with COVID-19. To improve return to practice and follow the effect of training on muscle mass, periodic evaluation of body composition and force production capacity is recommended.

Patients infected with COVID-19 also present central nervous system alterations. [START_REF] Li | The neuroinvasive potential of SARS-CoV2 may play a role in the respiratory failure of COVID-19 patients[END_REF] Excepted for asymptomatic, Urine markers Organ References Glomerular filtration rate Kidney dysgeusia, described as an highly frequent early or lone symptom of COVID-19, and hyposmia or anosmia have been reported, even by professional (basketball) athletes with mild symptoms. [START_REF] Li | COVID-19 patients' clinical characteristics, discharge rate, and fatality rate of meta-analysis[END_REF] However, the neuroinvasive potential of COVID-19 remains poorly understood, and the impact on neuromuscular function warrants further investigation. [START_REF] Sun | Understanding of COVID-19 based on current evidence[END_REF] From a practical point of view, it may specifically affect power and force production through modulation of the neural drive. [START_REF] Hortobágyi | The effects of detraining on power athletes[END_REF] The countermovement jump (CMJ) is one of the most popular tests to monitor an athlete's muscle power of the lower extremities. An athlete's CMJ performance is relevant in a variety of sports and commonly quantified by jump height or flight time, which have been considered as indicators of vertical jump performance 63 64 and used to measure training adaptations. [START_REF] Thorpe | Monitoring fatigue during the in-season competitive phase in elite soccer players[END_REF] The standardised drop vertical jump (DJ) from 30 cm height as a screening tool for evaluation of deceleration activities, eccentric strength, has also largely been reported in the literature. 66 67 To the best of our knowledge, there is no information about the impact of COVID-19 infection on neuromuscular performance in the literature. Given its potential effects of the central nervous system, evaluating muscle strength and power using simple and classical field-tests could be clinically relevant as a first intention.

To get further insights into the effects of COVID-19 on the motor drive, electromyography, twitch interpolation and evoked potentials may be implemented in athletes who have a history of severe COVID-19 infection and/or demonstrate a slow recovery of force/power production capacity.

Moreover, extra attention should be given to the possible post-viral fatigue syndrome, [START_REF] Fittipaldi | Cardiorespiratory performance and acute effect of high-intensity exercise on lipid profile in hypertensive sedentary older adults with and without diabetes mellitus[END_REF] which may also generate neuromuscular fatigue. The use of CMJ and DJ in isolation as a global indicator of muscle power of the lower limb muscles does not inform about the specific effect of neuromuscular fatigue on sports performance. [START_REF] Laffaye | Countermovement jump height: gender and sport-specific differences in the force-time variables[END_REF] Neuromuscular fatigue arises not only because of peripheral changes at the level of the muscle but also because the central nervous system fails to drive the motoneurons adequately. [START_REF] Gandevia | Spinal and supraspinal factors in human muscle fatigue[END_REF] Appropriate screening methods, combining stimulation methods, dynamometry and electromyography may be used to identify and evaluate the relative contributions of central and peripheral factors to peripheral fatigue. [START_REF] Laffaye | Countermovement jump height: gender and sport-specific differences in the force-time variables[END_REF] That could help to prescribe exercise modalities adapted to the level of fatigue and propose countermeasures, such as nutritional and/or specific training regimens, to help athlete to regain performance faster. [START_REF] Pickering | Caffeine and exercise: what next?[END_REF] Finally, the fatigue reported by patients with COVID-19 may have affected their flexibility. Indeed, fatigue leads to inactivity, which in turn may decrease flexibility. [START_REF] Caldwell | Seasonal variation in physiological fitness of a semiprofessional soccer team[END_REF] Inactivity affects different muscles and muscle chains depending on whether they are tonic or phasic, causing muscle shortening and/or hypertonia or laxity and/or hypotonia depending on the muscle type. The 'Y Balance Test' is an easy and reproducible method to assess the flexibility of hip and leg muscles. [START_REF] Overmoyer | Relationships between lower-extremity flexibility, asymmetries, and the Y balance test[END_REF] Including flexibility evaluation in the PPE would give insight into the flexibility status of athletes with COVID-19 and may ensure a safe return to activity. Managing the combined consequences of COVID-19 infection and lock-down policies on athletes is represented in figure 1. This combined approach based on the physical performance determinant and medical assessment has a dual interest for the sports medicine community, ensuring a return to practice without risk and advising athletes to accelerate their return to performance.

CONCLUSION

The COVID-19 infection is characterised by influenza-like symptoms, respiratory troubles, and in some cases, multiorgan failure may occur. These effects may combine with the detrimental effects of detraining induced by the lockdown policies. In the context of sport, this may translate into potential alterations of the cardiorespiratory, metabolic and neuromuscular responses to exercise. The present guidelines aim to propose a framework for an easy and accessible assessment of athletes to ensure a safe return to practice and optimise a quick recovery of performance. However, our knowledge of the effects of COVID-19 infection on physiological responses to exercise is very scarce. Research in this field will have to be carried to document this issue and update the proposed framework.

Figure 1

 1 Figure 1 Guidelines for athletes infected with COVID-19 before returning to sport. ICU, intensive care unit; VO 2 , oxygen consumption.

Table 1

 1 Biomarker assessment recommendation for athletes infected with COVID-19

	Haematological parameters Organ	References
	Interleukin 6	Inflammatory 13 14
	Complete base compte	Inflammatory 13 14
	Albumin	Liver	13 14 36
	Alanine aminotransferase	Liver	13 14
	Aspartate aminotransferase	Liver	13 14
	C reactive protein	Inflammatory 13 14 16 19 24
			36 39
	Serum creatinine	Kidney	13 14
	Lactate dehydrogenase	Liver	16 19 24
	Procalcitonin	Liver	48
	Ferrintin	Liver	45
	Troponin I	Heart	48
	Creatine kinase-MB	Heart	48
	N-terminal pro b-type	Heart	41 48
	natriuretic peptide

Fabre J-B, et al. BMJ Open Sp Ex Med 2020;0:e000849. doi:10.1136/bmjsem-2020-000849 by copyright. on February 22, 2021 by guest. Protected http://bmjopensem.bmj.com/ BMJ Open Sport Exerc Med: first published as 10.1136/bmjsem-2020-000849 on 17 October 2020. Downloaded from

by copyright. on February 22, 2021 by guest. Protected http://bmjopensem.bmj.com/ BMJ Open Sport Exerc Med: first published as 10.1136/bmjsem-2020-000849 on 17 October 2020. Downloaded from

by copyright. on February 22, 2021 by guest. Protected http://bmjopensem.bmj.com/ BMJ Open Sport Exerc Med: first published as 10.1136/bmjsem-2020-000849 on 17 October 2020. Downloaded from

Funding The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.

Author affiliations 1 Human Motion Analysis, Humanfab, Aix-en-Provence, France [START_REF] Pascarella | COVID-19 diagnosis and management: a comprehensive review FE[END_REF] Health and Safety Department, Aix-Marseille-University, Marseille, France 3 Physical Therapy, Humanfab, Aix-en-Provence, France [START_REF] Ren | Identification of a novel coronavirus causing severe pneumonia in human: a descriptive study[END_REF] Institut Universitaire de France (IUF), Paris, France [START_REF] Zhou | A pneumonia outbreak associated with a new coronavirus of probable bat origin[END_REF] Université Clermont Auvergne, AME2P, Clermont-Ferrand, France Twitter Jean-Bernard Fabre @jibfabre.

Collaborator Delphine Courtois-Renolleau.

Contributors All authors have contributed substantially to the preparation of this manuscript. J-BF conceived the original idea, carried out the initial literature searches and drafted the first version of the manuscript and realised the submission process. VM and LG redrafted parts of the manuscript and contributed to further literature searches and revised the manuscript critically several times and contributed intellectually its contents. WV gave highlight about the medical examination. JM and RM realised figure and table. J-BF takes responsibility for the integrity and accuracy of the information contained in the article.

Competing interests None declared.

Provenance and peer review Not commissioned; externally peer reviewed.


P Declerck

Critical subsystems in Time Interval Models Application to a baking process

Keywords: Time Interval Model, P-time Event Graph, cycle time, critical, linear programming

The aim of this paper is the generation of the min-critical and max-critical subsystems which determine the optimal cycle times. Considering a Time Interval Model which can describe Timed Event Graphs and P-time Event Graphs completely, each critical subsystem depends on the lower and upper bounds of the time durations. The proposed approach which is based on linear programming makes a classification of the relations which describe the system. The application to a baking process in a plant bakery shows that the min-critical and max-critical subsystems are not limited to the critical circuits of the Event Graph.

Introduction

Context and aim. In the field of discrete event systems, the study of time is a fundamental topic and the dater description is an efficient tool which can describe complex time phenomena as synchronizations with different specific semantics, token deaths and additional time constraints. The studies [START_REF] Declerck | Extremum Cycle Times in Time Interval Models[END_REF] [START_REF] Declerck | Cycle Time of a P-time Event Graph with Affine-Interdependent Residence Durations[END_REF] consider a general model, named Time Interval Model below, which is described by an algebraic model A.x ≤ b and can cover a large number of models such as Timed Event Graphs, P-time Event Graphs and P-time Event Graphs with Affine-Interdependent Residence Durations [START_REF] Declerck | Extremum Cycle Times in Time Interval Models[END_REF] [START_REF] Declerck | Cycle Time of a P-time Event Graph with Affine-Interdependent Residence Durations[END_REF]. As the number of rows of A is twice the number of places and, each row of matrix A can contain more than two entries which can be different from 1 and -1 in general (despite that all arc weights of the Event Graphs are equal to 1), matrix A is not the incidence matrix of the event graph. So, classical techniques determining the circuits (or equivalently P-invariants) and computing the maximum of the classical ratios (defined by the sum of temporizations to the sum of the number of the initial tokens, for each elementary circuit) cannot directly be used, and an adaptation must be made. In addition, a simple search of any circuit faces a combinatorial explosion of the number of circuits. Therefore, most general tools such as the linear programming must be exploited. Based on a Farkas' Lemma and Stiemke's theorem, the papers [START_REF] Declerck | Cycle Time of a P-time Event Graph with Affine-Interdependent Residence Durations[END_REF] [START_REF] Declerck | Extremum Cycle Times in Time Interval Models[END_REF] analyze the existence of the extremum cycle times and propose techniques allowing the computation of the optimal trajectories. This paper was motivated by the problem of finding the bottleneck subnets which determine the minimum and maximum production rates of the production system. Applications can be found in the food industry and the transportation networks as railway nets or road networks: Typically, public works in a transportation network can reduce the general speed of the entire system for some specific localizations, but, this consequence is not systematic as the effect can be null for some other localizations. For the management of these works, it is important to determine these critical localizations (resp. non-critical localizations) where a parameter variation as a longer journey time of a particular train implies (resp. does not imply) a degradation of the best production rate of the complete process. The reliability of these critical subsystems is also crucial to ensure an adequate production rate. Another application is the vulnerability analysis of a critical infrastructure, as the crude-oil distribution network of Saudi Arabia, whose aim is to identify the critical parts of the network which should be protected against coordinated attacks [START_REF] Bernardi | A min-max problem for the computation of the cycle time lower bound in interval-based Time Petri Nets[END_REF].

In this paper, we analyze the relations of the Time Interval Model which can describe the tasks expressed by the places of Timed Event Graphs or Ptime Event Graphs. We focus on the determination of the critical relations which determine the lower and upper bounds of the optimal cycle time and are decisive for the optimal cycle times. These critical relations are defined as follows: a relation is said critical if there is a small variation of a time duration which produces a degradation of the optimal cycle time of the complete system and reduces its optimality. In the paper, we show that a variation of a time duration for a unique critical relation can lead to unexpected variations of the cycle times such as an increase of the minimum cycle time (which is a slow-down of the entire system if we desire the fastest rate) or a decrease of the maximum cycle time (which is an acceleration of the whole system if we desire the slowest rate).

The paper is organized as follows: firstly, some related works are presented below. In Section 2, we describe the algebraic model of P-time Event Graphs which allows to illustrate the approach. Section 3 presents some preliminary results in subsection 3.1, gives the main objective in subsection 3.2 and describes the techniques to determine the critical relations in subsection 3.3. Finally, the approach is illustrated by a baking process in a plant bakery in Section 3.4.

Related works. By lack of place, we only give some related works. Time can be associated with places, transitions and arcs of the Petri nets. Many studies [START_REF] Giua | Optimal Token Allocation in Timed Cyclic Event-Graphs[END_REF] consider Timed Event Graphs where the time durations of the task is associated to places. The computation of the cycle time can be made by building a transfer function of formal power series in two variables γ and δ [START_REF] Baccelli | Synchronization and Linearity. An Algebra for Discrete Event Systems[END_REF], but this technique faces numerical difficulties as a non-polynomial time complexity and the existence of a transient period which can lead to an extremely long transfer function even for small systems. A generalization is the class of P-time Event Graphs where a time interval is associated with the places. In the same spirit, the paper [START_REF] Lee | Steady State Analysis of Timed Event Graph with Time Window Constraints[END_REF] considers a model named Negative Event Graph which corresponds to a P-time Event Graph; The earliest and latest feasible steady firing schedules for each of the minimum and maximum cycle times are examined and the liveness is discussed. The article [START_REF] Hanen | Periodic schedules for linear precedence constraints[END_REF] analyzes a linear precedence constraints graph where the labels of the arcs bring a complex form of numbering of the starting times; A difference with P-time Event Graphs is that the processing times present a minimum delay but not a maximum time duration.

Different from the class of the P-time Petri nets, another class of models is the Time Petri nets that assign time or a time interval to each transition enabled by the marking. Focusing essentially on the determination of some parameters as the average marking but not on the optimal trajectories, the quantitative analysis of these models basically consists in applying enumerative techniques based on the construction of graphs as the graph of state classes or the reachability graph which suffer the well-known state space explosion problem even in the case of bounded Petri nets. To tackle this drawback, several methods avoid the construction of the whole state space. In [START_REF] Murata | Petri Nets: Properties, Analysis and Applications[END_REF], the computation of the throughput bounds is made with linear programming. To compute more accurate bounds in Stochastic Event Graphs, the strategy in [START_REF] Rodríguez | Accurate Performance Estimation for Stochastic Marked Graphs by Bottleneck Regrowing[END_REF] is to add a cycle potentially more restrictive at each iteration. The different basic quantities computed in [START_REF] Bernardi | A min-max problem for the computation of the cycle time lower bound in interval-based Time Petri Nets[END_REF] depend on an experiment interval whose value can modify the results. Another point is that these studies have a "performance monotonicity property", that is a local pessimistic transformation, such as incrementing the average firing times, leads to a slower transformed net system. This property implies that these papers can use the extreme points of the transition interval firing times for the computation of their performance bounds contrary to this proposed paper where the considered models do not presents this property: in general, each optimal bound of the production rate depends on the lower and upper bounds of the places together.

Petri nets and algebraic model

In the paper, "resp." is an abbreviation of "respectively". The notation |E| stands for the cardinality of the set E while the notation A i,. corresponds to row i of matrix A. The transpose of the matrix A is denoted A t . We consider the 'dater' representation well-known in (max, +) algebra [START_REF] Baccelli | Synchronization and Linearity. An Algebra for Discrete Event Systems[END_REF]: each variable x i (k) over R with k ∈ N represents the date of the k th event associated with x i . A brief description of the Petri nets and P-time Event Graphs is now presented. A Petri net is a pair (GR, M 0 ), where GR = (V, AR) is a bipartite graph defined as follows: the set V is a finite number of nodes which are partitioned into disjoint sets of places P and transitions T R; the set AR consists of pairs of the form (p i ,x j ) and (x j ,p i ) with p i ∈ P and x j ∈ T R. The initial marking M 0 is a vector of dimension |P | whose elements denote the number of initial tokens in the respective places and M 0 i is the initial marking of the place p i ∈ P . The set • p i (resp., p • i ) is the set of input (resp., output) transitions of p i ∈ P . The set • x i (resp., x • i ) is the set of input (resp., output) places of the transition x i ∈ T R. For a Petri net with |P | places and |T R| transitions, the incidence matrix

W = [W ij ] is a |P | × |T R| matrix of

integers and its entry is given by

W ij = W + ij -W - ij
where W + ij is the weight of the arc from transition j to place i, and W - ij is the weight of the arc from place i to transition j. In a Petri net, a firing sequence from a marking M, implies a sequence of successive markings. The characteristic vector s of a firing sequence S is such that each component of s is a natural number corresponding to the number of firings of the corresponding transition. A marking M reached from an initial marking M 0 by the firing of a sequence S, can be calculated by the fundamental relation: M = M 0 + W.s where W.s is the usual matrix product of W by vector s. A Petri net is called an Event Graph (also named Marked Graph) if each place has exactly one input and one output transition: formally,

| • p| = |p • | = 1 (∀p ∈ P ).
All arc weights of the event graphs considered below are unitary. In Petri Nets (PNs), time can be associated with places, transitions or arcs. A Timed Event Graph is a triple (GR, M 0 , f ) where (GR, M 0 ) is an Event Graph and f : P → R + is the mapping that assigns a holding time to each place. This duration represents the time that a token must spend in the place before it can play its role in the enabling of the downstream transition. The transition firings are supposed to be instantaneous. For Time PNs, temporal intervals can be associated with places or transitions but the corresponding subclasses (P-Time PNs and T-Time PNs ) are fundamentally different. In T-Time PNs, a temporal interval is associated with each transition enabled by the marking while a temporal interval of availability is associated with each token which enters a place in P-Time PNs. In Time Stream PNs, temporal intervals are associated with arcs outgoing from places and the firing interval of transitions is defined by different semantics [START_REF] Giua | Optimal Token Allocation in Timed Cyclic Event-Graphs[END_REF] [START_REF] Schrijver | Theory of linear and integer programming[END_REF]. We now define P-time Event Graphs.

P-time Petri nets allow the modeling of discrete event systems with time constraints on tokens. We associate a temporal interval defined in R + × (R + ∪ {+∞}) with each place: each place p l ∈ P is associated with an interval

[T - l , T + l ],
where T - l is the lower bound and T + l is the upper bound. A P-time Event Graph is a triple (GR, M 0 , f ) where (GR, M 0 ) is an Event Graph, and the mapping f :

P → R + × (R + ∪ {+∞}) associates with each place p l an interval [T - l , T + l ] with 0 ≤ T - l ≤ T + l . The interval [T - l , T + l ]
is the static interval of duration time of a token in place p l . The token must stay in this place during the minimum residence duration T - l . Before this duration, the token is in a state of unavailability for firing the outgoing transition. The value T + l is a maximum residence duration after which the token must leave place p l (and can contribute to the enabling of the downstream transition). If the token does not leave the place before T + l , it dies. So, the token is available to fire the outgoing transition in the time interval [T - l , T + l ]. Each considered event is the firing of transition x i ∈ T R and for the sake of simplicity the relevant date is usually denoted x i (k) ∈ R with k ∈ N where x i (0) is the first firing date of x i . Let m ∈ N be the maximum number of the initial tokens:

m = max{M 0 l |l ∈ [1, |P |]}.
If we assume a FIFO functioning of the places which guarantees that the tokens do not overtake one another and the absence of token deaths, a correct numbering of the events can be carried out. A P-time Event Graph can be described by the following system

T - l ≤ x p • l (k) -x• p l (k -M 0 l ) ≤ T + l (1)
for each place p l ∈ P where each place p l links one upstream transition {x j } =

• p l and one downstream transition

{x i } = p • l . The difference x p • l (k) -x• p l (k - M 0 l )
is the time duration of the presence of a token generated at x• p l (k -M 0 l ) and deleted just after x p • l (k). Without reduction of generality, we can suppose that m = 1, that is M 0 l ∈ {0, 1} which can be obtained by the building technique given in appendix of [START_REF] Declerck | Cycle Time of a P-time Event Graph with Affine-Interdependent Residence Durations[END_REF]. Therefore, system (1) can be rewritten under the following algebraic model [START_REF] Bernardi | A min-max problem for the computation of the cycle time lower bound in interval-based Time Petri Nets[END_REF]

defined over R which is named Time Interval model (TIM). Each row of (2) corresponds to a pair (p l , T - l ) for i ∈ {1, . . . , |P |} or (p i-|P | , T + i-|P | ) for i ∈ {|P | + 1, . . . , 2.|P |}. So, G - G + . x(k) x(k + 1) ≤ -T - T + , (2)
for k ∈ N where: n = |T R| and q = |P | are natural numbers; x(k) and 1) for each place p l ∈ P is contained in the first rows of (2) relevant to G -and -T -while

x(k + 1) ∈ R n ; T -and T + ∈ (R ∪ {-∞,+∞}) q ; G -= G - 1 G - 0 and G + = G + 1 G + 0 ∈ R q ×2.n . Matrices G -and G + express time connections inside the P-time Event Graph. Relation T - l ≤ x p • l (k) -x• p l (k -M 0 l ) of (
x p • l (k) - x• p l (k -M 0 l ) ≤ T +
l appears in the last rows relevant to G + and T + . Precisely, the following property defining G -and G + makes the connections with the rows of the usual incidence matrix W.

Property [START_REF] Declerck | Cycle Time of a P-time Event Graph with Affine-Interdependent Residence Durations[END_REF] Let us assume that m = 1. For each place p l ∈ P , we have:

(G - 0 ) l,. = W l,. and (G - 1 ) l,. = 0 if M 0 l = 0 ; (G - 0 ) l,. = -(W -) l,. and (G - 1 ) l,. = (W + ) l,. if M 0 l = 1. Moreover, G + = -G -.
3 Critical subsystems

Preliminary results

Let

H = G - 1 + G - 0 G + 1 + G + 0 , N = G - 0 .u G + 0 .u and θ = -T - T + .
The dimensions of these matrices are respectively (2.q × n), (2.q × 1) and (2.q × 1). Let u be a unitary vector with |u| = |x(k)| = n, that is, u = (1 . . . 1)

t . The following definition describes a 1-periodic behavior of TIM (2).

Definition 1 TIM (2) follows a 1-periodic behavior when its trajectory satisfies equality x(k + 1) = λ.u + x(k) for k ≥ 0 where λ is the cycle time.

The introduction of x(k + 1) = λ.u + x(k) in TIM [START_REF] Bernardi | A min-max problem for the computation of the cycle time lower bound in interval-based Time Petri Nets[END_REF] gives the equivalent form

(G - 1 + G - 0 ) (G + 1 + G + 0 ) .x(k) + G - 0 G + 0 .λ.u ≤ -T - T + (3) or H.x(k) ≤ θ -N.λ for k ≥ 0. (4)
which is the starting point of the study: if a pair (x(k),λ) satisfies ( 4), then TIM (2) can follow the relevant 1-periodic trajectory.

An objective is to determine the minimum or maximum cycle time such that the TIM (2) follows a 1-periodic behavior starting from the finite starting point x(0) (i.e. the first firing dates of the transitions when time Petri nets are considered) which is limited by the addition of the constraint x(0) ≥ L in the problem of minimization (x(0) ≤ L in the problem of maximization) with L ∈ R n . These constraints present a practical point of view as we can consider the example of a company where the working day starts at 8.00. A current rule is that each task i must start after 8.00 as the company is closed before: we have x i (0) ≥ L i with L i = 8.00 for i = 1, . . . , n. A symmetrical rule is as follows: each task i must start before 8.00 and we have x i (0) ≤ L i .

Assuming the existence of a solution in (4), the problems of minimization and maximization of the cycle time λ can be written as follows. The optimal cycle time such that the TIM (2) follows a 1-periodic behavior starting from a finite initial starting point x(0) is the solution to the following linear programming problem.

Problem I min(λ) (resp., max(λ)) under constraint (4) with x(0) ≥ L (resp., x(0) ≤ L).

Deduced from a variant of the famous Farkas' Lemma (Corollary 7.1.e in [START_REF] Schrijver | Theory of linear and integer programming[END_REF]), the following theorem focuses on the existence of a pair (x(0), λ) in (4) without the computation of a starting point x(0) as in Problem I. Let

Y = {y ∈ R 2.q | y.H = 0 with y ≥ 0} with the following partition Y = Y -∪ Y = ∪ Y + exploited below. Y -= {y ∈ Y | y.N = -1} Y = = {y ∈ Y | y.N = 0} Y + = {y ∈ Y | y.N = 1} (5)
Theorem 1 [START_REF] Declerck | Cycle Time of a P-time Event Graph with Affine-Interdependent Residence Durations[END_REF] System (4) has a solution (x(0), λ) if and only if the two following conditions are satisfied:

(∀y ∈ Y = ) y.Θ ≥ 0 (6)
λ min = max y∈Y - (-y.Θ) ≤ λ max = min y∈Y + (y.Θ) . (7)
In other words, the two inequalities ( 6) and ( 7) express general conditions of existence of a pair (x(0), λ). The first one is a constraint independent of the cycle time while the second one gives an interval of possible values [λ min , λ max ] which can be taken for the cycle time. In the case of a Timed Event Graph, λ min is the value of the usual optimal cycle while λ max is infinite. In fact, the previous theorem and Problem I implicitly consider a fixed event number k which must be generalized to an infinite horizon: this generalization is made in Theorem 3 which exploits the following definition of homogeneity and Theorem 2.

Definition 2 TIM ( 2) is said to be forward-homogeneous (resp., backwardhomogeneous) if any trajectory of TIM ( 2) is invariant by shifting with any positive delay (resp., negative delay), that is: if a trajectory x(k) satisfies (2), then the trajectory x(k) + ω.∆.u with ω = 1 (resp., ω = -1) also satisfies [START_REF] Bernardi | A min-max problem for the computation of the cycle time lower bound in interval-based Time Petri Nets[END_REF] for any delay ∆ ∈ R with ∆ > 0. TIM ( 2) is said to be strictly-homogeneous when it is forward-homogeneous and backward-homogeneous.

The following theorem gives an algebraic way to analyze the homogeneity of the TIM (2).

Theorem 2 [4] TIM (2) is forward-homogeneous (resp., backwardhomogeneous) if and only if H.ω.u ≤ 0 with ω = 1 (resp., ω = -1) . TIM (2) is strictly-homogeneous if and only if H.u = 0 .

Main objective

Let us analyze the 2.q relations of TIM (2) where I = {1, . . . , 2.q} is the set of the relevant indexes. For simplicity, we do not distinguish a relation and its relevant index and, a relation whose index is i is directly written "relation i". The main objective of this paper is the determination of the min-critical (resp., max-critical) relations of the TIM (2). In Timed Event Graphs, each circuit where the ratio equals the maximum ratio is a critical circuit as a small variation of any time durations relevant to the corresponding critical places can modify the value of the minimum cycle time. In this paper, the concept of critical places relevant to the critical circuits is replaced by the notion of critical relations which is now defined.

Definition 3 The relation

G - G + i,. . x(k) x(k + 1) ≤ Θ i , (8)
for i ∈ I in TIM ( 2) is said min-critical (resp., max-critical) for the minimization (resp., maximization) if there is a small variation of Θ i which modifies the optimal value of the cycle time.

Therefore, the goal is now to specify the connections between variations of a time duration and a degradation of an optimal cycle time, and to provide the relevant exact variation signs, when this connection exists.

Approach

Before presenting the proposition and the algorithm determining the critical relations, Theorem 1 is now extended to an infinite horizon. The addition of an algebraic condition allows to avoid the development of the relations (4) on an infinite horizon k = 0, 1, 2, 3, . . . Theorem 3 When H.ω.u ≤ 0 with ω = 1, TIM (2) satisfies x(k + 1) = λ.u + x(k) for any k ≥ 1 for some given cycle time λ, if and only if constraints ( 6) and ( 7) are satisfied.

Proof.

Firstly, Theorem 2 shows that if H.ω.u ≤ 0 with ω = 1 the TIM ( 2) is forward-homogeneous. By definition, if an elementary trajectory {x(k), x(k + 1)} satisfies (2), we can deduce that {x(k) + ω.∆.u, x(k + 1) + ω.∆.u} also satisfies (2) for any ∆ ∈ R. Particularly, we can take ∆ = λ and if {x(0), x(1) = x(0) + λ.u} satisfies (2) which expresses the starting of the trajectory, then {x(1) = x(0) + λ.u, x(2) = x(0) + 2.λ.u} also satisfies [START_REF] Bernardi | A min-max problem for the computation of the cycle time lower bound in interval-based Time Petri Nets[END_REF] and the reasoning can be repeated for {x(k), x(k + 1)} with k ≥ 2. Therefore, if H.ω.u ≤ 0 with ω = 1, the existence of the elementary trajectory {x(0), x(1) = x(0) + λ.u} satisfying (2) implies that the TIM (2) can follow a relevant 1-periodic behavior on an infinite horizon starting from x(0). Secondly, the existence of the elementary trajectory {x(0), x(1) = x(0) + λ.u} satisfying (2) is equivalent to the satisfaction of (4) which has been obtained by elementary manipulations. Finally, the application of Theorem 1 gives the equivalence of the existence of a solution in (4) and the satisfaction of the two constraints ( 6) and [START_REF] Lee | Steady State Analysis of Timed Event Graph with Time Window Constraints[END_REF].

As theorem 3 allows a complete analysis on an infinite horizon, relations (7) can be exploited as each non-null component of vector y highlights a relation of TIM (2) which corresponds to a min-critical relation if y satisfies y.Θ = λ min (resp., max-critical relation if y satisfies y.Θ = λ max ). The following result gives an efficient way of determining the critical relations and specifies the suited sign of the variation of the time duration in definition 3. Note that the consideration of a vector y ∈ Y -which is not relevant to the minimum, that is, -y.Θ < λ min (resp., maximum, that is, λ max < y.Θ) cannot produce a variation of the optimal solution for a small variation of the time durations contained in Θ. The two cases in following proposition consider the same variation of the entry Θ i that is, a decrease.

Proposition 1. Assuming that the constraints H.ω.u ≤ 0 with ω = 1 and (6) are satisfied, relation i ∈ {1, . . . , 2.q} in TIM ( 2) is:

min-critical if any decrease of the entry Θ i implies an increase of λ min .

max-critical if any decrease of the entry Θ i implies a decrease of λ max .

Proof.

Below, the decrease of the entry Θ i is denoted by the positive value ∆ i : formally, ∆ i > 0. Applying theorem 3, we can now analyze the remaining constraint [START_REF] Lee | Steady State Analysis of Timed Event Graph with Time Window Constraints[END_REF]. Let us consider a row-vector y ∈ Y -leading to the optimal solution λ min : So, -y.Θ = λ min otherwise -y.Θ < λ min . Firstly, for any modification of Θ, vector y is still a solution of the constraints as the sets Y and Y -do not depend on the time durations. The expression -y.Θ = λ min shows that the decrease of Θ i implies an increase of λ min for y i = 0 as the relevant optimal solution λ min is the maximum of the different products. We obtain λ min = λ min + y i .∆ i . Precisely, as Θ = (-T -) t (T + ) t t , a decrease of a component i of Θ corresponds to an increase of an entry i of T -or a decrease of an entry i of T + . The modified problem gives a greater optimal solution λ min = λ min +y i .∆ i for a decrease of an entry i of T + (T + i -∆ i ) or for an increase of an entry i of T -(T - i + ∆ i ). The value of coefficient y i. shows the sensitivity to a variation of the time parameter T - i or T + i . For λ max = y.Θ, the reasoning is similar. For y ∈ Y + and a decrease of an entry of Θ, the modified problem gives a lower optimal solution λ max = λ max -y i .∆ i .

As the previous result shows the effects on the optimal solution λ min or λ max which restrict the interval [λ min , λ max ], the determination of the critical parts can be made by the following algorithm based on the checking of each time duration.

Algorithm of detection of min-critical and max-critical relations.

Let ∆ be an arbitrary small positive scalar (∆ > 0)

For each relation i ∈ I in TIM (2) End-for Based on the computation of the cycle time which can be made by any technique, the approach generates values of the cycle time inside the interval [λ min ,λ max ]. It can be seen as a slight restriction of the evolution as the modification of the time duration has led to slow down or to accelerate the process described by the TIM (2) respectively.

Θ i ← Θ i -∆ (that is, T - i ← T - i + ∆ or T + i ← T + i -∆ resp

Practical case study

Represented by a P-time Event Graph in Fig. 1, this example considers a baking process in a plant bakery composed of two processes giving a bread of higher quality (upper range 1) and of an ordinary quality (ordinary range 2) respectively. The two ranges follow the same production sequence on the four machines which are the kneading machine, the dividing machine, the molding machine and the oven: they are described by places p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 19 and p 7 , p 8 , p 9 , p 10 , p 11 , p 12 , p 13 , p 18 , respectively. The connections between the two production lines are expressed by places p 14 , p 15 , p 16 , p 17 . Taken from a semi-industrial plant bakery [START_REF] Gros | DEA report in "Biotechnologies et Industries Alimentaires[END_REF], the time durations specific for each range and an additional operation of proofing (place p 8 ) for the range 2 lead to two different qualities. The main operations are as follows. After mixing the ingredients, the dough is kneaded. This step of kneading is represented by places p 1 and p 7 . This operation lasts between 15 and 27 minutes. The following step is the first proofing which is the process of leaving the dough in the machine for 10 to 30 minutes at a steady temperature of 27 • C which leads to a fermenting. Only the process 2 contains this step (place p 8 ). Then, the dough is divided into pieces of about 900 grams each and placed in nets in the proofer (this step of dividing is represented by p 2 for the process 1 and p 9 for the process 2). The following step is the second proofing which lasts 180 minutes for the range 1 (p 3 ) and 15 to 20 minutes for the range 2 (p 10 ). Then each piece of dough is shaped with a molding machine (the molding step is represented by p 4 for the range 1 and p 11 for the range 2). Then the operation of final proofing provides the last fermenting (p 5 for the range 1 and p 12 for the range 2). Finally, Baking the bread takes up 20 to 25 minutes at a temperature of 250 to 280 • C in the oven and is identical for the two ranges (the baking corresponds to p 6 for the range 1 and p 13 for the range 2). The following operations as the cooling, the packing and the transport are not considered. Two dividing machines and two molding machines are available. We assume an upper limit of 300 minutes in the period of non-use of the dividing machines and molding machines otherwise another production line not described in the event graph can use these machines (places p 14 to p 17 ). The oven is assumed to be sufficiently large. The system contains 19 places and 15 transitions. By lack of place, the initial marking, the vectors and the matrices of the time interval model are not given but can directly be deduced from the figure.

The values of the minimum cycle time and maximum cycle time which are 145 and 190 respectively, are computed by applying the linear programming problems I with the function linpro() of the software Scilab. The CPU time of the execution of a linear programming problem is around 0.011s with a PC Intel Core 2.93GHz and 0.9s for the 78 problems of the complete approach.

For λ min = 145, the computation of the 38 problems based on a decrease of the component Θ i with ∆ = 0.1 (that is, an increase of T - i or a decrease of T + i resp.) leads to an increase of the cycle time where λ = 145.1 for 9 problems. Represented by the bold circle of the places in Fig. 1, the relevant set of min-critical pairs is )} while the remaining pairs yield the cycle time value λ = λ min = 145 despite the variation of the duration. For λ max = 190, the computation of the 38 problems based on the variation of each time duration with ∆ = 0.1 leads to a variations of the cycle time where λ = 189.9 for 9 problems. Represented by the grey places in Fig. 1 )} while the remaining pairs yield the cycle time value λ = λ max = 190 despite the variation of the duration. So, the set P crit min as well as P crit max depends on the lower and upper bounds of the time durations and does not correspond to some (oriented) circuits of the event graph.

P crit min = {(p 1 , T -
Let us illustrate a practical interest in the optimization of the process. We can desire to keep the best cycle time λ min which corresponds to the greatest production rate and so, to avoid a general slow-down of the whole process. It implies that a greater time duration of the kneading is not possible for the range 1 (as (p 1 , T - 1 ) is critical) but possible for the range 2 (as (p 7 , T - 7 ) is not critical). So, the choice of a reliable machine is necessary in the first case. The remark is similar for the two dividing steps (as (p 2 , T - 2 ) is critical but not (p 9 , T - 9 )) and the molding steps (as (p 4 , T - 4 ) is critical but not (p 11 , T - 11 )). Therefore, if we consider the kneading, the dividing and the molding for range 2 which are not relevant to critical pairs, an energy saving and so a financial economy can be obtained by reducing the speed of the relevant machines without modifying the best production rate of the entire process. Another possibility is the replacement of the relevant machines of the range 2 by less efficient machines. Therefore, we can make an adaptation of the algorithm of detection of the critical relations which computes the greatest ∆ with the condition that the optimal cycle time λ min is kept and the system remains consistent. For λ min , the results are as follows: the new values of time durations for T - 7 , T - 9 and T - 11 can be 20,15,20 while the initial values are 15, 10, 20, respectively. The same technique can be made if we desire to make an optimization of the machine speeds for the maximum cycle time λ max .

Conclusion

Contrary to the time-costly enumerative technique extracting every circuit, the proposed approach considers each time duration. Exploiting the variations of durations, the approach produces the min-critical and max-critical subsystems where each critical subsystem depends on the lower and upper bounds of the time durations together. The technique is effective as it only needs the application of polynomial algorithms of linear programming [START_REF] Schrijver | Theory of linear and integer programming[END_REF]. Showing that the graphical concepts must be surpassed, the practical example illustrates that the critical subsystems are not limited to the classical critical circuits of the event graph. As suggested by the practical case study in Section 3.4, a natural perspective is the cost optimization of the resources and machines without reducing the production rate and the service quality of the process.

Fig. 1

 1 Fig. 1 Plant bakery

  .) Compute the optimal solution λ min and λ max by any technique If λ min > λ min , then the relation i (the pair (p i ,T - i ) or (p i ,T + i ) resp.) is min-critical If λ max < λ max , then the relation i (the pair (p i ,T -

	i ) or (p i ,T + i ) resp.)
	is max-critical