Sebin Gracy

Irinel Constantin Morȃrescu

Vineeth S Varma

Philip E Paré

Analysis and On/Off Lockdown Control for Time-Varying SIS Epidemics with a Shared Resource

The paper studies the spread of a virus over a (possibly) time-varying graph, with the spread being (possibly) worsened by the presence of a shared resource. We propose a time-varying susceptible-infected-water-susceptible (SIWS) model, with the water compartment representing the contamination level in the shared resource. We say that the system is in the disease-free equilibrium (DFE) if none of the nodes (representative of sub-populations, such as cities, districts, etc.) are infected, and the shared resource is contaminantfree. We identify multiple sufficient conditions for exponential convergence to the DFE. Based on one of the aforementioned sufficient conditions, an on/off lockdown strategy that eradicates the infection spread is proposed. More specifically, we design a switching rule between lockdown and free (i.e., no lockdown) modes to guarantee exponential convergence to the DFE.

I. INTRODUCTION

Spreading processes such as epidemics, information in social networks, etc., have drawn the attention of several research communities ranging from physics and computer science to economics and sociology. The earliest work in the direction of studying epidemic spread is the model for the smallpox virus formulated and analysed by Daniel Bernouli [START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF]. Mathematical epidemiology, as a discipline, advanced rapidly during the 20 th century; see [START_REF] Becker | The uses of epidemic models[END_REF]- [START_REF] Hethcote | The mathematics of infectious diseases[END_REF], with [START_REF] Bailey | The mathematical theory of infectious diseases and its applications[END_REF] being one of the seminal works. Given that the destruction that epidemics leave in their wake is often times unprecedented, the fundamental questions of interest in the aforementioned works revolves around understanding how a disease spreads, what causes it to become persistent in the population, how can the spread be mitigated or eradicated, etc. While there are several models proposed in the literature, the present paper focuses on the susceptible-infected-susceptible (SIS) model, that was originally developed in [START_REF] Kermack | Contributions to the mathematical theory of epidemics. ii.-the problem of endemicity[END_REF].

Networked SIS models with static graphs (where the graphs represent the interaction between the various population nodes) have been well-studied; for continuous-time setting, see [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF], [START_REF] Khanafer | Stability of epidemic models over directed graphs: A positive systems approach[END_REF], whereas for discrete-time setting, see [START_REF] Ahn | Global dynamics of epidemic spread over complex networks[END_REF],

Sebin Gracy is with the Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, and Digital Futures, Stockholm, Sweden. Email: gracy@kth.se. This work was supported in part by the Knut and Alice Wallenberg Foundation, Swedish Research Council under Grant 2016-00861. I.C. Morȃrescu and V.S. Varma are with Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France. Emails: constantin.morarescu@univlorraine.fr; vineeth.satheeskumar-varma@univ-lorraine.fr. This work was supported in part by the ANR through the grant NICETWEET ANR-20-CE48-0009. Philip E. Paré is with the Elmore Family School of Electrical and Computer Engineering at Purdue University. Email: philpare@purdue.edu. This work was supported in part by the National Science Foundation, grants NSF-CNS #2028738 and NSF-ECCS #2032258. [START_REF] Wang | Epidemic spreading in real networks: An eigenvalue viewpoint[END_REF]. Given that modern societies involve mobile agents thus imposing a time-varying topology on the interconnection graph, it is natural to consider networked SIS models that also account for time-varying graphs. Such models have been proposed in [START_REF] Bokharaie | Spread of epidemics in time-dependent networks[END_REF], [START_REF] Paré | Epidemic processes over timevarying networks[END_REF]. The present paper focuses on timevarying networked SIS models.

A major drawback with traditional epidemic models is that these assume that the only way an epidemic could spread is through node-to-node interaction. However, other pathways also exist, for instance spread of viruses through a shared resource such as a water distribution network [START_REF] Tien | Multiple transmission pathways and disease dynamics in a waterborne pathogen model[END_REF], neighborhood supermarkets, infected hospital surfaces [START_REF] Weinstein | Contamination, disinfection, and cross-colonization: are hospital surfaces reservoirs for nosocomial infection?[END_REF], etc. Addressing this gap, a networked continuous-time SIS model that also accounts for the presence of a shared resource was first proposed by [START_REF] Liu | A networked SIS disease dynamics model with a waterborne pathogen[END_REF], referred to as susceptible-infectedwater-susceptible (SIWS) model. For the SIWS model, a sufficient condition for asymptotic convergence to the diseasefree equilibrium (DFE) (i.e., the state where each agent is healthy, and the shared resource is contamination-free) has been identified in [START_REF] Liu | A networked SIS disease dynamics model with a waterborne pathogen[END_REF], whereas a sufficient condition for the existence, uniqueness and asymptotic convergence to the endemic equilibrium (i.e., the state where each agent has a non-trivial infection level, and the shared resource is contaminated) has been identified in [START_REF] Janson | Networked multi-virus spread with a shared resource: Analysis and mitigation strategies[END_REF]. Building off of the SIWS model, a model that accounts for the presence of multiple shared resources, referred to as the layered networked SIWS model, has been proposed in [START_REF] Pare | Multi-layer SIS model with an infrastructure network[END_REF]. However, both the SIWS model and the layered networked SIWS model only account for static interaction graphs, and neither of these works propose a control scheme that involves manipulating the weights on the edges between the nodes (the so-called lockdown approach). The present paper aims to address these shortcomings.

Our main contributions are as follows: i) We develop a model that captures the spread of a virus in a population where the interconnection among the agents (possibly) are time-varying, and also accounts for the presence of a shared resource that is accessed by (possibly) the entire population. ii) Identify a sufficient condition for global exponential stability (GES) of the DFE, both for homogeneous spread, and for heterogeneous spread; see Theorem 1 and Theorem 2, respectively. iii) Identify a less restrictive sufficient condition, which can be checked easily, for exponential convergence to the DFE; see Theorem 3. iv) Finally, we design a switching rule between lockdown and free (i.e., no lockdown) modes to control the infection spreading. To be specific, the overall switching system satisfies the conditions in the previous item (i.e., item iii)) and consequently, the dynamics, regardless of the initial condition (i.e., infected or otherwise), converges to the DFE exponentially fast; see Lemma 2.

Paper Outline

The rest of the paper is organized as follows. We conclude the present section by listing the notations that will be used in the sequel. We introduce the model, and formally present the main problems of interest in Section II. The main results are spread out across two sections, namely in Section III we identify sufficient conditions for GES of the DFE. Building on the aforementioned sufficient conditions, a control strategy for efficient management of the epidemic is proposed in Section IV. We illustrate our theoretical findings in Section V. Finally, we summarize the paper, and highlight some problems of possible interest to the wider community in Section VI.

Notation

Let R, R ≥0 , and N denote the set of real numbers, nonnegative real numbers, and positive integers, respectively. For any positive integer n, we have [n] = {1, . . . , n}. Given a matrix A, supposing its spectrum is real, λ 1 (A) denotes the maximum eigenvalue of A; the largest real-valued part of the eigenvalues of A is denoted by r 1 (A). We use 0 and 1 to denote the vectors whose entries all equal 0 and 1, respectively, and use I to denote the identity matrix, while the sizes of the vectors and matrix are to be understood from the context. For any two real matrices A, B ∈ R n×m , we write

A ≥ B if A ij ≥ B ij for all i ∈ [n], j ∈ [m], and A > B if A ≥ B and A = B.

II. MODEL

Consider a population of individuals, divided into n subpopulation nodes in a network, with a resource W that is shared among (possibly) all of the population nodes. Suppose that a virus is active in the aforementioned population. The spread of the virus among the n nodes can be represented by a (possibly) directed time-varying graph G(t) = {V, E(t)}, where V = {1, 2, . . . , n} are the nodes (which represent subpopulations), with, at time t, existence of a directed edge from node j to node i if, at time t, individuals in node j can infect those in node i. We also assume that not only does a node get (possibly) infected due to contact with W but also that W could be contaminated whenever an infected population node comes in contact with it. Thus, at each t, each node in G(t) possibly has bidirectional connections with W .

Each population node i, contains N i individuals, at time t, has birth rate µ i (t), death rate μi (t), and, at time t, S i (t) denotes the number of susceptible individuals, while I i (t) denotes the number of individuals infected by the virus. For each node i, at time t ≥ 0, infected individuals have a recovery rate γ i (t). We denote the infection rate of node i, at time t, as β i (t). Let α ij (t)(≥ 0) denote, at time t, the strength of interconnection (could also be interpreted as the frequency of interactions) between an individual in node i and an individual in node j. Clearly, if node j is not connected to node i at time t, then α ij (t) = 0; otherwise, α ij (t) > 0. Let z(t) denote, at time t, the concentration of the virus in the shared resource W . The rate at which the virus decays in the shared resource, at time t, is denoted by δ w (t), while the concentration of the virus in the shared resource W grows at a rate proportional to the sum of I i (t) scaled by ζ i (t), where ζ i (t) denotes, at time t, the rate at which the resource gets contaminated due to node i. The resource-to-node infection rate for node i at time t is denoted by α iw (t).

Given the interactions between different individuals in a population node (resp. between different individuals in different population nodes), the fraction of infected individuals in a population node changes with time. More precisely, the evolution of the number of susceptible and infected individuals in node i can be represented as follows:

Ṡi (t) = µ i (t)N i -μi (t)S i (t) + γ i (t)I i (t) -α iw (t)W (t) - n j=1 β i (t)α ij (t) Ij (t) Ni S i (t), İi (t) = -(μ i (t) + γ i (t))I i (t) + α iw (t)W (t) + n j=1 β i (t)α ij (t) Ij (t) Ni S i (t), Ẇ (t) = -δ w (t)W (t) + n j=1 ζ j (t)I j (t). (1)
We define new variables to simplify the system. Let:

x i (t) = I i (t) N i , z(t) = δ w (t)W (t) n j=1 ζ j (t)N j , δ i (t) = γ i (t) + µ i (t), β ij (t) = β i (t)α ij (t) N j N i , β iw (t) = α iw (t) δ w (t) n j=1 ζ j (t)N j , and
c i (t) = ζ i (t)N i n j=1 ζ j (t)N j . (2)
Then, assuming that, at each time t ≥ 0, the birth rates and the death rates are equal, (1) can be rewritten as:

ẋi (t) = -δ i (t)x i (t) + 1 -x i (t) β iw (t)z(t) + n j=1 β ij (t)x j (t) , (3)
ż(t) = δ w (t) -z(t) + n i=1 c i (t)x i (t) . (4)
Observe that assuming that the parameters in system (3) and ( 4) are time-independent, we recover the time-invariant SIWS model proposed in [START_REF] Liu | A networked SIS disease dynamics model with a waterborne pathogen[END_REF]. Likewise, in the absence of the shared resource (i.e., setting z(t) = 0), we recover the time-varying SIS networked model studied in [START_REF] Paré | Epidemic processes over timevarying networks[END_REF]. The model from ( 3)-( 4) in vector form becomes:

ẋ(t) = -D(t) + B(t) -X(t)B(t) x(t) + I -X(t) b(t)z(t) (5)
ż(t) = -δ w (t)z(t) + c(t) x(t) , (6)
where 5)-( 6) can be written more compactly using

D(t) = diag(δ i (t)), B(t) = [β ij (t)] n×n , X(t) = diag(x(t))
y(t) := x(t) z(t) , X(y(t)) := diag(x(t)) 0 0 0 , B w (t) := B(t) b(t) c (t) 0 , and
D w (t) := D(t) 0 0 δ w (t) .
With the new notations in place, system ( 5)-( 6) can be rewritten as:

ẏ(t) = -D w (t) + (I -X(y(t)))B w (t) y(t). ( 7
)
Observe that if x = 0, and z = 0, then none of the nodes are infected nor is there any contamination in the shared resource. Furthermore, (0, 0), is an equilibrium of system [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF]. Hence, we call this equilibrium the disease-free equilibrium (DFE).

With the setup as given in [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF], the problems being investigated in this paper are as follows.

i) Assuming the graph G(t) is, for each t, symmetric, and that β i (t) = β(t) for all i ∈ [n] and for each t, identify a sufficient condition such that y(t) converges to 0 exponentially fast. ii) Accounting also for (possibly) directed graphs G(t), and also heterogeneous spread, identify a sufficient condition such that y(t) converges to 0 exponentially fast. iii) Ensure that the convergence to the DFE is exponential even in finite time and not just asymptotic to ensure that critical health constraints are not violated. Thus, for a given initial time t 0 and y(t 0 ), we want to ensure that

y(t) ≤ exp(-λ 0 (t -t 0 ))γ y(t 0 ) , (8)
for all t ≥ t 1 , for some finite t 1 ≥ t 0 and γ > 0. We require the following assumptions on the model parameters.

Assumption 1: Suppose that, for each t ∈ R ≥0 , δ i (t) > 0, and δ w (t) > 0. Suppose that, for each t ∈ R ≥0 , β ij (t) ≥ 0 for all i, j ∈ [n], and

β iw (t) > 0, c i (t) > 0, for all i ∈ [n].
Given that we are interested in identifying sufficient conditions for exponential convergence to the DFE, we recall the following definition.

Definition 1: Consider a system, described as follows:

ẋ(t) = f (t, x(t)), (9)
where f : R ≥0 × R n → R n is locally Lipschitz. The origin is a GES equilibrium point of ( 9) if there exist positive constants α and η, with 0 ≤ η < 1, such that

x(t) ≤ α x(t 0 ) η (t-t0) , ∀ t, t 0 ≥ 0, ∀ x(t 0 ) ∈ R n .
Since each x i represents the fraction of infected individuals in group i, it is immediate that the initial value of x i is in [0, 1], because otherwise the value of x i will lack physical meaning for the epidemic model considered here. Similarly, it is also natural to assume that the initial value of z measured, for instance, in milligrams per litre is nonnegative. Consequently, we can say that the DFE of system [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF] is GES if the condition in Definition 1 is satisfied for all x 0 ∈ [0, 1] n and z(0) ≥ 0. Hence, we can restrict our analysis to the set:

D := {y(t) : x(t) ∈ [0, 1] n , z(t) ∈ [0, ∞)}. ( 10
)
The following lemma establishes that the set D is positively invariant.

Lemma 1: Consider system (7) under Assumption 1. If

x i (0) ∈ [0, 1] for all i ∈ [n], and z(0) ≥ 0, then x i (t) ∈ [0, 1] for all i ∈ [n]
, and z(t) ≥ 0, ∀t ≥ 0. Proof: Suppose that, for some arbitrary i ∈ [n], at time τ ∈ R ≥0 , we have that x i (τ ) = 0, and z(τ ) ≥ 0. Therefore, from (3), since by Assumption 1 β ij (t) > 0 and β iw (t) > 0, ẋi (τ ) ≥ 0. If x i (τ ) = 1, then from (3), since by Assumption 1 δ i (t) > 0, ẋi (τ ) < 0. Putting the two cases together, it follows that, assuming z(τ ) ≥ 0, x i (t) ∈ [0, 1] for all t ≥ τ . Since the choice of node i is arbitrary, it follows that, for all i ∈ [n], x i (t) ∈ [0, 1] for all t ≥ τ . Next we show that, if, for the same τ as above, z(τ ) ≥ 0, then z(t) ≥ 0 for all t ≥ τ . Suppose that z(τ ) = 0. Then, from (4), since by Assumption 1 c i (t) > 0 for all i ∈ [n], and for all t ∈ R ≥0 , and since we have already shown that x i (t) ∈ [0, 1] for all t ≥ τ , it follows that ż(τ ) ≥ 0. Observe that, by setting τ = 0, we fulfill the hypothesis of the lemma, and, therefore, the claim follows.

III. CONVERGENCE TO THE DFE

In this section, we identify sufficient conditions for GES of the DFE. We first consider the case where the spread is homogeneous (i.e., all agents have the same infection rate, for all time instants) and the underlying graph is symmetric and undirected, then we consider the case where the spread could also be heterogeneous (i.e., not all of the agents have the same infection rate for all time instants) and the underlying graph could also be directed.

A. Homogeneous case

We consider the case where each node has the same infection rate, i.e., β i (t) = β(t) for all i ∈ [n], and for all t ∈ R ≥0 . Furthermore, we assume that the interconnection graph, represented by the matrix A(t), is undirected, and has symmetric weights, i.e., for each t ∈ R ≥0 , a ij (t) = a ji (t). We have the following result, which is partly inspired from [START_REF] Paré | Epidemic processes over timevarying networks[END_REF]Theorem 1].

Theorem 1: Consider system (7) under Assumption 1. Suppose that, for each t ∈ R ≥0 , β i (t) = β(t) for all i ∈ [n], A(t), b(t), c(t), D(t) and δ w (t) is piecewise continuous in t, and bounded. Suppose that, for each t ∈ R ≥0 , A(t) = A(t) , and b(t) = c(t). If sup t≥0 λ 1 (-D w (t) + B w (t)) < 0, then the DFE of ( 7) is GES.

Proof: Consider an arbitrary y(t) ≥ 0. For ease of notation, we will drop the time index in the rest of the proof. Define the Lyapunov function V (y) = 1 2 y y. Observe that V (y) = 0 if, and only if, y = 0. For y = 0, we have:

V (y) = y ẏ = y -D w (t) + (I -X(y))B w (t) y = y -D(t) + (I -X)B(t) (I -X)b(t) c(t) -δ w (t) y ≤ y -D(t) + B(t) b(t) c(t) -δ w (t) y (11) = y -D w (t) + B w (t) y (12) ≤ λ 1 (-D w (t) + B w (t))||y|| 2 (13) ≤ sup t≥0 λ 1 (-D w (t) + B w (t)) ||y|| 2 (14)
< 0.

Observe that, due to Assumption 1, B(t), b(t), and c(t) are nonnegative. Since, by Lemma 1,

x i (t) ∈ [0, 1] for each i ∈ [n]
, and for all t, it follows that (I -X)B(t) < B(t), and hence we obtain inequality [START_REF] Bokharaie | Spread of epidemics in time-dependent networks[END_REF]. By assumption, for each

t ∈ R ≥0 , A(t) = A(t) , β i (t) = β(t) for each i ∈ [n],
and b(t) = c(t). Therefore, the matrix (-D w (t) + B w (t)) is symmetric, and by applying Rayleigh-Ritz Quotient [START_REF] Horn | Matrix Analysis[END_REF] to [START_REF] Paré | Epidemic processes over timevarying networks[END_REF], inequality ( 13) is immediate, while from the definition of supremum we obtain inequality [START_REF] Weinstein | Contamination, disinfection, and cross-colonization: are hospital surfaces reservoirs for nosocomial infection?[END_REF]

B. Heterogeneous case

We now consider the case where the infection rates are not necessarily the same for every node; the interconnection graph could be directed.. We have the following result.

Theorem 2: Consider system (7) under Assumption 1, with B w (t) and D w (t) to be continuously differentiable. Suppose that i) there exists

L > 0 such that B w (t) -D w (t) ≤ L ∀t; ii) for some α 1 > 0 sup t≥0 r 1 (B w (t) -D w (t)) < -α 1 ; and iii) there exists κ > 0 such that sup t≥0 d dt B w (t) - D w (t) < κ.
If κ is sufficiently small, then the DFE of ( 7) is GES.

Remark 1: The proof technique is closely related to that of [START_REF] Paré | Epidemic processes over timevarying networks[END_REF]Theorem 2] and [START_REF] Rosenbrook | The stability of linear time-dependent control systems[END_REF]; in the interest of completeness, we provide the details here. Proof: By assumption, for some α 1 > 0, sup t≥0 r 1 (B w (t) -D w (t)) < -α 1 , which means that sup t≥0 r 1 (B w (t) -D w (t)) < 0. This implies, from the definition of supremum, that, for each t ∈ R ≥0 , r 1 (B w (t) -D w (t)) < 0, which further implies that, for each t ∈ R ≥0 , the matrix (B w (t) -D w (t)) is Hurwitz. Therefore, from [19, Theorem 4.12] it follows that, for each t, there exists a positive definite matrix

Q(t) such that Q(t)(B w (t) -D w (t)) + (B w (t) -D w (t)) Q(t) = -I. ( 15
)
The solution to ( 15) is, then, given by

Q(t) = ∞ 0
e (Bw(t)-Dw(t)) τ e (Bw(t)-Dw(t))τ dτ. [START_REF] Janson | Networked multi-virus spread with a shared resource: Analysis and mitigation strategies[END_REF] Consider the possible Lyapunov function V (y, t) = y Q(t)y. Since Q(t) is, for each t, positive definite, it is clear that V (y, t) > 0 for all y = 0, and for all t ≥ 0. Since Q(t) is positive definite, for all t Q(t) is symmetric, by definition. Moreover, since Q(t) is an integration of product of matrix exponentials, and by noting that the first term in a power series expansion of a matrix exponential is the identity matrix I n+1 , it follows that I n+1 ≤ Q(t) for all t ≥ 0. This implies that y 2 ≤ V (y, t). Since conditions i) and ii) are, by assumption, satisfied, from [START_REF] Desoer | Slowly varying system ẋ = A(t)x[END_REF] it follows that there exists a constant m, that depends on L and α 1 such that

e Bw(t)-Dw(t)τ ≤ m • e -α1τ ∀τ ≥ 0. ( 17
)
Hence, from ( 16) and ( 17) we obtain

Q(t) ≤ m 2 2α 1 . (18)
Since Q(t) is symmetric for all t ∈ R ≥0 , by applying the Rayleigh-Ritz Quotient (RRQ) Theorem [START_REF] Horn | Matrix Analysis[END_REF] we have:

λ min (Q(t))I ≤ Q(t) ≤ λ max (Q(t))I, which implies λ min (Q(t))||y|| 2 ≤ y Q(t)y ≤ λ max (Q(t))||y|| 2 ≤ ||Q(t)|| • ||y|| 2 (19) ≤ m 2 2α 1 ||y|| 2 , (20)
where [START_REF] Khalil | Nonlinear Systems[END_REF] follows from [18, Theorem 5.6.9], and ( 20) is due to [START_REF] Horn | Matrix Analysis[END_REF]. Combining with the lower bound on V (y, t) obtained previously, we have the following:

y 2 ≤ V (t, y) ≤ m 2 2α 1 y 2 , ( 21
)
Taking the time derivative of V (y, t), we obtain

V (y, t) = y Q(t)y + y Q(t) ẏ + ẏ Q(t)y = y Q(t)y + y Q(t) -D w (t) + (I -X(y))B w (t) y + y -D w (t) + B w (t) (I -X(y)) Q(t)y (22)
≤ y Q(t)y + y Q(t)(B w (t) -D w (t)) + (B w (t) -D w (t) Q(t) y (23)
= -y 2 + y Q(t)y, (24)
where inequality [START_REF] Gao | Unified stability criteria for slowly time-varying and switched linear systems[END_REF] comes from the fact that, since, by Assumption 1, B w (t) is nonnegative for each t, and by Lemma 1,

x i (t) ∈ [0, 1] for each i ∈ [n]
, and for all t, thereby implying that (I -X(y))B w (t) < B w (t), while ( 24) is due to [START_REF] Liu | A networked SIS disease dynamics model with a waterborne pathogen[END_REF]. Differentiating Q(t) with respect to time, we obtain:

Q(t)(B w (t) -D w (t)) + Q(t)( Ḃw (t) -Ḋw (t))+ (B w (t) -D w (t)) Q(t) + ( Ḃw (t) -Ḋw (t)) Q(t) = 0. (25)
Define

S(t) : = ( Ḃw (t) -Ḋw (t)) Q(t) + Q(t)( Ḃw (t) -Ḋw (t)). (26)
Therefore, in view of (26), equation ( 25) can be rewritten as

Q(t)(B w (t) -D w (t)) + (B w (t) -D w (t)) Q(t) = -S(t). (27)
The solution to (27) is, then, given by

Q(t) = ∞ 0 e (Bw(t)-Dw(t)) τ S(t)e (Bw(t)-Dw(t))τ dτ. ( 28
)
Taking norms on both sides of (28) we get

Q(t) = ∞ 0 e (Bw(t)-Dw(t)) τ S(t)e (Bw(t)-Dw(t))τ dτ = ∞ 0 e (Bw(t)-Dw(t)) τ S(t)e (Bw(t)-Dw(t))τ dτ (29)
≤ ∞ 0 e (Bw(t)-Dw(t)) τ S(t) e (Bw(t)-Dw(t))τ dτ

(30) = S(t)
∞ 0 e (Bw(t)-Dw(t)) τ e (Bw(t)-Dw(t))τ dτ

(31) = S(t) Q(t) (32)
≤ m 2 2α 1 S(t) (33)
≤ m 4 4α 2 1 Ḃw (t) -Ḋw (t) , (34)
where (29) comes from replacing the norm of the integral by integral of the norm, (30) comes from submultiplicativity of matrix norms, and (32) comes from [START_REF] Janson | Networked multi-virus spread with a shared resource: Analysis and mitigation strategies[END_REF]. The inequality (33) is a direct consequence of ( 17), whereas inequality (34) comes from taking norms on both sides of equation ( 26), and, again, leveraging the upper bound on the norm of Q(t) in [START_REF] Pare | Multi-layer SIS model with an infrastructure network[END_REF]. Plugging the inequality in (34) back into [START_REF] Gronwall | Note on the derivatives with respect to a parameter of the solutions of a system of differential equations[END_REF], we obtain the following:

V (y, t) ≤ (-1 + m 4 4α 2 1 Ḃw (t) -Ḋw (t) ) y 2 . ( 35
)
By assumption, there exists κ > 0 such that sup t≥0 d dt B w (t) -D w (t) < κ. Consequently, (35) could be rewritten as

V (y, t) ≤ (-1 + m 4 4α 2 1 κ) y 2 . ( 36
)
Observe that choosing κ = 2α 2 1 m 4 results in V (y, t) ≤ -1 2 y 2 , which further implies that V (y, t) < 0. Consequently, together in view of [START_REF] Desoer | Slowly varying system ẋ = A(t)x[END_REF], from [START_REF] Khalil | Nonlinear Systems[END_REF]Theorem 4.10] it follows that the DFE of ( 7) is GES, thus concluding the proof.

Remark 2: Particularized to the setting without a shared resource, the difference between Theorem 2 and [12, Theorem 2] is as follows: Theorem 2 does not assume the existence of a constant that upper bounds the Lyapunov function V (y, t); it turns out that the existence of such a constant is a direct consequence of the first two conditions in Theorem 2 being satisfied. The result [START_REF] Paré | Epidemic processes over timevarying networks[END_REF]Theorem 2], to the contrary, assumes that such a constant exists and is well-defined, and together with conditions in Theorem 2 establishes exponential convergence to the DFE. Therefore, even particularized to the setting without a shared resource, Theorem 2 is more general than [START_REF] Paré | Epidemic processes over timevarying networks[END_REF]Theorem 2].

Remark 3: The third condition in Theorem 2 could be slightly relaxed as follows while (assuming that the first two conditions are fulfilled) still guaranteeing GES [START_REF] Ioannou | Robust Adaptive Control[END_REF]:

t+T t Ḃw (s) -Ḋw (s) ds ≤ µT + η, ∀t ≥ 0, ∀T ≥ 0, (37)
where η > 0 is some scalar, and µ depends on L and α 1 .

Observe that both Theorem 2, and its relaxation in Remark 3, require B w (t) and D w (t) to be continuously differentiable. For linear time-varying systems, a sufficient condition for GES that does not insist on the state matrix being continuously differentiable has been identified in [START_REF] Gao | Unified stability criteria for slowly time-varying and switched linear systems[END_REF]Theorem 3]. Extending [START_REF] Gao | Unified stability criteria for slowly time-varying and switched linear systems[END_REF]Theorem 3] to also account for nonlinear time-varying systems of the kind in [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF] is beyond the scope of the present paper, and is left for future work.

Observe that Theorems 1 and 2 insist on the matrix -D w (t) + B w (t) being Hurwitz for each t ≥ 0. Consequently, it becomes extremely hard to verify this condition. As such, we are interested in seeking less restrictive sufficient condition(s). Towards this end, we need the following assumption.

Assumption 2: Suppose that i)

lim T →∞ 1 T t0+T t0 -D w (s) + B w (s) ds ≤ α < ∞, ∀t 0 ≥ 0. (38)
ii) for some v > 0, there exists an h > 0 such that

-D w (t + h) + B w (t + h) + D w (t) -B w (t) ≤ vh γ , (39)
for all t ≥ 0, and for some γ such that 0 < γ ≤ 1. iii)

1 T t0+T t0 r 1 (-D w (s) + B w (s))ds ≤ σ, (40)
for all t 0 ≥ 0, T ≥ T 0 and for some σ < 0 and T 0 > 0. Theorem 3: Consider system (7) under Assumptions 1 and 2. The DFE is GES, and Note that the maximum eigenvalue is always less than zero and the system converges to the DFE rapidly consistent with Theorem 1.

y(t) ≤ exp(-λ 0 (t -t 0 ))γ y(t 0 ) , (41)
if an agent arrives at a boundary of the box, the velocity of the agent in the corresponding dimension changes sign. Note that the simulation setup follows [START_REF] Paré | Epidemic processes over timevarying networks[END_REF] except with the shared resource added. For all the simulations we set r = 10. We assume there are 10 nodes (n = 10) that bounce around a box of dimension 5 × 5, centered at z c = (2.5, 2.5). A randomly infected set of nodes is initially infected (x i (0) = 1). We assume timeinvariant homogeneous viral spread, that is, the same β and δ for each node.

We first simulate a scenario that meets the assumptions of Theorem 1. We set β = 1, δ = 3, and δ w = 0.25, and z(0) = 0.5. The average infection level ( 1 n n i=1 x i (t)) and the resource contamination level (z(t)) are plotted in Fig. 1. The maximum eigenvalue for the simulation is plotted over time in Fig. 2. Note that the assumptions of Theorem 1 are met, with the largest eigenvalue always remaining below zero. Consistent with Theorem 1, the virus and the resource contamination die out quickly; see the blue line and red line, respectively.

We also simulated a system that violated the assumptions of all the results. We set β = 2, δ = 2, δ w = 0.5, and z(0) = 0.5. The average infection level and the resource contamination level are plotted in Fig. 3 with the corresponding maximum eigenvalues plotted over time in Fig. 4. Note that Fig. 4: Maximum eigenvalue of the simulation in Fig. 3. Note that the maximum eigenvalue is greater than zero most of the time and the virus persists in the network (see Fig. 3).

the eigenvalues are mostly greater than zero and the virus persists in the system.

In order to enact the on/off lockdown control policy proposed in Section IV, we devise the following social distancing protocol that is implemented for ten time steps. While not letting node i leave the square region,

χi (t) = j,aij (t)>0.001 χ i -χ j j,aij (t)>0.001 χ i -χ j , (51)
forcing the nodes to separate from each other. We implement this algorithm every 20 time steps, starting at time t = 10, for the endemic system from Fig. 3. The average infection level and the resource contamination level are plotted in Fig. 5 with the corresponding maximum eigenvalues plotted over time in Fig. 6. Note that the eigenvalues are almost always less than zero, the 20-time step average is always below zero, and the virus no longer persists in the system, dying out quite quickly, consistent with the result in Lemma 2.

VI. CONCLUSION

In this paper, we proposed a SIWS model that also accounts for time-varying interactions between the agents. We identified several sufficient conditions for exponential convergence to the DFE; a difficult-to-check condition, and, an easily checkable one. Both the conditions also account Note that the average is always below zero, satisfying the conditions of Lemma 2; the control strategy eradicates the virus (see Fig. 5).

for heterogeneous spread, and directed graphs. We have also proposed a control strategy guaranteeing that the DFE is GES. This control strategy is based on switching between a lockdown and a free (no-lockdown mode), and maintaining lockdown measures for sufficiently long time periods of time. We illustrated our theoretical findings using an extensive set of simulations.

There are several possible research directions. one could focus on studying the endemic behavior for the time-varying SIWS model. Another interesting problem would be to extend the model to also account for the presence of multiple shared resources. Finally, in the time-varying case, one could extend the SIWS model to account for competitive viruses, and identify conditions for the existence, uniqueness, and convergence to various equilibria.

Fig. 1 :

 1 Fig. 1: Average infection level ( 1 n

Fig. 2 :

 2 Fig. 2: Maximum eigenvalue of the simulation in Fig. 1.Note that the maximum eigenvalue is always less than zero and the system converges to the DFE rapidly consistent with Theorem 1.

Fig. 3 :

 3 Fig.3: Average infection level and z(t) in illustration of system with β = 2, δ = 2, δ w = 0.5, and z(0) = 0.5. The lack of lockdown measures causes the infection level to rise rapidly around t = 30.

Fig. 5 :Fig. 6 :

 56 Fig.5: Average infection level and z(t) in illustration of the endemic system from Fig.3with the social distancing policy from (51) every 20 time steps, starting at time t = 10. With the on/off lockdown strategy implementation added, the virus in Fig.3is eradicated.

  1 , b(t) is a column vector in R n with its i th element being β iw (t), and c(t) is a row vector in R n whose i th element is δ w (t)c

i (t). Defining B(t) := diag(β i (t)) and A(t) := [α ij (t) Nj Ni ] n×n , we have that B(t) = B(t)A(t). System (

When clear from context, for ease of notation, we will drop the time index, and use X instead of X(t).

for all t ≥ t 1 , for some finite t 1 ≥ t 0 and λ 0 , γ > 0. Proof: Note that, by Assumption 1, B w (t) is, for all t ∈ R ≥0 , nonnegative. Since by Lemma 1, for each i ∈ [n], x i (t) ∈ [0, 1] for all t ≥ 0, it follows that (I -X(y))B w (t) < B w (t). Therefore, system [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF] can be bounded from above by the linear system ẏ(t) = (-D w (t) + B w (t))y(t).

(42)

Consequently, by Gronwall's inequality [START_REF] Gronwall | Note on the derivatives with respect to a parameter of the solutions of a system of differential equations[END_REF], the solution of (7) will be bounded from above by the solution of the linear system (42). Observe that inequality (40) implies

for some negative scalar σ, and for all t 0 ≥ 0. Particularizing the result in [START_REF] Solo | On the stability of slowly time-varying linear systems[END_REF]Theorem 2] for the case with no perturbation, inequalities (38), (39) in Assumption 2 together with inequality (43) imply that system (42) is GES. The existence of constants λ 0 > 0 and γ > 0 such that (41) is fulfilled follows from [START_REF] Solo | On the stability of slowly time-varying linear systems[END_REF]. Hence, system (7) is GES, and (41) holds, thus concluding the proof.

IV. CONTROL STRATEGIES FOR AN EFFICIENT

LOCKDOWN

We study the case in which the epidemics is controlled by a lockdown procedure. In its natural state, i.e., without a lockdown, we consider that the population behaves in such a way that

for all t 0 ≥ 0, with σN > 0.

On the other hand, during lockdown, the behavior for the population is such that Assumption 2 is satisfied and

for all t 0 ≥ 0, with σL < 0 as long as T ≥ T min , with T min > 0 known. Assuming that switching between the two strategies preserves the properties (38)-(39), our goal is to find a switching rule which ensures that all the conditions in Assumption 2 are satisfied, and therefore, guarantees the DFE is GES and the system's trajectory satisfies [START_REF] Khanafer | Stability of epidemic models over directed graphs: A positive systems approach[END_REF].

For any given t 0 , let us first denote the sequence of switching times by {t k , k ∈ N}. We assume that a switching to a lockdown mode is done at any t 2k , k ∈ N, while a switching to free (no lockdown) mode is done at any t 2k+1 , k ∈ N. We also assume that the lockdown is initiated at t 0 to control the epidemic.

Lemma 2: Assuming (44) and (45) hold true, if the switching sequence {t k , k ∈ N} is chosen such that t 2k+1t 2k ≥ T min , ∀k ∈ N and

for all K ∈ N, then (40) holds true with T 0 = T min .

Proof: Note that for any T ≥ T min there exists a K ∈ N and a switching rule such that t 2K+1 < T ≤ t 2K+2 . Note also that without any loss of generality we can consider that T = t 2K+2 and the inequality below can be obtained straightforwardly when T ≤ t 2K+2 .

We have thus ensured that (40) holds for all T which are the switching instants.

Due to Lemma 2, we can minimize the time spent in lockdown as well as ensure an exponential convergence to the disease free equilibrium for all t > t 0 + T min by selecting a switching rule that satisfies (46) with the inequality replaced by an equality. The details on how to do so are left for future work.

V. SIMULATION

We now illustrate the analysis and control results via simulations. To capture the time-varying structure of the model, we model the adjacency matrix as the following:

where χ i (t) ∈ R 2 is the position of node i and can change as a function of time. We assume that each node has the same population size, thus implying that Nj Ni = 1, for all i, j ∈ [n]. Hence, a ij (t) = α ij (t) for each t. We also define

which is assuming that the location of the shared resource is fixed and located at (0, 0). We set b(t) = c(t). Also, note that, from (47), A(t) is symmetric. While these symmetry assumptions are only needed for Theorem 1, we factor the infection rates in a similar manner and employ the symmetric graph structure for simplicity in all the simulations. For movement of the nodes, we assume piece-wise constant drift, confining the nodes to a fixed region. The positional dynamics of each node i are given by χi (t) = φ i ,

where φ i ∈ R 2 with

for each dimension k = 1, 2, where z c is the center of a square that the nodes are bouncing around. In other words,


Sélim Yahia Coll
email: selim.coll@unige.ch

Francis Eustache
email: francis.eustache@unicaen.fr

Franck Doidy
email: franck.doidy@unicaen.fr

Florence Fraisse
email: florence.fraisse@unicaen.fr

Denis Peschanski
email: denis.peschanski@univ-paris1.fr

Jacques Dayan
email: jacques.dayan@unicaen.fr

Pierre Gagnepain
email: gagnepain@cyceron.fr

Mickaël Laisney
email: mickael.laisney@ephe.psl.eu

Avoidance Behavior Generalizes to Eye Processing in Posttraumatic Stress Disorder

Keywords: PTSD, eye-tracking, social cognition, eye avoidance, emotions

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

Introduction

Posttraumatic stress disorder (PTSD) refers to a set of symptoms that emerge following exposure to one or more traumatic events. One of these symptoms is avoidance, which involves the engagement of thoughts or behaviors to limit exposure to reminders of the stressful experience [START_REF] Sautter | A couple-based approach to the reduction of PTSD avoidance symptoms: Preliminary findings[END_REF]. Although it is a momentarily helpful reaction to trauma, avoidance becomes problematic as a long-term strategy [START_REF] Boden | Emotion regulation and posttraumatic stress disorder: A prospective investigation[END_REF], and is central to PTSD [START_REF] Salters-Pedneault | The role of avoidance of emotional material in the anxiety disorders[END_REF]. Attempts to avoid distressing situations may either precipitate or worsen PTSD symptoms [START_REF] Badour | A longitudinal test of the bi-directional relations between avoidance coping and PTSD severity during and after PTSD treatment[END_REF]. Most current PTSD treatments attempt to overcome avoidance of trauma-related elements that do not represent danger, and restore normal processing of these elements [START_REF] Brewin | Memory and forgetting[END_REF].

In addition to thoughts or external reminders, avoidance has also been shown to prevent uncomfortable trauma-related emotions (emotional avoidance), including subjective, physiological and expressive components of emotional responses [START_REF] Salters-Pedneault | The role of avoidance of emotional material in the anxiety disorders[END_REF].

Social cues conveying emotions, such as faces and gazes, may also trigger emotional experiences in the observer, causing discomfort [START_REF] Samson | Reading other people's mind: Insights from neuropsychology[END_REF]. The computation of others' emotions is fast and automatic, thereby facilitating the understanding and sharing of emotional states [START_REF] De Vignemont | The empathic brain: How, when and why?[END_REF]. The eyes play a critical role in social interactions and emotional contagion [START_REF] Land | Looking and acting: Vision and eye movements in natural behaviour[END_REF]. In PTSD, gaze processing leads to sustained activation of the brain's innate alarm [START_REF] Lanius | The innate alarm system in PTSD: Conscious and subconscious processing of threat[END_REF]. Avoiding eye contact may therefore reflect a coping mechanism for minimizing the emotional impact of others and reducing perceived threat (e.g., [START_REF] Moukheiber | Gaze avoidance in social phobia: Objective measure and correlates[END_REF]. Eye avoidance may also explain the impaired recognition of fear and sadness observed in PTSD [START_REF] Poljac | Reduced recognition of fear and sadness in post-traumatic stress disorder[END_REF]. While several studies have demonstrated an avoidance of threatening stimuli in PTSD (e.g., [START_REF] Schoorl | Attentional bias and attentional control in posttraumatic stress disorder[END_REF], other conclude to an increase hypervigilance towards these stimuli (e.g., [START_REF] Bardeen | Attentional control as a moderator of the relationship between posttraumatic stress symptoms and attentional threat bias[END_REF]. Various tracks are proposed by scientists to explain these discrepancies in the literature, such as the measures, tasks or timing investigated (see for example [START_REF] Armstrong | Eye tracking of attention in the affective disorders: A meta-analytic review and synthesis[END_REF][START_REF] Lazarov | Attention to threat in posttraumatic stress disorder as indexed by eye-tracking indices: A systematic review[END_REF] for a review).

According to [START_REF] Mogg | Time course of attentional bias for threat information in non-clinical anxiety[END_REF], these inconsistencies are mainly explained by the sequential nature of attentional effects in the face of threat. Thus, the presence of a negative stimulus automatically attracts the attention of anxious individuals. This hypervigilance phase is followed by an avoidance phase, when attention is shifted away from the stimulus. Interestingly, this model developed for trait and generalized anxiety is still debated for PTSD and more controlled studies are needed [START_REF] Felmingham | Eye tracking and physiological reactivity to threatening stimuli in posttraumatic stress disorder[END_REF].

The quality of the social environment and its relation to avoidance symptoms has been extensively investigated in PTSD (Stevens & Jovanovic, 2019). However, studies interested in the perception and understanding of social information (i.e., social cognition) conveyed by others in PTSD are scarce (Stevens & Jovanovic, 2019). Patients with PTSD have a social cognition deficit characterized by emotion recognition and mentalizing difficulties (e.g., [START_REF] Plana | A meta-analysis and scoping review of social cognition performance in social phobia, posttraumatic stress disorder and other anxiety disorders[END_REF].

Inference of a person's emotional state based solely on the eye area is also disrupted in PTSD (e.g., [START_REF] Schmidt | PTSD and impaired eye expression recognition: A preliminary study[END_REF]. However, the relationship between this social cognition deficit and avoidance behaviors remains poorly understood.

Moreover, although avoidance contributes to failure to recover from trauma, little is known about the extent to which the processing of unpleasant social cues contributes to adaptive behaviors and resilience in the aftermath of a trauma. Resilience is an adaptive process that is engaged to maintain mental health following a stressful event [START_REF] Kalisch | The resilience framework as a strategy to combat stress-related disorders[END_REF]. Facing an unpleasant stimulation may contrast with the goal of avoiding negative emotion, but it may also serve a longerterm purpose of maximizing wellbeing (Tamir & Ford, 2012).

In the present study, we investigated whether individuals with and without PTSD, but who had all been exposed to the same degree of traumatic exposure, behaved differently when processing eye cues during an attentional task specifically designed to require the visual processing of the eye area in face stimuli. In order to better understand how the processing of non-traumarelated social cues (eyes of faces expressing either an emotion or no emotion) may be related to distinct clinical trajectories in response to trauma, we designed a gaze cueing paradigm (see Fig. 1). The gaze cueing task is a well-established paradigm that is ideal for revealing any difficulty associated with eye processing and social skills [START_REF] Frischen | Gaze cueing of attention: Visual attention, social cognition, and individual differences[END_REF]. It was recently adapted to emotional faces, to show how emotions, particularly fearful expressions, enhance joint attention [START_REF] Lachat | Watch out! Magnetoencephalographic evidence for early modulation of attention orienting by fearful gaze cueing[END_REF]. In our adaptation of the task, we used sadness instead of fear, to avoid destabilizing traumatized individuals and prevent emotional overload from compromising task success. Using sad faces also allowed us to assess the hypothesis of a general reluctance of emotions associated with the same dysfunctional behavior as avoidance, independently of fear, which has a special resonance in PTSD. In this sense, sadness also represents a neater negative counterpart of happiness, allowing for cleaner comparisons between the two. Task performances were monitored in terms of both reaction times and response accuracy, even though these behavioral measures are often biased (for a review, see [START_REF] Armstrong | Eye tracking of attention in the affective disorders: A meta-analytic review and synthesis[END_REF] and lack sensitivity, especially when gaze direction is uninformative. Critically, fixations on the eye area were recorded throughout the task, using an eyetracker. This investigation technique is an ecological measure of attention selection performed via eye movements, i.e. overt attention. In between eye movements, attention selection can also be accomplished with covert attention, i.e. without reorienting the gaze. However, although the eye-tracking methodology cannot capture covert attention, it has been shown that overt attention is the primary means of attentional selection in natural viewing situation and that it is a central mediator of covert visual biases in anxiety [START_REF] Armstrong | Eye tracking of attention in the affective disorders: A meta-analytic review and synthesis[END_REF]. The eye-tracking methodology has recently been adopted to capture threat-related hypervigilance in PTSD (see [START_REF] Felmingham | Eye tracking and PTSD[END_REF], but so far, no visual scanpath studies have focused on eye processing and avoidance in PTSD. Here, we looked at the percentage of fixations made during gaze cueing (see Fig. 1) and whether PTSD avoidance symptoms generalized to the processing of the eyes of our stimuli, particularly those expressing negative emotions. We expected that being affected by a PTSD would be characterized by a significantly lower percentage of fixations on the eyes of sad faces. Finally, we investigated whether eye avoidance was negatively related to the severity of avoidance symptoms in individuals with PTSD, as measured with Cluster C from the PTSD Checklist for DSM-5 (PCL-5; [START_REF] Blevins | The posttraumatic stress disorder checklist for DSM-5 (PCL-5): Development and initial psychometric evaluation[END_REF].

Methods

Participants

We recruited 159 participants. Of these, 52 had been exposed to the terrorist attacks and had developed PTSD (PTSD+; 26 women; Mage = 36.33 years, SD = 7.60), 51 had been exposed to the same event but did not develop PTSD (PTSD-; 20 women; Mage = 36.27 years, SD = 7.07), and 56 were nonexposed controls who did not live in Paris recruited from a local panel of volunteers living in the vicinity of Caen, Normandy (32 women; Mage = 34.43 years, SD = 10.85). Exposed participants had experienced the traumatic event either as direct victims or witnesses, or as grieving relatives or first responders (in line with DSM-5 definition of exposure). They were recruited through the French 13-November transdisciplinary and longitudinal research program (http://www.memoire13novembre.fr/), a nationwide government-funded program supported by victim organizations. More demographic and clinical information about our participants can be found in Appendix F of the Supplemental material section and in [START_REF] Mary | Resilience after trauma: The role of memory suppression[END_REF], a study involving the same clinical research trial. No statistical methods were used to predetermine the sample size, given the novelty of the question and the effect tested, but the sample size used in this study (N = 159) was higher than in previous PTSD studies using eye-tracking methodology (for a review, see [START_REF] Felmingham | Eye tracking and PTSD[END_REF]. Participants were financially compensated for their participation in the study.

The study was approved by the regional research ethics committee (Comité de Protection des Personnes Nord-Ouest III, sponsor ID: C16-13, RCB ID: 2016-A00661-50, clinicaltrials.gov registration number: NCT02810197). All participants gave written informed consent before participation, in agreement with French ethical guidelines. Participants were asked not to consume psychostimulants, drugs, or alcohol before or during the experimental period.

PTSD diagnosis

Exposed participants were diagnosed using the semistructured clinical interview for DSM-5 [START_REF] First | Structured Clinical Interview for DSM-5: Research Version[END_REF], conducted by a trained psychologist and supervised by a psychiatrist. PTSD symptom intensity was measured using the PCL-5 [START_REF] Blevins | The posttraumatic stress disorder checklist for DSM-5 (PCL-5): Development and initial psychometric evaluation[END_REF]. The PCL-5 is a self-report measure that assesses the 20 symptoms of PTSD identified in the DSM-5. DSM-5 symptom cluster severity scores can be obtained by summing the scores for PCL-5 items within each cluster. Items 1-5 assess intrusion symptoms (Cluster B of the DSM-V), Items 6 and 7 avoidance symptoms (Cluster C), Items 8-14 negative changes in cognition and mood (Cluster D), and Items 15-20 changes in arousal and reactivity (Cluster E). In the present study, we used the PCL-5 to gauge the level of our participants' avoidance symptoms (Cluster C) and correlate it with our results for emotional gaze fixations. More information about participants' diagnosis can be found in [START_REF] Mary | Resilience after trauma: The role of memory suppression[END_REF].

Materials and stimuli

Participants were comfortably installed at a distance of 65 cm from a Dell P2213 22 monitor with 1680 x 1050 px resolution, connected to an HP Elite Intel Inside Core i3 computer. The task was designed and executed using the innate Experiment Center 3.6 software provided by SMI (SensoMotoric Instruments GmbH, Teltow, Germany). All the task items were randomized and then evenly divided into two blocks, to allow participants to take a break midway through. The stimuli were taken from the validated Radboud Faces Database [START_REF] Langner | Presentation and validation of the Radboud Faces Database[END_REF]. They consisted of color photographs of the faces of seven men and eight women wearing sad, happy and neutral expressions with a direct, left-oriented, or right-oriented gaze. A mask was added so that the faces were visible, but not the background, hair, ears, or neck. Faces (450 x 580 px) were displayed against a gray background in the center of the screen. The target was a checkerboard (450 x 540 px) displayed on the left or right side of the screen. It was vertically centered, at a horizontal distance of 390 px from the face.

Procedure

An overview of the task is provided in Figure 1. First, a fixation cross was displayed in the middle of the screen. In order to proceed to the next step, participants had to look at it for at least 400 ms.

The cross was then replaced by the first face (staring face), which remained on the screen for 1200 ms. This staring face could be that of a man or woman, expressing either a sad or happy emotion, or no emotion at all (i.e., neutral), and it looked directly at the participant. It was replaced by a second face (cueing face), which was displayed alone for 400 ms. This face depicted the same person, expressing the same emotion, but its eyes looked to the left or right, in order to direct participants' attention to the left or right part of the screen. Participants were told that the direction of the cueing face's gaze did not necessarily correspond to the location of the ensuing target, in line with [START_REF] Lachat | Watch out! Magnetoencephalographic evidence for early modulation of attention orienting by fearful gaze cueing[END_REF]'s paradigm. The cueing face's eyes indicated the correct location of the target in only 50% of the trials. This way, participants were allowed to use gaze information, but attentional bias and reorientation processes were limited. In 91% of the trials, a target appeared on the left or right side of the cueing face until the participant gave a response concerning its location, but for no longer than 1000 ms. Participants responded by pressing the 1 or 0 key on a keypad, using their left or right index finger. We created two experimental conditions. In one condition, the target appeared in the location cued by the cueing face's gaze (valid trials), and in the other, the target appeared on the opposite side (invalid trials). To discourage anticipatory responses, 9% of the trials were catch trials, in which no target was presented. Participants were instructed not to respond to them. Finally, a blank screen was displayed for 1500-3500 ms until the next trial.

The 144 trials of the gaze cueing task were a combination of 12 ID photographs (6 women), 3 emotions (sad vs. happy vs. neutral), 2 gaze directions (left vs. right) and 2 target locations (left vs. right). For the reason outlined above, we added 14 catch trials featuring 3 new ID photos (1 man) expressing the same 3 emotions as the other trials, and with the same 2 gaze directions as for the cueing face. These catch trials were excluded from the analyses.

Eye tracking

Eye movements were measured using the SMI RED 250 screen-based eyetracker. Movements were detected using corneal reflection with a detection algorithm standard in the SMI ® system.

Data were sampled at 250 Hz. BeGaze 3.6 SMI's software was used to overlay fixations (minimum fixation duration set at 100 ms) on the stimuli for subsequent offline analyses.

Each part of the gaze cueing task started with a calibration/validation sequence in which participants had to follow a dot presented randomly at 13 different locations on the screen. The system was calibrated on a per participant basis at the beginning of the task to a mean spatial accuracy of 0.61° (SD = 0.41). The calibration procedure was repeated up to three times if necessary, and the best of the three trials was retained. Once the calibration was completed, participants began the first trial.

To analyze how participants processed the eye area, we defined one area of interest (AOI) by contouring the eyes of our stimuli with the AOI editor function of SMI's BeGaze 3.6 software (Fig. 1). We were particularly interested in fixations on this AOI for sad and happy versus neutral stimuli. For information purposes, results for fixations on the mouth and nose areas can be found in Appendix E of the Supplemental Material Section. It should, however, be noted that the number of fixations on the mouth in particular, all participants and conditions taken together, was much lower than that of fixations on the eyes, making it difficult to compare the two areas (3444 vs. 16227 fixations). Moreover, about ten participants did not fixate the mouth at all in our task. To analyze fixations on the eye AOI, we binarized the fixation timecourses (1 during a fixation on the AOI of a given stimulus, and 0s outside the fixation period). A Gaussian-weighted moving average filter was then applied to this binary vector, using the smoothdata function and specifying the Gaussian method in MATLAB R2018b (Mathworks, Natick, MA, USA) in order to normalize it.

The smoothed timecourses were averaged across items in a given condition, producing a fixation timecourse percentage per participant and condition.

Preprocessing

Although almost all participants performed very well on the gaze cueing task (99.23% of correct responses), two participants (1 PTSD-and 1 nonexposed) only correctly responded to 45% (SD = 49.93%) and 46.85% (SD = 50.08%) of trials. Their data were therefore excluded from the analyses. Data from one nonexposed participant also had to be excluded, owing to extensive use of the wrong response keys during the task, generating missing data. Preprocessing of reaction times during the behavioral task involved excluding trials with incorrect (0.77% of total trials), late (> 2 SDs above the mean; 4.68% of total trials) or anticipatory (< 2 SDs below the mean; 0.18% of total trials) responses. This preprocessing was performed separately for each group and each experimental condition.

Concerning eye-tracking recordings, we ran a data quality check before the analyses. We projected all the fixations of participants who had an X and/or Y deviation of the right and/or left eye of more than 1° at the end of the calibration procedure (10% of the participants) to visually check whether the exploration patterns revealed systematic deviations from the fixation crosses and faces. Data from two PTSD+ participants, one PTSD-participant, and three nonexposed controls were excluded from the analyses as a result. Moreover, data from three PTSD+ participants and one PTSD-participant were not available, as their experiment had to be interrupted for schedule reasons. Following the data quality control, preprocessing of eye-tracking recordings also included the exclusion of outliers from the analyses. This preprocessing was performed on the fixation timecourse percentages (see above). First, participants with an unusual number of fixations regardless of the emotion (2 SDs above or below their group mean) were excluded. Second, unusual swings between the minimum and maximum fixation percentages, indicative of attentional lapses, were screened separately for each group and each emotion. In total, after these preprocessing steps, the data of five PTSD+, four PTSD-and six nonexposed participants were excluded from the final analyses. As a result of the preprocessing stages, 52 PTSD+ (23 men), 50 PTSD-(30 men) and 53 nonexposed (22 men) participants were included in the final analyses.

Behavioral analysis

We first sought to ensure that groups performed equally on the gaze cueing task, to check that eyetracking differences were not driven by behavioral differences. As accuracy of target direction detection revealed a ceiling effect; all participants made on average 99.23% correct responses (SD = 8.71%; PTSD+: M = 99.51%, SD = 6.98%; PTSD-: M = 98.56%, SD = 11.93%; nonexposed: M = 99.61%, SD = 6.24%); we particularly focused our behavioral analyses on the reaction times for target detection. A mixed analysis of variance was chosen, with group (PTSD+ vs. PTSD-vs. nonexposed), emotion (sad vs. happy vs. neutral) and validity (valid vs. invalid) as factors.

Eye-tracking analysis

Binary raw fixations were smoothed and weighted using a Gaussian-weighted moving average window before statistical analysis, producing fixation timecourse percentages (see Method and Fig. 2). Based on the literature (see Introduction) showing a reduction in the percentage of fixations on the eyes of faces expressing negative emotions in PTSD, and in agreement with our a priori hypothesis that individuals with PTSD make fewer fixations on the eye area of emotional faces, we ran statistical analyses of eye fixations using one-tailed t statistics, comparing sad and happy faces with neutral ones. In addition, we directly compared sad and happy faces. These analyses were performed for each timepoint in the cueing face window and using the mean percentage of fixations on the eye AOI (1200-1600 ms; see Section 2, "Methods"). Our primary focus was on cueing faces because attention was maximally directed toward the eye AOI during this time period.

For the sake of completeness, we also report findings for the staring face in Appendix B in the Supplemental Material section. However, processing of this face was biased by the initial central fixation cross, which automatically focused the gaze on the eye AOI, potentially masking intergroup differences, especially at early timepoints. Descriptive results concerning fixations on the eye AOIs of the staring and cueing faces can be found in Appendices C and D of the Supplemental Material section. The problem of correcting for multiple comparisons across the cueing face (1200-1600 ms) and staring face (800-1200 ms) timepoints to correct for initial bias was solved by using random field theory [START_REF] Brett | Introduction to random field theory[END_REF], which controls for the familywise error rate using a general expression that accounts for spatial dependences between datapoints. For the sake of completeness, we also calculated group-level inferences, using nonparametric random effects statistics, by bootstrapping the participant set with 2000 iterations and computing 90% confidence intervals on the means of the staring face and cueing face windows (and thus indicating significance when they did not overlap with zero). To account for confounding variables, namely age, sex, education level, type of exposure to the traumatic event, and level of depressive symptoms, as measured with the Beck Depression Inventory [START_REF] Beck | An inventory for measuring depression[END_REF], we included them as nuisance covariates in our analyses (results without covariates can be found in Appendix G of the Supplemental material section).

In order to correlate the eye-tracking results of the PTSD+ and PTSD-groups to the avoidance symptoms of PTSD, we performed r-skipped correlations [START_REF] Pernet | Robust correlation analyses: False positive and power validation using a new open source Matlab toolbox[END_REF] while scaling avoidance symptoms by total symptom severity, to ensure that this relationship was not confounded with PTSD intensity and other symptoms. Robust correlations were preferred over more classical association analyses, because they allow to take into account bivariate outliers in their calculations [START_REF] Pernet | Robust correlation analyses: False positive and power validation using a new open source Matlab toolbox[END_REF].

Results

Behavior

Behavioral results are summarized in Figure 1 and Appendix A in the Supplemental Materials.

There were two significant outcomes. First, we observed a main effect of validity, F(1, 151) = 263.02, p < .001, ηp 2 = .635. Participants were significantly faster in the valid condition than in the invalid one. Second, validity did not interact with group, F(2, 151) = 0.32, p = .729, showing that each group benefited equally from the gaze of the cueing face and processed the face efficiently.

It should, however, be noted that we observed a main effect of group, F(1, 151) = 3.18, p = .044, ηp 2 = .040, characterized by reaction times that were significantly longer overall (irrespective of conditions) for PTSD+ participants than for PTSD-participants, t(152) = 2.52, p = .013, but not for the nonexposed group, t(152) = 1.39, p = .168. PTSD-participants were not significantly different than the nonexposed group, t(152) = 1.16, p = .246.

Eyetracking

Eye-tracking results are summarized in Figure 2, as well as Tables 1 and2. Note. FWER = familywise error rate.

PTSD+ participants made significantly fewer fixations on sad faces than on neutral or happy faces. Regarding the sad versus neutral contrast, we observed a greater difference in the number of fixations in the PTSD+ group than in the PTSD-or nonexposed groups. There were no significant group differences for the sad versus happy contrast. Moreover, there was no significant difference in fixations on happy versus neutral faces in the PTSD+ group. In the PTSD-group, a late marginal difference was observed between fixations on sad versus neutral faces, but no significant difference was observed between fixations on sad versus happy faces. Despite the absence of a significant difference between fixations on sad versus happy faces in the PTSDgroup, there were significantly fewer fixations on happy versus neutral faces. This difference occurred at a late point in the 400-ms cueing face window, and was not obtained when we compared the PTSD+ and nonexposed groups. Finally, nonexposed individuals exhibited an intermediate pattern of findings. They made significantly fewer fixations on sad faces than on neutral faces, and marginally fewer fixations on sad faces than on happy faces. Critically, however, no differences were observed between the nonexposed and PTSD-groups for the sad versus neutral and sad versus happy contrasts. Moreover, we found no differences between fixations on happy versus neutral faces in the nonexposed group.

Interestingly, the smaller number of fixations on sad faces made by PTSD+ participants was specifically related to avoidance symptoms (r-skipped = -0.35, t(46) = -2.58, p = .007, bootstrapped 90% CI [-0.56, -0.12], 2 bivariate outliers detected; Fig. 2). This relationship was specific to sad faces, and was not observed for happy ones (r-skipped = -0.13, t(47) = -0.89, p = .190, bootstrapped 90% CI [-0.37, 0.14]; 1 bivariate outlier detected). Moreover, we did not observe it among PTSD-participants for either sad (r-skipped = -0.02, t(41) = -0.89, p = .440; bootstrapped 90% CI [-0.26, 0.22], 4 bivariate outliers detected) or happy (r-skipped = 0.08, t(41) = 0.54, p = .290; bootstrapped 90% CI [-0.19, 0.32], 4 bivariate outliers detected) faces.

Discussion

According to models of PTSD, avoidance behavior increases the severity of PTSD symptoms.

Avoidance becomes problematic when it affects the processing of elements that do not represent any danger. Using an eyetracker, we found that resilient individuals (i.e., PTSD-) did not display the normal tendency to avoid sad eyes, compared with those expressing happiness or no emotion, whereas this tendency was accentuated in PTSD+, in parallel with their avoidance symptoms.

These findings link eye avoidance to variations in responses following trauma, and cannot be attributed to poor behavioral performances of individuals with PTSD.

Our results suggest that PTSD is accompanied by difficulties in processing non-traumarelated social cues, such as the eyes, and that these difficulties relate to the same avoidance mechanism that targets trauma-related material (e.g., [START_REF] Salters-Pedneault | The role of avoidance of emotional material in the anxiety disorders[END_REF]. Although extensive research has emphasized the relationship between a healthy social environment and resilience and recovery from PTSD (Sharp et al., 2012), the potential role of social cognition remains poorly understood (Stevens & Jovanovic, 2019). Previous studies among individuals with PTSD have highlighted difficulty recognizing emotional facial expressions (e.g., [START_REF] Mazza | Social cognition disorders in military police officers affected by posttraumatic stress disorder after the attack of An-Nasiriyah in Iraq 2006[END_REF]. In the present study, we extended this framework, and our results suggest that emotional difficulties may also result from the effort to avoid negative social information, such as the eyes.

Gaze is an important component of social interactions [START_REF] Land | Looking and acting: Vision and eye movements in natural behaviour[END_REF]. A link between gaze avoidance and poor social functioning has been found in typically developing persons [START_REF] Hayward | Feature and motion-based gaze cuing is linked with reduced social competence[END_REF], and individuals with autism [START_REF] Pelphrey | Neural basis of eye gaze processing deficits in autism[END_REF] or social phobia [START_REF] Moukheiber | Gaze avoidance in social phobia: Objective measure and correlates[END_REF]. No studies, however, have previously investigated social functioning in PTSD through the lens of attention paid to gaze. Contrary to what is observed in other pathologies, our results indicate that PTSD is not accompanied by a general avoidance of the eyes, but by an avoidance of the eyes of negative faces (e.g., expressing sadness). Eye-tracking studies show that sadness recognition is mainly related to the eyes area, which receives more initial fixations and attention [START_REF] Eisenbarth | Happy mouth and sad eyes: Scanning emotional facial expressions[END_REF]. This suggests that individuals with PTSD avoid sad eyes because they perceive them to be aversive [START_REF] Emery | The eyes have it: The neuroethology, function and evolution of social gaze[END_REF]. This idea was corroborated by the significant correlation between avoidance symptoms and the percentage of fixations on sad eyes. Eye contact triggers affective processing through the activation of subcortical routes or higher mentalizing processes [START_REF] Hietanen | Affective eye contact: An integrative review[END_REF]. Activation of affective systems may be perceived as a threat in PTSD, triggering avoidance behaviors. However, it is worth mentioning that only sad faces were presented to participants in the present study and evidences are missing in order to fully generalize our results to other negative emotions.

Does eye avoidance reflect a preexisting factor or an acquired behavior exacerbated by chronic stress? We noticed that the percentage of eye fixations clearly differentiated PTSD+ from PTSD-participants, while results for the nonexposed participants were in between these two exposed groups. This suggests that PTSD+ and PTSD-participants lay at the two ends of a continuum. Interestingly, our observations are in line with PTSD studies about social cognition, which found the largest effect size when participants with PTSD were compared with trauma control participants, that is, the equivalent of our PTSD+ and PTSD-groups (Stevens & Jovanovic, 2019). A recent functional imaging study among the same participants as in the present study found a similar pattern of intergroup differences with respect to memory control processes [START_REF] Mary | Resilience after trauma: The role of memory suppression[END_REF]. Taken together, our findings suggest that a fundamental personality trait of resilient individuals is the ability to optimize the cognitive processing of trauma by facing up to unpleasant content [START_REF] Iacoviello | Psychosocial facets of resilience: Implications for preventing posttrauma psychopathology, treating trauma survivors, and enhancing community resilience[END_REF]. The sociocognitive model of PTSD also suggests that preexisting vulnerability factors in attachment schemas, which form the basis of representations of the self and others, can explain differences in responses to trauma across individuals (Sharp et al., 2012), and may predict resilience to or recovery from PTSD. One specificity of the sample of unexposed participants must be considered when comparing this group to exposed participants in our study. We choose to include people geographically distant from the place and the repercussions of the attacks that occurred in Paris and Saint-Denis as non-exposed participants. There are, however, socio-demographic differences between the people living in Caen and those exposed to the attacks living in Paris and its suburbs. Some studies highlighted the impact of sociodemographic factors in the variations of the response to trauma (e.g., [START_REF] Afana | The Prevalence and associated socio-demographic variables of post-traumatic stress disorder among patients attending primary health care centres in the Gaza strip[END_REF][START_REF] Gaviria | Socio-demographic patterns of posttraumatic stress disorder in Medellin, Colombia and the context of lifetime trauma exposure[END_REF]Zambaldi et al., 2011). Among the factors retained age, gender, ethnicity and immigration status are often highlighted. Groups of our study did not differ in gender, age, education level. However, inhabitants of Caen which is a medium-sized provincial city have a different lifestyle from that of the Parisians. Such environmental aspects may have played a role in the differences observed with the non-exposed group compared to the others. Further studies should investigate the importance of these factors, in particular the social support variables, for the functional status of individuals.

Although pre-existing factors may explain the largest difference in behaviors between the groups in response to social stimuli, this pattern may also reflect an acquired deficit associated with PTSD. Avoidance is a maladaptive coping mechanism that is closely linked to learning abnormalities, and leads to the heightening of the anxiety response to environmental cues [START_REF] Lissek | Learning models of PTSD: Theoretical accounts and psychobiological evidence[END_REF]. This in turn cascades into overgeneralization of fear to non-harmful stimuli [START_REF] Dunsmoor | Fear generalization and anxiety: Behavioral and neural mechanisms[END_REF]. Further studies will be required to assess whether this overgeneralization extends to other anxiety-provoking situations, resulting in more defensive behaviors. Moreover, the exposed participants of our study developed or not a PTSD after a terrorist attack, which is a fairly particular event, which has a unique (not repeated) and sudden (unlike those of a military for example who may be concerned about the occurrence of an event) character, and it might thus impact the generalizability of our results to PTSD from other causes. Another particularity of our study is that we integrated participants with different levels of exposure to the traumatic events.

We were careful to control for this variable in our analyses and the majority of our participants are direct victims, but it might be interesting in a future study to better characterize the effect of the type of exposure on the differences we have highlighted by trying to get a significant equivalent number of participants for each level of exposure.

The cross-sectional nature of our study meant that it could not shed any light on the origin of the eye processing distinction between the exposed subgroups. However, our findings suggest that gaze processing could help us to understand resilience to trauma and treat PTSD. In the presence of negative social stimuli, understanding others' emotions, and benefitting from a healthy social environment, may be the keys to resilience and prevention. To the best of our knowledge, none of the current therapies for PTSD focus on the attention paid to social stimuli, such as the information conveyed by eye contact, whereas sociocognitive behavioral therapies deal with dysfunctions in social support (Tarrier & Humphreys, 2004). In recent years, current PTSD therapies have come in for criticism, mainly because of their failure to fully address trauma-related concerns, and more particularly social indicators (Yehuda et al., 2016). As a result, clinicians have suggested developing more integrative therapies as a way of considering all aspects of PTSD (Yehuda et al., 2016). The present study therefore opens up a new avenue for research that goes beyond reducing social discomfort, and aims to target social cues (i.e., eyes) to reduce avoidance symptoms. Effective treatments for PTSD often involve overcoming avoidance to improve the processing of distressing trauma-related material (e.g., [START_REF] Brewin | Memory and forgetting[END_REF]. If avoidance spreads to other, non-trauma-related contexts, such as eye contact, it may be possible to treat it by administering tasks that stimulate and remediate eye processing. Treatment that does not reawaken trauma could be an important complement to existing therapies, whose success in overcoming trauma avoidance does, to some degree, required fresh exposure to problematic content.

90% CI), nor the Happy versus Neutral contrast (t-peak(47) = -1.73, pFWER = .142, peak-max = 112 ms; t-mean(47) = -0.96, p = .170, [-0.03; 0.007] bootstrapped 90% CI) were not significant.

As expected, the results of the staring face appear to be consistent with what was obtained in the cueing face, but probably for the reasons described above, the differences are less pronounced. The correlation between the mean percentage of fixation on sad eyes in the staring face in PTSD+ and their PTSD avoidance symptoms followed the same pattern. Indeed,

although

Figure 1 .

 1 Figure 1. Experimental design and behavioral results. (a) Overview of the displays and timing of

Figure 2 .

 2 Figure 2. Eye-tracking results. (a) Mean percentage of fixations throughout the trials on the eye

  the reduced eye fixation over sad faces observed in PTSD+ participants was significantly related to avoidance symptoms in the cueing face, this result was only marginal in the staring face (r-skipped = -0.23, t(45) = -1.59, p = .06; [-0.50; 0.08] bootstrapped 90% CI; 3 bivariate outliers detected). Once again, this relationship was not significantly observed for happy faces (r-skipped = -0.03, t(47) = -0.18, p = .430; [-0.28; 0.20] bootstrapped 90% CI; 1 bivariate outlier detected) and was not found in PTSD-for both sad (r-skipped = -0.01, t(41) = -0.05, p = .480; [-0.29; 0.32] bootstrapped 90% CI; 1 bivariate outlier detected) or happy (rskipped = 0.01, t(41) = 0.08, p = .469; [-0.28; 0.31] bootstrapped 90% CI; 1 bivariate outlier detected) faces.

Table 1 .

 1 Eye-tracking results summarizing contrasts between the experimental conditions for PTSD+, PTSD-and nonexposed participants.

	PTSD+
	contrast	t-peak

(df) pFWER peak-max t-mean(df) p-mean bootstrapped 90% CI

	Sad versus Neutral	-3.59(48) .002 24	-3.36(48) < .001 [-0.06; -0.02]
	Happy versus Neutral	-1.95(48) .092 400	-1.17(48) .123	[-0.04; 0.007]
	Sad versus Happy	-2.38(48) .038 164	-1.75(48) .043	[-0.05; -0.012]
	PTSD-
	contrast	t-peak

(df) pFWER peak-max t-mean(df) p-mean bootstrapped 90% CI

	Sad versus Neutral	-2.22(45) .069 384	-0.90(45) .185	[-0.03; 0.008]
	Happy versus Neutral	-3.45(45) .004 372	-1.86(45) .035	[-0.04; -0.002]
	Sad versus Happy	0.30(45) .681 4	1.08(45) .143	[-0.006; 0.03]

Nonexposed contrast t-peak(df) pFWER peak-max t-mean(df) p-mean bootstrapped 90% CI

	Sad versus Neutral	-2.57(47) .029 124	-2.10(47) .021	[-0.038; -0.005]
	Happy versus Neutral	-0.56(47) .521 156	-0.07(47) .474	[-0.018; 0.016]
	Sad versus Happy	-2.18(47) .072 316	-2.17(47) .018	[-0.04; -0.005]
	Note. FWER = familywise error rate.

Table 2 .

 2 Eye-tracking results summarizing group contrasts for sad versus neutral, happy versus neutral, and sad versus happy. It should be noted that these results take into account five

	Sad

nuisance covariates: age, sex, education level, type of exposure to the traumatic event, and level of depressive symptoms.

versus Neutral contrast t-peak(df) pFWER peak-max t-mean(df) p-mean bootstrapped 90% CI

	PTSD+ versus PTSD-	2.77(93) .016 36	2.73(86)	.004	[0.006; 0.06]
	Nonexposed versus PTSD+	2.29(95) .047 152	1.89(88)	.031	[-0.005; 0.05]
	Nonexposed versus PTSD-	1.65(92) .172 60	1.22(85)	.113	[-0.04; 0.01]
	Happy versus Neutral
	contrast	t-peak(df) pFWER peak-max t-mean(df) p-mean bootstrapped 90% CI
	PTSD+ versus PTSD-	0.97(93) .361 44	0.73(86)	.235	[-0.03; 0.03]
	Nonexposed versus PTSD+	1.73(95) .138 400	1.45(88)	.075	[-0.01; 0.05]
	Nonexposed versus PTSD-	2.20(92) .064 392	1.28(85)	.102	[-0.006; 0.04]

Sad versus Happy contrast t-peak(df) pFWER peak-max t-mean(df) p-mean bootstrapped 90% CI

	PTSD+ versus PTSD-	1.87(93) .110 188	1.79(86)	.038	[0.008; 0.06]
	Nonexposed versus PTSD+	1.02(95) .343 160	0.24(88)	.406	[-0.02; 0.03]
	Nonexposed versus PTSD-	-0.78(92) .461 72	-0.17(85) .432	[-0.05; -0.008]

Acknowledgments

We thank all the people who volunteered to take part in this study and the victim organizations that supported this project. We also thank all the medical doctors, researchers, psychologists, technicians and administrative staff at the Neuropsychology and Imaging of Human Memory research unit (Caen), the 13-November Program (Paris), and INSERM's northwest regional

Funding details

This study was funded by the French Commissariat-General for Investment (CGI) via the National Research Agency (ANR) and the Investments for the Future Program (PIA) (ANR-10-EQPX-0021-01). The study was conducted within the framework of the 13 November Program (EQUIPEX Matrice) headed by D.P. and F.E, and supported by a postdoctoral fellowship from the Swiss National Science Foundation to S.Y.C. (grant no. 178002).

Data availability statement

Data

presented in the manuscript are available for download at https://unicloud.unicaen.fr/index.php/s/B8QkyeifYt2WdZm.

division (Lille), and clinical research hub (PRC). We finally thank Miss Portier for proofreading our manuscript.

Disclosure statement

The authors report no potential conflict of interest.

Author contributions statement

F.E., D.P., P.G. and M.L. conceived and designed the study. M.L. programmed the study. F.F. A review and meta-analysis. Genes, Brain and Behavior, 18(1), e12518. Tamir, M., & Ford, B. Q. (2012). Should people pursue feelings that feel good or feelings that do good? Emotional preferences and well-being. Emotion, 12( 5 Women's Mental Health,14(5) In agreement with the results of the cueing face, PTSD+ participants showed a significant reduction in eye fixation of sad faces compared with neutral faces (t-peak(48) = -3.46, pFWER =

.003, peak-max = 400 ms; t-mean(48) = -3.19, p = .001, [-0.05; -0.02] bootstrapped 90% CI). This sustained reduction increased significantly compared with PTSD-individuals (t-peak( 93 Note. FWER = familywise error rate.