Denis Efimov

Rosane Ushirobira

On interval prediction of COVID-19 development in France based on a SEIR epidemic model

This paper aims to identify the parameters of an original modified SEIR model for the COVID-19 epidemic's course in France by using publicly available data and applying such an identified model for the prediction of the SARS-CoV-2 virus propagation under different conditions of confinement. For this purpose, an interval predictor is designed, allowing variations and uncertainties in the model parameters to be taken into account.

I. INTRODUCTION

The SEIR model is one of the most elementary compartmental models of epidemics [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF]; it is prevalent and widely used in different contexts [START_REF] Wang | Statistical physics of vaccination[END_REF]. This model describes the evolution of the relative proportions of four classes of individuals in a population of constant size; see a general scheme given in Fig. 1. Namely, the susceptible S, capable of contracting the disease and becoming infectious; the asymptomatic E and symptomatic I infectious (capable of transmitting the disease to susceptible); and the recovered R, immune (after healing or dying). Such a simple model represents well a generic behavior of epidemics (plainly as series of transitions between these populations), and a related advantage consists of a small number of parameters to identify (three transition rates in Fig. 1: σ , γ, andb). These model characteristics are an essential issue in a virus attack with a limited amount of data available. In March 2020, that was the case in France under the appearance of the SARS-CoV-2 virus.

There are many types and variations of SEIR models [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF] (e.g., in the simplest case, the classes E and I are modeled at once, leading to a SIR model). This paper proposes a new modified discrete-time SEIR model for the COVID-19 epidemic's evolution in France. Our model is inspired by the one proposed in [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF] used for the epidemic's trend of COVID-19 in China. Other similar SIR/SEIR-type models used recently for modeling SARS-CoV-2 virus are [START_REF] Ferguson | Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand[END_REF]- [START_REF] Peng | Epidemic analysis of COVID-19 in China by dynamical modeling[END_REF]. The selected model is proposed 1 as follows (we do not consider the influence of the natural birth and mortality, since for the short period of analysis considered here the population may be assumed quasi-constant):

S t+1 = S t -b (p C I t + r t E t ) N S t , (1a)
E t+1 = (1 -σ )E t + b (p C I t + r t E t ) N S t , (1b)
I t+1 = (1 -γ)I t + σ E t , (1c)
R t+1 = R t + γI t , (1d)
where t ∈ N (the set of non-negative integers) is the time counted in days (t = 0 corresponds to the beginning of measurements or prediction), N ∈ N denotes the total population, the parameter γ > 0 represents the recovery rate (from which the mortality rate can be deduced), b > 0 corresponds to the infection rate of the virus transmission from infectious/exposed to susceptibles during a contact, σ > 0 is the incubation rate by which the exposed develop symptoms and p C > 0 corresponds to the number of contacts for the infectious I (it is supposed that infected people with symptoms are in quarantine, then the number of contacts is reduced). The parameters γ, b, and σ are constant. Finally, p C ≤ r t < +∞ is the number of contacts per person per day for the exposed population E (in confinement and depending on its severity, r t is time-varying).

Therefore, the SEIR model (1) has only three parameters to be identified: σ , γ, and b, representing the rate of change between the states E to I, I to R and S to E (as in in Fig. 1). The parameter σ has a physical meaning: σ = 1 T S , where T s is the average duration of incubation period of the virus after contamination, which can be well identified in patients. The numbers of contacts p C and r t can be evaluated heuristically based on the population density and social practices. The identification of these parameters must be performed using statistics published by authorities. As a worthy remark, many research works devoted to the estimation and identification for SIR/SEIR models were developed by now, and several in the last few years, like [START_REF] Onofrio | The interplay of public intervention and private choices in determining the outcome of vaccination programmes[END_REF]- [START_REF] Ushirobira | Estimating the infection rate of a SIR epidemic model via differential elimination[END_REF], to mention a few.

Since the measured data and parameters contain numerous uncertainties and perturbations, it is not easy to make a reasonable prediction based on the simulation of such a model with fixed parameter values (also taking into account the model simplicity and generality). However, the interval predictor and observer framework [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF]- [START_REF] Efimov | Design of interval observers for uncertain dynamical systems[END_REF] allows a set of trajectories to be obtained corresponding to the interval values of parameters and inputs, increasing the model validity without augmenting its complexity. This approach has been applied to different SEIR models (see, e.g., [START_REF] Degue | Interval estimation of sequestered infected erythrocytes in malaria patients[END_REF]- [START_REF] Degue | An interval observer for discrete-time SEIR epidemic models[END_REF]). In this paper, we apply the interval predictor method for our SEIR model [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF] to improve its forecasting quality.

Remark 1. It is worth highlighting that the interval predictor framework used here is not the only method oriented on improving the reliability of prediction using SEIR models. Usually, as in [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF]- [START_REF] Peng | Epidemic analysis of COVID-19 in China by dynamical modeling[END_REF], stochastic and agent-based simulation approaches are used. In those cases, by assuming that the parameters and initial conditions are distributed with some given probability, multiple numeric experiments are performed to reconstruct the behavior of all possible trajectories of the system. As a first remark, such a methodology needs more computational effort for its realization. Second, additional information on the form of a probability distribution for all parameters and variables is necessary, demanding either extra hypotheses or more measured data for estimation. As it is currently demonstrated by SARS-CoV-2 virus attack, it is hard to obtain such data quickly during the epidemic development. Contrarily to these approaches, the interval predictor method does not use these additional assumptions on distributions. It has also been proposed to estimate a guaranteed interval inclusion of trajectories with a minimal computational effort by the cost of a more complicated mathematical analysis and design [START_REF] Efimov | Design of interval observers for uncertain dynamical systems[END_REF].

The outline of this paper is as follows. We describe the measured data applied for parameter identification in Section II, together with some hypotheses used in the sequel. The parameters obtained in [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF] for China are tested for France and validated on this data in Section III. The method for parameter identification is presented in Section IV, with validation and some experiments on the influence of confinement parameters on COVID-19 development. An interval predictor is designed in Section V, which allows us to evaluate the situation under the variation of parameters and initial states. Final discussions and remarks are provided in Section VI.

II. AVAILABLE DATA ON COVID-19 IN FRANCE

The current population in France is N = 670640002 . The incubation period T s , widely reported in the literature for COVID-19, is considered to be between 2 and 14 days [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF], or in a more specialized research, between 2 and 12 days [START_REF] Lauer | The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application[END_REF], so we assume 1 2 ≤ σ ≤ 1 12 . The data available from public sources 3 , and which is used in this work, for the time period between March 12th and July 30th can be found in GitHub.

I t = α 1 I t , R t = D t + α 2 H t , (2)
i.e., the deaths are reported exactly (see also [START_REF] Lourenco | Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic[END_REF]), but the actual number of infected cases and the related recovered individuals can be masked due to the complexity of examination and the actual confirmation of the virus presence. An additional reason is that in many cases, the virus symptoms result in a mild reaction of patients (approximately in 80% of cases, see the sources above), hence with no official virus confirmation in such a situation. In this work, we assume modest values for these parameters, α 1 = 5, α 2 = α 1 + 1, so roughly speaking, such a choice of α 1 corresponds to registration of all non-mild cases and α 2 is selected just to be a little bit higher. The techniques to identify α 1 from the measurements of I and D are described in the last footnote (for the data in France, these approaches provide α 1 = 2.418). So, by fixing α 1 and α 2 , the two variables of the model (1), I and R, are available from the beginning of the epidemics via [START_REF] Wang | Statistical physics of vaccination[END_REF]. We also select an average value for the incubation rate, σ = 1 7 , to simplify further identification (the variation in this value is taken into account later in the interval predictor). [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF] In [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF], using the data available for the Chinese provinces of Zhejiang, Guangdong and Hubei, which have been impacted by the virus differently, the following parameter bounds have been evaluated: Selecting the average values γ = 0.155 and b = 0.0525, we choose to decrease the number of contacts for France as:

III. NUMERICAL SIMULATION WITH PARAMETERS OF

p C = 2, p N = 12, p R = 6,
which is related with smaller population density in France, in comparison to the aforementioned Chinese provinces.

In [START_REF] Ferguson | Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand[END_REF], the theory of a cyclic application of quarantine regimes of different severity is evaluated for COVID-19. By iterating the periods of complete isolation for everybody (suppression), which decelerates the virus advancement, with a time of mild regulation (mitigation), which allows the economics to be maintained on an arguable level, and when only fragile parts of the population are isolated, it is possible to attenuate the material consequences of epidemics while decreasing the load on health services. Following this idea, for simulation we consider here two scenarios of confinement:

1) Six weeks of strict quarantine and two weeks of a relaxed one, which is further periodically repeated. 2) Twelve weeks of strict quarantine and four weeks of a relaxed one, which is further periodically repeated. Remark 2. For the chosen model, these scenarios impact only r t . In other words, r t can be considered as a kind of control for the virus propagation, by imposing different periods and strictness levels for the confinement. The results of simulation of the model (1) with the described above parameters for both scenarios, 1 and 2, are given in Fig. 2 (for a better visibility, all populations are plotted in the logarithmic scale in this work). As we can conclude from these results, these scenarios of confinement do not lead to a stabilization of COVID-19 development in France (the black dotted line in the top of the plots correspond to N (the total population), then according to these graphics, the epidemics is going to stop after a total infection of the country). However, after a short analysis on how close are the obtained curves to the measured data (see Fig. 3 where dashed lines correspond to the measurements), we recognize that the model with the parameters from [START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF] does not correspond well (overestimates) to the situation in France. Therefore, an identification of parameters is needed.

IV. PARAMETER IDENTIFICATION For the parameter identification, we assume that the incubation rate σ is fixed as above, and that the symptomatic infectious I t and the recovered people R t are measured for the first J days of the virus attack as in (2) for t = 0, 1, . . . , J. From (1d) we can identify the value of the parameter γ (for convenience, we write γ for its estimate as well):

γ = R t+1 -R t I t ,
whose least squares estimation is

γ k = ∑ J-k-1 t=0 I t (R t+1 -R t ) ∑ J-1 t=0 I 2 t
for k = 0, 1, . . . , K, where 0 < K < J -1 is the number of the last days used for identification (in this work we selected K = J -10). Then the average value is used for further analysis and design (with a mild ambiguity we are using the same symbol to denote the parameter and its estimate):

γ = 1 K + 1 K ∑ k=0 γ k = 0.0391.
Next, (1c) allows us to calculate the related number of asymptomatic infectious (σ is chosen and γ is estimated):

E t = 1 σ (I t+1 -(1 -γ)I t ) ,
while the number of susceptible individuals can be evaluated using the total population S t = N -I t -R t -E t . From (1b) we can derive the infection rate (for the selected p C and r t ):

b = N E t+1 -(1 -σ )E t (p C I t + r t E t ) S t ,
whose least squares estimation is

b k = N ∑ J-k-1 t=0 (p C I t + r t E t ) (E t+1 -(1 -σ )E t ) S t ∑ J-1 t=0 (p C I t + r t E t ) 2 S 2 t
for k = 0, 1, . . . , K, then the identified value is again the average of these estimates: By the time the final version of this paper was completed, the strict lockdown has ended in France (May 11th). We chose to show in Fig. 5, the simulation results of the model (1) with the identified values of parameters for the actual scenario. Notice that the missing points in the identification of b is caused by an additional rule of the identification algorithm to stop the calculation of b if the values of p C and r t are too small. Verification on the measured and reconstructed data is shown in Fig. 6. The model can approximate the virus propagation reasonably well since these results are more consistent with France's COVID-19 statistics.

b = 1 K + 1 K ∑ k=0 b k = 0.
Let us enlarge the validity of the prediction based on (1) by considering intervals of admissible values for parameters and initial conditions.

V. INTERVAL PREDICTION

In the previous section, to make a prediction, the model [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF] with identified values of the parameters b, γ and selected choices for α 1 , α 2 , σ was used. The initial values for the states of the model S 0 , I 0 , E 0 and R 0 were selected from measured/reconstructed sets. However, due to the generic structure of the model, uncertainties in the values of the auxiliary parameters, and noises in the measured information, it is evident that the reliability of the obtained prognosis is limited. A way to overcome such weakness is to consider the Fig. 6. Results of verification with identified parameters intervals of admissible values for all variables and parameters used for simulation, so enlarging the model's validity. In such a case, we calculate/evaluate the sets of the resulted trajectories. In this work, we use for this purpose the interval framework [START_REF] Efimov | Design of interval observers for uncertain dynamical systems[END_REF].

The idea of interval prediction can be illustrated in a simple scalar system:

x t+1 = a t x t + d t ,
where x t ∈ R + is a non-negative system state, whose initial conditions belong to a given interval x 0 ∈ [x 0 , x 0 ], a t ∈ R + and d t ∈ R are uncertain inputs, which also take values in known intervals a t ∈ [a t , a t ], d t ∈ [d t , d t ], for all t ∈ N. So, we assume that x 0 ≤ x 0 , 0 ≤ a t ≤ a t and d t ≤ d t are known for all t ∈ N. The imposed non-negativity constraints on x t and a t correspond to the case of the model [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF]. We wish to calculate the lower x t and upper x t predictions on the state x t of this system under the introduced hypotheses on all uncertain variables, which have to satisfy x t ≤ x t ≤ x t , ∀t ∈ N. The theory of interval observers and predictors [START_REF] Efimov | Design of interval observers for uncertain dynamical systems[END_REF], [START_REF] Leurent | Interval prediction for continuous-time systems with parametric uncertainties[END_REF] answers this question, and a possible solution (that utilizes the non-negativity of x t and a t ) is as follows:

x t+1 = a t x t + d t , x t+1 = a t x t + d t ,
which is rather straightforward. To substantiate the desired interval inclusion for x t by x t , x t , we can consider the lower e t = x t -x t and the upper e t = x t -x t prediction errors, whose dynamics take the form:

e t+1 = (a t x t -a t x t ) + (d t -d t ) , e t+1 = (a t x t -a t x t ) + d t -d t ,
then it is easy to verify that the terms d td t , d td t are non-negative by the definition of d t , d t , and the terms a t x ta t x t , a t x ta t x t have the same property for t = 0 by the definition of a t , a t and x 0 , x 0 , hence, e 1 ≥ 0, e 1 ≥ 0 (that implies x 1 ∈ [x 1 , x 1 ]) and the analysis can be iteratively repeated for all t ∈ N. Let us apply this method to the model (1) (clearly, each equation there has the form as above).

To this end, we assume that all parameters belong to the known intervals:

σ ∈ [σ , σ ], γ ∈ [γ, γ], b ∈ [b, b], r t ∈ [r t , r t ] ∀t ∈ N, (3)
together with the initial conditions in (1):

S 0 ∈ [S 0 , S 0 ], I 0 ∈ [I 0 , I 0 ], E 0 ∈ [E 0 , E 0 ], R 0 ∈ [R 0 , R 0 ], (4)
where non-negative values σ , σ , γ, γ, b, b, r t , r t , S 0 , S 0 , I 0 , I 0 , E 0 , E 0 and R 0 , R 0 are obtained from the ones used in the previous section by applying ±δ % deviation from those nominal quantities. Then applying the approach explained just above, we derive the equations of the interval predictor:

S t+1 = 1 -b p C I t + r t E t N S t , (5)
E t+1 = 1 -σ + b r t N S t E t + p C b I t S t N , I t+1 = (1 -γ)I t + σ E t , R t+1 = R t + γI t , S t+1 = min N, 1 -b (p C I t + r t E t ) N S t , E t+1 = min N, 1 -σ + b r t N S t E t + p C b I t S t N , I t+1 = min N, (1 -γ)I t + σ E t , R t+1 = min N, R t +
the interval predictor (5) guarantees the interval inclusions for the states of (1) for all t ∈ N:

S t ∈ [S t , S t ], I t ∈ [I t , I t ], E t ∈ [E t , E t ], R t ∈ [R t , R t ]
with boundedness of all predictions for all t ∈ N: and such a verification can be repeated for all t ∈ N. In the same way we can show that if the relations

S t ,
S 1 ∈ [S 1 , S 1 ], I 1 ∈ [I 1 , I 1 ], E 1 ∈ [E 1 , E 1 ], R 1 ∈ [R 1 , R 1 ],
S t ≤ S t , I t ≤ I t , E t ≤ E t , R t ≤ R t
are satisfied for some t ∈ N, then they also hold for t = t + 1 in (5). To substantiate boundedness of the state of the interval predictor, it is enough to guarantee that S t , I t , E t and R t do not exceed N (as it is done by construction in ( 5)), while non-negativity of S t , I t , E t and R t is ensured by [START_REF] Lourenco | Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic[END_REF]. Remark 4. The boundedness of the state of (5) established in Theorem 3 does not imply the stability of the internal dynamics of the interval predictor (it is also a reason to impose the explicit saturation in ( 5)), which is a frequent and challenging problem for the predictors [START_REF] Mazenc | Interval observers for discrete-time systems[END_REF], [START_REF] Leurent | Interval prediction for continuous-time systems with parametric uncertainties[END_REF]. Remark 5. The dynamics of lower and upper interval predictions are interrelated through the update equations of S t , S t . Thus, the dimension of the predictor ( 5) is twice higher than in the system [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF]. The values of the variables S t , S t can be evaluated using the population equation S t + E t + I t + R t = N:

S t = N -I t -E t -R t , S t = N -I t -E t -R t ,
which however does not isolate the dynamics of lower and upper interval predictions. In addition, preliminary simulations show that such a modification leads to more conservative results, so we keep (5) for all further utilization. The simulation results of the interval predictor (5) with δ = 10% are presented in Fig. 7 (the dashed and dotted lines represent, respectively, upper and lower interval bounds, the solid lines correspond to the results of simulation obtained in the previous section, the circles depict measured and reconstructed data points used for identification and validation). As we can conclude from this curve, under sufficiently significant deviations of the parameters (which correspond to the amount of data publicly available now), the lockdown slows down the epidemics. The measurements are nearly included in the obtained intervals validating the prediction (the value of δ was selected to ensure this property). There are two variants of epidemic development demonstrated in these results: optimistic, corresponding to the lower bounds of I and E, and pessimistic presented by the respective upper bounds. A further precision of the model and the parameters is needed, but as a direction after these preliminary simulations is that in March, an augmentation of the quarantine's severity was desirable. This suggestion is illustrated by Fig. 8 that presents the interval prediction for the infectious population I in the actual scenario with a deviation of all parameters. As previous, blue dashed and dotted lines correspond to the upper I and the lower bounds I (the bold lines are calculated using previous day initial conditions), and the magenta circles are the measured information, the red line is the average behavior. This graphic corresponds to the case given previously in Fig. 7 for p C = 0.5. In the optimistic scenario, the confinement constraints the virus, which is an important achievement representing a significant decay of the load to the public health services.

VI. CONCLUSION

A novel simple discrete-time SEIR epidemic model was identified and used to predict the quarantine's influence on the SARS-CoV-2 virus propagation in France. To enlarge the model prediction performance, an interval predictor method was also used to analyze the COVID-19 course. The prediction showed that a longer confinement may be a bit more efficient, but under the current uncertainty level, a more strict as possible confinement seems to be advisable.

Machine learning tools can be further used to identify and optimize the time profile for the confinement. Another possible direction of improvement is to consider a SEIR model with population separation either by age or by region (or both), but this implies increasing the number of parameters to be identified and needs a specially structured data.

Fig. 1 .

 1 Fig. 1. A schematic representation of SEIR model

  0.0721 ≤ γ ≤ 0.238, 0.05068 ≤ b ≤ 0.05429, p C = 3 (number of contacts in quarantine), p N = 15 (number of contacts in normal mode), p R = 10 (number of contacts in relaxed quarantine).

Fig. 2 .

 2 Fig. 2. Simulation results with parameters of [3] for scenarios 1 and 2

Fig. 3 .

 3 Fig.3. Results of verification with parameters of[START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF]

  0387. The obtained values γ k , b k (solid lines) together with the selected estimates γ, b (dot lines) are shown in Fig. (4). The identified values for γ and b are not included in the confidence intervals obtained for China in [3] (they are

Fig. 4 .Fig. 5 .

 45 Fig. 4. Identified parameters

Fig. 7 .

 7 Fig. 7. Simulation results of (5) under ±10% variation of all parameters

Fig. 8 .

 8 Fig. 8. Prediction of the growth of I for scenario 1 with p C = 2 or p C = 1 under deviations of values of all parameters

  γI t , where S t , S t , I t , I t , E t , E t and R t , R t are the lower and upper interval predictions for S t , I t , E t and R t , respectively.

	Theorem 3. For the model (1) satisfying the relations (3)
	and (4) with
	2b sup

t∈N r t ≤ 1, σ ≤ 1, γ ≤ 1,

  S t , I t , I t , E t , E t , R t , R t ∈ [0, N]. ≤ σ E t ≤ σ E t , γI t ≤ γI t ≤ γI t

	Proof: By direct calculations we can check that
	b	(p C I t + r t E t ) N	≤ b	(p C I t + r t E t ) N	≤ b	p C I t + r t E t N	,
		b	r t N	S t -σ ≤ b	r t N	S t -σ ≤ b	r t N	S t -σ ,
			p C b	I t S t N	≤ p C b	I t S t N	≤ p C b	I t S t N	,
	σ E t due to (3) and (4) for t = 0, and
		1 ≥ b		p C I t + r t E t N	, 1 + b	r t N	S t ≥ σ
	due to (6) (recall that r t ≥ p C , I t + E t ≤ 2N, thus S t ≥ 0),
	then as we demonstrated above

Denis Efimov and Rosane Ushirobira are with Inria, Univ. Lille, CNRS UMR 9189 -CRIStAL, F-59000 Lille, France, Denis.Efimov, Rosane.Ushirobira@inria.fr[START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF] Compared to the model in[START_REF] Yang | Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions[END_REF], we do not consider the inflow/outflow variables for each state in our analysis.

www.en.wikipedia.org/wiki/Demographics of France.

See www.santepubliquefrance.fr/, www.geodes.santepubliquefrance.fr and www.data.gouv.fr.

We denote here by I, D and H the number of detected infected, deceased and recovered individuals, respectively.Obviously, not all cases can be detected and documented by the public health services, then there is a ratio between populations I and I, R and D + H as well, which is denoted in this work by α, whose interval of admissible values is estimated from different sources

as 5 ≤ α ≤ 20. Formally, such a ratio α has to be time-varying and different for I and R. Due to the strict requirements of France Health Services, in this paper, we take the following hypotheses:

Data from France.

See, for example, Coronalinks-3-19-20, or a dedicated analysis in CMMID or University of Melbourne.


Understanding the role of scholarly journals

Registration

Dissemination

Peer review

Archival record

The current scholarly journal landscape is locking the whole process as one single workflow Main functions of scholarly journals (Mabe, 2010) • Registration: used to claim priority

• Dissemination: receiving feedback, getting fame

• Peer review: canvassing content to fit into the expected model

• Archival record: obliviating a personal management of publication assets

Consequences on scholars' attitude/objectives:

Issues with the current scholarly system

• Ambiguity of the term publication

• Making something public

• Cf. use of online documents in patent examination

• Having it peer-reviewed

• "How many publications do you have?"

• Reflected in the terminology: preprint -postprint • Raises a lot of critical IP issues

• Who has the rights on a given document? Author vs. publisher

• Variety of peer-reviewed (or not) objects

• Journal paper, conference proceedings, books, book chapters, blog posts, data sets, educational objects (cf. DARIAH campus) etc.

• What counts as a "publication"?

• The scholarly publishing business

• Price and copyright • Open access publishing has not necessarily contributed to improving the landscape

• Bias towards publishing more (not just predatory journals), to earn more, not to improve science

General issues related to peer review

• A slow and inefficient process?

• Time to publica-on • Subjec-vity and biased judgement • Small number of reviewers, one stop shop for being accessible

• ORen leads to desk rejec5on (by editor, prior to any peer review)

• Conserva-sm

• Journal topics, editorial cau5ousness, level of exper5se ("I don't understand this new idea")

• Leads to cascading submissions and possibly to research invisibility

• Benjamin List, director at the Max Planck Ins.tut für Kohlenforschung in Mülheim an der Ruhr, Germany, and the editor-in-chief of Synle&.

• "When it works -and that's much of the -me -peer review is a wondrous thing. But all too o^en, it can be an exercise in frustra-on for all concerned. Authors are on tenterhooks to learn of poten-ally career-changing decisions. Generous peer-reviewers are overwhelmed. And editors are condemned to doggedly sending reminders weeks a^er deadlines pass."

• Source: (List, 2017)

Escaping the increasing journal desperation

• Tough figures 2015) reported that across a range of journals, the supply of submitted papers exceeded reviewer availability by between 15 and 249%

• 63.4 million hours were devoted to peer review, of which 18.9 million hours were undertaken by the top 5% of contributing reviewers

• (Villar, 2019) "why not practise immediate acceptance, too?"

• "is it time to consider whether a paper needs review on submission in the first place?"

• "[I]t seems that the strong coupling between journals and peer review as a requirement to protect scientific integrity seems to undermine the very foundations of scholarly inquiry."

Decoupling publication and peer review with PPPR

General principles of post publication peer review

• By definition: providing an assessment mechanism after a scholarly document has been "published", i.e. made publicly available • Two basic building blocks

• A publication repository where the document is made available (host) • A certification platforms that allows reviewers to carry out their assessment (overlay service)

• Beyond these basic concepts, a variety of approaches

Various forms of PPPR -1

• Un-peer-reviewed material vs already reviewed material

• PPPR as the first assessment process Various forms of PPPR -2

• Levels of formality (free vs. channelled PPPR)

• Formally invited reviewers, after publication of the un-reviewed article • Volunteer reviewers, after publication of the un-reviewed article

• Credits associated to potential reviews

• E.g. Science Open, The Winnover

• Cf. commenting system already existing in traditional journals

• Comments on a 3rd party site (e.g. Blog or dedicated commenting platform)

• Towards a free commenting space

• PubPeer, Pubmed Commons, ResearchGate, Academia

Channelled peer-review as a process

• "journals could institute periodic post-publication review, in which the journal would solicit formal review of the article, focusing on how well its methods and results have held up, given the research that has been published in the intervening period. Such reviews would provide valuable historical perspective." (T.A. Gibson, Nature-Correspondance, 2007)

• Note: PPPR seen as an opportunity to question the traditional peerreview process

A disruptive view on scholarly publishing?

• Contributes to getting a disruptive view on publishing in any case…

• Beyond "open access"

• Services

• Better identification of the services needed for scholarly publication • Better specification of the workflows/interfaces between them (cf. COAR initiative)

• Better compensation models for peer reviewers

• Better tracing, better acknowledgements (cf. Publons),

• Business models

• Contributing to associating precise costs to services:

• to hosting, certification, long-term archiving • Cf. PEER project economic report

• Getting a better estimate of what we pay with commercial offers

• Caveats

• Theoretical decoupling?

• E.g. F1000, myScienceWorks (not to mention RG or Academia)

• Not so disruptive because not so novel… So in a way, this is not new…

• Papers presented at a conference before publishing a journal full version, or an edited book • Comments from colleagues, scholarly correspondence • Ensuring an open scholarly debate (see Pubpeer) The controversies over anonymity in post-publication peer review"

  We see the point where open science and post publication peer review may come into the game Back to basics: the role of scholarly publication (Tennant et al. 2019) • Makes science public • Ensures the quality of science • Defines anteriority of results • Makes articles searchable/findable • Archives for the future Melanie Imming, Jon Tennant. (2018). http://doi.org/10.5281/zenodo.128557 Do the current scholarly publication mechanisms fulfil these requirements? The scholarly publishing tension (Tennant et al. 2019) (A) TradiVonal peer review publishing workflow. (B) Preprint submission establishing priority of discovery.

•

  Cf. "LMF and TEI crosswalks": hVps://hal.inria.fr/hal-00762664v4 • LeWers to the Editor • Formal reviews as publicaYons • Recensions in the SSH domain • Comments on a blog post Challenges for post-publication peer review • Ensuring participation • Limited corpus of papers gaining interest, bias towards controversial ones (cf. Science Open, The Winnower) • Traditional reviewers' selection proves more efficient • Controlling reviewers' expertise • Science Open requires reviewers to have five articles in ORCID, PubMed Commons requires one article in PubMed. • No check of expertise in the field of the commented article • Preventing the fragmentation of discussions

•PubPeer•

  'gotcha' mentality • Encouraging online junk… or fostering responsibility (online persona) .1 Practices, evaluation and mapping: Methods, tools and user needs. http://openup-h2020.eu/wp-content/uploads/2017/01/OpenUP_D3.1_Peer-review-landscape-report-1.pdf Post (peer-reviewed) publication peer review Conceived for the biological domain • Objective of the creators: develop critical thinking • Workflow • Based upon already published papers (DOI, entry in ArXiv as identification means) • Open comments à la Amazon => automatic notification of authors • Possible answers from authors • Commenters remain anonymous • Not part of the initial design, but as a request from younger scholars • Shift in practices: whistle blowing and/or harassment • Cf. Torny, 2018: "Pubpeer: vigilante science, journal club or alarm raiser?

  • A general issue of social networks: comments on the article itself (if that feature is available), in PubPeer, on PubMed Commons, on Research Gate, on blogs, on Twitter, on F1000Prime…

Failed attempts -PubMed Commons

• Platform launched in 2013 (NIH's National Library of Medicine (NLM))

• Posting comments on any published paper indexed by PubMed (Registered users)

• "responsible post-publication peer review,"

• Comments that can be viewed by anyone accessing the paper

• Closes door in 2018:

• Low levels of engagement

• Alternate venues for public discussion

• PubPeer, journals' sites (BMJ, PLOS, eLife)

• "it just wasn't really useful, let alone essential"

Cf. Dolgin, 2018

Post publication peer-review of authors' manuscripts The baseline: a publication repository • Long-term archiving

• Advantages for the user • Article submission in F1000 repository ("publication") • Peer review by invited referees

Episciences -lessons learnt

• Variable acceptance of the post-publication peer-review model

• Candidate journals may be frightened by the feature

• Sustainability

• Decoupling the sustainability of content from that of the certification platforms

• Towards even more certification models

• Open peer review, author's initiative, community review

• "Transitioning" from the existing journal eco-system

• Candidate journals are reassured to feel "at home"

• Raises a few questions

• Community and traditional review in parallel

• Authors ask friends to provide feedback during the review process

• Multiple branding The Peer Community in (PCI) project

How does this work?

Source: https://peercommunityin.org "We would value the recommendations seriously and may even use them for handling without further peer review (only peer review by handling editors)" etc.

PCI and journals

Generalising PPPR plaGorms

The COAR initiative Modelling Overlay Peer Review Processes with Linked Data Notifications (Walk, 2020)

• An initiative by COAR (cf. Pubfair)

• Cf. report: Next Generation Repository (NGR)

• Resource-centric network (metadata and content)

• Connected with resource-oriented networked services Where do we stand?

• Building blocks of a generalized architecture for PPPR

• Covering a wide variety of scholarly objects

• Contributing to the evolution of peer review

• Decoupling prevents peer review from being a go/no-go activity • Scholarly relevance seen as a long term activity

• Lessons for DARIAH

• Understanding

• Looking at existing platforms, matching features to SSH needs

• Recommending

• As part of DARIAH's open science policy (e.g. preprints, early publication of results)

• Experimenting

• Data journals, assessment of educational material (DARIAH campus)

• Need for a cultural revolution?

Merci pour votre aIenJon

Some references • Mark Ware and Michael Mabe, 'The STM Report: An Overview of Scientific and Scholarly Journal Publishing', 2015.

• "Why Figshare? Choosing a New Technical Infrastructure for 4TU.ResearchData." 2020. Open Working (blog). August 18, 2020. https://openworking.wordpress.com/2020/08/18/whyfigshare-choosing-a-new-technical-infrastructure-for-4tu-researchdata/.


Dermas Moffo
email: dermas.moffo@femto-st.fr

Philippe Canalda
email: philipe.canalda@femto-st.fr

Relative and Absolute Positioning in Ultra Dense Mems System

Keywords: 2D Micro-localization, relative and global positioning, Micro-systems, localization algorithm, simulation

Mems Microrobot applications evolve in ultradense contexts. They are at the stage of the simulation to obtain 2-Dimension or shapes for the deployment of individual or collective intelligent programs, and collectively achieve miniaturized and flexible workshops. Now-on, manufacturers and academicians projects are at the stage of modeling and prototyping Mems, and simulating Mems systems applications.

To tackle these objectives, addressing individual positioning of Mems inside very dense micro-systems becomes strategic. In many situations, it is more useful to know the relative positioning between Mems and their orientation, than the knowledge of absolute positioning of each Mems.

Works achieved on microrobots micropositioning are either stochastic or deterministic. The formers are based on probabilistic approaches, that provides better results on a small scale but produce greater error accumulation with a large sample. The latters are based on geometric considerations to accurately compute the position of each Mems, and then distribute it to all the elements.

We propose a model of a smart-grid (orthogonal and hexagonal lattice) of microrobots with regular geometry, and their connectors (actuators and sensors for moves and other actions) communicating by contact. Based on this model, we propose a mixed positioning algorithm (absolute and relative) in 2D without mobility of Mems in a group ranging in size from thousands to millions of items, based only on neighborly relations. Then by simulation we perform a functional validation of our algorithm, and a validation of the scalability of our algorithm on orthogonal grid of over 1 million node.

I. INTRODUCTION

The miniaturization of the automated systems continues inexorably. With her, the densification of the systems reaches the million items in one m 3 , requiring new strategies of coordination. In such systems communications aspects between Mems, the individual and collective positionings of Mems, and finally the modelling of these systems constitute so many challenges in the academic and industrial communities.

In the sector of the communications for example, the development of new materials such as graphene, a leaf of the thickness of an atom, compound of atoms of carbon arranged according to a hexagonal motive, allows the manufacturing of new nano-devices. The frequency expected from nanotransmitters and nano-antennas made of graphene is of the order of the terahertz, giving rise to nano-wireless communications between microrobots [START_REF] Josep | The internet of multimedia nano-things[END_REF] Microrobots endowed with these ways are still at the stage of the design. The current microrobots however communicate by contact and in real time when the classic robots communicate without contact but for very low densities (of the order of 2 in 6 robots by room). Micro-robots are used in various applications, as for the construction of intelligent 2D shape ( programmable matter -claytronics [START_REF] Seth | Claytronics: A scalable basis for future robots[END_REF]), or still within the framework of miniaturized flexible workshops where elements collectively realize actions. All these applications require a precise knowledge of the position of the microrobot In many cases, it is more useful to have a local information about the relative positions of microrobots and their orientation, than a global position of every robot.

A. Best poosition microrobots in systems with very high densities

The two main issues of work on Mems systems relate, first carrying out a 2D shape with constraints resources or not (calculation time, completion time, energy consumption, percentage of achievement of shape goal), and then a succession of taken shape over time. Each taken shape can be completed with actions that can lead each Mems for individual or collective goal. In this context, the form of construction requires the knowledge of the positioning of Mems, or microrobots.

For the remainder of this work, we focus on the first issue, in other words the "static" and "instant" phase of the positioning of a micro-robots system, as opposed to a second, more dynamic phase that requires time synchronization of system actions. So many sub-issues appear. Among them, there is that of the knowledge of global and local (or relative) positioning of all Mems microrobots of a system, of all related components of a shape. There is also that of determining the position in an ultra-dense system of communicating Mems, which requires very reactive and efficient computational strategies.

B. Summary of our contributions of modeling algorithm and validation

In the remainder of this paper, we propose at first a model of a microrobots smart-grid (orthogonal and hexagonal lattice) with regular or irregular geometry, and of their connectors (actuators and sensors for displacements and other actions) communicating through contact. Secondly, starting from this model, we propose a mixed positioning algorithm (absolute and relative) in 2D and 3D without mobility of Mems in a group ranging in size from thousands to million of items, based on neighborly relations depending on exchange links between microrobots. Thirdly, we perform a functional validation of our algorithm by simulations and a validation of the scalability of our algorithm on orthogonal grid of over 1 million nodes. Fourthly we analyze the results and present prospects for this work.

II. THE POSITIONING A PRELIMINARY HYPOTHESIS

Table I of the page 3 offers an analysis of the most representative works on Mems microrobots and Mems systems of the decade and who tackled the problem of individual massive positioning. This works are exclusively academic. They are structured from state projects or communities of states. The nature of work is still something of simulation and scaling faces on one hand the number of micro-robots that have to communicate and cooperate, and on the other hand, a quite relative functional and geometrical simplicity. As show by the data in the table, the maximum size of the robots grid does not exceed 10000. In other works which cover the simulation of application of systems reconfiguration, the number may reach 400000 [START_REF] Boillot | Using nano-wireless communications in micro-robots applications[END_REF], but the calculation of positioning is collective (in groups of 3 catoms). It is handled in an implicit way applied in the context of a virtual territory discovery. The first target of this work [START_REF] Boillot | Using nano-wireless communications in micro-robots applications[END_REF] is the simulation of the Nano-wireless communications.

The data in the table also show that the size of Mems or microrobots elements considered range from a few mm 3 to a few dm 3 , that communication technologies are moving towards a wireless infrared communication (whereas previously the preferred communications technology between the robots were a matter of Bluetooth or WiFi). The data also reveal that the techniques of positioning are either probabilistic, either exact by neglecting the inherent uncertainties in the transmission of signal either still in the errors of mechanical or physical adjustments errors when switching to experiment with real prototypes. More important for our choices of orientations of works, positioning types are never mixted, that is to say, combining absolute positioning and the relative. Furthermore, the communication by contact is not understood in academia, the responsiveness of the positioning calculation is not measured in these dense systems of microrobots.

III. CONNECTORS, MEMS, AND MEMS SYSTEMS

In this section we present the modeling of a smart-grid (orthogonal and hexagonal lattice) of Mems with regular or irregular geometry, and their connectors (actuators and sensors for displacement and other actions) communicating through contat. We start by modeling the components and eventually those of the Mems system.

A. In contact thanks to connectors

A connector is a sensor/actuator enabling Mems to dock with each other. These connectors are also used for communication by contact, the energy transfer, solidification and dislocation of the material, and the movement of Mems. For a Mems M i , we model its j -th connector as follows:

Connector i,j = (C i,j , P osR(C i,j , M i ), Etat i,j , F onct i,j )
In this formula, C i,j is the identifier of the connector j of the Mems M i and j given such as 0 ≤ j ≤ (C Mi -1); where C Mi the set of connectors of Mems M i and C Mi is the total number of connectors of the Mems M i . P osR(C i,j , M i ) is the relative position of the connector C i,j in the landmark R i = (O Mi , --→

x Mi , --→ y Mi ) of the M ems i and it's coordinates are given by:

P osR(C i,j , M i ) = (d i,j (O Mi , C i,j ), α i,j ) polar (X Ci,j , Y Ci,j [, Z Ci,j ]) cartesian
Etat i,j the state of the connector C i,j . A connector has a number of states that define its operation and the possible exchanges he has with the connectors of other Mems around him. We distinguish the following states: Let,s be M i a Mems and

R i = (O Mi , --→ x Mi , --→ y Mi ) the landmark associated to M i ; let's be a Mems M k and R k = (O M k , --→ x M k , --→ y M k ) the landmark associated to it. • In contact: if it is in contact with another connector. Two connectors C i,j ∈ C Mi and C k,l ∈ C M k are in contact if they are in physical contact.
In this case, we can consider that they are at the same position.

P osR(C i,j , M i ) = P osR(C k,l , M i ) in R i P osR(C i,j , M k ) = P osR(C k,l , M k ) in R k • contactless: if it is not in contact with another connec- tor. • connected: if connected to another connector. The con- nector C i,j ∈ C Mi is connected to connector C k,l ∈ C M k if C i,

B. Regular and irregular Mems with mixed positions

A Mems is a micro-electromechanical system comprising one or more mechanical components using electricity as energy source in order to perform a function of sensor and/or actuator, with at least one structure having micrometric dimensions.

Physically, a Mems can have a shape more or less regular. We find Mems cubic, cylindrical, or spherical. In this work, we consider regular or irregular shapes Mems, with on their surface C Mi connectors. We illustrate our work by 2D circular Mems having on their surface C Mi regularly distributed connectors as shown in Figure Fig. 1.

Communication allows Mems to exchange messages, information, and ensure overall coordination of the Mems system. Several types of communication exist for Microrobots. According to the used medium, we find the wired, molecular, acoustic, electromagnetic, or contact communications.

Mems studied here, communicates by contact of neighbor with neighbor. They have connectors that are actuators also used for communication. We give below the modelling of a Mems M ems i :

M ems i = (M i , O Mi , --→ x Mi , --→ y Mi , Geo Mi , C Mi , β Mi , F onct Mi , P os Mi , L Mi , N Mi )
With :

* id Mi : the identifier of M ems i . * O Mi : the center of the M ems i * R i = (O Mi , --→ x Mi , --→ y Mi ) : the local landmark associ- ated with M ems i .
To define an orientation, we associate with every Mems M ems i a local landmark

R i = (O Mi , - → x i , - → y i ).
We assume that the landmarks of all Mems have the same orientation.

Hypothesis: As represented on the figure Fig. 2, we formulate the following hypothesis:

∀M ems r , M ems s , r, s ∈ [0, M ems -1], r = s, R r = (O Mr , -→ x r , - → y r ) R s = (O Ms , - → x s , - → y s )
, then

( -→ x r ) Rr = ( - → x s ) Rr = ( - → x s ) Rs = ( -→ x r ) Rs ; ( - → y r ) Rr = ( - → y s ) Rr = ( - → y s ) Rs = ( - → y r ) Rs
M ems i . With C Mi = {C i,j , } 0≤j≤C M i -1
; where C Mi is the total number of connectors of every M ems i and C i,j the j -th connector of M ems i . In the case of regular 2D circular Mems, the number j of the connector allows to define its angular position

P osR(C i,j , O Mi ) = (r Mi , α i,j ) with α i,j = 2 * j * π C M i
* β Mi Is the orientation angle of M ems i . It is defines by the angle of rotation of the connector of origin

C i,0 in the landmark (O Mi , --→ x Mi , --→ y Mi ) of M ems i β Mi = ( x Mi , u i,0 ) with -→ u i,0 = -----→ O Mi C i,0
* F onct Mi the features of the M ems i . A Mems has several features: communicate, move, run, load/unload, store. * P os Mi the position of M ems i . Each Mems has an absolute position and a relative position. The absolute position is the position of its center in a common reference to all of our Mems system. We note it P osA Mi . The absolute position of a Mems M ems i is given by

P osA Mi = (d i , α i ) polar coordinates (X Mi , Y Mi ) cartesian coordinates
The position of Mems is a structure made up of an absolute position (when it exists) and of a list (empty or not) of relative positions. * L Mi the set of M ems i links. There is a link between two Mems if they have at least one pair of connector in contacts and connected. the Mems M ems i il linked to the M ems k , if ∃C i,j ∈ C Mi , ∃C k,l ∈ C M k such as C i,j and C k,l are in contact and connected. All the links of the Mems M ems i are note :

L Mi = {(C i,j , C k,l , M k ) with C i,j ∈ C Mi , C k,l ∈ C M k ,
and C i,j in contact and connected to C k,l } * N Mi All the nearby Mems of M ems i .

N Mi = {M k /∃(C i,j , C k,l , M r ) ∈ L Mi , M r = M k } C. Mems system
The Mems, because of their small size, cheap price and the fact that they are mass produced, millions of units can be used in very small spaces. The Distributed intelligents Mems systems consist of a large number of Mems, capable of fulfilling advanced tasks (shape construction), result of the addition of elementary actions of each of its constituent. Distributed Mems systems can be classified into two categories according to their topology: static or dynamic. Static distributed Mems have a fixed physical topology that will not change over time, while dynamic are composed of mobile units, making them dynamic logical topology. Mems systems can be categorized into different architectures based on the geometric arrangement of their nodes [START_REF] Yim | Modular self-reconfigurable robot systems [grand challenges of robotics[END_REF].

Lattice architectures have nodes which are arranged and connected in a regular manner, forming an orthogonal or hexagonal structure. This architecture allows an easy reconfiguration, bound to the nodes facility of movement in a finite set of positions of neighboring cells. The chained architectures or in tree possess nodes that are connected in a chain topology or tree. The mobile architectures possess nodes which lean on the environment to form more or less complex structures.

The relative arrangement of a node and its neighbors allows to define a regular grid structure. The prosition of a node and its neighbors to define a regular grid structure. The circular geometry of node forced to a maximum of 6 neighbors. When each node can have six neighbors, the set of possible positions of the nodes form a hexagonal lattice. In the situation where each node has at most 4 neighbors, an orthogonal lattice is obtained. This grid then defines all the possible positions that may occupy the nodes.

IV. ALGORITHM OF ABSOLUTE AND RELATIVE HYBRID

POSITIONING

Our proposed algorithm uses the communication by contact between Mems and cooperation between nearby Mems to spread information being used to the calculation of positions. It allows when in a related component, there is Mems having an absolute position, to calculate an absolute position for the set of Mems of this related component; and when there is not at least one Mems having an absolute position, to lean on Mems having a relative position, or just on the relation of neighborhood and the cooperation to define the relative positions of Mems.

Our positioning system works in the following way: neighborhood. The table of neighborhood of given Mems is so constituted by all the Mems which are neighbors to it physically, and in contact with at least one of its connectors through one of their connectors. 3) Calculation of the position of Mems of the neighborhood For every element of neighborhood we are going to calculate its relative position. The difficulty of this calculation lives on the fact that there is no global mark shared by all. Every mark of every Mems M ems i being made of an origin situated in the center of Mems and made of vectors x i , y i . In this landmark, every connector has a relative position. We formulated hypotheses of simplification which allow us to calculate the relative position. Resting on the hypothesis of contact and considering that two connectors in contact occupy the same position. The calculation of the relative position is made as follows:

a) P osR Mi,M k = ( ------→ O Mi O M k ) Ri in the landmark R i = (O Mi , - → x i , - → y i ) associated with M ems i b) ( ------→ O Mi O M k ) Ri = ( -----→ O Mi C i,j + ------→ C i,j OM k ) Ri using Ci, j applying the Chasles relationship c) ( ------→ C i,j OM k ) Ri = ( ------→ C k,l OM k ) Ri = ( ------→ C k,l OM k ) R k R k ,
the landmark associated with Mems M ems k ; C i,j and C k,l being in contact we can consider them as being has the same position, thus we consider that the point C i,j is equal to the point

C k,l d) ( ------→ C k,l OM k ) R k = (- ------→ OM k C k,l ) R k = -P osR k,l e) ( -----→ O Mi C i,j ) R k = P osR i,j f) P osR i,k = P osR i,j -P osR k,l

4) Calculation return of the position of nearby Mems.

This is made in the same ways the stage 3. has been done. 5) Calculation of the Mems absolute position. The calculation of the absolute position leans on the broadcasting of information. Because of the weak resources which Mems has, our choice is the one not to use the usual techniques (dead recogning, ordometry) to calculate the absolute position. We consider that we possess within our system at least one Mems that the absolute position is known. Thanks to the previously calculated relative positions, Mems can determine its absolute position as follows: When it receives a message with the field absolute position not empty, having to determine the position relative as in [START_REF] Funiak | Internal localization of modular robot ensembles[END_REF], it uses it to calculate its absolute position as follows :

P osA k = P osA i + P osR(k, i)

By correctly handling the removal of all the Mems that the absolute position have been calculated, we determine the absolute position of all Mems in the related component.

Thereafter our algorithm manipulates the following structures:

•

Mi = LM.get(i) VM i = {M k /∃(Ci,j, C k,l , Mr) ∈ LM i , Mr = M k } for each M k ∈ VM i do 12: ∃Li,j ∈ Li, Li,j = (Ci,j, C k,l , M k ) if ( (Mi.P osR i,k )) then calculP ositionR(Mi, M k ) 15: end if end for i = i + 1 18: end while ABSOLUTE POSITION i = 0 21: while (i < slLM wAP ) do Mi = slLM wAP.get(i) N uAPi = {M k , ∃(Ci,j, C k,l , M k ) ∈ LM i , M k = Mr and P osAM k = } 24:
for each M k ∈ N uAPi do P osA k = P osAi + P osR(M k , Mi) slLM wAP.add(M k )

27:

end for i = i + 1 end while Figure .4 present an example of our Mems distributed system, on an orthogonal grid, with each Mems having a maximum of 4 neighbours. The initial configuration present our system at the begining, in each ralated component, we have a Mems with an absolute position. In the final configuration, each Mems now have and absolute position, wich have been calculated for those who did not previously have an absolute position.

V. REALIZATIONS AND TESTS

In this section we present experimental results that illustrate the validation of our algorithm. We have developed Java simulations, which model 2D MEMS. Each Mems is modeled in 2D circular, With on its surface 4 connectors regularly spaced. This number of 4 connectors allows to define up to 4 physical neighbors for each Mems. When there is a connection between two nearby Mems, they can then exchange information. The communications between Mems are made by contact. Our simulations tests are in 2D dimension. The sets are generated in a random way. Due to the circular geometric structure and the number of neighbors, we perform our tests on an orthogonal grid where each cell is a square of length equal to the diameter of a circular MEMS, and contains at most a MEMS. In case every Mems possesses 6 neighbors, we can also have a hexagonal grid in the form of honeycomb, where every Mems is registered in a regular hexagon. All tested MEMS are homogeneous and the determined positions are in cartesian coordinates. The positioning of elements is relative and/or absolute, depending on the initial state of the system.

The tests of number of configuration 1 -3 of of Table III are functional validation tests. They validate: The successful completion of the relative or absolute positioning algorithm of a homogeneous system of Mems communicating by contact. The good realization consists of the ending and the correctness of the calculated result. Accepted configurations correspond to all combinations of relations with the immediate neighborhood for the coherent types of links with the relative positions. The calculation of relative positions between elements in contact and connected (by link) The spread of the calculation of the absolute position within a connected component The validation of the consistency of the positions calculated with regard to links The validation of the ending and the calculation of its convergence The test of n of configuration 4 represents a test of the coherence of the connections by reports to the relative positions. This test has the same parameters as the test of number of configuration 3; with the only difference of one relative position incoherence, of this fact, the number of global positions at the end the test is 18 and not 22 because of the inconsistency which limits the distribution of the global position to the other Mems. Table TABLE III and figures Fig. 5 present the result of passage on the scale of our algorithm. Unlike existing studies that position few micro-robots, our work goes beyond [START_REF] Funiak | Internal localization of modular robot ensembles[END_REF] which positions up to 10,000 micro-robots. We evaluate our algorithm on sets of Mems whose number varies from 10 to 700,000 for densities ranging from 0.1 to 0.95.

VI. CONCLUSION

In this work, we proposed a first approach of mixed positioning, absolute (global coordinate system) and relative ( local landmark), in systems of micro-robots with strong density. The systems of considered coordinates are mixed by manipulating the Cartesian coordinates and also those polar.

The model systems address the grid and honeycomb structures in 2D or 3D. Modeled Mems are heterogeneous or homogenous and have irregular or regular structures, like their connectors. The irregular structure concerns the geometry and specific functionalities of connectors of a Mems (or micro-robot) or specific and intelligent features of Mems (memorization, communication, movement, activations). The algorithm of proposed positioning calculates the individual positionings of heterogeneous micro-robots communicating by contact. These micro-robots have knowledge of the physical contact with their immediate neighborhood and manipulate in a hierarchical way, and the reception of messages and those positions relative and absolved from the same immediate neighborhood.

The hierarchy considered by the neighborhood in contact, by the link made by received messages, the superiority of the knowledge of the global positioning on that of the local positioning, it leads to the proposal of a convergent algorithm with a linear complexity.

The realized tests validate functionally the heterogeneousnesses of modelling in the various scales of the system of micro-robots, via Mems and their associated connectors. The calculated positioning is absolute and relative and expresses himself in polar and Cartesian coordinates.

The reached scale reaches the million positioned intelligent micro-robots. According to the density testing between 10% and 70% of a space, the respective calculations of 100000 and 700000 micro-robots do not exceed 75s.

The tracks in this work are numerous. The obtained results, very encouraging, lead us to pursue this work and propose a distributed and collaborative computing positioning and integrating these libraries of positioning systems for micro-simulation systems such as VisibleSim.

Besides, all the irregular or regular geometrical configurations, and those functional were not estimated in this work. It's the same for the three-dimensional configurations of the system of Mems, to which we could add the temporal dimension necessary for the dynamic reconfigurations of system.

Figure 1 .

 1 Figure 1. Circular geometrical shape of homogeneous micro-robot with n-connectors -bi-dimensional projection

Figure 2 .

 2 Figure 2. The vectors of local landmarks of neighboring robots are equal

Figure 3 .

 3 Figure 3. Configuration of orthogonal and hexagonal dense neighborhood of a Mems with circular geometry

  List < M EM S > LM : the list FIFO containing all Mems • List < M EM S > nT sl : the sub-list of LM containing untreated Mems • List < M ems > slLM wAP : the sub-list of LM containing Mems for which the absolute position is known • List < M ems > slLM wORP : the sub-list of LM containing Mems for which a relative position is known and the absolute position is not known • List < M ems > N i : the sub-list of nearby Mems of the M ems i • List < M EM S > N uAP i : the sub-list of nearby Mems of the M ems i having no known absolute posi-tion • P ositionR posR(M i /M j ): The relative position of the M ems i in the landmark associated with the M ems j • P ositionA posA(M i ) the absolute position of M ems i • M ems i : the current item (in process) • M ems k : the neighbor element being processed Algorithm 1 Absolute and relative positioning algorithm of heterogeneous Microrobots slLM wAP = Mi, i ∈ [0, M ems -1], and Mi.P osAi = slLM wRP = Mi, i ∈ [0, M ems -1], and Mi.P osRi = 3: slLM wORP = slLM wRP \slLM wAP Ni = subList(LM, Mi) N uAPi = Ni\{Ni ∩ slLM wAP } 6: RELATIVE POSITION i = 0 while (i < LM ) do 9:

Figure 4 .

 4 Figure 4. Initial an final configuration of 22 Mems distributed on an orthogonal grid, in 4 connected components, each having initially at least one Mems with and absolute position. Applying our algorithm aims to calculate the absolute position of all Mems.

Figure 5 .

 5 Figure 5. Curves of the calculation time of the relative and absolute positions in a MEMS system for densities of 0.1 and a number of MEMS ranging from 10 to 100,000, and a density of 0.7 for a number of MEMS ranging from 70-700000

  j can send and receive messages from C k,l .• disconnected: if there is no messages exchange with other neighboring connectors.• active: if it is in a state allowing it to send or receive messages.

	Ref	Positioning strategy		Work category	Work's type	System scalabil-	Size of ele-ment		Positioning	communication
						ity			Method	Type	Reactivity
	[5]	use of the max-	Millibots	simulation,	measures	-	-	-	-	infrared, without
		imum likelihood	project	prototypage	made	2	-			contact
		on a group of			2	between
		robot				several robots
	[9]	positioning	a	scientific	simulation	3 robots	undefined	statistics	absolute calculated infrared, without
		triangle of three	project					and		contact
		robots on a map						particles
									filters
	[8]			scientific	simulation,	¡ 2m		28*28*30	exact cal-	relative calculated infrared, without
				project	prototypage			cm	culation		contact
	[3]	using module		projet	simulation,	10000		millimeter	probabilistic relative calculated infrared, without
				Claytronics	prototypage						contactt
	[10]			Millibots	simulation,			7x7x7 cm	exact cal-		infrared, without
				projet	prototypage				culation		contact
							Table I
			PRINCIPAL PROJECTS AND ACADEMIC WORKS TACKLING POSITIONING WITHIN MICRO-ROBOT SYSTEMS

• inactive: if it can not send nor receive messages. F onct i,j the features of the connector C i,j . A connector is able to perform several functions: sense, act, attract, repelling, send messages, receive messages, hang, lift, dock, undock.

Table II BOARD

 II OF THE FUNCTIONAL VALIDATION TESTS: SHOWS THE BEHAVOUR OF OUR ALGORITHM IN DIFFERENTS SITUATIONS. WE VALIDATE THE CALCULATION OF ABSOLUTE AND RELATIVE POSITIONS, THE NORMAL BEHAVIOUR IN CASE OF INCOHENRENT LINKS, THE GOOD SPREAD IN A RELATIVE COMPONENT, AND MESURE OF CALCULATION TIME.

				Distribution				Complexity
	Test	Numbers	Numbers	Numbers	Numbers	Numbers	Numbers	processing	maximum	Actual	occupation
	Id	of	of	of	of po-	of	of	time	number of	number	rate
		global	relative	global	sitions	Mems	related		elements	of
		posi-	posi-	and	neither	with	compo-			elements
		tions	tions	relative	global	links	nents
				posi-	nor	incon-
				tions	relative	sistent

Table III TABLE

 III SHOWING CALCULATION TIMES OF THE RELATIVE AND ABSOLUTE POSITIONS OF MEMS FOR DENSITIES 0.1 AND 0.7 FOR A NUMBER OF MEMS RANGING FROM 100 TO 1000000.

	Number of MEMS for a density of 0.1	10	50	100	500	1000	5000	10000	50000	100000
	Calculation time of the positions of all Mems (ms)	0	0	5	11	26	73	141	838	1445
	Number of MEMS for a density of 0.7	70 350 700	3500 7000	35000 70000 350000	700000
	Calculation time of the positions of all Mems (ms)	0	5	3	14	23	180	660	19612	73802

ACKNOWLEDGMENT

This project has been performed in cooperation with the Labex ACTION program (contract ANR-11-LABX-0001-01). It has been also supported by the Ministère de l'Enseignement Suprieur et de la Recherche.


Dipayan Chaudhuri

David Granton

Dominic Xiang Wang

Karen E A Burns

Yigal Helviz

Sharon Einav

Vatsal Trivedi

Tommaso Mauri

Jean-Damien Ricard

Jordi Mancebo

High flow nasal cannula in the immediate post-operative period: a systematic review and meta-analysis

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


Florian Galinier

Éric Bourreau

Annie Chateau

Adel Ferdjoukh

Clémentine Nebut

Genetic Algorithm to Improve Diversity in MDE

Introduction

Model Driven Engineering (MDE) provides tools for systematization and automation of models. Model transformations are nowadays widely used either in a maintenance context or in a more traditional development contexts. It allows realization at many stages during life cycle for a software: code generation, reverse engineering, etc. These models transformations are for example a way to change a model from a specific domain -e.g. a UML class model -to another domain -e.g. a Java model. Domains of models are specified by models of higher abstraction level: Meta-Models (MM).

Generation of models from MM as a testing purpose is a quite complex task. Several combinatorial approaches have already been proposed. In [START_REF] Ferdjoukh | Instantiation of Meta-models Constrained with OCL: a CSP Approach[END_REF][START_REF] Cabot | Verification of UML/OCL class diagrams using constraint programming[END_REF][START_REF] Pérez | EMFtoCSP: A tool for the lightweight verification of EMF models[END_REF], the authors propose a generation based on Constraint Satisfaction Problems (CSP). These methods provide encouraging results, creating models that conform to meta-models and moreover, encoding constraints (OCL) that can be added to meta-models. Recently, in [START_REF] Cadavid | Searching the Boundaries of a Modeling Space to Test Metamodels[END_REF], simulated annealing generate a set of models which are the more dissimilar and which cover the meta-model. Meta-heuristics seem to provide interesting results in term of diversity, and, in our opinion, diversity is now a major issue to complete our generation process.

In this paper, we explain how to generate diverse models conform to a meta-model, using genetic algorithms.

Evolutionary algorithm

Evolutionary algorithm (EA) have been largely adopted to provide good solutions to combinatorial problems. In this section we will describe the differents components of our EA [START_REF] Holland | Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence[END_REF].

Chromosome representation

Representing models as a sequence in a chromosome is complex, moreover if we want to allow the possibility to crossover models. We base our approach on the CSP paradigm used by the authors in [START_REF] Ferdjoukh | Instantiation of Meta-models Constrained with OCL: a CSP Approach[END_REF]. In this work, Ferdjoukh et al. propose to describe a model as a constraint network composed by variables, domains associated to these variables and a set of constraints. A valid model (i.e. which respect the meta-model and constraints) is an instantiation of variables that respect CSP constraints. We first choose to use vectors of instanciated values as chromosome representation. In a second time, we prefer to express in a chromosome a set of models (a list of size p) by concatenating p consecutive vectors representing p different models.

Initial Population

In [START_REF] Chateau | A Model-Driven Approach to Generate Relevant and Realistic Datasets[END_REF], Ferdjoukh et al. introduce probabilistic simulation to generate realistic datas from a MM, based on a CSP instantiation. Their tool (named Grimm) is able to generate a set of valid models, conforms to meta-models, with variety in domain range instantiation due to probability distribution. A first set of 100 models, conforms to a specific meta model, is generated. Due to the value of p, we respectively considered 50 chromosomes (p=2), 25 chromosomes (p=4) or 20 (p=5).

Fitness function

Diversity between models is a rich problem. Indeed, it could be linked to a metrix representing similarity/dissimilarity. Some previous works in model driven engineering [START_REF] Falleri | Metamodel matching for automatic model transformation generation[END_REF] used semantics data of models to provide a similarity measure. In our case, semantics data seems not sufficiently realistic, models are generated. Instead, we choose to focus on structure to define fitness.

As previously explained we reduced the problem by representing a graph as a vector including for each artefact target, relations and attributes. A simple Hamming distance or Levenshtein distance can be then used to compared two models. We choose to used a more recent metric from Data Mining domain: the cosine dissimilarity [START_REF] Singhal | Modern information retrieval: A brief overview[END_REF]. This metric is usualy applied to vectors that represent occurences of word in a text. It is defined as:

D C ( -→ V 1 , -→ V 2 ) = 1 -S C ( -→ V 1 , -→ V 2 ) (1)
with

S C ( -→ V 1 , -→ V 2 ) the cosine similarity of two vectors -→ V 1 and -→ V 2 defined as : S C ( -→ V 1 , -→ V 2 ) = -→ V 1 • -→ V 2 -→ V 1 -→ V 2 = n i=1 V 1 [i]V 2 [i] n i=1 V 1 [i] 2 n i=1 V 2 [i] 2 (2)
For the simple case where one chromosome contain two models, this distance is quite easy to compute, but in other cases, we obtain a distance matrix. We follow different approaches to try to increase diversity. Indeed, we consider several fitness for the "more than two models" case:

(1) a fitness based on the lowest value of the distance matrix: with this approach, we consider that increase the minimal value will normally increase all the distances; (2) a fitness based on mean value of the distance matrix: with this approach, we hope that trying to increase the mean will increase all the distances; (3) a fitness based on a sum of ( 1) and ( 2), trying to increase both values; (4) two different fitness functions: the (1) and the (2), trying to increase one of the two values.

If the three first approaches can be realized with all genetics algorithms, the last approach need the use of a multi-objective evolutionary algorithm. We choose the widely used NSGA-II [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: NSGA-II[END_REF][START_REF] Deb | An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints[END_REF].

Crossover, mutation, feasibility repair and selection

We randomly pick couples of chromosomes and apply a simple one point crossover on them to generate new chromosomes. We determine that a mutation chance of 0.5 for each gene is enough to avoid a too quick convergence and allow diversification. To check feasibility, first we store extra-data, like domains and constraints, to quickly check if new vectors (generated by crossover or mutation) are still valid. Our modified vector is reintroduced in CSP tool to check satisfiability and if necessary, repair it by looking valid instantiation in a classical depth first search exploration.

As selection criterion, when we generating 100 new chromosomes, we keep the 100 best over the overall 200 chromosomes

Computational results

We first consider a population with individual composed only by 2 models. In Fig. 2, we can observe that our method provides better results than the CSP approach. Indeed, population at generation 100 has a mean distance score better than the original population. Moreover, we can note that for both distances the lowest value of our final population is better than the higher value of initial one. The lowest values for cosine distance are inherent to the fact that some values of vectors are Fig. 3 shows the different fitness approaches when p = 5. Trying min value (1) or mean value (2) brings quite similar results. The combination of two objectives, with sum aggregation (3) and even more with bi-objective vision (4), provides even better results. We also observe that our method provides better results than original one at the cost of an increased computation time to reach convergence.

Conclusion

Diversity of artefacts in model generation is an important point. The evolutionary meta-heuristic provide a tool to increase distance between models. With this approach, we are able to increase mean score for more than 50% for considered metrics. As perspectives, we will look at the case where chromosome are mono-model by considering a population fitness, and preliminaries computations seem to provide better results in terms of scalability.

Fig. 1 .

 1 Fig. 1. One point Crossover between two chromosomes composed by 2 models.

Fig. 2 .

 2 Fig.2. Evolution of mean distance for population with chromosome with 2 models too constrained -the dashed line indicates the best possible pairing at last generation, computed with a brute force algorithm.Fig.3shows the different fitness approaches when p = 5. Trying min value (1) or mean value (2) brings quite similar results. The combination of two objectives, with sum aggregation (3) and even more with bi-objective vision (4), provides even better results. We also observe that our method provides better results than original one at the cost of an increased computation time to reach convergence.

Fig. 3 .

 3 Fig. 3. Evolution of mean distance for population with chromosome composed by 5 models