Sasan Amini
email: sasan.amini@tum.de

Mostafa Ameli
email: mostafa.ameli@univ-eiffel.fr

Benjamin Winzenrieth
email: benjamin.winzenrieth@developpement-durable.gouv.fr

Fritz Busch
email: fritz.busch@tum.de

Klaus Bogenberger
email: klaus.bogenberger@tum.de

Ludovic Leclercq
email: ludovic.leclercq@univ-eiffel.fr

A BRAESS PARADOX EXPLOITATION FRAMEWORK FOR OPTIMAL TRAFFIC BANNING STRATEGY

Keywords: Network design problem, Braess paradox, User equilibrium, System optimum, Optimal banning

The transportation operators seek for control and management strategies to push the system towards the system optimum in spite of selfish drivers who make decisions in a user optimal manner. Motivated by the Braess paradox, we propose a framework for developing optimal traffic banning strategies to improve the total travel time (TTT) of a network. More specifically, our goal is to identify the links whose closure cause the Braess paradox. This problem belongs to the family of discrete network design problem (DNDP), often formulated as a bi-level mixed-integer program.

While the literature offers several algorithms to solve the mathematical problem, additional efforts are required to provide a solution method that guarantees global optimality in a reasonable computation time for real-world urban networks. We propose a heuristic optimization method based on link filtering to reduce the solution space for an exact method and evaluate its effectiveness on two sub-areas of the Chicago-Sketch network. We have designed two experiments to validate our approach and to evaluate the trade-off between computation time and solution optimality. Our results show that the proposed framework converges to the same solution as the exact method in most cases. Also, under a fixed computation time budget, the exact method could analyze only a tiny fraction of all feasible solutions, while our method not only converged significantly fast but also provided the same solution as the exact method. Lastly, we show that by implementing an optimal traffic banning strategy, the system performance can be improved up to 40%.

INTRODUCTION

The transportation network design problem (NDP) concerns with optimal decision-making on modifying a network under cost constraints. The objective function of the NDP is to optimize a system performance measure, e.g., minimizing the total travel time (TTT), while accounting for the route choice behavior of drivers. The route choice behavior by itself is an optimization problem known as traffic assignment problem (TAP), which is solved to optimize one or more objectives.

In road networks, fully rational drivers choose their path in a way to minimize their own travel cost.This principle is called user equilibrium (UE) (Wardrop,[START_REF] Wardrop | Some theoritical aspects of road traffic research[END_REF]. The network under the UE is a situation in which no driver can change her path to improve her travel cost. By definition, the UE does not tend to optimize the network performance compared to a situation wherein drivers cooperate with each other in order to reach a social optimum condition (also known as the system optimum (SO) principle). The ratio of TTT between these two principles is known as the "price of anarchy" (PoA) (Koutsoupias and Papadimitriou, 2). Clearly, the ultimate goal of transportation operators is to keep the value of PoA close to 1. This fact has motivated many studies to propose traffic management measures and strategies to ensure the selfish behavior of drivers results in the SO, i.e., decreasing the PoA, while the drivers follow the UE principle (Aashtiani and Poorzahedy, Ma et al., Yao et al.,[START_REF] Aashtiani | Braess' phenomenon in the management of networks and dissociation of equilibrium concepts[END_REF][START_REF] Ma | Link Restriction: Methods of Testing and Avoiding Braess Paradox in Networks Considering Traffic Demands[END_REF][START_REF] Yao | Paradox links can improve system efficiency : An illustration in traffic assignment problem[END_REF].

To achieve this goal, in this paper, we employ a rather counter-intuitive but well-known phenomenon in transportation referred to as the Braess paradox (Braess,[START_REF] Braess | Über ein Paradoxon aus der Verkehrsplanung[END_REF]. In his remarkable work, Braess demonstrated that adding a new link to a network degrades its performance and increases the TTT. There is a large number of theoretical and empirical implications of this paradox, see e.g. ( Bagloee et al.,[START_REF] Aashtiani | Braess' phenomenon in the management of networks and dissociation of equilibrium concepts[END_REF][START_REF] Yao | Paradox links can improve system efficiency : An illustration in traffic assignment problem[END_REF][START_REF] Pas | Braess' paradox: Some new insights[END_REF][START_REF] Roughgarden | On the severity of Braess's Paradox: Designing networks for selfish users is hard[END_REF][START_REF] Gisches | Degrading network capacity may improve performance: Private versus public monitoring in the Braess Paradox[END_REF][START_REF] Ameli | Simulation Modelling Practice and Theory Improving traffic network performance with road banning strategy : A simulation approach comparing user equilibrium and system optimum[END_REF][START_REF] Colini-Baldeschi | When is Selfish Routing Bad ? The Price of Anarchy in Light and Heavy Traffic[END_REF][START_REF] Bagloee | Is it time to go for no-car zone policies ? Braess Paradox Detection[END_REF]). In this study, we aim to answer the following question: in a given network with a given Origin-Destination (OD) matrix, which roads should be closed to obtain the minimum TTT under the UE principle. In other words, the goal is to find links whose closure cause the Braess paradox. Within the scope of this study, we rely on static traffic assignment models. Despite their limitations in representing traffic dynamics, they have proved useful particularly for design and management of road networks with the merit of simplicity and computational efficiency Wang et al. [START_REF] Wang | Dynamic traffic assignment: A review of the methodological advances for environmentally sustainable road transportation applications[END_REF].

In our problem, the decision variable is binary, i.e., closing a link or not. This problem can be categorized as discrete NDP (DNDP). It is a mixed-integer problem by taking into account the route choice behavior of drivers under the UE principle. This problem is often formulated via bi-level programming (Yang and Bell,[START_REF] Yang | Models and algorithms for road network design: a review and some new developments[END_REF]. Due to their intrinsic complexity, bi-level models (particularly when a discrete decision variable exists in either the upper or the lower level problem) are considered as one of the most challenging global optimality problems in transportation (Ben-ayed and Blair, Farvaresh and Sepehri,[START_REF] Ben-Ayed | Computational Difficulties of Bilevel Linear Programming[END_REF][START_REF] Farvaresh | A single-level mixed integer linear formulation for a bi-level discrete network design problem[END_REF]. Nevertheless, several methods have been proposed to solve it that can be categorized as exact or heuristics. The exact methods aim at finding the global optimum. For example, Leblanc (17) proposed a branch and bound algorithm, which he solved by a sequence of shortest path problems to improve computation time. Farvaresh and Sepehri (16) revised the branch and bound algorithm and added a Steiner minimal tree inequality to reduce the computation time. In a more recent study, Bagloee et al. [START_REF] Bagloee | A Hybrid Branch-and-Bound and Benders Decomposition Algorithm for the Network Design Problem[END_REF] employed a generalized Benders' decomposition to speed up the branch and bound algorithm. Using the Karush-Kuhn-Tucker (KKT) conditions to solve the lower-level problem is another approach that has been used extensively, see for example (Farvaresh and Sepehri,Fontaine and Minner,[START_REF] Farvaresh | A single-level mixed integer linear formulation for a bi-level discrete network design problem[END_REF][START_REF] Fontaine | Benders Decomposition for Discrete-Continuous Linear Bilevel Problems with application to traffic network design[END_REF]. The application of exact methods to real-world networks is restricted (Rey,[START_REF] Rey | Computational benchmarking of exact methods for the bilevel discrete network design problem[END_REF]. Therefore, many researchers have devel-oped heuristic methods to find nearly optimal solutions for medium-size networks in a reasonable duration of time by relaxing some of the properties of the problem. For instance, Friesz et al. [START_REF] Friesz | Simulated annealing approach to the network design problem with variational inequality constraints[END_REF] proposed simulated annealing and Poorzahedy and Abulghasemi, Poorzahedy and Rouhani [START_REF] Poorzahedy | Application of Ant System to network design problem[END_REF][START_REF] Poorzahedy | Hybrid meta-heuristic algorithms for solving network design problem[END_REF] developed a meta-heuristic approach based on ant colony systems. The main challenge these algorithms face is that they may converge to a local optimum solution that is only not worse than a solution in which the Braess paradox occurs. Rey [START_REF] Rey | Computational benchmarking of exact methods for the bilevel discrete network design problem[END_REF] has conducted a comprehensive comparison of the computation time for some of these methods. He argues that DNDP is NP-hard and even medium-size instances of this problem are computationally challenging. Thus, there is a need for developing stable and fast exact methodologies. In this paper, we propose a framework to address this need.

Traditionally, the NDP in transportation involves selecting the best solution from a limited number of possible expansion projects for an existing network under budget constraints. In our case, we are interested in the inverse problem in which some links must be closed. In the literature, only very few studies have formulated the NDP in this way. Bagloee et al. [START_REF] Bagloee | Is it time to go for no-car zone policies ? Braess Paradox Detection[END_REF] proposed a surrogate-based algorithm for identifying the links that need to be closed in a network. The authors have used supervised machine learning to train a multivariate linear regression model that treats the original objective function, TTT, as a black box. While such an approach may provide optimal results, it lacks in-depth explanatory power to be accepted as a generic solution. Ma et al. ( 4) have suggested link restrictions as a traffic management measure to cope with the Braess paradox under various demand scenarios. They have adapted the SO-relaxation method of Wang et al. [START_REF] Wang | Global optimization methods for the discrete network design problem[END_REF] to find links that cause Braess Paradox. However, their method cannot be applied to real-world networks because of the extensive computation time required by the SO-relaxation method. Moreover, they have used the interdiction constraint from the original method, which imposes a minimum number of two changes between the possible network designs, whereas the intended solutions should seek for only one change. Later in section 3 of this paper, we put this issue under scrutiny. Wu and Zhu (25) have also looked into the features of the links that may cause the Braess paradox. By running series of experiments, they have concluded that the minimum saturation rate is a notable common feature among such links. Sun et al. [START_REF] Sun | Locating inefficient links in a largescale transportation network[END_REF] have come to a very similar conclusion by studying the network of San Francisco and show that low value of volume over capacity is an indicator for a link to be inefficient in the network, and its removal could have positive impacts on the TTT. Ameli et al. [START_REF] Ameli | Simulation Modelling Practice and Theory Improving traffic network performance with road banning strategy : A simulation approach comparing user equilibrium and system optimum[END_REF] have developed a simulation-based framework to compare OD path flows under different principles, including UE and SO. Furthermore, by introducing the concept of demand breakpoints at which routing decisions are significantly changed, the authors propose a method that could detect the Braess paradox and close these roads to improve the network performance.

Our main goal in this paper is to extract the best subnetwork out of a given network that has the minimum TTT under the UE principle compared to the original network. Mathematically speaking, this problem can also be formulated as a 0-1 Knapsack optimization problem. The family of Knapsack problems considers a set of items, each having an associated profit and weight. The problem is then to select a subset of items such that the profit is maximized without exceeding the capacity of the knapsack Pisinger et al. [START_REF] Pisinger | Knapsack Problems[END_REF]. In 0-1 Knapsack problems, each item may be selected at most once. Here, items are the set of all built traffic network roads (links), and the Knapsack is a subset of links to be banned. The Knapsack has no capacity limit. The optimal sub-network obtained from banning the links belong to the most profitable Knapsack should serve the demand (i.e., keep all ODs connected), while providing minimum TTT under the UE conditions. Note that the number of candidates (binary decision variables) in this problem is equal to the number of links in the original network. Thus, regardless of the complexity of the constraints, the problem becomes computationally expensive by increasing the number of links. To overcome this curse of dimensionality, we propose a framework to identify a set of candidate links, which has the most potential to be banned and contribute to the problem. In other words, we aim to decrease the number of items in the 0-1 Knapsack problem in order to solve it faster.

In this study, we propose a filtering methodology to reduce the cardinality of the set of links whose closure cause the Braess paradox. This enables us to solve the optimal banning strategy problem within a reasonable computation time for even medium-size real-world networks. To the best of our knowledge, such an approach to solving the NDP has not been proposed in the literature.

First, we carry out a feature selection procedure to identify relevant link features. Second, we build a framework to analyze links features evolution under the UE and SO principles to highlight and extract the potential candidates. Third, we solve the problem with an exact method for the potential candidates. The application of the proposed framework to a real network shows that by considering a small subset of links (less than 5% of the total number of links) as potential candidates, the computation time decreases dramatically. Finally, we compare our solution with the global optimum solution obtained by the exact method without any filtering. The results show that the impact of the roads that are not detected by our method on the system performance is neglectable in comparison to the global optimum solution found by the exact method. Besides the proposition of a new methodology, we review the most advanced exact method in the literature and rectify an error in the solution method, which is used in some studies.

The remainder of this paper is structured as follows: in the next section we first review the fundamentals of the DNDP and its bi-level programming formulation and then present our solution. In section case study, we introduce our study network in which we evaluate the efficiency of our methodology and compare it with an exact solution approach. In section results, we present our results from the case study and discuss our findings, followed by some concluding remarks in the last section.

METHODOLOGICAL FRAMEWORK

In this section, we first formulate the original problem as a leader-follower game. This game can be formulated as a bi-level programming model. The upper-level problem is to find the best subnetwork in which the difference between the sub-network TTT under the UE assignment and the original network TTT under the SO assignment is minimized. We formulate this problem as a 0-1 Knapsack problem. The lower-level problem deals with solving the TAP under the UE. Before we continue with the problem formulation, we provide an overview of the notations used in this paper in table 1.

Problem formulation

Let us denote a given transportation network with a directed graph G = (V, E) where V and E are the sets of nodes and links, respectively. We define the 0-1 Knapsack problem as follows: for each link (i, j) ∈ E we define an associated weight and profit. The objective is to choose a subset of links from the set of all links E to maximize the profit without exceeding the capacity i.e. the number of links in the network |E|.

For the sake of simplicity, in this paper, we assume the weight of all links are equal to one.

The difficulty of the solution lies in the calculation of the value of the profit for each link. One main difference of our 0-1 Knapsack problem with the classic one is that the value of profit of each item Link from node i to node j; i, j ∈ V ; (i, j) ∈ E. rs OD pair of a trip from node r to node s; r ∈ V ; s ∈ D. u i j Binary decision variable for the link from node i to j to indicate whether it is opened or not. ūi j Binary decision variable for the link from node i to j to indicate whether it is banned or not.

t i j

Travel time function of link (i,j).

t 0 i j

Free flow travel time of link (i,j). q rs

Traffic demand between origin r and destination s.

Q The OD matrix, Q = [q rs ], ∀r, s ∈ V . x i j
Traffic flow on link (i,j).

x SO i j
Traffic flow on link (i,j) under the SO principle.

x UE i j Traffic flow on link (i,j) under the UE principle.

x i j,s Traffic flow on link (i,j) travel to destination node s.

Ω u

Set of all feasible network designs.

Ωu

Set of all explored network designs.

Ω x

Set of all feasible traffic flows.

Ω f

Set of all feasible path flows.

M

An arbitrarily large number.

λ θ i j
Value of feature θ for link (i, j) ∈ Λ. ξ i j

Normalized difference between traffic flow on link (i,j) under the UE and SO principle.

Θ

Vector of all considered link features.

Z(x)

Objective function of the original problem.

Z RP (x)

Objective function of the relaxed problem. f rs k Traffic flow on path k between OD pair (r, s).

δ rs i j,k
Binary variable, 1 if link (i, j) is part of path k between r and s, 0 otherwise. solution RP Set of tuple of banned links and links flows obtained form the solution of the relaxed problem. varies depending on the other selected items. More specifically, because of the interdependencies among the links of a network, the value of profit for banning a link depends on which other links in the network are also banned. In our problem, the profit is defined as how much the TTT in the solution subnetwork under the UE gets closer to the TTT under the SO of the original network. We can obtain this profit by solving the lower-level problem which is a classic UE traffic assignment problem.

Let D be the set of destination nodes and Q the OD matrix for this network with elements q rs that show the demand travelling from node r ∈ V to destination s ∈ D ⊆ V . Let x i j and t i j represent the traffic flow and travel time on link (i, j) ∈ E where t i j (.) represents the volume-delay function, which is usually modelled as a strictly increasing and convex function of x i j to make sure the traffic flow on each link under UE has a unique solution (Ameli et al.,[START_REF] Ameli | Day-to-day multimodal dynamic traffic assignment : Impacts of the learning process in case of non-unique solutions[END_REF]. In this paper, we use the well-known BPR function:

t i j (x i j ) = t 0 i j (1 + β i j ( x i j c i j ) η i j ) (1)
where t 0 i j and c i j are the free-flow travel time and capacity of link (i, j), respectively. β and η are the model parameters. Let us denote a feasible solution subnetwork with Ḡ = (V, Ē). We can then rewrite the classical profit maximization Knapsack problem as the following minimization problem:

min Z(x) = ∑ (i, j)∈ Ē x i j t i j ((x i j )) -∑ (i, j)∈E xi j t i j ( xi j ) (2)
where x i j is obtained from the UE assignment on the designed subnetwrok and xi j from the SO assignment on the original network. Ē ⊆ E is the set of all links that remain open in the designed subnetwork i.e. Ē := {(i, j) | u i j = 1; ∀(i, j) ∈ E}. u i j is the binary decision variable indicating if

link (i, j) is open or not.
Since demand is given and fixed, the second term in the objective function ( 2) is positive and constant. Thus, the objective function becomes simply to minimize the first term. We can then re-write the upper-level program as follows:

arg min x ∑ (i, j)∈ Ē x i j t i j (x i j ) (3)
subject to:

t i j (x i j ) = t 0 i j (1 + β i j (( x i j c i j ) η i j ) (4)
u i j ∈ {0, 1}; ∀(i, j) ∈ E (5)
where x i j is the solution of the lower-level (UE) problem (Beckmann et al.,[START_REF] Beckmann | Studies in the Economics of Transportation[END_REF]:

arg min x ∑ (i, j)∈ Ē x i j 0 t i j (ω)dω (6)
subject to:

∑ j∈V :(i, j)∈E x i j,s - ∑ j∈V :( j,i)∈E x ji,s = q is ∀i ∈ V, ∀s ∈ D (7) ∑ s∈D x i j,s = x i j ; ∀(i, j) ∈ E (8)
q ss = -∑ i∈V q is , ∀s ∈ D (9)
x i j,s ≥ 0; ∀(i, j) ∈ E [START_REF] Ameli | Simulation Modelling Practice and Theory Improving traffic network performance with road banning strategy : A simulation approach comparing user equilibrium and system optimum[END_REF] where constraints ( 7) and ( 9) assure flow conservation and that the demands of all destinations will be served. Besides, they also balance the inflow and the outflow in each node. Constraint ( 8) For that, we cut the link flow x i j into m i j segments between values α i j,n-1 and α i j,n (n ∈ [1, m i j ]). The goal is to find α L i j,n and α R i j,n such that:

x i j = m i j ∑ n=1 α i j,n λ L + α i j,n λ R (11)
Figure 1 provides an illustration of the linear approximation of the BPR function. Moreover, we need to introduce an arbitrary large number M in the function to ensure the travel time on closed links is high enough that no traffic will be assigned to them. As a result, we rewrite the travel time function t i j (.) as follows:

t i j (x i j ) = t 0 i j [1 + β i j c α i j i j m i j ∑ n=1 (α i j,n λ L + α i j,n λ R )] + (1 -u i j )M (12)
The exact solution method: SO-relaxation approach

Solving the lower-level program is relatively straightforward because the link travel times and the constraints are linear and convex. Therefore, the solution procedure is to enumerate the finite set of all feasible subnetworks Ω u to find the optimal solution u * . However, practically, this approach is computationally prohibitive as the cardinality of Ω u increases exponentially. In order to rectify this problem, the SO-relaxation approach was proposed by Wang et al. [START_REF] Wang | Global optimization methods for the discrete network design problem[END_REF]. The idea is to use the SO principle as a good approximate of the optimal solution for the UE principle and sort all the solutions successively in the order of increasing TTT. Optimal solution is guaranteed when the TTT for the unexplored solutions under the SO in not smaller than an already explored solution under the UE principle. The authors have used this methodology for adding new lanes and links to a network under budget constraints.We adapt their exact method and use it as the reference solution for our problem to develop an optima traffic banning strategy. The principle of the SO-relaxation method is described below.

In the first step, a relaxed problem (RP) is defined instead of the original problem in which the UE assignment constraint is replaced with the SO assignment. This single-level optimization problem is much easier to solve and yields a lower bound on the optimum of the original problem.

Such a lower bound can serve as a starting point of an iterative process. More specifically, all solutions u ∈ Ω u are sorted in the order of increasing based on their SO travel time. Then, from the top of that list, the corresponding UE traffic assignment is solved. The travel time under the UE assignment of each iteration serves as an upper bound for the next possible solution. This iterative process is repeated until the resulting SO travel time of a solution (lower bound) is larger than the UE travel time of the previous solution (upper bound). Wang et al. [START_REF] Wang | Global optimization methods for the discrete network design problem[END_REF] proved that the final solution derived from this procedure is the global optimum. In other words, there is no need to explore any of the remaining feasible solutions because the UE-based travel time of all the remaining unexplored designs cannot be less than that SO-based travel time. We formulate the objective function of the relaxed problem referred to as Z RP as follows:

arg min

u∈Ω u ,x∈Ω x Z RP = ∑ (i, j)∈E x i j t i j (x i j ) (13)
where Ω x and Ω f denote the set of link and path flows, defined by:

Ω x := {x | x i j = ∑ r ∑ s ∑ k f rs k δ rs i j,k ; ∀(i, j) ∈ E, ∀ f rs k ∈ Ω f } (14)
Ω f := { f | ∑ k f rs k = q rs ; ∀r, s ∈ V }. (15)
Note that the constraints of the UE assignment are already incorporated in the domain x ∈ Ω x and u ∈ Ω u . While iterating this process, an interdiction constraint is imposed in order to avoid exploring an already tested solution in the process:

∑ (i, j)∈E [(1 -u i j ) ūi j + (1 -ūi j )u i j ] ≥ 1, ∀ ū ∈ Ωu ( 16
)
where Ωu is the set of already explored solutions. The pseudo-code of this iterative process is presented in Algorithm 1.

It is worth to mention that in Wang et al. [START_REF] Wang | Global optimization methods for the discrete network design problem[END_REF] the right side of the inequality in the constraint ( 16) is set to 2 instead of 1. This implies that the current network that is being explored must have at least two differences with all of the already explored networks, which is not intended by the proposed algorithm. Apparently, this error roots in the numerical example provided wherein a hypothetical scenario, the addition of two potential links to the network is being investigated and the constraint of the example is used for the general scenario as well. Some of the existing literature such as (Ma et al., Rey, 4, 20) have used the same constraint which implies that their solutions are not guaranteed to be global optimum .

Proposed solution method: identifying candidate links

Despite the significant reduction in computation time, the SO-relaxation method is still impractical for large-scale networks. Therefore, we propose a heuristic approach to further reduce the size of the solution space that the SO-relaxation algorithm must explore. We postulate that the optimal or nearly optimal solution is found by considering only a small fraction of the links in a network. In our approach, the problem formulation and the solution algorithm remain the same as before.

The main difference is that we provide a list of candidate links that are more likely to exhibit the Braess paradox and reduce the solution space to these candidate links only. In order to identify such candidate links, we use some of the key features of the links in the network. The goal is to significantly decrease the computation time to make the solution method practical, while the final solution remains very close to that of the global optimum.

In order to identify which features are important for our problem, like any other learning process, we need to collect a reasonable number of Braess occurrences. For a given network and demand profile, we do this in an iterative process as depicted in the flowchart of Figure [START_REF] Koutsoupias | Worst-case Equilibria[END_REF]. We scale the given demand by a global factor at each iteration step and solve the SO and UE traffic assignments. Afterwards, we run algorithm 1 to identify the set of links that cause the Braess paradox, i.e., E -Ē. At this point, we can calculate the value of each feature. In this paper, we refer to each feature by θ and denote its value for link (i, j) with λ θ i j . We define λ θ i j as the relative ranking position of link (i, j) after sorting in the order of increasing all links in the network based on the value of θ . The value of each feature for link (i, j) as λ θ i j , ∀θ ∈ Θ. We define the value of θ as the relative ranking position of link (i, j) after sorting in the order of increasing all links in the network based on the value of θ . In order to decide which features are important, we compare the summation of λ θ with a predefined threshold f . Only features that are above this threshold are considered for the link filtering, which is the last step in our procedure. Here, we set a cut-off score, i.e., p th percentile for each of the important features and filter links that are only above this score.

Consequently, in our method, the solution space, i.e., E in objective function [START_REF] Wang | Dynamic traffic assignment: A review of the methodological advances for environmentally sustainable road transportation applications[END_REF], is replaced with Λ defined as:

Λ := {(i, j) | λ θ i j ≥ P 10% ∀θ ∈ Θ} (17)

CASE STUDY

In this section, we present the toy networks that we considered in order to evaluate the efficiency of our method. There are several transportation networks with demand provided by Stabler et al. [START_REF] Stabler | Transportation Networks for Research[END_REF] that have been extensively used in transportation studies Amini et al., Tilg et al. (31,[START_REF] Tilg | Simulation-based design of urban bi-modal transport systems[END_REF]. From this repository, we consider two medium-size areas in the network of the Chicago-Sketch that allow us to conduct our benchmark experiments. Because, as mentioned before, calculation of the SO-relaxation is still very expensive on these networks but feasible by normal computer resource.

Therefore, we can have a reference point (the global optimum solution) in order to evaluate the quality of our solution. The two considered areas of the Chicago-Sketch are depicted in Figure 3 highlighted by the red (hereafter called network 1) and the green (hereafter called network 2)

rectangles. Some general information regarding the size of each of these areas is provided in table 2. We perform two types of experiments on these two networks. But before explaining the details of these experiments, we first need to discuss the settings of our feature selection and link filtering prcedure.

Settings of the feature selection procedure

As mentioned in the methodology, we design a feature selection process intending to find the most relevant link features to our problem. After identifying the feature(s), we apply the filtering procedure and pre-select a set of links. These selected links make the set of candidate links that we believe it contains most likely links belongs to the final optimum solution based on the learned feature(s). In this part, we specify the details of that general procedure for the current case study.

First, we select network 1 and the provided demand as the inputs. The next step is to define the initial set of features Θ that include free flow travel time (fftt), capacity, flow under the UE (x UE i j )

and the SO assignment (x SO i j ) and their difference as well as an indicator that reflects the centrality of a link -by counting the number of paths that a certain link is part of them. Please note that this set can be extended by considering different aspects. Then, we basically run the algorithm of flowchart (2) until we collect 50 occurrences of Braess paradox. Please note that, 50 refers to the number of links and not the number of iterations. Lastly, we set the feature value threshold and the cut-off score, which depending on computation resources and network size can be adjusted, to 75% and 10 th percentile, respectively.

Experiments setup

We conduct two different experiments on both networks. In the first experiment, the exact method is executed to obtain the global optimum solution, as the reference, in a number of different demand scenarios. These demand scenarios are derived by scaling up the demand by a global factor as provided in table 2. The number of demand scenarios and the factor for which we observed the Braess paradox are different for each network. This is due to the fact that the Braess paradox occurs only when demand lies within an intermediate range of values (Pas and Principio, 7). Once the results from the exact method are ready, we re-run the SO-relaxation method on the set of candidate links, obtained from the previous step. Finally, we compare the optimality of the obtained solution with the reference solution. In this experiment, we investigate the effectiveness of our filtering method in terms of providing a solution with an objective value, TTT, closed to the exact method.

Basically, the exact method has to explore a large number of feasible solutions (maximum the total number of possibilities (2 |E| ) to evaluate different traffic banning strategies for each scenario). Note that the number of evaluations cannot be predicted until the algorithm converges to the optimum solution; however, typically, on large real-world networks, this could take up to weeks or months if a solution exists. For the second experiment, the performance of the proposed solution method is evaluated in front of the the exact method. We impose a computation time limit of three hours and compare the solution found by each method. The motivation for this experiment is to account for some reasonable computation time that the transportation planner may consider while making a decision which algorithm to implement. With this experiment, we aim to investigate the trade-of between computation time and the accuracy of the solution methods.

RESULTS

In this section, we first present the results of our feature selection procedure that is used to identify the relevant features for link filtering. Then, we provide the results of the first experiment in which we show the validation of the proposed methodology. After that, we discuss the second experiment and evaluate the performance of both solution methods under computation time budget. Lastly, we discuss the effectiveness of traffic banning strategy as a traffic management measure to improve the overall system performance.

Analysis of the link features

After a careful analysis of the 50 (link) occurrences of the Braess paradox, we found only one feature to be relevant among all the selected features. In Figure 4a, we present the value of each feature for every link in the solution of the exact method i.e. links taht exhibited the Braess paradox.

We can easily see, in the figure, that only two features seem to be relevant. By calculating the λ θ for these two features it becomes clear, as shown in Figure 4b, that only one feature namely the flow difference between the UE and SO (x UE i jx SO i j ) should be considered for link filtering. Note that this feature is significantly correlated with PoA; however, we are going to investigate whether all links in the optimal banning solution always belong to the top candidate links ranked by this feature or not. Note that, according to the characteristics of a network, banning a link with a high value of x UE i jx SO i j does not necessarily improve the system. It can even provide an infeasible sub-network, e.g., disconnecting some OD pairs. Therefore, we consider this feature for further investigation and validation in the next subsections through the filtering method. Regarding the second-best feature, intuitively, travel time on a link depends on the link travel time function t i j (.)

in which the fftt is only one parameter. In our study, other parameters of this function are identical for all links (η and β in BPR function) which can be another reason that why the fftt is not a representative feature for the filtering procedure. In order to be able to compare the value of x UE i jx SO i j in different scenarios and links with different levels of congestion, we normalize its value by dividing it by the x UE i j . Consequently, we consider a single feature ξ i j =

x UE i j -x SO i j

x UE i j in our link filtering procedure. In fact, ξ i j shows the degree to which removal of a link brings the system closer to the SO principle. ξ i j = 1 implies that link (i, j) is used under the UE principle but not under the SO.

Moreover, we plot the value of λ for each network-demand scenario as presented in table the boxplot (the red dots in the scatter plot), which confirms our previous conclusion on the feature selection process. In Figure 5b we observe some red dots i.e. links that are in the solution of the exact method in the bottom of the boxplot which means they have a very low value for the selected feature. This motivates us to evaluate the contribution of each link to the objective value of the optimal solution. In other words, how much these links further reduce the TTT compared to our solution.

The first experiment: Validation

In order to prove the effectiveness of the selected feature, we compare the results of the obtained solution by both methods in the first experiment. Each solution method will return a set of links that must be closed to the traffic. While the exact method requires several days to converge, our method needs only a few minutes to deliver the results. Figure 6 show the results for network 1 and network 2 respectively. The red links are the ones found by both algorithms, while the yellow links are only found by the exact method. The blue links are the set of candidate links obtained from our filtering process. Clearly, the results are very similar. Regarding network 2, our method finds only one of the two links that must be closed. After a quick check, it turned out that the second link (the yellow link in Figures 6f-6h) does not carry any traffic. This problem arises due to the fact that in the solution algorithm 1 when two solutions have the same objective value, the last one is returned as the optimal solution. Regardless of that, the calculated TTT in both methods for all three demand scenarios of this network are identical. Figure 7 depicts the differences in the calculated TTT with both solution methods.

Network 1 is twice as big as network 2, and therefore, it is more challenging to find a solution for it. As expected, the offset between the solutions found by the algorithms will increase as the size of the network grows. We can see in Figures 6a-6e that for lower demand scenarios in this network, our method finds the same solution as in the exact method. However, for the higher demand scenarios, one link with non-zero traffic flow is not found by our algorithm. The main reason is that these links are not in the set of candidate links. We argue that these links are trivial to the traffic banning strategy. We can clearly see this fact in Figure 7 that by adding the yellow links, only 0.04% improvements in the TTT can be achieved. The second experiment: Performance evaluation

The results of the second experiment are also very promising. After three hours, which was set as a reasonable computation time limit, the exact method managed to analyze only 583 possible solutions, that is only a fraction of maximum number of feasible solutions i.e. 2 136 it must explore.

Interestingly, our method converged to the same solution in only 12 minutes, which implies that the exact method had found the near-global optimum solution but still had to test thousands of feasible solutions. Even though we cannot guarantee our solution is the global optimum solution, but we can be up to a certain level confident that this solution is very close to the global optimum solution, and it is not worth it to spend more time to find those trivial links that may reduce the TTT only by a negligible fraction. Figure 8 illustrates the solutions from both methods on the toy network. Similar to the previous results, red links represent the solution of the algorithm. We can see that these links are identical in both solutions. The blue links in Figure 8b are the set of candidate links.

Remarks on the value of the traffic banning strategy

Another crucial observation we can make is that the solution is relatively robust to small changes in demand. This is crucial to the real-world implementation of such an idea using traffic management measures. A sudden change of the location of the optimal links with slight changes in demand would have required a bang-bang controller, which is not efficient for such complex problems. Our results on the effectiveness of traffic banning as a management measure are also very encouraging.

CONCLUSION

In this paper, we proposed a heuristic optimization framework based on network link features in order to find the links that cause the Braess paradox. The framework improves the network performance by imposing a traffic banning strategy. This improvement by banning link(s) is occurred because of the difference in link cost and flow distribution of the network status at the users optimum (i.e., UE) and system optimum (SO). The methodology consists of three main steps: (i)

Identify the feature of candidate links by selecting the link features on the UE and the SO principles for different demand profiles on a traffic network, (ii) Filter the candidate links which are the most potential link(s) to ban, and (iii) Find the optimal set of links to be banned by an exact solution method. The last step is essentially based on the SO-relaxation method first for discrete network design problem (DNDP). First, we performed a complete analysis on this solution method and fixed the original interdiction cut constraint in order to account for one change between the network designs when seeking the optimum solution. This correction guarantees the global optimum solution for DNDP. Second, we adapt the solution method to find the optimal network design from an existing network under the UE principle. Third, we added a link filtering procedure based on the first step of our methodology to reduce the solution space. The filtering procedure is based on a single feature ξ , which was found to be very effective. This feature, in essence, reflects the importance of a link during the UE in comparison to the SO assignment. The value of 1 basically means the link is only used in the UE and not SO, which is a sign for the Braess paradox.

In order to examine the effectiveness of the proposed solution, we conducted a case study on the network of Chicago-Sketch. We designed two experiments, one without a computation time limit in order to let the exact method converge, and the second experiment in which we imposed a reasonable computation time limit to account for the trade-off between computation time and accuracy of the solution. The results of the first experiment confirmed our initial postulation regarding the proposed feature for link filtering. ξ i j =

x UE i j -x SO i j

x UE i j

was the most influential feature to distinguish a link that may exhibit the Braess paradox among other features we investigated. Additionally, the first experiment results proved that for low-demand scenarios, our method converges to the same solution as in the exact method. We also showed that for higher levels of demand our method returns a near-global solution. In other words, the difference between the global optimum and our solution is trivial, and we conclude that it is not worth the computation time required to find the global optimum solution. We used the second experiment to test this statement. After three hours, the exact method could only evaluate a fraction of the solution space, whereas our method converged in 12 minutes. The solutions were identical by both methods, which implies that the exact method found a good solution but had to explore the rest of the feasible solutions, which for real-world networks can take up days or weeks.

Despite the promising results of the proposed method, we suggest some improvements for future work. First and foremost, it is essential to perform a sensitivity analysis on different demand levels in other realistic network settings. In this paper, we only scaled the demand by a factor on two networks. It would be interesting to see how stable the solution is in different OD relations, such as peak hours versus non-peak hours. This is very similar to the idea of defining demand breakpoints as discussed in (Ameli et al.,[START_REF] Ameli | Simulation Modelling Practice and Theory Improving traffic network performance with road banning strategy : A simulation approach comparing user equilibrium and system optimum[END_REF]. Another possibility to enrich our study is to add operational budget constraints. While closing a link may not need high monetary costs in comparison with building a new road, but there are other aspects in terms of easiness to implement a link closure, e.g., if the banning infrastructure exists.

  Aashtiani and Poorzahedy, Yao et al., Pas and Principio, Roughgarden, Gisches and Rapoport, Ameli et al., Colini-Baldeschi et al.,

FIGURE 1 :

 1 FIGURE 1: Piecewise linear approximation of BPR. function

FIGURE 2 :

 2 FIGURE 2: The flow chart of the proposed feature selection procedure and filtering process

FIGURE 3 :

 3 FIGURE 3: The targeted areas from the network of Chicago-Sketch (Stabler et al., 30): Network 1 boundaries are highlighted by red, and the green rectangle defines the boundaries of Network 2.

FIGURE 4 :

 4 FIGURE 4: (a) Ranking position of the feature with respect to the rest of the links in the corresponding network and (b) Number of times a feature appears in the top 10%.

FIGURE 5 :

 5 FIGURE 5: The value of the selected feature ξ for different scenarios. The red dots indicate the links in the global optimum solution whereas the blue dots are all other links in the network.

(a) Network 1 , 1 FIGURE 6 :

 116 FIGURE 6: Traffic banning strategy in each scenario for network 1 and network 2: Red links are in both solutions, yellow links are only in the exact method solution, and blue links are the candidate links considered by our method but eventually remain open.

FIGURE 7 :

 7 FIGURE 7: Difference of the TTT between the solution obtained by the exact solution method and the solution obtained by the proposed filtering approach.

Figure 9a and 9b

  Figure9aand 9b illustrate the TTT under both the UE and SO principle and Figure9cdepicts the improvements in the PoA after applying the traffic banning strategy for all network-demand scenarios. We looked into the PoA in the original network and the network obtained after banning the links. It turns out that implementing the proposed traffic banning strategy can reduce the PoA on average about 20% in network 1 and up to 40% in small and less congested scenarios in network 2. This is a very motivating observation that should be indeed further investigated in more realistic settings.

FIGURE 8 :

 8 FIGURE 8: Identical traffic banning found by both solution methods.

FIGURE 9 :

 9 FIGURE 9: The effectiveness of traffic banning strategy on reducing the PoA.

TABLE 1 :

 1 NomenclatureESet of all links in the original network.ĒSet of all open links in the solution subnetwork. Λ Set of candidate links to be considered for solution algorithm; Λ ⊆ E.VSet of all nodes in the network.

D

Set of all destination nodes; D ⊆ V . (i, j)

  Algorithm 1: Pseudo-code for the SO-relaxation solution method

	Ωu ← [ ] ;
	u * , u * temp ← [ ];
	UB ← +∞;
	LB ← 0;
	Z RP ← +∞;
	converged ← False ;
	while converged == False do
	solution RP = solve((13), s.t. inequality (14)-(16));
	if |solution RP | ≥ 0 then
	return u * ;
	break;
	else
	(u * temp , xi j ) ← solution RP ;
	Z RP ← ∑ (i, j)∈E xi j t i j ( xi j );
	x i j = solve((6), s.t. (7)-(10));
	LB ← ∑ (i, j)∈E x i j t i j (x i j );
	if LB ≤ UB then
	UB ← LB;
	u * ← u * temp ; Ωu = Ωu ∪ u * temp
	else
	Ωu = Ωu ∪ u * temp
	end
	if Z RP ≥ UB then
	converged == True;
	return(u * );
	break;
	end
	end
	end

TABLE 2 :

 2 Test

networks characteristics Network No. nodes No. links No. OD pairs Scenarios [Demand factors] 1 48 136 210 [2.8, 3.0, 3.5, 4.0, 4.2] 2 26 66 72 [8.9, 9.0, 9.1]

(a) 582 iterations in 180 minutes. (b) Converged after 12 minutes.

ACKNOWLEDGMENT

This work has received funding from the Department of Science and Technology at the French embassy in Berlin via the Procope Mobilität 2021 Grant, No 0185-DEU-21-016.

AUTHOR CONTRIBUTION STATEMENT

All the authors have contributed to all aspects of this study, ranging from the conception and design of the methodology, analysis and interpretation of the results and discussion, and the manuscript preparation.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.


Mostafa Ameli

Negin Alisoltani

Ludovic Leclercq

Lyon Metropolis realistic trip data set including home to work trips with private vehicles during the morning peak

This

Description of the demand profile:

The data contains the home to work trips traveling by private vehicles, including departure time, arrival time, and trip length of all users in Lyon metropolis traffic network. Lyon is the second-largest urban area of France with an area of 80 km 2 . The city is located in the south of Paris and is close to other mega-cities in France and Switzerland (Marseille, Grenoble, and Geneva). The trip set contains all trip come from outside and travel to outside the city. This data set represents the demand dynamics, which is typical for a peak period from 6:30 AM to 12 PM. The number of trips during this period is 99,138. The geographical coordinates system is RGF93 / Lambert-93. This dataset is used in [1] and [2].

Data set and references:

Referencing: You can find the data set, DOI and BibTex in the following link and please cite the following papers for the description: Link to the data: https://research-data.ifsttar.fr/dataset.xhtml?persistentId=doi:10.25578/MLIDRM

Figure 1 :

 1 Figure 1: Lyon city in France.

  Ameli, Mahdi Zargayouna, and Ludovic Leclercq. "Space-time clustering-based method to optimize shareability in real-time ride-sharing." Plos one 17, no. 1 (2022): e0262499. [2] Alisoltani, Negin, Ludovic Leclercq, and Mahdi Zargayouna. "Can dynamic ride-sharing reduce traffic congestion?" Transportation research part B: methodological 145 (2021): 212-246. doi: 10.1016/j.trb.2021.01.004.

Funding for this project: European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 646592 --MAGnUM project).

Funding for this project: European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 646592 --MAGnUM project).

data set is generated based on raw data collected by

Data files:

The index of trips are determined by the row number.


V Le Moigne

J.-L Gaillard

J.-L Herrmann

Vaccine strategies against bacterial pathogens in cystic fibrosis patients

Keywords: mucoviscidose, vaccination, Mycobacterium abscessus, Pseudomonas aeruginosa, Burkholderia spp cystic fibrosis, vaccination, Mycobacterium abscessus, Pseudomonas aeruginosa, Burkholderia spp. Résumé

De nombreux pathogènes associés à la mucoviscidose, tels que les bactéries du complexe Burkholderia cepacia, Pseudomonas aeruginosa ou Mycobacterium abscessus posent des problèmes thérapeutiques complexes en raison de leur multirésistance intrinsèque aux antibiotiques. De plus, aucun vaccin n'est actuellement disponible contre ces pathogènes. Les approches vaccinales représentent donc une arme clé pour combattre ces bactéries multirésistantes dans un certain nombre de cas cliniques, dont celui de la mucoviscidose. Différentes stratégies peuvent être envisagées pour développer ces vaccins. Certains facteurs de virulence similaires sont exprimés au cours de l'infection par différents pathogènes et pourraient ainsi être utilisés comme antigène pour évaluer une protection croisée. De nombreux essais sont en cours pour tenter de générer une prophylaxie dans le cadre de la mucoviscidose.

and viruses can infect the patient's respiratory system. Bacterial colonization occurs very early in the disease progression [START_REF] May | Bacterial Infection in cystic fibrosis[END_REF]. Patients first present with Haemophilus influenzae and Staphylococcus aureus colonization. Months or years later patients are colonized with Pseudomonas aeruginosa. Burkholderia cepacia is the fourth bacterium mainly responsible for CF patients' lung infection. The administration of an antibiotic treatment is the only effective strategy to combat the infection. However, bacteria become resistant to antibiotics when repeatedly used. A single bacterium can have many different strains and can easily mutate. Developing effective treatment strategies is therefore difficult.

Resistance to antibiotics is a major problem for CF patients. Multidrug-resistant bacteria such as B. cepacia, P. aeruginosa, or Mycobacterium abscessus lead to therapeutic difficulties and are responsible for fatal infections [START_REF] Lopez | The global burden of disease, 1990-2020[END_REF].

Guidelines focusing on prophylaxis must therefore be drawn and therapeutic strategies for respiratory tract infections must be developed. Such therapeutic strategies should be integrated into the overall disease management.

The Burkholderia cepacia complex (Bcc) consists of 18 species responsible for opportunistic infections that can be life-threatening in CF patients. Burkholderia cenocepacia and Burkholderia multivorans are most frequently identified. These environmental and biofilm-forming intracellular bacteria are highly resistant to antibiotics. Bcc infections developing in CF patients are rarely eliminated once a patient is colonized. Pseudomonas aeruginosa is another environmental pathogen responsible for opportunistic infections. It is the most frequent bacterium isolated from CF patients. P. aeruginosa colonization and chronic infections affect up to 80% of CF adult patients [START_REF] Gibson | Pathophysiology and management of pulmonary infections in cystic fibrosis[END_REF]. This pathogen is responsible for chronic endobronchial infections and increases morbidity and mortality rates. P. aeruginosa is resistant to antibiotics. It is therefore a dangerous pathogen as once patients are colonized the pathogen is rarely, or even never, eliminated. P. aeruginosa colonization usually affects the lungs of CF patients. The bacterium forms a biofilm on the lungs and reduces the patient's immune response, thus contributing to the bacterium high level of resistance to antibiotics [START_REF] Drenkard | Antimicrobial resistance of Pseudomonas aeruginosa biofilms[END_REF].

Mycobacterium abscessus is the most recently identified bacterium to be highly resistant to antibiotics. It is a rapidly-growing mycobacterium belonging to the Mycobacterium abscessus complex [START_REF] Leao | Proposal that Mycobacterium massiliense and Mycobacterium bolletii be united and reclassified as Mycobacterium abscessus subsp. bolletii comb. nov., designation of Mycobacterium abscessus subsp.abscessus subsp. nov. and emended description of Mycobacterium abscessus[END_REF]. Mycobacterium abscessus is responsible for a wide variety of human diseases, especially in CF patients [START_REF] Olivier | Nontuberculous mycobacteria[END_REF][START_REF] Roux | Multicenter study of prevalence of nontuberculous mycobacteria in patients withcystic fibrosis in France[END_REF]. Person-to-person transmission has recently been reported in CF patients [START_REF] Aitken | Respiratory outbreak of Mycobacterium abscessus subspecies massiliense in a lung transplant and cystic fibrosis center[END_REF][START_REF] Bryant | Wholegenome sequencing to identify transmission of Mycobacterium abscessus between patients with cystic fibrosis: a retrospective cohort study[END_REF]. M. abscessus is associated with major therapeutic difficulties because of its natural resistance to most antibiotics [START_REF] Ripoll | Nonmycobacterial virulence genes in the genome of the emerging pathogen Mycobacterium abscessus[END_REF][START_REF] Nash | A novel gene, erm(41), confers inducible macrolide resistance to clinical isolates of Mycobacterium abscessus but is absent from Mycobacterium chelonae[END_REF]. Severe and even fatal infections have already been reported in CF patients due to the lack of therapeutic strategies [START_REF] Sanguinetti | Fatal pulmonary infection due to multidrug-resistant Mycobacterium abscessus in a patient with cystic fibrosis[END_REF]. Several countries consider that patients presenting with a M. abscessus infection cannot be eligible for lung transplant [START_REF] Taylor | Mycobacterium abscessus chest wall and pulmonary infection in a cystic fibrosis lung transplant recipient[END_REF]. CF patients presenting with such infection are therefore left with no therapeutic option.

Acute or chronic bronchial infections and superinfections progressively deteriorate the patient's respiratory function. They are treated with antibiotics in light of the bacteriological examination results. Sputum culture (sputum cytobacteriological examination) or blood samples (blood cultures) allows for identifying the involved bacterium, evaluating the extent of the colonization, and determining the most effective antibiotics. The most frequently observed bacteria (S. aureus, P. aeruginosa, and B. cepacia) are rapidly resistant to antibiotics. The most effective doses are still unclear but they are usually higher than the ones recommended in the agents' marketing authorization. The administration of two intravenous antibiotics is, for instance, often combined with an inhaled maintenance antibiotic treatment.

Inhaled antibiotics must be administered after chest physical therapy (CPT) and after having administered beta-2-agonists and rhDNase.

For S. aureus infections, a primary prophylactic treatment is not recommended for infants and children. For methicillin-susceptible S. aureus exacerbations, the recommendation is to first administer oral beta-lactams as first-line treatments with or without fusidic acid. Treatment duration is at least 14 days. For methicillin-resistant S. aureus exacerbations, it is recommended to administer a combination of pristinamycin and rifampicin. There is currently no recommendation for secondary antibiotic prophylaxis (or maintenance treatment) as there is no consensus on that matter.

For P. aeruginosa infection, it is recommended to first administer two intravenous bactericidal antibiotics (14-21 days) to patients presenting with a primary colonization (betalactams + aminoglycoside). Inhaled colistin may then be prescribed for 3 to 6 months.

Exacerbations of patients presenting with chronic infection should be treated with a combination treatment to prevent the emergence of resistant strains: beta-lactam and tobramycin for at least 14 days. For a multidrug-resistant strain, a combined treatment with three antibiotics should be administered including oral ciprofloxacin or intravenous colistin.

Although there is currently no guideline recommending such treatment, a maintenance antibiotic treatment (inhaled) administered for 28 days or IV treatments administered every three months, preferably at home, may be considered.

A significant association between previous intravenous antibiotic treatments and M. abscessus isolation in the lungs of CF patients has recently been reported. Such association highlights the role of a broad spectrum antibiotic treatment in the occurrence of M. abscessus infection [START_REF] Catherinot | Mycobacterium avium and Mycobacterium abscessus complex target distinct cysticfibrosis patient subpopulations[END_REF].

Vaccination strategies

Pathogens can be divided into two groups: vaccine-preventable pathogens and non-vaccinepreventable pathogens. There is currently no human vaccine against most antibiotic-resistant pathogens previously mentioned. It would thus be interesting to develop a prophylactic vaccination strategy to improve the prevention of those infections. Reverse-vaccinology is interesting as it would help target antigens associated with strong vaccine effectiveness. A better understanding of the regulation of bacterial gene expression helps in developing new strategies to combat such bacteria [START_REF] Saade | Technologies for enhanced efficacy of DNA vaccines[END_REF].

Various vaccination strategies can be considered once a potential target is identified: conventional vaccination using a recombinant protein or vaccination using a plasmid DNA encoding the antigen.

Various pathogens may contain highly similar antigens acting as virulence factors. Those antigens could be used to ensure a potential cross-protection. For instance, M. abscessus phospholipase C (PLC) could be a joint vaccine target against PLC-producing CF pathogens such as P. aeruginosa. This would be an interesting strategy as the serum of P. aeruginosainfected patients contains antibodies that recognize M. abscessus recombinant PLC [START_REF] Moigne | Bacterial phospholipases C as vaccine candidate antigens against cystic fibrosis respiratory pathogens: the Mycobacterium abscessus model[END_REF].

Conventional vaccination schedule

CF patients are advised to comply with the recommended vaccination schedule (diphtheria, tetanus, poliomyelitis, acellular pertussis, Haemophilus influenzae type B, Streptococcus pneumoniae, hepatitis B, measles-mumps-rubella) as well as with hepatitis A and influenza vaccination.

Haemophilus influenzae often colonizes the respiratory tract of CF patients. The prevalence of non-typable Haemophilus strains is increasing [START_REF] Razvi | Respiratory microbiology of patients with cystic fibrosis in the United States, 1995 to 2005[END_REF]. The vaccination for Haemophilus influenzae type b has proved effective in preventing invasive diseases [START_REF] Malfroot | Immunisation in the current management of cystic fibrosis patients[END_REF]. However, no study has so far been conducted to assess its benefits for CF patients.

Streptococcus pneumoniae is the fourth bacterium frequently isolated from respiratory samples of CF patients after S. aureus, P. aeruginosa, and H. influenzae. However, invasive infections due to Streptococcus pneumoniae are rare in those patients [START_REF] Lahiri | Preimmunization anti-pneumococcal antibody levels are protective in a majority of patients with cystic Fibrosis[END_REF]. The population of patients infected by Streptococcus pneumoniae has changed since the introduction of the pneumococcal conjugate vaccines (PCV7 and PCV13) [START_REF] Pimentel De Araujo | Characterization of Streptococcus pneumoniae clones from paediatric patients with cystic fibrosis[END_REF]. Although CF patients rarely contract pneumococcal disease, the United Kingdom and United States advise CF patients to get the pneumococcal vaccine [START_REF] Burgess | Pneumococcal vaccines for cystic fibrosis[END_REF]. No study has so far been published on the effectiveness of such vaccine in that population of patients. However, Browning et al. [START_REF] Browning | Pneumococcal polysaccharide vaccine responses are impaired in a subgroup of children with cystic fibrosis[END_REF] observed that a subgroup of CF children poorly responded to a pneumococcal polysaccharide vaccine (Pneumovax II). This may be due to disease severity.

Influenza-related respiratory tract infections in CF patients deteriorate lung function and contribute to disease progression. Yearly influenza vaccination is strongly recommended for CF patients aged above 6 months. However, there is no evidence based on randomized studies on the benefit of such vaccine for those patients [START_REF] Dharmaraj | Vaccines for preventing influenza in people with cystic fibrosis[END_REF]. Recent data does not suggest that influenza develops more frequently in CF patients than in healthy individuals [START_REF] Patria | Influenza vaccination in children at high risk of respiratory disease[END_REF]. The influenza vaccine is however recommended for CF patients. Launay et al. [START_REF] Launay | Factors associated with humoral immune response to pandemic A/H1N1(v) 2009 influenza vaccine in cystic fibrosis[END_REF] observed that CF patients presenting with malnutrition and receiving the non-adjuvanted vaccine had a lower immune response to the pandemic influenza vaccine. Their data also suggests that CF patients had an inadequate immune response to the influenza vaccine. One may thus wonder whether a different vaccination strategy is necessary.

All other vaccinations are based on a specific vaccination schedule and are not subject to specific recommendations for CF patients.

Vaccines currently being developed

Vaccine development is currently focusing on some infections frequently associated [START_REF] Sagel | Impact of pseudomonas and staphylococcal infection on inflammation and clinical status in young children with cystic fibrosis[END_REF][START_REF] Pillarisetti | Infection, inflammation, and lung function decline in infants with cystic fibrosis[END_REF] with respiratory failure in CF patients: Pseudomonas aeruginosa, Staphylococcus aureus infections, or respiratory syncytial virus (RSV) infections.

Many researchers are focusing on developing a vaccine against P. aeruginosa even though no clinical trial has so far been successful in supporting the benefits of a specific vaccine. Many vaccines have been tested against P. aeruginosa. The results of recent studies are useful to get an overview of the current state of vaccine development for CF patients [START_REF] Grimwood | Vaccination against respiratory Pseudomonas aeruginosa infection[END_REF][START_REF] Savoia | New perspectives in the management of Pseudomonas aeruginosa infections[END_REF][START_REF] Johansen | Vaccines for preventing infection with Pseudomonas aeruginosa in cystic fibrosis[END_REF][START_REF] Sharma | Recent developments for Pseudomonas vaccines[END_REF]. Major target antigens include the lipopolysaccharide O-polysaccharides, cell-surface alginate, flagella, components of the Type III secretion system, and outer membrane proteins [START_REF] Pier | Application of vaccine technology to prevention of Pseudomonas aeruginosa infections[END_REF]. The flagellin protein (FliC) is thought to be a virulence factor and is a major component of flagella. The protein induces inflammation on respiratory epithelial cells. Just like in Burkholderia cepacia, flagellin was one of the main targets for vaccine trials and especially for CF patients. In 1995, P. aeruginosa flagella were already developed as a P. aeruginosa vaccine. Results of phase I clinical trials showed that intramuscular immunization of healthy adults resulted in high and long-lasting serum IgG antibody titers against flagella antigens and also elicited specific antibodies to flagella of the IgG, IgA, and secretory IgA isotypes in the secretory immune system of CF patients [START_REF] Döring | Parenteral application of a Pseudomonas aeruginosa flagella vaccine elicits specific anti-flagella antibodies in the airways of healthy individuals[END_REF]. The results of a phase III trial then revealed that the administration of a bivalent P. aeruginosa flagella vaccine, exhibiting flagella subtypes, significantly decreased the risk of P. aeruginosa initial infection in CF patients [START_REF] Döring | Flagella vaccine trial study, a double-blind randomized placebo-controlled phase III study of a Pseudomonas aeruginosa flagella vaccine in cystic fibrosis patients[END_REF]. Other proteins have been used in human clinical trials such as the OprF and OprI proteins. They are both outer membrane proteins able to induce a specific antibody response in the lungs after nasal or oral vaccination. Both of these proteins are good candidates for a P. aeruginosa vaccine [START_REF] Bumann | Systemic, nasal and oral live vaccines against Pseudomonas aeruginosa: a clinical trial of immunogenicity in lower airways of human volunteers[END_REF]. A fusion protein from a part of P. aeruginosa OprF, OprI, and FliC helped in P. aeruginosa clearance in a pulmonary challenge model [START_REF] Weimer | A fusion protein vaccine containing OprF epitope 8, OprI, and type A and B flagellins promotes enhanced clearance of nonmucoid Pseudomonas aeruginosa[END_REF]. Another clinical trial evaluated the role of these two outer membrane proteins in a mucosal vaccine. Its results indicated respiratory tract immunogenicity against the pathogen with better effectiveness than systemic vaccination [START_REF] Sorichter | Immune responses in the airways by nasal vaccination with systemic boosting against Pseudomonas aeruginosa in chronic lung disease[END_REF].

One of the most recent clinical trials evaluating antigens and immunization strategies used a virulence factor produced by mucoid strains known as surface exopolysaccharide alginate [START_REF] Farjah | Immunological evaluation of an alginate-based conjugate as a vaccine candidate against Pseudomonas aeruginosa[END_REF]. The clinical trial was conducted with mice: vaccinated mice were protected when challenged intranasally with P. aeruginosa. The virulence factor therefore seems to be effective as a therapeutic vaccine. A human trial had previously been conducted with Opolysaccharide-toxin A conjugate P. aeruginosa vaccine. The results indicated that immunized children presented with fewer chronic P. aeruginosa lung infection than nonimmunized patients [START_REF] Zuercher | Antibody responses induced by long-term vaccination with an octovalent conjugate Pseudomonas aeruginosa vaccine in children with cystic fibrosis[END_REF][START_REF] Zuercher | Cellular immunity in healthy volunteers treated with an octavalent conjugate Pseudomonas aeruginosa vaccine[END_REF]. The authors of another study conjugated a part of the alginate (polymannuronic acid) with flagellin and obtained a high level of protection in lung-infected mice [START_REF] Campodónico | Efficacy of a conjugate vaccine containing polymannuronic acid and flagellin against experimental Pseudomonas aeruginosa lung infection in mice[END_REF]. Some P. aeruginosa antigens have also been conjugated with bovine serum albumin and tested on mice [START_REF] Rodrigues | Bovine serum albumin nanoparticle vaccine reduces lung pathology induced by live Pseudomonas aeruginosa infection in mice[END_REF]. A human vaccine targeting P. aeruginosa is, however, still not available despite more than 50 years of research.

With regard to Burkholderia cepacia complex, various virulence factors associated with the occurrence of the infection in humans have been tested as potential vaccines [START_REF] Casey | Exploiting molecular virulence determinants in Burkholderia to develop vaccine antigens[END_REF]. That same strategy had already been considered by a group of researchers looking for immunoreactive proteins expressed by both Burkholderia cenocepacia and Burkholderia multivorans [START_REF] Shinoy | Immunoproteomic analysis of proteins expressed by two related pathogens, Burkholderia multivorans and Burkholderia cenocepacia, during human infection[END_REF]. The authors of a recent literature review took a detailed inventory of vaccination studies performed in relation to the Burkoldheria genus [START_REF] Choh | Burkholderia vaccines: are we moving forward?[END_REF]. Within the related species Burkholderia pseudomallei, the flagellar protein (FliC) can be considered as a virulence factor and has been used as an antigen in a recent trial. The protein had a protective effect on mice [START_REF] Musson | CD4+ T cell epitopes of FliC conserved between strains of Burkholderia: implications for vaccines against melioidosis and cepacia complex in cystic fibrosis[END_REF]. The results of the study also revealed that B. pseudomallei FliC epitopes of interest cross-reacted with orthologous FliC sequences from Burkholderia multivorans and Burkholderia cenocepacia. Those epitopes could thus be integrated into vaccination programs. Other proteins such as outer membrane proteins from various Burkolderia species can also induce protection in mice models [START_REF] Makidon | Induction of immune response to the 17 kDa OMPA Burkholderia cenocepacia polypeptide and protection against pulmonary infection in mice after nasal vaccination with an OMP nanoemulsion-based vaccine[END_REF][START_REF] Bertot | Nasal immunization with Burkholderia multivorans outer membrane proteins and the mucosal adjuvant adamantylamide dipeptide confers efficient protection against experimental lung infections with B. multivorans and B. cenocepacia[END_REF]. Although research is still focusing on finding the adequate target for a prophylactic treatment, vaccine development research has not yet managed to go beyond human clinical trials [START_REF] Choh | Burkholderia vaccines: are we moving forward?[END_REF].

Staphylococcus aureus is usually the first bacterium isolated from the respiratory secretions of CF patients [START_REF] Marks | Clinical significance of Staphylococcus aureus in cystic fibrosis[END_REF] and its prevalence is currently increasing [START_REF] Lipuma | The changing microbial epidemiology in cystic fibrosis[END_REF]. The affinity of Staphylococcus aureus for cystic fibrosis mucus, mucociliary abnormalities, and other unknown factors contributes to persistent colonization with this bacterium. It causes progressive pulmonary damage and may facilitate Pseudomonas infections [START_REF] Marks | Clinical significance of Staphylococcus aureus in cystic fibrosis[END_REF]. Initial effort to develop a Staphylococcus aureus vaccine was based on conventional immunologic mechanism: a humoral-based vaccine [START_REF] Spellberg | A new view on development of a Staphylococcus aureus vaccine: insights from mice and men[END_REF]. The first attempts were however unsuccessful in reducing the risk of contracting invasive Staphylococcus aureus infections [START_REF] Release | Biopharmaceuticals Announces Results of StaphVAX(R) Confirmatory Phase III Clinical Trial[END_REF][START_REF]Inhibitex Reports Top-Line Results in Phase III Study of Veronate(R). PR Newswire[END_REF]. A new immunologic strategy is therefore needed to develop a Staphylococcus aureus vaccine. This may be done by inducing memory T cells which are capable of increasing the rapidity and strength of phagocyte recruitment to infection sites, thus facilitating clearance of the bacterium from tissues [START_REF] Spellberg | A new view on development of a Staphylococcus aureus vaccine: insights from mice and men[END_REF].

One may also mention the role of the RSV. RSV infections can indeed deteriorate the respiratory function of children presenting with CF. RSV might also be involved in the initial respiratory tract infection by P. aeruginosa [START_REF] Peterson | Respiratory infections in cystic fibrosis patients caused by virus, chlamydia and mycoplasma-possible synergism with Pseudomonas aeruginosa[END_REF]. Vaccines have been tested [START_REF] Piedra | Immunogenicity of a new purified fusion protein vaccine to respiratory syncytial virus: a multi-center trial in children with cystic fibrosis[END_REF] but there is once again no available vaccine yet.

Vaccination and mycobacteria: benefits of the BCG vaccine

Nontuberculous mycobacteria are identified in 7 to 13% of respiratory samples isolated from CF patients [START_REF] Olivier | Nontuberculous mycobacteria[END_REF]. However, M. tuberculosis infections are not common in industrialized countries [START_REF] Morand | Mediastinal tuberculosis in an adult patient with cystic fibrosis[END_REF][START_REF] Asherova | Cystic fibrosis complicated by multiresistant tuberculosis[END_REF]. The BCG vaccine is currently recommended for children at risk of tuberculosis exposure. The vaccine is effective on nontuberculous mycobacteria and should always be given to CF children.

Mycobacterium avium and Mycobacterium abscessus are mainly responsible for mycobacterium infections in CF patients. Mycobacterium avium is a slow-growing mycobacterium (SGM) while Mycobacterium abscessus is a rapidly-growing one (RGM).

Mycobacterium abscessus infections are more severe and the mycobacterium is resistant to antibiotics. Vaccination strategies have so far not been studied for CF patients presenting with such infections. Comparing whole-genome sequencing may help in the research for target vaccines [START_REF] Hoft | Optimizing vaccine development[END_REF]. A reverse-vaccinology strategy using whole-genome sequencing was considered to develop a vaccine against that bacterium [START_REF] Ripoll | Nonmycobacterial virulence genes in the genome of the emerging pathogen Mycobacterium abscessus[END_REF]. A first antigen was selected on the basis of its pathogenic role: MAB_0555, encoding PLC [START_REF] Bakala | Mycobacterium abscessus Phospholipase C expression is induced during coculture within amoebae and enhances M. abscessus virulence in mice[END_REF]. Further antigens were then selected and are currently being tested. An in silico subtractive genome analysis allowed for identifying MAB_0555 in the genome of M. abscessus and for confirming its absence from both M. smegmatis and M. chelonae genomes, which are two RGM respectively lesspathogenic or non-pathogenic. M. abscessus is the only RGM to be a major respiratory pathogen in CF patients. CF mice vaccination (F508) [START_REF] Van Doorninck | A mouse model for the cystic fibrosis delta F508 mutation[END_REF] against this antigen was performed either using the recombinant protein or a coding plasmid for that antigen to perform a DNA vaccination [START_REF] Moigne | Bacterial phospholipases C as vaccine candidate antigens against cystic fibrosis respiratory pathogens: the Mycobacterium abscessus model[END_REF]. Similar results were observed with both formulations: a decreased bacterial load in the lungs after three weeks of aerosolized M. abscessus challenge [START_REF] Moigne | Bacterial phospholipases C as vaccine candidate antigens against cystic fibrosis respiratory pathogens: the Mycobacterium abscessus model[END_REF]. PLC are also present in other CF pathogens such as P. aeruginosa. Crossed-immune reaction between M. abscessus PLC and that of P. aeruginosa could help develop a protective vaccine against both mycobacterium infections and Gram-negative infections in CF patients.

Conclusion

Vaccines are effective in preventing infections. Vaccination is strongly recommended for CF patients, especially to prevent respiratory tract infections. In addition to conventional vaccinations, evaluation studies aiming to show the immune and clinical effectiveness of new vaccines targeted against multidrug-resistant bacteria are still needed for CF patients.

Conflict of interest

The authors report no conflict of interest

Authors' contribution

All authors contributed to writing the article.