Ektoras Chandakas
email: ektoras.chandakas@transamo.com

ON DEMAND FORECASTING OF DEMAND-RESPONSIVE PARATRANSIT SERVICES WITH PRIOR RESERVATIONS

The present study investigates the determinants of the volatility of passenger demand for paratransit services and explores the feasibility of a data-driven model for medium-term forecast of the daily demand. Medium-term demand forecasting is a significant insight to optimise resource allocation (staff and vehicles) and reduce operations costs. Using operational data from the reservation platform of the paratransit services in Toulouse, France, and enriching them with exogenous information, the study derives statistical and deep learning models for medium-term forecast. These models include a seasonal ARIMAX model with rolling forecast, a Random Forest Regressor, a LSTM neural network with exogenous information and a CNN neural network with independent variables. The seasonal ARIMAX model yields the best performance, suggesting that when linear relationships are considered, econometric models and deep learning models do not have significant differences in their performance. All the models show limited ability to grasp unique events with multi-day impacts such as strikes. Albeit a highly volatile demand and limited knowledge ahead of the forecast, these models suggest the volume of early reservations is a good proxy for the daily demand.

INTRODUCTION

The United Nations (UN, 2018) project that by 2050 the world population living in urban areas will grow to 68% compared to 2018 where the urban population of 4,2 billion accounts to 55% of world population. This increase is believed to increase urban passenger transport by 60-70% by 2050 (OECD-ITF, 2018), while the car remains the main mode of transport. Various initiatives exist to develop mass transit systems, mainly in dense areas where high passenger flows can justify the investment and operation expenditures.

To provide an adequate alternative to private cars it is essential to complement the high performance transportation network with services which are better adapted to these particular cases: in low-density areas, during certain periods of the day and week (night hours, weekend, …) and for users with specific needs (seniors, reduced mobility, …). That can be addressed with demand-responsive transportation (DRT) services, where small-and medium-sized vehicles adapt the services to the specific demand (Alonso-Gonzalez et al., 2018). The sanitary context in 2020 urged various transit networks replace the fixed-route services with demand-responsive ones. Changing mobility patterns and caution with high-occupancy vehicles lays ground for the emergence of Mobility as a Service and demand-responsive services.

In addition, current technology trends new information technologies and app-based reservation platforms increase the service reactivity with real-time reservation and vehicle location tracking. [START_REF] Attard | The technology behind a shared demand responsive transport system for a university campus[END_REF] suggest the three-part IT architecture for the deployment of efficient demand responsive transit. These technologies, combined with Mobility as a Service (MaaS), drive the reinvention of transit services as a whole [START_REF] Shaheen | service (MaaS): early understanding of shared mobility impacts and public transit partnerships[END_REF]. The combination of public policy towards shared mobility and technological development makes DRT service ideal to fill the gap of First-Mile-Last-Mile (FMLM) and paratransit services, providing an almost seamless inclusive urban transportation system. Coutinho et al (2020) compare fixed routes and DRT services in a real case study to show that the latter offer a better operational performance as of vehicular kilometers, operational cost and greenhouse gas (GHG) emissions per passenger.

Demand Responsive Transportation services are subject by definition to a volatile demand, more so if they are focused on non-constraint journeys, such as the ones motivated by leisure activities or purchases. [START_REF] Rahimi | System modelling of demand responsive transportation services : Evaluation cost efficiency of service and coordinated taxi usage[END_REF] establish a link between the cost efficiency of large-scale DRT operations and the demand, the service area and the demand density. [START_REF] Palmer | Factors influencing productivity and operating cost of demand responsive transit[END_REF] suggest a positive impact on productivity when using automatic grouping of the requests. A DRT operator, in regards to the Level of Service Agreement with its clients (B2C) or public authorities (B2G), needs to anticipate the size of the operational fleet and the number of the employees needed to meet the demand. Overestimating them, leads to slack resources and increased production costs. An underestimation of the daily demand may lead to non-captured revenues, a loss of the customer image or specific penalties from the transportation authorities. Having a better knowledge on travel demand behavior of the potential customers and improving the forecast tools of passenger demand allows a near to real-time optimization of the vehicle fleet, along with the vehicle maintenance planning and the drivel and personnel scheduling.

The paper seeks to shed some light to the main variables affecting demand variability. In addition, it seeks to examine the feasibility of an operational model able to provide, via endogenous information, reservation trends and exogenous information (holidays, weather, etc.), an adequate forecasting of the number and type of reservations 7 days in advance or prior, when crew scheduling is done according to the transit network collective agreement. The direct and indirect impacts of an improved forecast are translated operationally to:

• A reduction of production costs through a reduction of the number of idle drivers • An increase of commercial revenue, by reducing the refusal rate for late reservations • A guarantee of a good image to the local transport authorities, therefore better chances to maintain the contract The present paper is structured in 6 parts as follows: After a brief introduction, section 2 reviews existing research on short-term forecast in traffic flow and passenger transportation networks. Section 3 describes the site and the demand data used for the modelling, as well as the pre-processing and feature engineering processes. Section 4 formalizes the modelling architecture, while section 5 explores the specification and validation of the set of predictive models used for the short-term forecast. In conclusion, section 6 interprets and compares the main outcomes of the models and sets out some areas of future research.

BACKGROUND ON SHORT TERM FORECASTING MODELS

The short-term forecasting in transportation is a vivid research area, especially in recent years. The main topics address flow and travel time forecast in road traffic and public transportation context. In Vlahogianni et al (2014) the authors do an extensive review of neural network approaches in road traffic modelling and they set the main challenges in that area. The authors suggest the model interpretability, or lack of thereof, is one of the barriers for adopting more sophisticated deep learning models in traffic operations. [START_REF] Karlaftis | Statistics versus neural networks in transportation research: differences, similarities and some insights[END_REF] lay out a thorough comparison between statistical and neural networks in transportation, focusing mainly in road traffic. [START_REF] Karlaftis | Memory properties and fractional integration in transportation time-series[END_REF] examine long-term processes for ARIMA models captured through fractional integration. Their study points out that over-differentiation tends to over-inflate the MA component of the model, thus weakening the quality of the results. Novel approaches in short-term forecasting of traffic flow speeds include [START_REF] Polson | Deep Learning for short-term traffic flow prediction[END_REF] who develop a deep learning approach based on a regularization process to reduce the non-linearity of the traffic state transitions and Wang et al (2014) who propose Bayesian combination approach factoring the prediction errors in order to improve the prediction. In the public transportation setting, [START_REF] Farid | Evaluation of Short-Term Bus Travel Time by using Low-Resolution Automated Vehicle Location Data[END_REF], among multiple data-driven models, suggest the Support Vector Regression (SVR) model is more accurate in the short-term forecast of bus travel times. Lin et al (2018a) introduces a Particle Swarm Optimization and Extreme Learning Machine (PSO-ELM) neural network to quantify the uncertainty in short-term traffic prediction by examining the reliability and the sharpness of the Prediction Interval (PI).

Transit systems, TNC and micromobility operators pursue the digitalization of the production and client outreach processes, accumulating large amount of data. These are often used in the development of short term demand prediction models for mass transit services and micromobility schemes. Lin et al (2018b) explore various Graph Convolutional Neural Networks with Data-driven Graph Filter (GCNN-DDGF) models to predict station-level hourly demand in a large-scale bikesharing network. The graph approach of these models with the associate neural network architecture captures efficiently hidden correlations among bike-sharing stations.

The use of smart card data for demand forecasting is widely explored. [START_REF] Van Oort | Short-Term Prediction of Ridership on Public Transport with Smart Card Data[END_REF] propose a smartcard data-driven model for the analysis of transit service variations. [START_REF] Dai | Short-Term Origin-Destination Based Metro Flow Prediction with Probabilistic Model Selection Approach[END_REF] build on smartcard tap-on and tap-off date to develop short-term metro passenger flow forecast through a combination of a probabilistic model selection and random forest classification model. In [START_REF] Gong | Sequential Framework for Short-Term Passenger Flow Prediction at Bus Stop[END_REF] the authors develop a three step model based on seasonal ARIMAX model, an event-based method and a Kalman Filter based method to forecast short-term passenger flow at bus stops. Moreira-Matias and Cats (2016) propose a model through local constrained regression to infer passenger loads through automatic vehicle location data (AVL). Ma et al. (2013) specify an Interactive Multiple Model-based Pattern Hybrid approach for short-term demand forecast by developing pattern model based on historical demand data and dynamically optimize the interaction between them using real-time observations. [START_REF] Noursalehi | Real time transit demand prediction capturing station interactions and impact of special events[END_REF] propose a two-level approach for the short term prediction of passenger flows in transit stations. On the station level the authors opt for a univariate state-space model. A hierarchical clustering algorithm identifies similarities between stations and regroups them. Within each cluster, a dynamic factor model grasps the interdependencies between the stations. Toque et al. (2017) use LSTM and Random Forest models for short-term (15 and 30 minutes windows for rail and buses) and long-term (one-year) for demand forecasting in multimodal transportation facilities. [START_REF] Guo | Short-Term Abnormal Passenger Flow Prediction Based on the Fusion of SVR and LSTM[END_REF] use an approach based on the fusion of Support Vector Regression (SVR) and long short-term memory (LSTM) neural network to predict abnormal passenger flows in a metro station. [START_REF] Ke | Short-term forecasting of passenger demand under on-demand ride services: A spatio-temporal deep learning approach[END_REF] propose a fusion convolutional long short-term memory architecture (FCL-Net), integrating spatio-temporary and non-spatial time-series variables for short-term forecasting of passenger demand of demand-responsive services.

Short-term prediction of demand-responsive transit schemes is a topic most frequently overlooked. However, the digitalization of the operations is a major driver for the creation of new categories of models in a real-time setting. [START_REF] Koffman | Forecasting Demand for Paratransit Required by the Americans with Disabilities Act[END_REF] make the argument of the importance of adequately forecasting the demand for the optimal design of paratransit services. Benjamin et al (1998) explore the determinants of choice of paratransit services and develop an econometric model to address mode choice. Furthermore, Deka and Gonzales (2014) provide a spatial structure of paratransit trips, by identifying potential trip generators.

The present study builds on previous scientific contributions both in short-term predictive models and in demand-responsive paratransit services. It expands the scope of the existing models, and provides a comparison of existing and novel models for short-term demand forecast. These contributions improve the understanding of the determinants of the demand variability and contribute on the understanding of theoretical and practical implications of machine learning and deep learning models on demand-responsive transit operations.

SITE AND DATA DESCRIPTION

The paper is focused in the paratransit services (Tisseo, Mobibus) in Toulouse, France. This service is operated through a Public Service Obligation Contract. The company has 58 vehicles of various sizes (from 3 to 12 passengers) and 75 employers (of which 60 drivers). It is operating during weekdays from 7:00 am to 0:30 am and weekends and bank holidays from 9:00 am to 0:30 am. The reservations open 28 days prior (D-28) and can be made at the latest 2 hours (H-2) before the departure. No reservation is accepted if the period between the reservation and the trip is shorter than 2 hours. Furthermore, once a reservation is made (from D-28 to H-2), the client can modify or cancel it for free up to 2 hours (H-2) before the departure. These reservations are subject to grouping, in order to optimize the services and reduce production cost. The drivers' collective agreement stipulates the drivers' shifts are determined 7 days in advance (D-7). Further tuning of drivers schedules up to D-1 is based on negotiation and mutual agreement, but usually incur high marginal costs.

The initial database

The data acquired correspond to the data of reservation platform of the paratransit operations. This relational database stocks all data relevant to reservations management, staff and vehicle scheduling and service optimisation. The initial operational database may contain personal data subject to the General Data Protection Regulation (GDPR). However, the data extracted are anonymous. Further analysis and treatment is grouped in such ways as to not be able to determine any personal data of the clients.

The relevant demand data are further extracted into a plain table, where each observation corresponds to a reservation, completed or cancelled. The data concern a period of roughly 4 years (1478 days from 25/05/2015 to 10/06/2019) and total 736 019 observations. Further transformation of the database includes the identification of the service category (mode), through the tariff rate. The structure of the reservation data is illustrated in Table 1.

The spatial distribution of the data

The density of the departures and arrivals per ZIP code is further explored. Furthermore, 22% of the trips are made within the same ZIP code. This ratio approaches 49% if the municipality boundary is considered as a level of aggregation. That illustrated the short distance of the trips which makes the trip aggregation harder. Figure 1 illustrates the ratio of departure and arrival addresses per ZIP code. That highlights the volumes of the Toulouse municipality in the center, as it corresponds to 61% of departing trips. Furthermore, the data show some symmetry between departures and arrivals for each ZIP code area.

The temporality of the data

Due to the inherent volatility of the demand, the data are explored with respect to their temporal structure. The weekly demand, as illustrated in Figure 2(a), suggests a positive linear trend of the demand, as well as some seasonality, accounting for Christmas, summer holidays and spring holidays in May, along with a significant effect of bank and school holidays. The day-to-day variation of the demand (Figure 2(b)) during a 4 month period shows to a greater extent the variability during weekday and weekend. Since the initial data at the moment of the extraction included future dates of trips, the analysis is limited to the period until 30/04/2019. Further analysis seeks to identify whether this profile is sensible to the type of service and type of day (weekday, weekend, … ). The data analysis focuses on within-day dynamics of reservation. Figure 3 illustrates reservation volumes per 15 minute time-step per day of the week. It suggests significant difference with respect to the day of the week and an important variability per type of day throughout the years. Nevertheless, some within-day patterns of the demand per type of 5 day are visible.

The qualitative attributes of the data

Addressing the qualitative characteristics of the demand, the acceptance ratio is roughly 71%, while a slight variation of this ratio is observed throughout the year and within the day, as illustrated in Figure 4. The mode variable accounts for the service type demanded are illustrated below. The "Le Lien" service has a very low number of observations, making their characteristics non-representative in statistical sense at that level of aggregation. The journey duration with 15 minute time steps is further described, with respect to the volume of departures. Travel times have an average value of 00:27:24 and standard deviation of 00:15:48. The travel time per reservation seems to be sensible to morning and evening peak hours, since vehicles are subject to general traffic. Nevertheless, the travel times do not seem to be impacted by the number of reservations. The average travel times seem to be lower during the noon off-peak period compared to the evening off-peak one. Differences in travel times may be attributed to longer route distances. Nonetheless, travel time is an appropriate proxy of service production per reservation, since the size of the service is marginal and it does not impact general car traffic.

The average travel time per commercial trip (reservation travel time) per service type and day of the week are illustrated in Table 3. The D2D service has a more important travel time than A2A type. The 4 minutes difference can be accounted to the additional time needed for accompanying the client from the door of the origin and destination to the vehicle. Furthermore, the commercial trip travel time during the weekend are lower than those from weekdays. Characteristics of the "Le Lien" service should be considered with extreme caution due to their limited sample. The trips cover a variety of purposes. The main trip purposes for the use of the service, of the users that informed on the relevant trip purposes when making the reservation by phone of internet,, are medical visits, professional trips (to and from work accounting to 25%) and leisure (45% of trips with known purpose). As of the concentration of the reservations to some clients, the volume of reservations per unique client is analyzed. It is observed that 30% of the most frequent clients, make for 50% of all requested reservations. On the other hand 40% of the least frequent clients make for only 20% of all requested reservations. That suggests a frequent use of the service by a given set of clients, although the frequent clients are not a dominant group. Hence, that should have a straightforward impact on the volatility of the demand.

This service accepts reservations up to 28 days in advance to the trip date. In addition, it is possible to pre-plan reservations that are repetitive for certain days per week or a period. These reservations are injected into the system 21 days before. Once the reservation is created, it can be subject to modifications, cancellation or rejection up to the trip day. Figure 5(a) illustrates the part of all the reservations, with respect to the number of days in advance they were made. Indeed more than half the reservations are made 14 days prior to departure. Roughly 2/3 of the reservations are made 7 days before the departure.

Current cancellation policy accepts free cancellations up to 2 hours before the departure, adding to the volatility of the final demand. Figure 5(b) illustrates the cancellations per day prior to the trip. Whereas the cancellations amount to 30% of total reservations, roughly 70% of them occur during the last week, while 35% occur the last two days prior to the trip.

The reservation policy in place, lacking incentives for early reservation and for avoiding late cancelations, is a major source of demand volatility and cost inefficiency. However, this must be interpreted by the desire of the transport authority for a responsive service, thus facilitating mobility and inclusion for this particular client group.

The exogenous information

Additional features are added to the days, namely related to weather, bank and school holidays. In addition to bank and school holidays, the "pont", or extensive weekend, variable is determined: this dummy variable characterizes a Monday followed by a bank holiday, or a Friday following a bank holiday, where the demand is assumed to be closer to that of a bank holiday rather than of a weekday. As of the type of day, the following 5 categorical variables were added:

• TypeOfDay : categorical, corresponding to the day of the week : 1, …, 7 for Monday through Sunday, That is further transformed into three one-hot encoders : o isMoWeFr o isTuTh o isSa • IsWE : categorical, corresponding to weekend • IsJF: categorical, corresponding to bank holidays • isShoolHoliday, categorical, corresponding to school holidays • isPont : categorical, corresponding to extended weekends Historic hourly weather data of Toulouse are retrieved starting from 01/09/2016 from on-line weather archives. These data are transformed and the information on the mean, max and min temperature during each day and the number of hours with a rain fall is introduced as an exogenous information.

Figure 6. Local weather data and main exogenous information 4 MODELLING PIPELINE

The various models are compared within a standard modelling pipeline, as illustrated in Figure 7.

The following section details the data preparation process and the evaluation metrics used for the calibration and comparison of the models.

Data preparation

The final dataset is obtained from the combination of the endogenous and exogenous information for the period from 01/09/2016 to 30/04/2019. Each observation accounts to a specific date of that period. Two datasets are further specified into two distinct datasets:

• Y (972,1) contains the time-series volumes to be predicted In addition to the exogenous information, the input variables include the volume of reservations made 14, 10 and 7 days prior to the day of the trip, referring to the ResBef_14d, ResBef_10d and ResBef_7d respectively. These values do not take into consideration further modifications or cancellations.

The datasets (Y, X_exogenous) are separated into three distinct datasets. To ensure comparability of the results, the following datasets are used by all the algorithms:

• A training dataset, used for the training of the models • A validation dataset, used for the selection of the best model • A test dataset used for the unbiased appreciation of the model precision

Evaluation metrics

The quality of the models is evaluated through it is proximity to observed data. The evaluation metrics, summarized in Table 5, allow for the comparison of the forecast accuracy. While the RMSE is the most usual metric, used when large errors are undesired, the MAE is most robust to the outliers. The MASE allows for the comparison of the forecast with the MAE produced by a naïve forecast. Finally, the MAPE metrics address the forecast error as a proportion of the actual values.

Table 5. Description of the evaluation metrics

Evaluation metrics Loss function Mean Absolut Error

(MAE) = 1 | -| Root Mean Square Error (RMSE) = 1 - Mean Average Scaled Error (MASE) = ! Mean Average Percentage Error (MAPE) " = 1 | -| . 100%
statsmodel [START_REF] Seabold | Statsmodels: Econometric and statistical modeling with python[END_REF], tensorflow [START_REF] Abadi | TensorFlow: Largescale machine learning on heterogeneous systems[END_REF] and keras [START_REF] Chollet | keras[END_REF] for comparing the models. The present study explores different model families, related to forecasting univariate timeseries of as a standard regression model. The exploration phase of the present study included a multitude of relevant machine learning and deep learning models. For simplicity, it was chosen to pursue with a limited number of representative models from each category, with the selection criteria covering both practical and theoretic aspects. In addition it provides a naïve model to serve as a baseline forecast. The following table resumes the main families examined.

Univariate Time Series Standard Regression

Machine Learning

Seasonal ARIMA model with exogenous information (SARIMAX)

Random Forest Regression (RFR)

Deep Learning

Long Short Term Memory neural network (LSTM)

Convolutional Neural Network (CNN) 5.1 A naïve model for the baseline forecast

A naïve forecast model is built to serves as a baseline estimate. The naïve model assumes for each day the forecast of the reservations corresponds to the average values of the training set values, with respect to the type of day within the week. The evaluation metrics suggest that the forecast cannot grasp the difference on the level of reservations due to the within-year seasonality and the unique events.

An Autoregressive model to address a univariate time series

Addressing the dataset as a univariate time series with exogenous data, an Autoregressive Integrated Moving Average (ARIMA) model [START_REF] Box | Distribution of residual autocorrelations in autoregressiveintegrated moving average time series[END_REF] is used. It is a model that is adapted to short-term real-time forecasting provided addressing issues such as complexity of model specification, estimation and maintenance [START_REF] Williams | Multivariate Vehicular Traffic Flow Prediction[END_REF]. The analysis of the data and their seasonal decomposition suggest seasonality based on the period of the year and the day of the week. In addition exogenous data relevant to the type of day and the temperature are added to a seasonal ARIMA model.

As long as the time-series data are considered, the Dickey-Fuller Test suggests the time series is stationary at a 1%. The autocorrelation function suggests a strong autocorrelation for the autoregressive part for a small lag and some strong seasonality every 7 time-steps. The partial autocorrelation function shows a lag greater than a week.

The seasonal ARIMAX (SARIMAX) is characterized by non-seasonal (p, d, q) and seasonal (P, D, Q, s) hyper-parameters as well as the use of the exogenous data, The non-seasonal hyperparameters are:

• The autoregression (p) denoting the past values included in the regression • The differencing (d) defining the differencing transformations • The moving average (q) denoting the lag of the error The trend parameter can take the following values: none, constant, linear, linear & constant. The seasonal parameters (P, D, Q) represent the same hyper-parameters for the seasonal part of the time series. Finally, s corresponds to the number of timesteps in the season.

The calibration of the SARIMAX model needs the exploration of the space around 8 hyperparameters, with a walk forward validation, more adapted to a time-series context. The final dataset used for the gridsearch is limited to the 292 days, of which the forecast is limited to 73 days. For each time-step a forecast of 7 days ahead is made, to account for the operating lag between schedule optimization and operations.

An exhaustive Gridsearch of the hyper-parameters suggestions the parameter combination [ (1, 0, 1), (1, 0, 0, 7), linear trend ] yields the optimal score. The first results indicate that for the best fitted model the variables relative to the weather conditions are not significant. The non-significant variables are excluded from the final model. Hence the results of the final SARIMAX are summarized in Table 6. ------------------------------------------------------------------------------------------------- These results suggest that weather conditions are not significant to the prediction of the volume of reservations. That may be attributed to the presence of constrained trip purposes (work, medical visits …). Furthermore, the volume of reservations 10 (Res_Bef10d) and 14 (Res_Bef14d) days before is not significant. It shows that the presence of the variable Res_Bef7d, for the volume of reservation 7 days prior to the trip, is probably a sufficient proxy for this evaluation. Finally, the nonsignificance of the isHolidays variable, integrating school holidays periods, may be attributed to the characteristics of the service's clients, with a large proportion of seniors, and the presence of variables as the volume of reservations 7 and 14 days.

The exploration of the residuals, plotted in Figure 8, shows non-auto-correlated and normally distributed residuals. The plot of the standardized residual seems to take random values around 0. However, it may be showing some cyclical structure of the residuals. The forecast errors seem to be normally distributed around a zero mean. The Q-Q plot with residual quantiles vs the theoretical normal quantiles shows a slight tail, Nevertheless, it is deemed satisfactory. Finally the autocorrelation of the residual error does not show any significant correlation between the residuals. The test process of the final model is made through an adaptation of the previous algorithm and the use of a different set of data. The test set covers an entire year prior to the data used for the selection of the best model. The model forecasts 73 days, corresponding to 20% of a year. A rolling forecast of the 7 th day ahead for each timestep is made on the basis of the previous reservations and the exogenous variables.

The analysis of the residuals of the fitted model on the test dataset allows to show noncorrelated and fairly normal distribution of the residuals. The Q-Q plot with the residual vs the theoretical quartiles shows a slight trail. However, the autocorrelation of the residual error seems to show some correlation with 7 day lag of the residuals. That could be attributed to the weekly periodicity of the initial demand.

A Random Forest Regressor

A Random Forest regressor (Ham, 1995) is designed to address the demand forecast as a regression problem addressing in an independent way each daily prediction. In such case, the input variables are limited to the exogenous information of the dataset. The regression model is based on the scikit-learn algorithm [START_REF] Pedregosa | Scikit-learn : Machine Learning in Python[END_REF]). An exhaustive Gridsearch algorithm explores the main parameters of the Random Forest Regression: number of decision trees and max depth. The training step suggests 1000 decision trees with up to 5 splits.

The analysis of the residuals of the fitted model of the test dataset shows non-auto-correlated and normally distributed residuals, corresponding to a white noise. The plot of residuals vs fitted values suggests the residuals take random values. A Long Short-Term Memory neural network [START_REF] Hochreiter | Long short-term memory[END_REF]) is specified for the regression problem of a time-series with exogenous information. This architecture includes a pre-processing phase necessary for the adaptation to the data structure of the LSTM model. The data are transformed into 3D ndarrays with the following dimensions [nb_of_samples, look_back timesteps, nb of features], as well as the output data which are transformed into 3D ndarrays with the following dimensions [nb_of_samples, look_back values] (Figure 11). The target data, as well as input data relevant to reservations, are scaled to a (0,1) range by a MinMax scaler. The design of the model follows a standard architecture. In addition to the LSTM layer, dense layers with a time distributed wrapper are considered. Table 7 resumes the design of the model with 500 epochs. A rolling forecast is used to predict the reservation values. For each step throughout the validation and test dataset, the historic data of current forecast are added to the input data and used for the forecast of the following time-step.

The analysis of the residuals on the test datasets suggests non-auto correlated and normally distributed residuals. The residuals vs fitted values plot suggests constant variance and the presence of ) is designed for the forecast of reservation values on the basis of a particular day's features. The particularity of this model is that it is not modelled as a time-series; rather each observation (day) is considered independently from the others. The CNN model is built with:

• An input layer: a 1D convolutional layer • Four Hidden layers, a 1D convolutional layer, followed by 1D average pooling and flattening, and 3 dense layers, • An output dense layer with one unit In addition a dropout ratio of 0.25 is chosen between the dense layers to avoid overfitting during the training phase. The activation layers correspond to relu, sigmoid and linear functions. The CNN model uses the Adam optimization algorithm [START_REF] Kingma | Adam : A method for stochastic optimization[END_REF]. A Gridsearch suggests the use of 500 epochs for the final CNN model. The characteristics of the layers are as follows: The input data is reshaped with respect to the 2D ndarary [sample, nb of features] needed and a MinMax scaler is used to scale target date to a (0,1) range. The futures included correspond to the input variables used in all the models. With respect to the insight of the SARIMAX model, the input data omit the weather-related features (min, average and max temperature and rain duration).

The analysis of the residuals of the test dataset suggests non-auto-correlated and normally distributed residuals. The plot of the residuals vs the fitted values, suggests some outliers. Nevertheless, they seem to have a zero mean and a constant variance. The Q-Q plot of the residuals vs theoretical normal quantiles shows a slight tail.

MODEL COMPARISON AND CONCLUDING REMARKS

The paper explores the determinants of the volatility of passenger demand of paratransit services and the feasibility of a data-driven model to accurate forecast future passenger demand and to improve operations scheduling. In addition to a naïve model, four models are developed: a seasonal ARIMAX model with a rolling forecast, a Random Forest Regressor with independent variables, a LSTM neural network with exogenous variables and a rolling forecast and a CNN model with independent variables. The results on the test dataset are illustrated in Figure 14. The evaluation metrics, summarized in Table 9, suggest the SARIMAX model performs well on most metrics, suggesting it is more robust to outliers, while large errors are limited. The SARIMAX underperforms on the MAPE metric, where the Random Forest Regressor performs best. This may be attributed to less accuracy when forecasting days with lower volumes. SARIMAX slightly outperforms the Deep Learning models, although the precision level is similar. This comes in line with [START_REF] Karlaftis | Statistics versus neural networks in transportation research: differences, similarities and some insights[END_REF] who suggest that statistical and deep learning models have similar performance when addressing phenomena with linear relationships. A qualitative interpretation of the results of the fitted models suggests they adequately forecast the volume of demand for typical days and predictable events, such as weather episodes. However, all the models present a significant error of roughly 10% on March 19 th , which was the day of an interprofessional strike in Toulouse.

The time-series based simulations, as well as the other models, suggest that only a limited number of variables are necessary for an accurate forecast of daily demand. For a DRT service with prior reservation, these variables can be limited to the number of reservations seven days ahead of the departure, as well as the type of the day (day of the week and holidays). Additional variables, such as specific events (strikes etc.) could improve the forecast. These variables do not include weather related variables, such as the temperature, and the rain fall, as they do not seem significant for forecasting the demand. However, they may be significant for measuring the operational performance of the service.

Future work will address the specification and maintenance issues involved in including this SARIMAX architecture to the production chain. It will include model variation for D-10 and D-4 in order to provide some insight for resources planning. Future research will include the exploration of additional features, with respect to exceptional events, such as strikes, and will focus on the forecasting of multivariate time series, in order to address the within-the-day temporality of the demand. It will also seek to expand the medium-term forecasting to the human and vehicle resources needed. Additional model will include CNN-LSTM model architectures and a variation of the SARIMAX model with Kalman filters in order to grasp the hidden Markov chains of the time-series.

Figure 1 .

 1 Figure 1. Density of departures and arrivals per ZIP code

Figure 2 .

 2 Figure 2. Volume of (a) weekly completed reservations during the simulation period (top) and (b) daily completed reservations during October-2018 -February 2019 period (bottom)

Figure 3 .

 3 Figure 3. Reservation volume per 15 minute time-steps per day of the week for individual days (grey)and mean (red)

Figure 4 .

 4 Figure 4. Reservation acceptance rate per day

Figure 5 .

 5 Figure 5. (a) Volume of reservations based on the number of days reserved prior to trip -left (b) Volume of reservations cancelled with respect to the day before the trip a cancellation occurs

Figure 7 .

 7 Figure 7. The modelling pipeline used for the model specification and comparison

Figure 8 .

 8 Figure 8. Final SARIMAX model residual evaluation

Figure 9 .

 9 Figure 9. Evaluation of the residuals of the test dataset of the SARIMAX model

Figure 10 .

 10 Figure 10. Evaluation of the residuals of the test dataset of the Random Forest Regressor

Figure 11 .

 11 Figure 11. Representation of the shapes of the X and Y arrays used with keras

  /target array# sample -#days some outliers. The Q-Q plot shows a slight tail on both extremities. Overall, it shouldn't impact the quality of the model.

Figure 12 .

 12 Figure 12. Evaluation of the residuals of the test dataset of the LSTM model

Figure 13 .

 13 Figure 13. Evaluation of the residuals of the test dataset of the CNN model

Figure 14 .

 14 Figure 14. Predicted vs, observed value of reservation with (a) top-left: naïve model (b) center-left: SARIMAX model, (c) center-right: Random Forest regression model (d) bottom-left : LSTM model (e) bottomright: LSTM model

Table 1 .

 1 Data structure

	Column	Type	Description
	REFCON	Int	Reservation Unique ID
	JOUR (DAY)	Datetime Trip date
	IDCLIENT	Int	Client Id
	IDSERVICE	Int	Id of the service
	IDMAITRE (IDPRIMARY)	Int	Id of primary client (in case of
			grouped trips)
	ADRDEB_ID (ADRDEP_ID)	Int	Id of the departure address (closest
			node of the route graph)
	ADRDEB_CP (ADRDEP_ZIP)	Int	ZIP Code of departure.
	ADRDEB_VILLE (ADRDEP_CITY)	Char	Departure borough name
	ADRFIN_ID (ADRARR_ID)	Int	Id of the arrival address (closest node
			of the route graph)
	ADRFIN_CP (ADRARR_ZIP)	Int	ZIP Code of arrival. Float
			transformed to Integer
	ADRFIN_VILLE (ADRARR_CITY)	Char	Arrival borough name
	HDEB_PLANIFIEE (TOD_PLANNED)	Datetime Planned time of departure
	HFIN_PLANIFIEE (TOA_PLANNED)	Datetime Planned time of arrival
	DATE_CREATION (DATE_INIREQUEST)	Datetime Date the initial reservation was made
	DATE_MODIFICATION	Datetime Date of last modification of the
			request
	IDAUTORISATION	Int	Tariff Id
	IDCATEGORIE (IDCATEGORY)	Int	Trip purpose
	KMS_MISSION_DIRECT	Float	Distance of a direct trip
	KMS_MISSION_REEL	Float	Distance of planned trip (with
	(KMS_MISSION_REAL)		grouping)
	DUREE_MISSION_DIRECT	Float	Direct trip length in minutes
	(DUR_MISSION_DIRECT)
	DUREE_MISSION_REEL	Float	Planned trip length in minutes
	(DUR_MISSION_REAL)
	COMPLETED	Boolean	The reservation is completed
	MODE	Char	Service category

Table 2 .

 2 Service characteristics

	ID	Name	Characteristics	Price			# Observations
							(Tot = 736 069)
	A2A Address-to-	Pick-up and drop-off are	1,10 -1,35 whether it's	239 116
		Address	made at the desired address	peak or off-peak
	D2D Door-to-Door	Like A2A, where the driver	Like	A2A	with	494 983
			accompanies the client up to	additional charge for
			the door	accompany person
	Le	Metro	Connection to metro or BRT	Free of charge		1 920
	Lien	connection	transit services

Table 3 .

 3 Average travel time (in minutes) per commercial trip with respect to the service type and the day of the week

	Service	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday	Mean
	Type	(0)	(1)	(2)	(3)	(4)	(5)	(6)
	A2A	25.6	25.4	25.6	25.3	24.4	21.5	22.6	24.9
	D2D	22.2	30.2	28.9	29.1	28.5	26.2	27.0	28.9
	Le Lien	23.5	23.3	26.5	22.8	27.7	22.0	24.4	25.3
	Mean	27.9	28.6	27.8	27.8	27.2	24.7	25.9	26.7

Train data Validation data Test data Model Loss fit Fitted Model Best Fitted Model Model Loss evaluate evaluate GridSearch•

  X_exogenous (972, 13) contains the exogenous data. These are described in the following table. The TypeOfDay is further transformed into a one-hot encoding (isMoWeFr, isTuTh, isSa) combined with the isWE variable.

Table 4 .

 4 Descriptive statistics of the input variables

	Variable	Type	Descriptive Statistics
	ResBef_14d	Continuous	Mean=226.2 stddev=132.0
	ResBef_10d	Continuous	Mean=259.9 stddev=141.1
	ResBef_7d	Continuous	Mean=299.1 stddev=156.6
	isMoWeFr	One-hot	F(1) = 0.43
	isTuTh	One-hot	F(1) = 0.29
	isSa	One-hot	F(1) = 0.14
	isWE	One-hot	F(1) = 0.28
	isJF&Pont	One-hot	F(1) = 0.03
	isSchoolHoliday	One-hot	F(1) = 0.31
	MeanTemp	Continuous	Mean=13.6 stddev=6.8
	MaxTemp	Continuous	Mean=18.3 stddev=7.7
	MinTemp	Continuous	Mean=9.5 stddev=6.4
	RainDuration	Continuous	Mean=1.7 stddev=3.2

Table 6 .

 6 Summary of the results of the final SARIMAX model.

	==============================================================
	Dep. Variable:		y		No. Observations:	292
	Model:	SARIMAX(1, 0, 1)x(1, 0, 0, 7) Log Likelihood	-1224.286
					AIC		2468.572
					BIC		2505.340
	Sample:		0 -292	HQIC		2483.300
	Covariance Type:	opg
	==============================================================
		coef	std err	z	P>|z|	[0.025	0.975]

Table 7 .

 7 Description of the LSTM model

	_____________________________________________________
	Layer (type)	Output Shape	Param #
	=====================================================
	lstm_1 (LSTM)	(None, 100)	43200
	_____________________________________________________
	repeat_vector_1 (RepeatVecto (None, 7, 100)	0
	_____________________________________________________
	lstm_2 (LSTM)	(None, 7, 100)	80400
	_____________________________________________________
	time_distributed_1 (TimeDist (None, 7, 200)	20200
	_____________________________________________________
	time_distributed_2 (TimeDist (None, 7, 50)	10050
	_____________________________________________________
	time_distributed_3 (TimeDist (None, 7, 1)	51
	=====================================================
	Total params: 153 901
	Trainable params: 153 901
	Non-trainable params: 0
	_____________________________________________________

Table 8 .

 8 Description of the CNN model

	____________________________________________________
	Layer (type)	Output Shape	Param #
	====================================================
	conv1d_1 (Conv1D)	(None, 2, 64)	448
	____________________________________________________
	average_pooling1d_1 (Average(None, 1, 64)	0
	____________________________________________________
	flatten_1 (Flatten)	(None, 64)	0
	____________________________________________________
	dense_1 (Dense)	(None, 400)	26000
	____________________________________________________
	dropout_1 (Dropout)	(None, 400)	0
	____________________________________________________
	dense_2 (Dense)	(None, 80)	32080
	____________________________________________________
	dense_3 (Dense)	(None, 1)	81
	====================================================
	Total params: 58 609
	Trainable params: 58 609
	Non-trainable params: 0
	____________________________________________________

Table 9 .

 9 Evaluation metrics

	Model type	MAE	RMSE	MASE	MAPE
	Naïve Model	49.80	69.11	0.3537	14.87 %
	SARIMAX model	14.03	17.85	0.1034	4.24 %
	Random Forest Regressor	14.16	18.36	0.1006	3.89 %
	LSTM model	13.73	18.44	0.0974	4.08 %
	CNN model	14.63	17.87	0.1039	4.27 %

MODELLING FRAMEWORKVarious families of models can be specified to deal with modelling time-series with exogenous data. The experiment platform uses Python 3.7 with scikit-learn[START_REF] Pedregosa | Scikit-learn : Machine Learning in Python[END_REF]

ACKNOWLEDGEMENT

The author would like to thank Transdev for supplying the data used in this study, and in particular Thierry CHIOCCA for his valuable remarks in the analysis of the T-PRM dataset and the inspiration behind this paper.


Julien Dumurgier
email: julien.dumurgier@aphp.fr

Christophe Tzourio

Epidemiology of neurological diseases in older adults

Keywords: Epidemiology, Older adults, Alzheimer's disease, Aging, Neurological diseases

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

Introduction

More than 600 diseases affecting the nervous system, i.e. the brain, the spine or the nerves, have been reported, ranging from benign affections, like the common migraine, to severe life-threatening diseases, such as primitive brain tumor or lateral amyotrophic sclerosis [START_REF] Ropper | Adams and Victor's principles of neurology[END_REF]. Neurological diseases are so frequent in the general population, particularly in elderly individuals, that they pose a massive global health problem. A recent report from the Rotterdam Study estimated than one in two women and one in three men will develop dementia, stroke or parkinsonism during their life [START_REF] Licher | Lifetime risk of common neurological diseases in the elderly population[END_REF]. Worldwide, the Global Burden of Disease Study [START_REF] Gbd | Global, regional, and national burden of neurological disorders, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF] estimated that in 2016, there were more than 80 million stroke survivors in the world [START_REF] Gbd | Global, regional, and national burden of stroke, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF], 43.8 million people with dementia [START_REF] Gbd | Global, regional, and national burden of Alzheimer's disease and other dementias, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF], 45.9 million patients with an active epilepsy [START_REF] Gbd | Epilepsy Collaborators. Global, regional, and national burden of epilepsy, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF], and 6.1 million individuals with Parkinson's disease [START_REF] Gbd | Parkinson's Disease Collaborators. Global, regional, and national burden of Parkinson's disease, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF]. Globally, in 2016, neurological disorders were the leading cause of disability (276 million disability-adjusted life-years) and the second leading cause of deaths (9 million) in the world [START_REF] Gbd | Global, regional, and national burden of neurological disorders, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF].

The etiology and determinants of neurological diseases remain poorly understood, although most of these diseases are considered to be "complex", involving both multiple genetic and environmental risk factors. The concept of "cognitive reserve" or "brain reserve" has been developed to explain individual differences in the ability to cope with the same level of brain neuropathological lesions [START_REF] Stern | Whitepaper: Defining and investigating cognitive reserve, brain reserve, and brain maintenance[END_REF]. Increasing age is by far the most important risk factor for most neurological diseases. The global prevalence of dementia, stroke or Parkinsonism has been multiplied by two or three during the 1990-2016 period [START_REF] Gbd | Global, regional, and national burden of neurological disorders, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF], mainly due to the marked aging of the world's population during this period. Rapid aging of population across the world will lead the number of persons aged 80 years or over to triple, from 143 million in 2019 to 426 million in 2050 [START_REF]World Population Prospects: the 2019[END_REF]. By 2050, 25% of persons living in Europe and Northern America could be aged 65 or over [START_REF]World Population Prospects: the 2019[END_REF]. In this review, we illustrate the role of age in the most common neurological diseases: dementia and cognitive disorders, Parkinson disease and related disorders, stroke, and epilepsy.

Dementia in the elderly

Alzheimer's disease

Alzheimer's disease (AD) is the most common cause of dementia in older adults; it is characterized by lesions including extracellular β-amyloid peptide (Aβ) accumulations, neurofibrillary tangles composed of intra-neuronal abnormally phosphorylated Tau, and neuronal and synaptic losses [START_REF] Hyman | National Institute on Aging-Alzheimer's Association guidelines for the neuropathologic assessment of Alzheimer's disease[END_REF]. A European collaborative study of population-based cohorts found that prevalence of AD increased continuously with age, from 0.6% in the 65 to 69 years age group to 22.2% at the age of 90 years and older [START_REF] Lobo | Prevalence of dementia and major subtypes in Europe: A collaborative study of population-based cohorts. Neurologic Diseases in the Elderly Research Group[END_REF]. Prevalence of AD increases substantially with age in both men and women, approximately doubling every five years between the ages of 50 and 80 years, after which the increase may slow in the oldest age groups [START_REF] Gbd | Global, regional, and national burden of Alzheimer's disease and other dementias, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF]. Because of this very strong age-related incidence and prevalence, the number of persons living with dementia which is currently estimated to be around 44 million in the world [START_REF] Gbd | Global, regional, and national burden of Alzheimer's disease and other dementias, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF] is predicted to increase up to 135 million in 2050 according to the WHO [START_REF]Global action plan on the public health response to dementia 2017 -2025[END_REF].

Development of biomarkers of AD, especially non-invasive positron emission tomography (PET) radioligands, offers new opportunities for studying AD lesions in vivo [START_REF] Koric | Molecular imaging in the diagnosis of Alzheimer's disease and related disorders[END_REF]. A large meta-analysis of older adults free of dementia reported that the prevalence of cerebral amyloid pathology, estimated by amyloid PET imaging or cerebrospinal fluid (CSF) assessment, increased from age 50 to 90 years from 10% to 44% [START_REF] Jansen | Prevalence of cerebral amyloid pathology in persons without dementia: a meta-analysis[END_REF]. Postmortem studies found however that AD neuropathological lesions are common in the elderly [START_REF] Braak | The preclinical phase of the pathological process underlying sporadic Alzheimer's disease[END_REF], even in cognitively normal people [START_REF] Santacruz | Alzheimer disease pathology in subjects without dementia in 2 studies of aging: the Nun Study and the Adult Changes in Thought Study[END_REF]. The association between neuropathological lesions common in AD and the clinical expression of the disease is weaker in the oldest age groups [START_REF] Savva | Age, neuropathology, and dementia[END_REF], and the ability of biomarkers to discriminate between AD and non-AD dementia patients decreases with age [START_REF] Dumurgier | Exacerbated CSF abnormalities in younger patients with Alzheimer's disease[END_REF].

Another important finding is that the effect of some risk factors of AD is modified by age. The APOE ε4 allele, the most important genetic risk factor for late-onset AD, has a less important role after than before the age of 80 [START_REF] Frisoni | The prevalence of apoE-epsilon4 in Alzheimer's disease is age dependent[END_REF]. While mid-life hypertension is a risk factor for AD [START_REF] Abell | Association between systolic blood pressure and dementia in the Whitehall II cohort study: role of age, duration, and threshold used to define hypertension[END_REF], late-life hypertension could be protective against the risk of dementia in the elderly [START_REF] Iadecola | Impact of Hypertension on Cognitive Function: A Scientific Statement From the American Heart Association[END_REF]. These complex patterns suggest that dementia could be the final endpoint of lifetime exposure to several factors, some of them being modifiable. In order to conceive and set up future preventive strategies, we need to have a better understanding of these mechanisms through lifetime observational studies.

Several epidemiological studies have reported a decrease in the incidence of dementia over the course of the last three decades [START_REF] Grasset | Trends in dementia incidence: Evolution over a 10-year period in France[END_REF][START_REF] Satizabal | Incidence of Dementia over Three Decades in the Framingham Heart Study[END_REF]. While the exact mechanisms of this phenomenon remain to be elucidated, better care for cardiovascular risk factors as well as increasing mean level of education in the general population may play a role [START_REF] Chibnik | Trends in the incidence of dementia: design and methods in the Alzheimer Cohorts Consortium[END_REF]. The development of peripheral biomarkers such as plasma amyloid markers for early diagnosis and synaptic markers to determine the progression of the cognitive slope will be useful in the future to detect biological-positive AD prior to onset of cognitive symptoms [START_REF] Olsson | CSF and blood biomarkers for the diagnosis of Alzheimer's disease: a systematic review and meta-analysis[END_REF].

Suspected non-Alzheimer's pathophysiology (SNAP)

The recent development of biomarkers of neurodegeneration -Tau in CSF, fluorodeoxyglucose PET imaging, or volumetric magnetic resonance imaging -has led to the identification of individuals with abnormal levels of neurodegeneration markers but normal levels of Aβ peptide. This condition has been called "suspected non-Alzheimer's pathophysiology" (SNAP) [START_REF] Jack | An operational approach to National Institute on Aging-Alzheimer's Association criteria for preclinical Alzheimer disease[END_REF], and may contribute to identifying neuropathologicaldefined cases of primary age-related tauopathy or PART [START_REF] Jack | PART and SNAP[END_REF]. SNAP appears to be very common in the general population, with a prevalence around 25% in the population older than 65 years; whether it represents a pathological condition remains highly controversial and debated [START_REF] Jack | Suspected non-Alzheimer disease pathophysiology--concept and controversy[END_REF].

Vascular dementia

Vascular dementia (VD) is a clinical pathological condition in which cerebral lesions of vascular origin lead to cognitive impairment. It represents one of the most common causes of dementia, accounting for around 15% of all cases [START_REF] O'brien | Vascular dementia[END_REF][START_REF] Wolters | Epidemiology of Vascular Dementia[END_REF]. VD is an heterogeneous entity with several subtypes corresponding to different vascular mechanisms: small-vessel dementia, multi-infarct dementia, strategic-infarct dementia, hemorrhagic dementia, hypoperfusion dementia [START_REF] O'brien | Vascular dementia[END_REF] . Rates of VD are known to increase with age, with an incidence doubling every class of five years after the age of 60 [START_REF] Jorm | The incidence of dementia: a meta-analysis[END_REF]. Co-existence of brain VD and AD lesions are frequent in patients with mixed dementia [START_REF] Zekry | Vascular dementia and mixed dementia[END_REF], while the Aβ peptide is involved in vascular lesions among patients with cerebral amyloid angiopathy [START_REF] Greenberg | Cerebral amyloid angiopathy and Alzheimer disease -one peptide, two pathways[END_REF].

Limbic-predominant age-related TDP-43 encephalopathy (LATE)

Tar DNA-binding protein 43 (TDP-43) proteinopathy, along with peptide beta-amyloid, protein Tau, and alpha-synuclein, is one of the most common findings in the neuropathological examination of patients with dementia -present in around 15% of these patients [START_REF] Kovacs | Non-Alzheimer neurodegenerative pathologies and their combinations are more frequent than commonly believed in the elderly brain: a community-based autopsy series[END_REF]. TDP-43 is known for being associated with some forms of amyotrophic lateral sclerosis and fronto-temporal dementia [START_REF] Couratier | ALS and frontotemporal dementia belong to a common disease spectrum[END_REF], which are rare pathologies in the general population [START_REF] Berning | The Pathobiology of TDP-43 C-Terminal Fragments in ALS and FTLD[END_REF]. Recently, it has been shown that TDP-43 is commonly observed in limbic brain structures of subjects past 80 years of age. This proteinopathy can be associated with cognitive impairment that mimics AD clinical syndrome. The term LATE (limbic-predominant age-related TDP-43 encephalopathy) has been proposed for this disease [START_REF] Nelson | Limbicpredominant age-related TDP-43 encephalopathy (LATE): consensus working group report[END_REF]. Although robust epidemiological evidence is lacking for this newly described disease, LATE could be present in more than 20% of people after the age of 80, and the overall public health impact of LATE could be on the same order as AD [START_REF] Nelson | Limbicpredominant age-related TDP-43 encephalopathy (LATE): consensus working group report[END_REF].

Lewy body disease

Lewy body disease (LBD) is the second most common cause of dementia at older ages [START_REF] Walker | Lewy body dementias[END_REF]. The proportion of LBD in neuropathological studies ranges from 4% to 24.7% of demented people [START_REF] Jellinger | Prevalence and pathology of dementia with Lewy bodies in the oldest old: a comparison with other dementing disorders[END_REF]. The hallmarks of LBD are diffuse neocortical α-synuclein neuronal inclusions (Lewy bodies, and Lewy neurites) and marked neuronal death [START_REF] Sorrentino | alpha-Synuclein and astrocytes: tracing the pathways from homeostasis to neurodegeneration in Lewy body disease[END_REF]. Revised diagnostic criteria were published in 2017 [START_REF] Mckeith | Diagnosis and management of dementia with Lewy bodies: Fourth consensus report of the DLB Consortium[END_REF]; LBD is thought to be underdiagnosed currently in clinical practice [START_REF] Mok | Clinicopathological concordance of dementia diagnoses by community versus tertiary care clinicians[END_REF][START_REF] Toledo | Clinical and multimodal biomarker correlates of ADNI neuropathological findings[END_REF]. It is proposed that dementia with Lewy body should be diagnosed when dementia occurs before or concurrently with parkinsonism, while the term "Parkinson disease dementia" should be used for dementia occurring in a context of already diagnosed Parkinson disease [START_REF] Mckeith | Diagnosis and management of dementia with Lewy bodies: Fourth consensus report of the DLB Consortium[END_REF]. The epidemiology of LBD remains under-documented. In systematic reviews, the proportion of individuals with LBD has ranged from 0 to 23% in cohorts of patients with dementia [START_REF] Jones | The prevalence and incidence of dementia with Lewy bodies: a systematic review of population and clinical studies[END_REF], and prevalence estimates has ranged from 0.02 to 63.5 per 1000 persons in the general population [START_REF] Hogan | The Prevalence and Incidence of Dementia with Lewy Bodies: a Systematic Review[END_REF]. In the French PAQUID study, incidence of LBD was found at 112 cases/100.000 person-years [START_REF] Perez | A 15-year population-based cohort study of the incidence of Parkinson's disease and dementia with Lewy bodies in an elderly French cohort[END_REF]. A Swedish multicentric study found that patients with LBD represented 15 to 20% of all residents in elderly nursing homes [START_REF] Zahirovic | Prevalence of Dementia With Lewy Body Symptoms: A Cross-Sectional Study in 40 Swedish Nursing Homes[END_REF].

Neuropathological studies reported that α-synuclein deposition is common with age, up to 25% of older adults with normal cognition are alpha-synuclein-positive in one or more brain regions [START_REF] Markesbery | Lewy body pathology in normal elderly subjects[END_REF].

Parkinson's disease and related disorders

Parkinson's disease

Parkinson's disease (PD) is the second most common neurodegenerative disorder after AD [START_REF] Alexoudi | Parkinson's disease pathogenesis, evolution and alternative pathways: A review[END_REF]. Increasing age is the main known risk factor [START_REF] De Lau | Epidemiology of Parkinson's disease[END_REF][START_REF] Lee | Epidemiology of Parkinson Disease[END_REF]; Parkinson's disease affects around 1% of the population over the age of 60 and 5% of the population over 85 years [START_REF] Nussbaum | Alzheimer's disease and Parkinson's disease[END_REF]. A French national study reported, according to the age groups 55-64, 65-74, 75-84 and ≥85 years, that incidence of PD was respectively 33-46, 139-172, 301-363 and 442-560 per 100,000 person-years amongst men and 32-55, 81-117, 203-270 and 251-313 per 100,000 person-years amongst women [START_REF] Blin | Parkinson's disease incidence and prevalence assessment in France using the national healthcare insurance database[END_REF]. Incidence of PD in France has been reported to be 263 per 100 000 person-years [START_REF] Perez | A 15-year population-based cohort study of the incidence of Parkinson's disease and dementia with Lewy bodies in an elderly French cohort[END_REF].

Several studies have shown that age modifies the clinical progression of PD [START_REF] Levy | The relationship of Parkinson disease with aging[END_REF].

Advancing age is associated with faster rate of motor progression [START_REF] Diederich | Parkinson disease with old-age onset: a comparative study with subjects with middle-age onset[END_REF], decreased responsiveness to levodopa [START_REF] Durso | Age influences magnitude but not duration of response to levodopa[END_REF], more severe gait and postural impairment [START_REF] Hely | Age at onset: the major determinant of outcome in Parkinson's disease[END_REF], and more severe cognitive impairment [START_REF] Dubois | Age-induced cognitive disturbances in Parkinson's disease[END_REF]. The risk of dementia increases with duration of PD, reaching up to 50% ten years after PD diagnosis [START_REF] Ch | The CamPaIGN study of Parkinson's disease: 10-year outlook in an incident population-based cohort[END_REF][START_REF] Svenningsson | Cognitive impairment in patients with Parkinson's disease: diagnosis, biomarkers, and treatment[END_REF]. Increasing age, rather than the age of onset, appears to be the main risk factor for developing dementia in patients with PD [START_REF] Aarsland | The effect of age of onset of PD on risk of dementia[END_REF].

Multiple-system atrophy

Multiple-system atrophy (MSA) is characterized by progressive autonomic failure, parkinsonian features, and cerebellar and pyramidal features in various combinations [START_REF] Fanciulli | Multiple-system atrophy[END_REF]. Depending on the predominance of parkinsonism or cerebellar symptoms, patients are further classified into MSA-P or MSA-C subtypes. The hallmark of the disease is the accumulation of aggregated alpha-synuclein in oligodendrocytes, forming glial cytoplasmic inclusions [START_REF] Laurens | Multiple System Atrophy -State of the Art[END_REF]. MSA is a rare disease, with a global incidence estimated to be around one to five cases per 100,000 person-years [START_REF] Bjornsdottir | Incidence and prevalence of multiple system atrophy: a nationwide study in Iceland[END_REF][START_REF] Bower | Incidence of progressive supranuclear palsy and multiple system atrophy in Olmsted County, Minnesota, 1976 to 1990[END_REF][START_REF] Wenning | The natural history of multiple system atrophy: a prospective European cohort study[END_REF]. Disease usually begins in the sixth decade of life and the median survival from symptom onset is approximately 10 years [START_REF] Low | Natural history of multiple system atrophy in the USA: a prospective cohort study[END_REF][START_REF] Poewe | Neurodegenerative disease: Multiple system atrophy-new insight from prospective studies[END_REF]. Clinical diagnosis of MSA remains challenging and it is suggested that many affected patients are not diagnosed [START_REF] Miki | Improving diagnostic accuracy of multiple system atrophy: a clinicopathological study[END_REF]. Recent genome-wide association studies enabled the identification of several potentially interesting gene loci, including the MAPT locus [START_REF] Gu | Analysis of GWAS-linked variants in multiple system atrophy[END_REF][START_REF] Sailer | A genome-wide association study in multiple system atrophy[END_REF].

Stroke

Stroke is defined as a neurological deficit attributed to an acute focal injury of the central nervous system by a vascular cause, including cerebral infarction, intracerebral hemorrhage (ICH), and subarachnoid hemorrhage (SAH) [START_REF] Sacco | An updated definition of stroke for the 21st century: a statement for healthcare professionals from the American Heart Association/American Stroke Association[END_REF]. Of all strokes, 87% are ischemic, 10% are ICH strokes, and 3% are SAH in studies based in the US [START_REF] Benjamin | Heart Disease and Stroke Statistics-2019 Update: A Report From the American Heart Association[END_REF], but the proportions of subtypes vary strongly across populations and ethnic groups [74]. The main causes of ischemic strokes include large-artery atherosclerosis, cardioembolic pathologies including atrial fibrillation, and occlusion of small vessels [START_REF] Adams | Classification of subtype of acute ischemic stroke. Definitions for use in a multicenter clinical trial. TOAST. Trial of Org 10172 in Acute Stroke Treatment[END_REF].

After taking age into account, the incidence of stroke ranges from 90 to 350 cases per 100,000 person-years [START_REF] Gbd | Global, regional, and national burden of stroke, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF]. The highest age-standardized incidences of stroke are observed in east Asia, especially China (354 per 100 000 person-years), followed by eastern Europe, while the lowest incidences are in central Latin America and western Europe [START_REF] Gbd | Global, regional, and national burden of stroke, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF]. The lifetime risk of stroke for a middle-aged person is around 20% [START_REF] Licher | Lifetime risk of common neurological diseases in the elderly population[END_REF][START_REF] Seshadri | Lifetime risk of stroke and dementia: current concepts, and estimates from the Framingham Study[END_REF].

Strokes are the second leading cause of death in the world, accounting for 5.5 million deaths in 2016 [START_REF] Gbd | Global, regional, and national burden of neurological disorders, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF], and the first cause of serious long-term disability, with 116 million disability-adjusted life-years [START_REF] Gbd | Global, regional, and national burden of stroke, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF].

Age has a strong influence on the incidence of stroke [START_REF] Chen | Ischemic stroke in the elderly: an overview of evidence[END_REF]. The incidence doubles with each decade after the age of 45 years, and nearly 75% of all strokes occur above the age of 65 years, and 30% above the age of 85 [START_REF] Kelly-Hayes | Influence of age and health behaviors on stroke risk: lessons from longitudinal studies[END_REF][START_REF] Feigin | Stroke epidemiology: a review of populationbased studies of incidence, prevalence, and case-fatality in the late 20th century[END_REF]. The incidence is low below the age of 45 (10 cases per 100.000 person-years) and continuously increases with age to reach 1% or more after the age of 75 years [START_REF] Benjamin | Heart Disease and Stroke Statistics-2019 Update: A Report From the American Heart Association[END_REF]. The prevalence is around 2% between 40 and 60 years old, and nearly 12% after the age of 80 [START_REF] Benjamin | Heart Disease and Stroke Statistics-2019 Update: A Report From the American Heart Association[END_REF].

The impact of risk factors of stroke is also influenced by age [START_REF] Chen | Ischemic stroke in the elderly: an overview of evidence[END_REF]. The impact of high blood pressure (systolic≥140 mmHg or diastolic≥90 mmHg) seems to be lower with increasing age, although it remains the main risk factor for stroke. The HYVET randomized trial in individuals aged ≥ 80 years has shown a 39% reduction of fatal stroke [START_REF] Beckett | Treatment of hypertension in patients 80 years of age or older[END_REF]. The risk of stroke attributable to atrial fibrillation increases substantially with age, rising from 1.5% for individuals aged between 50 and 60 years to 23.5% for people aged between 80 and 90 years [START_REF] Wolf | Atrial fibrillation as an independent risk factor for stroke: the Framingham Study[END_REF]. Physical inactivity has also been found to be a risk factor for stroke, but only in people older than 80 years [START_REF] Willey | Physical inactivity is a strong risk factor for stroke in the oldest old: Findings from a multi-ethnic population (the Northern Manhattan Study)[END_REF]. Older patients with stroke have more severe initial deficits compared to younger patients [START_REF] Kammersgaard | Short-and longterm prognosis for very old stroke patients. The Copenhagen Stroke Study[END_REF], and they recover more slowly [START_REF] Marini | Burden of first-ever ischemic stroke in the oldest old: evidence from a population-based study[END_REF]. Furthermore, age greater than 75 years is associated with higher immediate mortality after stroke [START_REF] Carter | Predictive variables for mortality after acute ischemic stroke[END_REF].

Epilepsy

Epilepsy is a chronic disorder of the brain characterized by recurrent episodic epileptic seizures. Seizures are the clinical manifestations of a process involving networks of neurons with focal or diffuse distribution. Epilepsy has a bimodal distribution with peaks in the youngest individuals and in older adults, with the higher prevalence in the latter group than in any age [START_REF] Beghi | Aging and the Epidemiology of Epilepsy[END_REF][START_REF] Brodie | Epilepsy in elderly people[END_REF]. Prevalence of epilepsy is approximately 0.5% in younger adults versus 1.5% in community-dwelling older adults [START_REF] Leppik | Epilepsy in the elderly[END_REF]. In the elderly nursing home population, the prevalence of epilepsy can reach 7% of residents [START_REF] Birnbaum | Prevalence of epilepsy/seizures as a comorbidity of neurologic disorders in nursing homes[END_REF]. The annual incidence of epilepsy rises from 90 per 100,000 in people between the ages of 65 and 69 to more than 150 per 100,000 in those over 80 years [START_REF] Brodie | Epilepsy in elderly people[END_REF].

Causes and presentation of epilepsy differ with age. At older ages the presentation of epilepsy is often less specific, and misdiagnosis is frequent [START_REF] Kwan | Clinical trials of antiepileptic medications in newly diagnosed patients with epilepsy[END_REF]. Focal seizures are more common than generalized seizures [START_REF] Brodie | Epilepsy in elderly people[END_REF]. Post-seizure confusion is more prolonged in older adults, and can sometimes last several days [START_REF] Acharya | Epilepsy in the elderly: Special considerations and challenges[END_REF]. The majority of epileptic older patients have symptomatic epilepsy, and the most frequent etiology appears to be sequelae of stroke and cerebrovascular diseases [START_REF] Quirins | Epilepsy after stroke: Definitions, problems and a practical approach for clinicians[END_REF]. From 30% to 40% of epileptic seizure cases in older adults occur in persons with a history of stoke [START_REF] Leppik | Epilepsy in the elderly[END_REF]. AD and other neurodegenerative diseases are also common causes of epilepsy in older patients [START_REF] Friedman | Seizures and epilepsy in Alzheimer's disease[END_REF][START_REF] Cretin | Do we know how to diagnose epilepsy early in Alzheimer's disease?[END_REF]. Other common causes include brain tumors and sequelae of trauma [START_REF] Beghi | Aging and the Epidemiology of Epilepsy[END_REF]. Hippocampal sclerosis, characterized by severe volume loss of the hippocampus, severe neuronal loss, and reactive gliosis, is also a frequent cause of epilepsy in elderly people, in the presence or not of AD [START_REF] Dutra | Update on Hippocampal Sclerosis[END_REF].

Co-pathologies

While neurological diseases are reported and described separately, co-pathologies are frequent, especially among elderly people. Large-scale neuropathological studies have found that around 50% of dementia cases exhibited pathological lesions associated with more than one neurodegenerative disease [START_REF] Kovacs | Mixed brain pathologies in dementia: the BrainNet Europe consortium experience[END_REF][START_REF] Rahimi | Prevalence of mixed pathologies in the aging brain[END_REF], with the frequent co-existence of neuropathological lesions of AD, VD, argyrophilic grain dementia, or LBD. Co-existence of AD and LBD lesions [START_REF] Leverenz | Lewy body pathology in familial Alzheimer disease: evidence for disease-and mutation-specific pathologic phenotype[END_REF], or AD and vascular lesions [START_REF] Marchant | The aging brain and cognition: contribution of vascular injury and abeta to mild cognitive dysfunction[END_REF] appears specially frequent, and the presence of such mixed pathologies increases with age [START_REF] Jellinger | Prevalence of dementia disorders in the oldest-old: an autopsy study[END_REF].

Conclusion

The prevalence of most of disabling neurological disorders increases sharply with age.

At the origin of 9 million of deaths and 276 million disability-adjusted life-years every year in the world [START_REF] Gbd | Global, regional, and national burden of neurological disorders, 1990-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF], neurological disorders are the leading cause of disability and the second leading cause of mortality in the world. As populations age, especially in lowincome countries, our societies will face increasing challenges for supporting treatments, rehabilitations, and the consequences of these neurological disorders. Large scale epidemiological studies can contribute to a better understanding of the particularities of neurological diseases at older ages and help in designing future preventive strategies. In particularly, as the number of nonagenarians and centenarians continues to increase in the population, epidemiological studies are required to better identify the particularities of neurological diseases in this population. Collaboration and dialogue between geriatrics and neurologic teams is needed more than ever in order to optimize the care of patients with neurological diseases.

Disclosure of interest

The authors declare that they have no competing interest.