Huu-Tri Nguyen

Pascale Domingo

Luc Vervisch
email: luc.vervisch@insa-rouen.fr

Phuc-Danh Nguyen

Machine learning for integrating combustion chemistry in numerical simulations

Keywords: Combustion chemistry, Micro-mixing modeling, Principal component analysis, Artificial neural network, Chemistry reduction

A strategy based on machine learning is discussed to close the gap between the detailed description of combustion chemistry and the numerical simulation of combustion systems. Indeed, the partial differential equations describing chemical kinetics are stiff and involve many degrees of freedom, making their solving in three-dimensional unsteady simulations very challenging. It is discussed in this work how a reduction of the computing cost by an order of magnitude can be achieved using a set of neural networks trained for solving chemistry. The thermochemical database used for training is composed of time evolutions of stochastic particles carrying chemical species mass fractions and temperature according to a turbulent micro-mixing problem coupled with complex chemistry. The novelty of the work lies in the decomposition of the thermochemical hyperspace into clusters to facilitate the training of neural networks. This decomposition is performed with the Kmeans algorithm, a local principal component analysis is then applied to every cluster. This new methodology for combustion chemistry reduction is tested under conditions representative of a non-premixed syngas oxy-flame.

Introduction

Despite of the continuous progress in supercomputing methodologies, all the degrees of free-2 dom required to fully describe the detail of combustion chemistry cannot be introduced in the nu-3 merical simulations of large-scale combustion systems, such as boilers and furnaces. Several methods have been proposed in the literature to reduce and optimize large chemical kinetic schemes, such as Quasi-Steady State Approximation (QSSA) and partial equilibrium [START_REF] Peters | Systematic reduction of flame kinetics: Principles and details[END_REF], Directed Relation Graph (DRG) [START_REF] Lu | A directed relation graph method for mechanism reduction[END_REF], Directed Relation Graph with error propagation [START_REF] Pepiot-Desjardins | An efficient error-propagation-based reduction method for large chemical kinetic mechanisms[END_REF], DRG-aided sensitivity analysis (DRGASA) [START_REF] Karalus | A skeletal mechanism for the reactive flow simulation of methane combustion[END_REF] or again unimportant reaction elimination [START_REF] Lu | Strategies for mechanism reduction for large hydrocarbons: n-heptane[END_REF] (not exhaustive list).

Even after reducing the number of differential equations solved, because of the very stiff character of differential systems associated to combustion chemistry, the CPU time devoted to their integration remains quite significant and in most cases, too large to allow for performing the multiple simulations needed to complete the quest of the optimal design for a combustion system. These computing limitations motivated the introduction of artificial neural networks (ANN) to deal with chemistry reduction and its time integration [START_REF] Chi | On-the-fly artificial neural network for chemical kinetics in direct numerical simulations of premixed combustion[END_REF] and also to develop data driven turbulent combustion modeling [START_REF] Nikolaou | Progress variable variance and filtered rate modelling using convolutional neural networks and flamelet methods[END_REF], or to analyse experimental measurements [START_REF] Wan | Combustion regime identification from machine learning trained by Raman/Rayleigh line measurements[END_REF] and to setup digital-twins for process control [START_REF] Aversano | Application of reduced-order models based on PCA & Kriging for the development of digital twins of reacting flow applications[END_REF]. Among these works, ANN and convolutional neural networks (CNN) were trained on a representative dataset to replace chemistry integration [START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF] and the method was applied to perform direct numerical simulation (DNS) of a syngas oxy-flame. Nevertheless, the training of the full dataset using a single or two ANNs required specific treatments, such as data augmentation through manufactured intermediate solution points for dealing with some fast reacting species [START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF]. These operations requiring fine tuning by the user were found necessary to secure good accuracy over the whole domain covering from chemically frozen-flow mixing up to the equilibrium state and pollutant emissions, with fast ignition and fuel oxidation in between.

To avoid relying on these additional more or less ad-hoc operations, thus allowing for having a computerised and fully automatic procedure, a novel method is proposed in the present study for turbulent non-premixed flames in which techniques such as data clustering with K-means [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] and dimension reduction using LPCA [START_REF] Alqahtani | A data-based hybrid model for complex fuel chemistry acceleration at high temperatures[END_REF], are combined with ANN training to secure, by construction, accuracy over the entire composition space domain (mixing, ignition, combustion and equilibrium state).

The paper is organised as follow, in the subsequent section the operating conditions considered and the novel methodology proposed are discussed. After that, the obtained ANN is tested in a posteriori manner for chemistry integration, to study the robustness of the method and the CPU time reduction.

Chemistry integration with machine learning

Thermochemical conditions

The operating condition considered in this study is taken from previous works on syngas, with a chemical composition representative of those found in the steel industry in recycled exhaust gases [START_REF] Wan | Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF]. This non-premixed syngas oxy-flame operates under a pressure of 341.3kPa, with the two inlets: (i) syngas fuel injected at 1223K for a mass flow rate of 0.74kg/s and the composition in mass fraction H 2 : 0.0085, CO: 0.7852, CO 2 : 0.0514, N 2 : 0.1549; (ii) pure oxygen at 298K with a mass flow rate of 0.69kg/s.

Database generation

A database of chemical evolutions is first generated by solving for the mass fractions and the enthalpy of a set stochastic particles featuring at initial condition the concentration in the fuel or in the oxidizer inlet. Then, the species mass fraction Y p i (t) and sensible enthalpy h p s (t) of each stochastic particle evolve according to the following equations:

dY p i (t) dt = MIX p i (τ T ) + ωp i , (1)
dh p s (t) dt = MIX p h s (τ T ) + ωp h s , (2)
where 'MIX' denotes the stochastic turbulent micro-mixing closure for the diffusive budget in a non-premixed system. τ T denotes the micro-mixing time and ωp i and ωp h s are the species and enthalpy chemical sources, respectively. In the present study, the Euclidean minimum spanning tree (EMST) micro-mixing model [START_REF] Subramaniam | A mixing model for turbulent reactive flows based on euclidean minimum spanning trees[END_REF] is used to model the turbulent micro-mixing.

This approach to generate thermochemical conditions representative of those observed in a real system for studying its chemical response, without explicitly solving for the flow, has been shown effective for partial oxidation of natural gas [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF], aircraft engines combustion chambers [START_REF] Bouaniche | Vitiated high Karlovitz n-decane/air turbulent flames: Scaling laws and micro-mixing modeling analysis[END_REF], large-scale furnaces with urea DeNOx automated control [START_REF] Locci | Selective non-catalytic reduction (SNCR) of nitrogen oxide emissions: A perspective from numerical modeling[END_REF], micro-and meso-scale combustion systems [START_REF] Bioche | Premixed flame-wall interaction in a narrow channel: Impact of wall thermal conductivity and heat losses[END_REF] and also to build the database for machine learning application to sooting flames [START_REF] Seltz | Solving the population balance equation for non-inertial particles dynamics using probability density function and neural networks: Application to a sooting flame[END_REF].

Here a total number of 1429 stochastic particles is employed with 740 particles (corresponding to the mass flow rate of 0.74kg/s) assigned to Fuel inlet and 689 particles (corresponding to the mass flow rate of 0.69kg/s) to O 2 inlet. From the evolution of these particles, the first step consists of identifying, from a detailed chemical scheme, the important species to be retained and whose evolutions shall be learned by the ANNs. This is done using the direct relation graph with error propagation method (DRGEP) [START_REF] Pepiot-Desjardins | An efficient error-propagation-based reduction method for large chemical kinetic mechanisms[END_REF].

For the present operating conditions, this step was previously performed and reported in [START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF].

Starting from the detailed scheme GRI3.0 [20]

Data pre-treatment

To secure an appropriate analysis of the multivariate nature of the problem, the thermochemical variables obtained from the solving of Eqs. 1 and 2 are going through pre-treatments, such as centering and rescaling and unsupervised machine learning, like clustering with K-means [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF],

and dimension reduction with LPCA [START_REF] Kambhatla | Dimension reduction by local principal component analysis[END_REF], before the ANN training is to be applied. This strategy is expected to close the gap between the previous works [START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF], in which artificial data augmentation was required to secure accuracy, and the introduction of complex chemistry in the numerical simulation of large scale furnaces.

Following the recent study by D'Alessio et al. [START_REF] D'alessio | Adaptive chemistry via pre-partitioning of composition space and mechanism reduction[END_REF] in the context of adaptive chemistry for reacting flow simulation, the original dataset is first pre-processed with centering and auto-scaling (standardizing) to insure reliable and robust results. The different steps in K-means used for the thermochemical quantities (species mass fractions and temperature) are illustrated in Figure 1 and summarized as follows:

• Definition of the number of clusters. • Selecting randomly an initial cluster centroid: To ensure the quality of clustering and improve the convergence speed, the K-means++ initialization algorithm [START_REF] Arthur | K-means++: The advantages of careful seeding[END_REF] is applied. The algorithm assigns randomly a first centroid, a second centroid is then chosen as farthest as possible to the first centroid, a third centroid is as farthest as possible to the first two centroids, and so on.

• Assigning data point to cluster with shortest Euclidean distance.

• Updating centroid locations by averaging all data points in each cluster.

• Repeat the process until the centroid locations are converged by evaluating the shifting tolerance, which is chosen at 1e-20.

Defining an adequate number of clusters is crucial. The usual K-means++ initialization algorithm may lead to poor clustering in our case, where the majority of clusters concentrate in the ignition and combustion stages. To overcome this difficulty, an additional hierarchical clustering strategy was adopted; the first clustering leads to only 2 clusters (named parentCluster0 and parent-Cluster1), which separates the low and high temperature zones. The first parent containing the data of mixing, ignition and combustion stages is then clustered into 14 child clusters (childCluster0 0 to childCluster0 13). The subset of data in the second parent, representing the end of combustion followed by the equilibrium state, requires a special treatment to guarantee the accuracy of ANN training and prediction. For this reason, the second parent is classified into only 3 child clusters (childCluster1 0, childCluster1 1 and childCluster1 2), and then these 3 child clusters will have, respectively, 20, 10, 10 grandchildren (grandChild1 0 0 to grandChild1 0 19, grandChild1 1 0 to grandChild1 1 9, grandChild1 2 0 to grandChild1 2 9). Each child cluster of the first parent and each grandchild cluster of the second parent benefits from its own ANN training.

The effectiveness of K-means algorithm applied to the flame turbulence/chemistry interaction (Eqs. 1 and 2) is illustrated in figures 2 and 3. Figure 2 shows how K-means automatically classifies data in parentCluster1 (representing the end of combustion and the equilibrium state) into 3 distinct subsets, i.e. childCluster1 0, childCluster1 1 and childCluster1 2. The childCluster1 1 contains the particles from O 2 inlet, while the childCluster1 2 gathers the particles from Fuel inlet (Figure 3); both childClusters represent the end of combustion. The particles from O 2 and Fuel inlets finally approach the equilibrium state and their data points are classified in a single common cluster childCluster1 0.

Local principal component analysis (LPCA)

Principal component analysis (PCA), also known as a dimensional reduction technique, is used to reduce a large number of correlated variables (species mass fractions and temperature, in our case) to a smaller number of uncorrelated variables (principal components) [START_REF] Jolliffe | International Encyclopedia of Statistical Science[END_REF]. Because of the strongly non-linear character of combustion chemistry, it is usually recommended to adopt a locally linear approach called 'local PCA' (LPCA) [START_REF] Kambhatla | Dimension reduction by local principal component analysis[END_REF], to properly describe such dynamical system.

In our case, each cluster represents a piece-wise variation of species mass fractions and temperature during a specific combustion stage. The application of LPCA in each cluster transforms correlated variables of high variance into the most important uncorrelated principal components (PCs), which will be carefully addressed during the ANN training process. The choice of the number of PCs is essential to secure a good quality of ANN prediction, and a quite high number of PCs is usually required dealing with chemistry in reacting flows [START_REF] D'alessio | Adaptive chemistry via pre-partitioning of composition space and mechanism reduction[END_REF].

In practice, it was observed that reducing from 11 PCs to 6-8 PCs would still allow for keeping more than 98% of the relevant information. However, here the full set of control variable is transformed into PC (i.e., 10 chemical species and temperature). Therefore, this does not reduce the size of the problem to be learned, but the parametrisation of the data from the PCs considerably eases the ANN learning process. Indeed, species and temperature are usually well-correlated, the PCA analysis transposes these variables to non-correlated PCs and the performances of ANNs then become much greater with these non-correlated inputs.

Two heat maps are presented in Figure 4 to show the correlation of original variables in com-

Artificial neural network

The ANN regression structure (Table 1) used in this study is adopted from our previous works [START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF]. The ANN consists of 1 input layer, 4 dense hidden layers with the rectified linear activation function (ReLU) and 1 output layer. The total number of parameters (weights and biases) is 179851. The training process is performed using Tensorflow 2 with GPU support (NVIDIA GeForce GTX 1080 Ti) and the Adam optimizer in default setting. To prevent the overfitting, the early stopping callback is adopted and set as 200 epochs, which means that the training will be halted after 200 epochs without improvement. Additionally, the check point callback saves the best model with the lowest mean squared error (MSE) during the training process. The total training time for all ANNs is about 11 hours.

Results

The methodology adopted in the study is summarised in Figure 5. To perform a posteriori tests of the model, the evolutions of the stochastic particles are recomputed replacing the integration of chemistry by the ANN, as summarised in Figure 6.

The time evolutions of temperature and species mass fractions averaged over the stochastic particles issued from one of the inlets and computed with and without ANNs, are shown in figures 7-9.

The case without any pre-treatment of the data (dashed-line) and a single ANN rapidly deviates from the reference solution (symbols). Without pre-treatment, the ANN captures the very first part The solution with data pre-treatment (solid lines in figures 7-9) is in good agreement with the reference solution obtained using a CPU consuming stiff-solver for chemistry (symbols). This is the case for temperature and all species considered, even radicals and intermediate ones such as O, H, OH, HO 2 and H 2 O 2 . All the evolution stages (mixing, ignition, combustion and equilibrium) are well predicted.

Figure 7 shows that the ignition delay of 0.08 ms is correctly predicted by the ANN model.

After the ignition, the temperature profiles feature a sharp increase to attain high values due to oxycombustion (the highest value is 3081K for the particles from the fuel side). Then, the temperature proceed towards its equilibrium state at 3037K. The trajectories of reactants (H 2 , CO, O 2 ), and main combustion products (CO 2 , H 2 O) are also perfectly captured by the ANNs-based model during their evolution in all specific stages, i.e. mixing, ignition, combustion and equilibrium (Figure 8).

Regarding minor and radicals species (Figure 9), the shape and peak of the trajectories are also correctly predicted except a very slight discrepancy for H 2 O 2 species departing from the oxygen inlet.

The computation time is drastically reduced by using the ANN model as given in Table 2. The simulation with ANNs (single processor calculation) is 8 times faster than the simulation for the reduced scheme, and 80 times faster than the simulation with the detailed GRI3.0 scheme. The CPU time reduction is measured here solving for micro-mixing and chemistry (Eq. ( 2)). This relative CPU reduction thus differs from our previous works [START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF], also relying on ANN for chemistry, in which the speed-up was measured solving for the aerothermochemical equations in threedimensional flows.

This quantification of the speed-up does not account for the preprocessing time needed to train the neural networks. However, the CPU effort required for training stays much smaller than the time spent performing the numerous operations required to generate a reduced chemical scheme (analysis of the most important chemical species and reaction paths and optimisation of the rate constants). What could slightly increase the CPU cost with ANN is the introduction of more degrees of freedom which may become mandatory to reproduce very specific combustion characteristics (cool flame effects, etc.).

Chemistry

Speed-up GRI3.0 detailed scheme 1 Reduced scheme 10 ANNs (present work) 80

Table 2: Speed-up between chemistry descriptions.

Conclusion

A novel methodology based on the training of neural networks was proposed to reduce combustion chemistry and applied to a non-premixed syngas oxy-flame canonical configuration. The unsupervised learning technique K-means was used to cluster the entire dataset, obtained from the solution of stochastic particles in a turbulent micro-mixing system, into multiple subsets. In the case of a two-inlet problem, with one of the streams bringing the energy necessary for ignition, the proposed decomposition in clusters is likely to be robust and generic. Indeed, after some turbulent mixing, the mixture will always ignite to then evolve towards equilibrium. Some level of finite rate chemistry may appear, but full quenching is very unlikely, especially in the case of oxy-combustion studied here. However, the addition of heat losses and of non-adiabatic recirculating burnt gases may require further examination to determine the best compromise in terms of clustering, still following a similar approach.

Local Principal Component Analysis (LPCA) was then applied on each subset to reduce the dimension of the problem and most important principal components were retained for the ANN training process. Neural networks were finally trained with the aim to replace expensive chemistry integration. The simulation results showed that the proposed method is able to predict the evolution of the thermochemical variables (temperature and species concentration) with excellent accuracy over the whole domain covering from mixing to ignition, combustion and finally chemical equilibrium. All of those with a drastically reduced simulation time.

This promising result is paving the way for the introduction of reduced chemical schemes based on ANN training in reacting flow simulations, in order to mitigate CPU times without accuracy loss.

Figure 1 :

 1 Figure 1: Clustering with K-means

2 Figure 2 :

 22 Figure 2: Data clustering for parentCluster1. Temperature [K] (a) and species mass fraction [-] (b)-(j) vs H 2 mass fraction. Red: childCluster1 0. Black: childCluster1 1. Blue: childCluster1 2.

2. 3 . 1 .

 31 Data clusteringK-means[START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] is a widely used and well-established unsupervised machine learning algorithm, which consists in grouping similar data into different clusters. K-means is particularly suited for the classification of the micro-mixing stochastic particles, which evolve through a large variety of stages from chemically-frozen flow mixing, rapid ignition, fuel oxidation and combustion up to the chemical equilibrium state. Moreover, the particles from Fuel and O 2 inlets feature different time histories, which benefit from being classified in separate clusters for chemistry ANN integration. After clustering into separate groups, every individual subset of data has similar and regular distribution to ensure effectiveness of the subsequent operations.

Figure 3 :

 3 Figure 3: Repartition of data points in childCluster1 0, childCluster1 1 and childCluster1 2. Abscissa is cluster numbering, ordinate is number of data points. Orange: particles from O 2 inlet. Blue: particles from Fuel inlet.

Figure 4 :

 4 Figure 4: Heat map showing correlation coefficients of data in grandChildCluster1 1 5. (a) Original variables in composition space. (b) PCs in PCA space.

Figure 5 :

 5 Figure 5: Chemistry reduction with trained ANN.

Figure 6 :Figure 7 :

 67 Figure 6: Flowchart of the a posteriori test. ('ORCh': Optimised and Reduced Chemistry.)

2 Figure 8 :

 28 Figure 8: Time evolution of major species mass fractions averaged over stochastic particles. Symbols: Reference chemical scheme (Table 2 of [13]). Continuous line: Kmeans-PCA-ANN. Dashed-line: ANN without data pretreament. Green: O 2 inlet. Red: Fuel inlet.

Table 1 :

 1 Structure of ANNs[START_REF] Wan | Chemistry reduction using machine learning trained from nonpremixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF][START_REF] Wan | Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF]

	Layer	Activation function Output shape
	Input	-	(None,11)
	Dense	ReLU	(None,512)
	Dense	ReLU	(None,256)
	Dense	ReLU	(None,128)
	Dense	ReLU	(None,64)
	Dense (Output)	-	(None,11)

Table 2

 2 Figure 9: Time evolution of radicals and intermediate species mass fractions averaged over stochastic particles. Symbols: Reference chemical scheme (Table2of[START_REF] Wan | Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF]). Continuous line: Kmeans-PCA-ANN. Dashed-line: ANN without data pre-treament.Green: O2 inlet. Red: Fuel inlet.

	of [13]). Continuous line: Kmeans-PCA-ANN. Dashed-line: ANN without data pre-
	treament. Green: O 2 inlet. Red: Fuel inlet.
	of ignition, but fails to reach the equilibrium condition, see for instance the temperature response in
	figure 7. (Notice that one does not make use here of the introduction of manufactured intermediate
	solutions for H 2 O 2 , as discussed in [10].)

Acknowledgments

The PhD of the first author is funded by ANRT (Agence Nationale de la Recherche et de la Technology) and ArcelorMittal under the CIFRE No. 2019/0056.

Stochastic particles undergoing turbulence/chemistry interaction

Data pretreatment for efficient machine learning chemistry integration K-means clustering

Composition space

Variable1

Variable2

PCA space

Principal component analysis

ANN training for chemistry integration

operating condition considered in this study is taken from previous works on syngas, ustry [26,27]. This non-premixed syngas oxy-flame operates under a pressure of and the composition in mass fraction H 2 : 0.0085, CO: 0.7852, CO 2 : 0.0514, N 2 :

ii) pure oxygen at 298K with a mass flow rate of 0.69kg/s. abase generation tabase of chemical evolutions is first generated by solving for the mass fractions and alpy of a set stochastic particles featuring at initial condition the concentration in or in the oxidizer inlet. Then, the species mass fraction Y p i (t) and sensible enthalpy each stochastic particle evolve according to the following equations:

3

Database generation


Dr Fadia Dib
email: fadia.dib@inserm.fr

Philippe Mayaud

Laetitia Longfier

Pierre Chauvin

Odile Launay

Effect of Internet use for searching information on vaccination on the uptake of human papillomavirus vaccine in France: a path-analysis approach

Keywords: HPV vaccine, vaccine hesitancy, Internet, sources of information

Internet is a popular source of information regarding vaccination. This study aimed to determine whether there is a negative association between Internet use among French vaccine-hesitant mothers and HPV vaccine uptake by their daughters, and to gain insight into the pathways that would link Internet use to the lack of HPV vaccine uptake. We conducted a pooled cross-sectional analysis across the 2015, 2016, 2017 and 2018 Vaccinoscopie® Survey. Multivariate logistic regression and path models were used in the analysis. The study sample included a total of 2038 respondent mothers. Of those, 89 (4.4%) declared having never been in the situation of searching for information regarding a vaccination they had hesitated about, leaving 1949 mothers for the present analysis.

Approximately 24% (466/1949) of the mothers declared using the Internet as a source of vaccine information. In multivariate logistic regression adjusted for physician recommendation of HPV vaccination, attitudes towards vaccines in general, perception of HPV vaccine usefulness, maternal level of education, region of residence, and the survey year, the use of Internet by the mothers was significantly associated with a lower HPV vaccination among their daughters (adjusted odds ratio (aOR), 0.66; 95% confidence interval (CI), 0.47-0.91). Path analysis further confirmed the negative effect of Internet use (β =-0.10, standard error (SE)=0.02, P<0.0001), highlighting how the Internet plays a detrimental role in HPV vaccine uptake through a lower perceived level of HPV vaccine usefulness, a lower perceived level of information on childhood vaccination, and unfavorable attitudes towards vaccination in general.

INTRODUCTION

Since its introduction in the French immunization program in 2007, the initiation rate of HPV vaccination in girls aged 15 years only reached 29.4% at its highest point, in 2018 [START_REF]Données de couverture vaccinale papillomavirus humains (HPV) par groupe d'âge[END_REF]. The same year, the coverage rate for a two-dose regimen among 16-year-olds girls was estimated at 23.7% [START_REF] Fonteneau | Évolution de la couverture vaccinale du vaccin contre le papillomavirus en France -2008-2018[END_REF]. This figure represents less than half of the target set for a two-dose regimen [START_REF]Plan Cancer[END_REF]. The HPV vaccine appears among those viewed less favorably in 2016 in France [START_REF] Gautier | FRA. Adhésion à la vaccination en France : résultats du Baromètre santé 2016. Vaccination des jeunes enfants : des données pour mieux comprendre l'action publique[END_REF], against the general background of a noticeable rise in vaccine hesitancy [START_REF] Larson | The State of Vaccine Confidence 2016: Global Insights Through a 67-Country Survey[END_REF], which may be viewed as pretty astonishing in Pasteur's country.

Internet has become a popular source of information regarding health-related topics, and HPV vaccine makes no exception. Indeed, this vaccination is a popular topic among "googling" searches for vaccine-preventable infectious diseases [START_REF] Bragazzi | How often people google for vaccination: Qualitative and quantitative insights from a systematic search of the web-based activities using Google Trends[END_REF]. The Eurobarometer Survey on the European citizens' digital health literacy conducted in 2014 showed that around one out of two respondents in France have used the Internet to search for health-related information within the last year, 40% of whom indicated looking on behalf of their children. The French National Health Barometer Survey conducted the same year showed that as many as 69 % of the French population used the Internet to search for health-related topics [START_REF] Spf | Quelle utilisation d'Internet dans la recherche d'informations santé ?[END_REF].

Whilst the Internet represents a remarkable tool for health information dissemination, it also provides a medium where misinformation (misinformed advice) and disinformation (deliberate falsehoods) [START_REF] Grimes | Establishing a taxonomy of potential hazards associated with communicating medical science in the age of disinformation[END_REF] are easily introduced, accessed and spread. This has become more problematic as we live in an era of "post-truth" (defined by Oxford dictionary as "circumstances in which objective facts are less influential in shaping public opinion than appeals to emotion and personal belief" [START_REF] Dib | Online mis/disinformation and vaccine hesitancy in the era of COVID-19: Why we need an eHealth literacy revolution[END_REF]). in The use of social media (such as Facebook) and messaging platforms (such as WhatsApp) allows instant global dissemination of false information [START_REF] Larson | The biggest pandemic risk? Viral misinformation[END_REF]. This comes with its own share of challenges in the health information field. For example, one study analyzing the reach of online disinformation in Europ [START_REF] Dib | Online mis/disinformation and vaccine hesitancy in the era of COVID-19: Why we need an eHealth literacy revolution[END_REF]e showed that the most popular false news website in France, a website claiming to "vulgarize information on health and wellness", had received approximately as many interactions on social media (shares, comments, reactions) as five prominent and reliable French news websites combined [START_REF] Fletcher | Measuring the reach of "fake news" and online disinformation in Europe[END_REF]. The spread of misinformation on the benefits and risks of vaccines has certainly been fueling vaccine hesitancy [START_REF] Mckee | Information wars: tackling the threat from disinformation on vaccines[END_REF] [START_REF] Carrieri | Vaccine hesitancy and (fake) news: Quasi-experimental evidence from Italy[END_REF] [START_REF] Basch | What do popular YouTubeTM videos say about vaccines?[END_REF] [START_REF] Hoffman | It's not all about autism: The emerging landscape of anti-vaccination sentiment on Facebook[END_REF], although it remains to be seen to what extent this translates into the decision to vaccinate. The World Health Organization has listed the "uncontrolled dissemination of misinformation" -including in the field of vaccination -among its urgent health challenges for the next decade [START_REF]Urgent health challenges for the next decade[END_REF].

HPV vaccination in France provides an interesting case study of challenges facing vaccination in an era of online mis/disinformation, as the introduction of the HPV vaccine in the French vaccine immunization program has coincided more or less with the rise of social media utilization. Furthermore, HPV vaccination in France is delivered opportunistically through healthcare providers, with no dedicated school-based vaccination program. Therefore, parents who have not heard about the HPV vaccine from their healthcare providers may have questions to be answered about this vaccination.

In this context, we aimed to determine whether there was a negative association between Internet use among French vaccine-hesitant mothers and HPV vaccine uptake by their daughters, and to gain insight into the pathways that would link Internet use to a lack of HPV vaccine uptake. Such an understanding is required to fully appreciate the impact of the Internet on the uptake of HPV vaccination in France, and provide relevant data for policymakers and practitioners to take necessary action.

METHODS

Data source

This study was carried out using the data acquired from the 2015, 2016, 2017 and 2018 Vaccinoscopie® Survey modules targeting mothers of girls aged 14 and 15 years. The methodology of this survey commissioned by GlaxoSmithKline (GSK) and conducted by IDM Families, an independent market research and polling company, has been published in detail elsewhere [START_REF] Stahl | Vaccination against meningococcus C. vaccinal coverage in the French target population[END_REF] [START_REF] Denis | Papillomavirus vaccination in France according to 2008 to 2012 Vaccinoscopie® data[END_REF]. Briefly, Vaccinoscopie® is a pluriannual web-based survey among French representative quota samples of mothers, with respect to the socio-professional category (SPC) of the reference person in the household, the number of children, and the geographic region of residence, per child's years of age. It has been designed to monitor the dynamics of childhood vaccine coverage, and perception and attitudes towards vaccination in France. A new sample of study participants is interrogated from one year to another.

Study variables

The dependent variable was HPV vaccine initiation (i.e., the receipt of at least one dose of HPV vaccine) in the 14-15 year-old daughter. The exposure of interest was "Internet use by the mother" (yes/no), given the mother's answers to the question "When you hesitate about a vaccine, what source(s) of information do you turn to to decide whether or not to have your child vaccinated?". This question had a multiple-choice response format. For the purpose of this analysis, we derived a dummy variable for each source of information.

Guided by reviews of the literature on the factors associated with HPV vaccine uptake [START_REF] Kessels | Factors associated with HPV vaccine uptake in teenage girls: a systematic review[END_REF] (20) [START_REF] Newman | Parents' uptake of human papillomavirus vaccines for their children: a systematic review and meta-analysis of observational studies[END_REF], we took into account the following relevant factors in the association between the exposure of interest and the dependent variable: 1) mother's demographics and socioeconomic stratus: age, level of education, household income and geographical region of residence; 2) maternal attitudes towards vaccination in general (response to the single answer question "How do you feel about vaccines?": in favour of vaccinating against all serious diseases if there are vaccines/ in favour of minimizing the number of vaccinations/ opposed to all vaccines/ no opinion), and perception of HPV vaccine usefulness (response to the single answer question "For each of these diseases (cervical cancer), supposing there was a vaccine, would vaccinating your child seem: indispensable/ useful/ not very useful/ useless/ do not know"); 3) physician recommendation of HPV vaccination (response to the single answer question "Has your doctor advised you to have your child vaccinated against HPV": yes/ no/ do not remember); 4) perceived level of information regarding childhood vaccination (response to the single answer question: "Do you think you are well informed about child vaccinations?": not at all wellinformed/ rather not well-informed/ rather well-informed/ fully informed and); and 5) daughter's usual medical follow-up setting (response to the single answer question : "Your child is followed by : a general practice/ a pediatrician practice/ a free preventive clinic for mother and infants").

Statistical analysis

We conducted a pooled cross-sectional analysis across the four aforementioned years of the Vaccinoscopie® Survey. We combined these data sets in order to achieve a large enough sample size to perform multivariate analysis. The data sets included data weighted according to the French general population census of the French National Institute for Statistics and Economic Studies. Although the data were generated each year from a quota sample and not from a random probability one, we decided to conduct inferential statistical analysis commonly used to make inferences about the larger population from which the sample was drawn. Since it is difficult to guarantee that the study sample is representative for characteristics other than those for which quotas units have been set, inferential statistical measures (95% confidence intervals (95%CI) and P-values) are meant only for indicative purposes in the present study.

Logistic regression analysis

We conducted a descriptive analysis of study variables followed by a bivariate analysis, where HPV vaccine initiation was regressed on the covariates, which were were assessed for multicollinearity beforehand. We then estimated a multivariate logistic regression model using a backward selection process, with a threshold of P=0.20 for a variable to stay in the model. The full model included the year of the Vaccinoscopie® Survey, in addition to the aforementioned exposure variable and covariates. For ease of interpretation, we used the grouping of response modalities required by the path analysis described below. The goodness-of-fit of the final model was tested using the information matrix test.

Crude and adjusted odds ratios were calculated with their 95%CI.

Path analysis

We formulated a general hypothetical model for how Internet use, attitudes towards childhood vaccination, perceived usefulness of HPV vaccine, and the other above mentioned factors might be interrelated to HPV vaccine uptake. The hypothesized interrelationships between the variables are depicted in a conceptual model in Figure 1. To test our proposed hypothetical model, we applied path analysis, a form of statistical modeling consisting of a set of linear equations that simultaneously assess the relationships between the measured variables. We conducted these analyses using the PROC CALIS procedure in SAS with the weighted least squares (WLS) estimation method, which handles categorical variables [START_REF] Browne | Asymptotically distribution-free methods for the analysis of covariance structures[END_REF]. Nominal variables with more than two categories of response modalities (geographic area of residence, perceived usefulness of HPV vaccine, attitudes towards childhood vaccination, and perceived level of information) were dichotomized by grouping responses modalities, as the PROC CALIS procedure handles continuous, binary and ordinal variables, but not nominal variables.

We first tested the hypothesized conceptual model, which included all possible associations between variables (Figure 1). Afterward, we reviewed the path coefficients to see if any of the paths in the initial model should be deleted, and we dropped non-significant associations from the original hypothetical model. The resulting model, called the "final model", was then reestimated (Figure 2). We used standardized regression coefficients (β) to assess the strength of association between the variables (effect). Total, direct, and indirect effects of each variable on HPV vaccine uptake were estimated (a direct effect represents a path coefficient, an indirect effect represents the product of sequential path coefficients, and a total effect represents the sum of direct and indirect effects).

We used Cohen's recommendations to interpret the relative size effect of the standardized coefficients (a β value varying around 0.1 is considered as low, a value varying around 0.3 as moderate, and a value greater than 0.5 as large) [START_REF] Cohen | A power primer[END_REF]. To evaluate the goodness-of-fit between the final model and the data, we used the comparative fit index (CFI), the standardized root mean square residual (SRMR), and the root-mean-square error of approximation (RMSEA) including 90%-confidence interval (90%CI). Values for CFI >0.94 suggest a good fit between data and path models, whereas SRMR and RMSEA values less than 0.090 suggest acceptable fit, and values less than 0.055 suggest good model fit [START_REF] Hatcher Phdl | Step-by-Step Approach to Using SAS for Factor Analysis and Structural Equation Modeling[END_REF].

All statistical analyses were performed using SAS (version 9.4, SAS Institute Inc., Cary North Carolina), and p-values below 0.05 were considered statistically significant.

Ethics statement

The Vaccinoscopie® Survey is a healthcare market research undertaken by professional market researchers (IDM families) on behalf of a pharmaceutical company (GSK). It has been conducted in accordance with the legal and ethical guidelines issued jointly by the European Society for Opinion and Market Research (ESOMAR) and the International Chamber of Commerce (ICC), as well as the French law on computer data and freedom. It does not require ethical committee review. It warrants the total confidentiality of collected data, which, in any case, cannot be used to any other end than the ones defined for the survey. Furthermore, each participant may stop answering the questionnaire at any time and refuse that some pieces of data to be processed. Study participants consented to IDM terms of use and privacy policy, which indicated that their data would be used anonymously. The Vaccinoscopy® data file was approved by the French National Committee for Data Protection (n°1551077, dated 08/12/2011).

RESULTS

Study sample

The global Vaccinoscopie® Survey data set (2015 to 2018) included in total 2038 different respondent mothers. Of those, 89 (4.4%) declared having never been in the situation of searching for information regarding a vaccination they had hesitated about, leaving 1949 records for the present study. The mean (+/-SD) age of the survey respondents was 43.5 (±4.9) years. Approximately half of the participants had a level at high school diploma (baccalaureate) or two-year post high school diploma (49.6%) and had a monthly household income of over 2700 Euros (48.0%). The full characteristics of the study participants, together with their responses to the survey questionnaire, can be found in Table 1.

HPV vaccine coverage

The mean uptake of the first dose of HPV vaccine over the study period was 27.7% (

Internet and other sources of vaccine information

Approximately 24% of the mothers declared using the Internet as a source of vaccine information. This medium was the third most frequently cited source of information reported by the mothers, after the family physician (91.8%) and the relatives/family/friends (26.1%), and before the pharmacist (14.9%) (Table 1).

Logistic regression analysis

The uptake of the first dose of HPV vaccine was lower among daughters of the mothers who indicated using the Internet as a source of vaccine information compared to daughters of those who indicated not using it: 18% compared to 31%, respectively (P<0.0001). In multivariate analysis adjusted for physician recommendation of HPV vaccination, attitudes towards vaccines in general, perception of HPV vaccine usefulness, maternal level of education, geographical region of residence, and the year of the survey, the use of the Internet as a source of vaccine information was significantly associated with a lower HPV vaccination (aOR=0.66; 95% CI, 0.47-0.91) (Table 2).

Path analysis

The results of the path analysis modeling are depicted in Figure 2. The estimated « final model » shows a good overall fit: CFI=0.97, SRMR=0.03 and RMSEA=0.02, 90%CI (0.02-0.03). Overall, 26 % of the variance in HPV vaccine uptake was explained by the model. Table 3 shows the direct, indirect, and total effects of Internet use and other sources of information on HPV vaccine uptake (effects of other variables are reported in the Supplementary Table ). The total effect of Internet use on HPV vaccine uptake was proved to be significant and negative. It is considered as low (β=-0.10, Standard Error (SE)=0.02, P<0.0001), and represents in absolute value 27% of the highest total effect on HPV vaccine uptake, which is observed with the physician recommendation of HPV vaccination (β=0.37, SE=0.01, P<0.0001) (Table 3

and Supplementary Table).

A total of 4 paths were significant to explain the relationship between Internet use and HPV vaccine uptake: Internet use was also found to mediate the negative effect of the « family and friends » source of information on HPV vaccine uptake (β=-0.10, SE=0.02, P<0.0001), as Internet use was positively associated with this source of information (β=0.10, SE=0.02, P<0.0001). In reverse, Internet use was negatively associated with the «family physician» source of information (β=-0.28, SE=0.03, P<0.0001), which was found to have a total positive effect on HPV vaccine uptake (β=0.08, SE=0.01, P<0.0001).

DISCUSSION

The spread of false information online and its influence on vaccination have been frequently addressed in the literature. However, the quantification of the effect of Internet use as a resource for answering questions on vaccination on actual vaccine uptake is much less documented. We found that maternal search for vaccine information in the Internet was associated with a lower HPV vaccination initiation by their daughters. This result is congruent with the findings of the 2016 French National Health Barometer, in which parents of children aged 1 to 15 who relied uniquely on the Internet for information on vaccination were less likely to vaccinate their children against diphtheria, tetanus, and pertussis [START_REF] Gautier | Sources d'information, opinions et pratiques des parents en matière de vaccination en France en[END_REF]. Path analysis further confirmed the negative effect of Internet use on HPV vaccine uptake (β=-0.10, SE=0.02, P<0.0001). The analysis further revealed how Internet mediated its effect through a lower perceived level of information on childhood vaccination (which could be explained by the confusion resulting from contradicting pieces of information found on the Internet), unfavorable attitudes towards vaccination in general, and a lower perceived level of HPV vaccine usefulness. This effect runs contrary to the positive effect of obtaining information from a family physician (β=0.08, SE=0.01, P<0.0001), through more favorable attitudes towards vaccines and higher perception of HPV vaccine usefulness.

The Internet affords unprecedented opportunities for finding answers to health questions, but also poses some challenges for its users. Parents should take complete responsibility to base their vaccination decision on accurate information, wisely and consciously choosing their sources of information whenever questions arise as to the benefit, efficacy or safety of any given vaccine. Until each and every parent is equipped with the necessary knowledge and skills to interrogate the Internet safely, healthcare providers remain the most suitable and accurate sources of information on the topic of vaccination, and parents should be encouraged to seek information primarily from them. Interventions should be designed to empower parents to better recognize mis/disinformation, and consult only reputable sources of information if they wish to go online. In any case, the need to question the veracity of the information retrieved online should be emphasized, as it is sometimes not easy to distinguish between trustworthy and questionable websites.

Strengths and limitations

To the best of our knowledge, this is the first study investigating the association between Internet use as a source of information by vaccine-hesitant mothers and the uptake of HPV vaccine by their daughters. A key strength of this study is the use of path analysis to examine specific pathways by which Internet use among vaccine-hesitant mothers is associated with a lack of HPV uptake by their daughters, beyond the traditional logistic regression approach which only allows quantification of the association between two variables, all others things being equal.

The following limitations should be considered when interpreting the results of this study. First, the exposure measure was related to the behavior of seeking online information for vaccines in general, rather than HPV vaccine specifically. Nevertheless, only HPV vaccination is recommended in the target's age group of the daughters in the surveyed sample, hence most probably the source of questioning among their mothers. Second, the quota sampling approach employed for the recruitment of the mothers in the Vaccinoscopie® Survey allows a study sample that is representative of the general population with respect to specific criteria (SPC, geographic region of residence and number of children), but does not ensure that it is representative of the general population with regard to other criteria. Therefore, caution is warranted when generalizing the results of the responding sample to the broader French population. In particular, confidence intervals and P-values have to be very carefully interpreted: they would apply if the data were obtained from a random probability sample.

Furthermore, there is a possibility of selection bias through self-selection of participants, and also because the Internet population might not be fully representative of the general population (excluding the most disadvantaged groups and/or the non-French-speaking immigrants). Third, some factors possibly intervening in the process unfolding between Internet use and HPV vaccine uptake were not questioned in Vaccinoscopie® Survey and could not be introduced in our model. In a Delphi survey based on the World Health Organisation Strategy Advisory Group of Experts framework of vaccine hesitancy, we have shown that the factors possibly determining HPV vaccine uptake in France are numerous (26). These include, for example, vaccine knowledge, perception of the safety of the vaccine, and trust issues towards the health system. This may explain the direct effect observed of Internet use on HPV vaccine uptake, which probably indicates a residual effect not captured by the model variables, rather than a genuine direct effect. Fourth, this analysis was based on cross-sectional data. Therefore, causal inferences cannot be drawn, and reverse links between some variables cannot be completely ruled out. For example, the association between Internet use and perceived level of information on childhood vaccination may be due to reciprocal effects. Despite these limitations, this study seems to confirm the detrimental impact of Internet use as a source of vaccine information in vaccine-hesitant mothers. Additional studies are needed using prospective study designs among a random sample of parents.

CONCLUSION

In conclusion, this study found a weak but significant association between vaccine-hesitant mothers' use of the Internet as a source of vaccine information and a lack of HPV vaccine uptake by their daughters, through a lower perceived level of HPV vaccine usefulness, a lower perceived level of information on childhood vaccination, and unfavorable attitudes towards vaccination in general. While further studies are needed to confirm this link, there is no doubt on the need to raise awareness about the importance of seeking information from a healthcare provider, and consulting only reputable sources of information on the Internet.

Figures

1 )

 1 Internet use -> perceived level of information regarding childhood vaccination-> attitudes towards vaccines in general -> perception of HPV vaccination usefulness -> HPV vaccination; 2) Internet use -> perceived level of information regarding childhood vaccination -> attitudes towards vaccines in general -> HPV vaccination; 3) Internet use -> attitudes towards vaccines in general -> perception of HPV vaccination usefulness -> HPV vaccination; 4) Internet use -> attitudes towards vaccines in general -> HPV vaccination (Figure 2).

Figure 1 .

 1 Figure 1. Conceptual model with the hypothesized interrelationships between the variables.

Figure 2 .

 2 Figure 2. Final model of associations with standardized path coefficients.

  23.7% in 2015, 23.5% in 2016, 30.3% in 2017 and 33.1% in 2018). The full 2-dose regimen was completed in 18.7% of their daughters (14.4% in 2015, 16.1% in 2016, 21.3% in 2017 and 22.6% in 2018).

Table 3 .

 3 Table1. Distribution of study variables among 1949 vaccine hesitant mothers (Vaccinoscopie® Survey 2015 to 2018). : CI, confidence interval; HPV, human papillomavirus; OR, odds ratio; Ref., reference category for odds ratio, SD, standard deviation. Adjusted odds ratio determined by backward multivariate logistic regression of HPV vaccination. Total, direct, and indirect effects of Internet and other sources of information on HPV vaccine uptake (weighted data).

	Unweighted data	Weighted data
	Total		Total
	(N=1949)		(N=1835)
	n or mean	(% or SD)	n or mean (% or SD)

Abbreviationsa 4 missing values. b 36 missing values. *

Funding

Financial support for the study was provided by a PhD fellowship from the French National Institute of Cancer awarded to FD. The other authors received no specific funding for this work.

HebdHors-Sér Vaccin. 2017;28-35.

26.

Dib F, Mayaud P, Launay O, Chauvin P, FSQD-HPVH Study Group. Design and content validation of a survey questionnaire assessing the determinants of human papillomavirus (HPV) vaccine hesitancy in France: A reactive Delphi study. Vaccine. 2020 Sep 3;38(39):6127-40.

Tables

Supplementary table. Total, direct, and indirect effects of variables other than sources of information on HPV vaccine uptake (weighted data).

Author's contributions

FD and PC had designed the study. FD had performed the data analysis, the redaction of the manuscript, and the revision process of this paper based on Vaccinsoscopie Survey data acquired by OL. PC, OL, PM and LL have participated in the revision process and the redaction of the manuscript. GSK and IDM had no role in the study design nor in the data analysis. All authors have approved the final manuscript which was reviewed by GSK before submission.

Conflict of interest

Supplementary table. Total, direct, and indirect effects of variables other than sources of information on HPV vaccine uptake (weighted data).