Guillaume Cantin
email: guillaumecantin@mail.com

Non identical coupled networks with a geographical model for human behaviors during catastrophic events

Keywords: PCR system, coupled network, global stability, synchronization

The purpose of this work is to study coupled networks of non identical instances of the PCR system (Panic-Control-Reflex), which is a geographical model for human behaviors during catastrophic events. We endow the subsequent graph with superposed linear and quadratic couplings, and explore the effect of the topology of the network on the dynamics of each node. Especially, we investigate the possibility of controlling the panic level in the network by a clever disposal of the connections. We establish a necessary and sufficient condition for synchronization, without any reductive assumption on the nature of the network, and study the global stability of the trivial equilibrium. We illustrate our theoretical results by numerical simulations of randomly generated networks.

§1 Introduction

In this paper, we aim to present the continuation of a recent work [START_REF] Cantin | Mathematical modeling of human behaviors during catastrophic events: Stability and bifurcations[END_REF], concerning the PCR system (Panic-Control-Reflex), which is a geographical model for human behaviors during catastrophic events, established in 2013 in order to better understand and predict behavioral reactions of individuals facing a brutal disaster, which can have a natural or an industrial origin [START_REF] Provitolo | Un exemple d'effets de dominos : la panique dans les catastrophes urbaines[END_REF][START_REF] Verdière | Mathematical modeling of human behaviors during catastrophic events[END_REF][START_REF] Provitolo | Les comportements humains en situation de catastrophe : de l'observation à la modélisation conceptuelle et mathématique[END_REF]. In this previous work, a complete stability and bifurcation analysis of that dynamical system was proposed, exhibiting a degeneracy case of a saddle-node bifurcation at infinity, with two possible asymptotic behaviors for the solutions. The first possible asymptotic behavior corresponds to a favorable return of all individuals affected by the catastrophe to a daily way of life, while the second possible behavior of the solutions presents a persistence of panic.

The present paper is devoted to the study of coupled networks of non identical PCR systems, with the concern of modeling the geographical relief of the area impacted by the catastrophe, as well as the individuals displacements and contacts. In a recent conference paper [START_REF] Cantin | Control of panic behavior in a non identical network coupled with a geographical model[END_REF], a concrete example prepared with the collaboration of geographers, concerning the particular risk of tsunami on the Mediterranean coast, has been studied. It is shown in this latter article that the evacuation of high risk zones corresponding to the beach places, towards the refuge zones situated in the city center, plays a very decisive role. For instance, a plugged corridor can provoke a persistence of panic. At the opposite, an additional evacuation path from the beach towards the city-center can help individuals return to the daily behavior. For the sake of generality, we aim to prove for any abstract PCR network that a lightened organization of the connections, on the disaster zone, is a crucial key for a global return of the affected population to a daily behavior. Indeed, we shall first investigate patterns emerging from basic configurations, before looking ahead to larger networks, since it is a subtle question to relate the global dynamic of the network to the internal dynamic of each of its cells. The generality of our study gives a rigorous framework for better understanding the impact of the connections disposal in any situation concerned with our model, and it might be adapted for various non identical coupled networks.

A huge literature tackles the subject of coupled networks, with a mathematical approach. Many questions are discussed, such as synchronization of chaotic systems or oscillators [START_REF] Chua | Experimental chaos synchronization in Chua's circuit[END_REF][START_REF] Pecora | Master stability functions for synchronized coupled systems[END_REF][START_REF] Belykh | Synchronization and graph topology[END_REF][START_REF] Aziz-Alaoui | Synchronization of chaos[END_REF][START_REF] Arenas | Synchronization in complex networks[END_REF], bifurcation analysis and global normal form calculation [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF][START_REF] Rink | Coupled cell networks: semigroups, Lie algebras and normal forms[END_REF]. Different topologies are considered, with symmetry or orientation assumptions. Many applications to physical or biological phenomena are studied, such as electric circuits networks or neural networks [START_REF] Thieme | Mathematics in population biology[END_REF][START_REF] Zhang | Multiple Hopf bifurcations of symmetric bam neural network model with delay[END_REF][START_REF] Zhang | Multiple Hopf bifurcations of three coupled van der pol oscillators with delay[END_REF][START_REF] Steur | Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology[END_REF].

Our analysis of PCR systems networks will deal with non identical nodes coupled networks, with an asymmetric topology of variable density, built with the aim to model the variety of human behavioral reactions, according to the nature of the disaster, and to the geographical relief of the impacted zone. As a tool, we will use graph theory path finding algorithms [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF][START_REF] Gallo | Shortest path algorithms[END_REF], yet it is not the subject of our analysis.

The outline of the paper is the following. We shall begin with preliminaries about non identical coupled networks. After recalling some basics about the PCR system, we shall define PCR networks and study two-nodes configurations, with a linear or quadratic coupling form. Our main result is presented in the third section, and establishes a link between on the first hand the possible evacuation of panic nodes, that can be detected using graph theory algorithms, and on the other hand the research of a global return to a daily behavior among the considered population. The global stability of the trivial equilibrium will be analyzed in the general case, without any reductive assumption on the topology of the network. Finally, we shall present numerical simulations in order to illustrate our qualitative results, with randomly generated PCR networks. §2 Non identical coupled networks

Preliminaries

Let us consider a dynamical system

ẋ = f (t, x, λ), t ≥ t 0 , x ∈ Ω ⊂ R n , λ ∈ R p , ( 1
)
where n and p are positive integers, Ω is a subset of R n , f is a smooth function defined on R×Ω×R p , and t 0 is a given initial time. We suppose that the system (1) admits a bifurcation at λ 0 , that is a change in the stability and structure of its invariant sets for λ < λ 0 and for λ > λ 0 (see [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF] for instance, or [START_REF] Langa | Stability, instability, and bifurcation phenomena in non-autonomous differential equations[END_REF] for the case of non autonomous systems).

Next, we consider a graph G formed with a finite set V of N vertices {1, . . . , N } (N ≥ 2), and two finite sets E L and E Q of oriented edges, corresponding to linear and quadratic couplings respectively. An element (i, j) of E L or E Q corresponds to a directed connection from vertex i towards vertex j. For short, we will name the elements of E L and E Q linear edges and quadratic edges respectively. We associate to each vertex i ∈ V an instance of system (1) with one parameter value λ i , and consider a coupled network of the form

ẋi = f (t, x i , λ i ) + N j=1 L i,j (t)Hx j + N j=1 Q(t, X) i,j , 1 ≤ i ≤ N, (2)
where x i ∈ R n is the vector representing the state of vertex i and X = (x 1 , . . . , x N ) T . Here, we consider the matrix of linear connectivity L(t) = (L i,j (t)), which is symmetric if the linear edges of the graph are non oriented. More precisely, if (i, j) ∈ E L , i = j, we have L j,i (t) > 0, for all t ≥ t 0 . If (i, j) / ∈ E L , i = j, we have L j,i (t) = 0, for all t ≥ t 0 . Moreover, the diagonal elements of matrix L satisfy

L i,i (t) = - N j=1 j =i L j,i (t), (3)
thus L(t) is a zero column sum matrix. The matrix H is of order n, and determines which components of vertices x i and x j are coupled. Similarly, we define the matrix of quadratic connectivity Q(t, X) as follows.

If (i, j) ∈ E Q , i = j, we set Q(t, X) j,i = -Q j,i (t)κ(x i , x j ), Q(t, X) i,j = +Q j,i (t)κ(x i , x j ), (4)
where Q j,i (t) is a smooth real valued function of t with positive values. The function κ is defined on R n × R n by a quadratic expression

κ(x i , x j ) =   1≤α,β≤n κ α,β l x (α) i x (β) j   T , 1 ≤ l ≤ n, (5) where x (α)
i , 1 ≤ α ≤ n denote the components of x i , and κ α,β l ∈ R for all l, α and β such that 1 ≤ l ≤ n, 1 ≤ α ≤ n and 1 ≤ β ≤ n. The matrix of coefficients Q j,i is of order N and anti symmetric. We precise that the function κ is not necessarily symmetric.

For convenience, we introduce the function Q(t, X), defined as a R n×N vector valued function, by setting

Q(t, X) = Q i (t, X) T =   N j=1 Q(t, X) i,j   T , 1 ≤ i ≤ n. ( 6
)
In the sequel, we will omit the time dependence of coefficients in matrices L and Q, in order to lighten our notations. Furthermore, we will assume that there exists a minimum coupling strength

η 0 > 0 such that L i,j (t) ≥ η 0 , Q i,j (t) ≥ η 0 , (7)
for all t ≥ t 0 and i, j such that 1 ≤ i, j ≤ n, L i,j (t) = 0 or Q i,j (t) = 0. We will refer to the vertices of G with the word nodes. The term f (t, x i , λ i ) determines the internal dynamic of each node, while the coupling terms contained in L and Q(X) depend on the topology of the network. For convenience and clarity, we will associate to each parameter value λ i one color. Two nodes x i and x j of the same color will then admit the same parameter value, that is λ i = λ j , thus the same internal dynamic.

Example 1. Assume n = 2, N = 4, and consider the graph G depicted in Figure 1. The set

x 1 x 2 L2,1 x 3 L3,2 x 4 L4,2 Q3,1 Q4,3 Q2,4
E L of linear edges is {(1, 2), (2, 3), (2, 4)}. Thus L =     -L 2,1 0 0 0 L 2,1 -L 3,2 -L 4,2 0 0 0 L 3,2 0 0 0 L 4,2 0 0     .
The set E Q of quadratic edges is {(1, 3), [START_REF] Aziz-Alaoui | Synchronization of chaos[END_REF][START_REF] Belykh | Synchronization and graph topology[END_REF], (4, 2)}. Thus

Q(X) =     0 0 -Q 3,1 κ(x 1 , x 3 ) 0 0 0 0 Q 2,4 κ(x 4 , x 2 ) Q 3,1 κ(x 1 , x 3 ) 0 0 -Q 4,3 κ(x 3 , x 4 ) 0 -Q 2,4 κ(x 4 , x 2 ) Q 4,3 κ(x 3 , x 4 ) 0     .
We consider three distinct parameter values λ 1 , λ 2 , λ 3 , and the coupled network given by

         ẋ1 = f (t, x 1 , λ 1 ) -L 2,1 Hx 1 -Q 3,1 κ(x 1 , x 3 ) ẋ2 = f (t, x 2 , λ 2 ) + L 2,1 Hx 1 -(L 3,2 + L 4,2 )Hx 2 + Q 2,4 κ(x 4 , x 2 ) ẋ3 = f (t, x 3 , λ 1 ) + L 3,2 Hx 2 + Q 3,1 κ(x 1 , x 3 ) -Q 4,3 κ(x 3 , x 4 ) ẋ4 = f (t, x 4 , λ 3 ) + L 4,2 Hx 2 + Q 4,3 κ(x 3 , x 4 ) -Q 2,4 κ(x 4 , x 2 ). (8) Let x (1) i , x (2) i denote the components of x i ∈ R 2 , i ∈ {1, . . . , N }.
If the first component of x i is coupled, but not the second one, we set:

H = 1 0 0 0 , thus Hx i = x (1) i , 0 T .
The following definition is classical for identical synchronization. For X 0 in Ω and t 0 ∈ R, we denote by X(t, t 0 , X 0 ) ∈ R n×N the orbit of system (2) passing through X 0 at t 0 , and x i (t, t 0 , X 0 ) ∈ R n the projection of X(t, t 0 , X 0 ) corresponding to the state of node i for any i ∈ {1, . . . , N }. Definition 1. Two nodes i and j of a coupled network [START_REF] Arenas | Synchronization in complex networks[END_REF] are said to identically synchronize if:

lim t→+∞ x i (t, t 0 , X 0 ) -x j (t, t 0 , X 0 ) R n = 0, (9)
for any initial condition X 0 ∈ Ω. We say that the whole coupled network (2) synchronizes if any pair of nodes identically synchronizes.

The main question we are interested in, is to determine the global dynamic of the network. In particular, we would like to know if one node can drive the rest of the network to its internal dynamic, and to analyze the effect of the topology on the bifurcation identified in the dynamical system (1).

PCR system

The PCR system is a model for human behaviors during catastrophic events, established in 2013 to better understand and predict human reactions facing a brutal disaster [START_REF] Provitolo | Un exemple d'effets de dominos : la panique dans les catastrophes urbaines[END_REF][START_REF] Provitolo | Les comportements humains en situation de catastrophe : de l'observation à la modélisation conceptuelle et mathématique[END_REF][START_REF] Verdière | Mathematical modeling of human behaviors during catastrophic events[END_REF][START_REF] Cantin | Mathematical modeling of human behaviors during catastrophic events: Stability and bifurcations[END_REF] . It is given by the following adimensional system of ordinary equations: [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF] for all t ≥ t 0 with a given t 0 ∈ R, together with an initial condition (r 0 , c 0 , p 0 , q 0 , b 0 ) ∈ (R + ) 5 that satisfies the property

               ṙ = γ(t)q(1 -r) -(B 1 + B 2 )r + F (r, c)rc + G(r, p)rp ċ = B 1 r + C 1 p -C 2 c -F (r, c)rc + H(c, p)cp -ϕ(t)c(1 -b) ṗ = B 2 r -C 1 p + C 2 c -G(r, p)rp -H(c, p)cp q = -γ(t)q(1 -r) ḃ = ϕ(t)c(1 -b),
r 0 + c 0 + p 0 + q 0 + b 0 = 1, (11)
corresponding to the fact that the population densities that we study have been adimensioned. The unknown functions r, c, p, q, b are real valued functions, and denote respectively the densities of individuals in reflex, control, panic, daily and back to daily behaviors, among a population concerned with the catastrophe. It is worth noting that the daily behavior group is divided into two subgroups corresponding to the unknowns q and b, that respectively correspond to the daily behavior before the beginning of the catastrophe, and the daily behavior after the catastrophe. Of particular interest is the initial condition (0, 0, 0, 1, 0), which corresponds to the situation of a catastrophic event without any alert to the population, and consequently means that all individuals are in a daily behavior before the disaster. Since the sum of the 5 equations in system [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF] is equal to 0, we can reduce the system as follows:

         ṙ = γ(t)q(1 -r) -(B 1 + B 2 )r + F (r, c)rc + G(r, p)rp ċ = B 1 r + C 1 p -C 2 c -F (r, c)rc + H(c, p)cp -ϕ(t)c(r + c + p + q) ṗ = B 2 r -C 1 p + C 2 c -G(r, p)rp -H(c, p)cp q = -γ(t)q(1 -r). ( 12
)
The expected asymptotic behavior of the orbits of system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] corresponds to a return of all individuals to a daily behavior after the catastrophe, that is

lim t→+∞ b(t) = 1, or equivalently lim t→+∞ r(t) + c(t) + p(t) + q(t) = 0.
The functions γ and ϕ model respectively the beginning of the disaster, and the return to a daily behavior. They both satisfy the properties

γ(t) > 0, ϕ(t) > 0, ∀t > t 0 , ( 13
)
and

γ(t) = ϕ(t) = 1, ∀t ≥ t 1 , ( 14
)
for a given t 1 > t 0 . Property ( 14) implies that the system ( 12) is non autonomous only for a finite time. This will be useful in the sequel, especially for the study of the global asymptotic stability of the trivial equilibrium, presented in the section ( §3). The parameters B i > 0, C i ≥ 0, i ∈ {1, 2}, model the behavioral changes of each individual, while the functions F , G and H model the interaction phenomena that act in parallel.

Remark 1.

The values of those parameters are partially chosen by a qualitative approach, in collaboration with geographers. Because of the lack of data, the fine numerical calibration of the model is a work in progress, which implies a narrow working association with secure services.

The next proposition summaries the qualitative results of the mathematical analysis.

Proposition 1. For any initial condition (r 0 , c 0 , p 0 , q 0 ) ∈ (R + ) 4 , the system (12) admits a unique global solution whose components are non negative and bounded. If C 1 > 0, O(0, 0, 0, 0) is the only equilibrium point, and it is locally asymptotically stable. If C 1 = 0, the system presents a persistence of panic behavior, that is:

lim t→+∞ p(t) = p > 0,
for any initial condition (r 0 , c 0 , p 0 , q 0 ) ∈ (R + ) 4 satisfying r 0 + c 0 + p 0 + q 0 > 0.

The complete proof is presented in [START_REF] Cantin | Mathematical modeling of human behaviors during catastrophic events: Stability and bifurcations[END_REF], where the authors show that the bifurcation occurring in the system, when the evolution parameter C 1 passes through 0, is a degeneracy case of a saddlenode bifurcation at infinity. They also focus on the inhibition role of quadratic terms in the case of a persistence of panic, and highlight the decisive potential role played by the total population density involved in the disaster, that is T = r + c + p + q, which is considered in order to build a Lyapunov function for the stability analysis. This potential role will appear again in the next section. §3 Coupled networks of non identical PCR systems

PCR networks

In what follows, we will consider a PCR network, that is, a graph whose nodes are coupled with non identical instances of the PCR system, whose state equations can be rewritten

ẋ = f (t, x, C 1 ), t ≥ t 0 , x ∈ R 4 , C 1 ≥ 0, (15)
where x = (r, c, p, q) T , and f (t, x, C 1 ) is defined by:

f (t, x, C 1 ) =     γ(t)q(1 -r) -(B 1 + B 2 )r + F (r, c)rc + G(r, p)rp B 1 r + C 1 p -C 2 c -F (r, c)rc + H(c, p)cp -ϕ(t)c(r + c + p + q) B 2 r -C 1 p + C 2 c -G(r, p)rp -H(c, p)cp -γ(t)q(1 -r)     .
Definition 2. The nodes which are coupled with an instance of system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] where C 1 = 0 will be called panic nodes or nodes of type (1), while those which are coupled with an instance of system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] where C 1 > 0 will be called control nodes or nodes of type (2).

Remark 2. The value of the evolution parameter C 1 is chosen according to geographical considerations, among them are the nature of the catastrophe, the position of the nodes with respect to the impact zone of it, but also human behavioral aspects. The example of an earthquake in Japan is studied in [START_REF] Verdière | Understanding and simulation of human behaviors in areas affected by disasters: From the observation to the conception of a mathematical model[END_REF]. Since in Japan, the risk culture is well established, the population is formed to react quickly, thus the causality process from reflex to control is important with respect to the other processes, which implies a positive value for C 1 , even for a node which is closed to the epicenter of the earthquake. At the opposite, a tsunami on the Mediterranean coast [START_REF] Cantin | Control of panic behavior in a non identical network coupled with a geographical model[END_REF] presents a non homogeneous structure. Because of the effect of surprise and fear, reactions are more instinctive in the heart of the disaster zone, namely the beach places. But the particular urban landscape of Mediterranean coastal cities, with dense centers, narrow streets, and elevation over the sea level, is known to hide the arrival of such an event, and represents a quality refuge zone. For those reasons, beach places are modeled by panic nodes, and city center zones by control nodes.

Next, we study a network made of n nodes x i , 1 ≤ i ≤ n of type (1), and m nodes

y j C j 1 , 1 ≤ j ≤ m of type (2), with C j 1 > 0, 1 ≤ j ≤ m. The whole network system reads Ẋ = Φ(t, X, C) + L X + Q(X), (16)
where the vectors X, X ∈ R 4(n+m) and C ∈ R n+m are defined by

     X = (x 1 , . . . , x n , y 1 , . . . , y m ) T X = (Hx 1 , . . . , Hx n , Hy 1 , . . . , Hy m ) T C = 0, . . . , 0, C 1 1 , . . . , C m 1 ,
and Φ corresponds to the internal dynamic of each node, and is defined by

Φ(t, X, C) = f (t, x 1 , 0), . . . , f (t, x n , 0), f t, y 1 , C 1 1 , . . . , f (t, y m , C m 1 ) T .
The matrix H determines which components are coupled:

H =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0     . ( 17
)
The latter definition of H means that we consider that only individuals in a catastrophe behavior (reflex, control or panic, but not daily behavior) are concerned with migrations from one node to another. The matrix L contains the coupling terms, and can be written as a block matrix:

L = L 1 L 2 L 3 L 4 , ( 18
)
where

L 1 = L 1 i,j , 1 ≤ i, j ≤ n, corresponds to the couplings among the nodes of type (1), L 2 = L 2 i,j , 1 ≤ i ≤ n, 1 ≤ j ≤ m, and L 3 = L 3 i,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n,
to the couplings between nodes of type (1) and nodes of type (2), and L 4 = L 4 i,j , 1 ≤ i, j ≤ m, to the couplings among the nodes of type (2) (see Figure 2).

The quadratic couplings are stored in

Q(X) = Q 1 (X), Q 2 (X)
, which is defined by the coefficients of the matrix Q(X) (see ( 4) and ( 6)), and the function κ, given by

κ x, x = x 1 (x 2 + x3 ), x 2 (x 1 + x3 ), x 3 (x 1 + x2 ), 0 T , ( 19
)
for all x, x ∈ R 4 . This expression models the situation when individuals of node i imitate individuals of node j of another behavior group, except those who are in the daily behavior. Let us consider the concrete example of an earthquake in a low density zone in Japan. Let (r 1 , c 1 , p 1 ) and (r 2 , c 2 , p 2 ) denote the reflex, control and panic behaviors on two nodes 1 and 2 respectively. We suppose that the node 1 is located at the epicenter of the catastrophe, and thus can present a persistence of panic in absence of coupling, whereas the node 2 corresponds to a refuge zone which favors the return to a daily behavior. In that case, the equations of ṗ1 , ṗ2 , ċ2 read as follows:

x i x k L 1 i,k y k L 2 i,k x l L 1 l,i y l L 3 l,i y j x k L 3 j,k y k L 4 j,k x l L 2 l,j y l L 4 l,
     ṗ1 = f 1 (r 1 , c 1 , p 1 ) -ε L p 1 -ε Q p 1 c 2 ṗ2 = f 2 (r 2 , c 2 , p 2 ) + ε L p 1 ċ2 = f 3 (r 2 , c 2 , p 2 ) + ε Q p 1 c 2 ,
where f 1 , f 2 , f 3 correspond to the internal dynamic in each equation (see [START_REF] Krasovskij | Stability of motion: applications of Lyapunov's second method to differential systems and equations with delay[END_REF]). The linear coupling ±ε L p 1 models the displacements of individuals in panic behavior from node 1 towards node 2. During this displacement, no behavioral evolution occurs. In a different manner, the quadratic coupling ±ε Q p 1 c 2 is the result of an interaction, say a message which is spread by rescue services through loudspeakers, and leads simultaneously to a displacement and a behavioral evolution: individuals in panic near the epicenter are calmed down by the rescue message, thus arrive at the refuge zone in a control behavior.

For clarity, we adopt the coloring representation that associates red to the panic nodes, and green to the control nodes (see Figure 2). The PCR network ( 16) can be rewritten in an expanded form:

                       ẋi = f (t, x i , 0) -     n k=1 k =i L 1 k,i + m l=1 L 3 l,i     Hx i + n k=1 k =i L 1 i,k Hx k + m l=1 L 2 i,l Hy l + Q 1 i (X), ẏj = f t, y j , C j 1 -     n k=1 L 2 k,j + m l=1 l =j L 4 l,j     Hy j + n k=1 L 3 j,k Hx k + m l=1 l =j L 4 j,l Hy l + Q 2 j (X), ( 20
) where x i ∈ R 4 , 1 ≤ i ≤ n and y j ∈ R 4 , 1 ≤ j ≤ m.
In order to distinguish the components of the nodes of type ( 1) and ( 2), we introduce the following notation:

x i = r (1) i , c (1) i , p (1) i , q (1) i T , 1 ≤ i ≤ n, y j = r (2) j , c (2) j , p (2) j , q (2) j T , 1 ≤ j ≤ m.

Remark 4.

The PCR networks that we study are inspired by some geographical configurations [START_REF] Provitolo | Les comportements humains en situation de catastrophe : de l'observation à la modélisation conceptuelle et mathématique[END_REF], which determine the values of the parameters of each node, according to their spatial position, and the coupling strengths, according to the dimensions of the communication paths. Thus, the geographical metric structure of the zone affected by the catastrophe is implicitly taken into account in the parameters of the network. More precisely, the coupling strength along one edge is inversely proportional to the geographical distance between the two corresponding nodes. Proposition 2. For any initial condition X 0 ∈ (R + ) 4(n+m) , there exists τ > 0 such that the Cauchy problem defined by [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF] and X(t 0 ) = X 0 admits a unique solution X(t, t 0 , X 0 ) defined on [t 0 , t 0 + τ [, whose components are non negative.

We skip the proof, since it is very similar to the proof of positiveness of the solution of the PCR system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF], detailed in [START_REF] Cantin | Mathematical modeling of human behaviors during catastrophic events: Stability and bifurcations[END_REF]. In the absence of coupling, the panic nodes x i , 1 ≤ i ≤ n will exhibit a panic persistence, while the control nodes y j , 1 ≤ j ≤ m will present a return to daily behavior.

Next, we consider the total population density T involved in the disaster, defined by:

T (X) = n i=1 r (1) i + c (1) i + p (1) i + q (1) i + m j=1 r (2) j + c (2) j + p (2) j + q (2) j . ( 21
)
Proposition 3. The function T is non negative and decreasing on the interval [t 0 , t 0 + τ [ along the solution X(t, t 0 , X 0 ) of the PCR network starting from an initial condition X 0 ∈ (R + ) 4(n+m) . Assume that the solution is global, that is τ = +∞. If T converges to 0, then the PCR network [START_REF] Prigogine | Self organization in non-equilibrium systems[END_REF] synchronizes towards the trivial equilibrium.

Proof. The sum of the 4(n + m) equations in [START_REF] Prigogine | Self organization in non-equilibrium systems[END_REF] makes the coupling terms vanish and leads to

Ṫ (X(t, t 0 , X 0 )) = -ϕ(t)   n i=1 c (1) i (t) r (1)
i + c

(1)

i + p (1) i + q (1) i (t) + m j=1 c (2) j (t) r (2) j + c (2) j + p (2) j + q (2) j (t)   ,
thus we have Ṫ ≤ 0. Next, the non negativity of the components (see Proposition 2) implies that (2) j , 1 ≤ j ≤ m. Thus, if T converges to 0, then necessarily all the components of the nodes of type (1) and (2) converge to 0, that is the PCR network synchronizes towards the trivial equilibrium.

0 ≤ r (1) i (t) ≤ T (t), 0 ≤ c (1) i (t) ≤ T (t), 0 ≤ p (1) i (t) ≤ T (t), 0 ≤ q (1) i (t) ≤ T (t), 1 ≤ i ≤ n, ( 22
The total population density T is also likely not to converge to 0, but to a positive limit. In that case, the network is susceptible to exhibit a persistence of panic. The complete answer to that question will be given at the end of the section, as a consequence of Theorem 1. Let us consider the energy function

V = 1 2 T 2 . ( 23
)
It is non negative, with a non positive orbital derivative by virtue of Proposition 3, and it enables us to construct an invariant region.

Proposition 4. The compact set Ω = X ∈ (R + ) 4(n+m) , V (X) ≤ 1 is a positively invariant region for the flow induced by the PCR network [START_REF] Prigogine | Self organization in non-equilibrium systems[END_REF].

Proof. Let us consider X 0 ∈ Ω. Since V X(t, t 0 , X 0 ) ≤ 0, then V X(t, t 0 , X 0 ) is decreasing. Therefore, X(t, t 0 , X 0 ) remain in Ω for all future time.

Consequently, the solution of the PCR network starting from X 0 ∈ Ω is global. In the rest of the paper, we study the effect of the coupling in the network. Definition 3. We will say that the PCR network [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF] presents a global return to daily behavior if:

lim t→+∞ X(t, t 0 , X 0 ) R 4(n+m) = 0, ( 24
)
for any X 0 ∈ Ω. We will say that a panic node (x) is evacuated towards a control node (y) by a linear chain, if there exists an oriented path in the subsequent graph made of linear edges that connects (x) to (y).

Two nodes converging to the trivial equilibrium O ∈ R 4 for any initial condition obviously synchronize in the sense of definition (1), while two nodes exhibiting a panic persistence are susceptible not to synchronize. The global return of a PCR network to daily behavior, which is the expected dynamic, is a sufficient condition for synchronization, since the solution is attracted to the trivial equilibrium O ∈ R 4(n+m) . When a panic node (x) is evacuated towards a control node (y), there may be multiple paths connecting (x) to (y). The length of one path is the number of edges along it (see Figure 6).

Patterns emerging from two-nodes configurations

In this section, we study two-nodes PCR networks (see Figure 3), considering a non symmetric coupling, with a linear or quadratic form, and show which patterns emerge from those basic configurations.

x y

η (a)
x y

η (b)
x y We first consider a two-nodes PCR network with a panic node (x) connected to a control node (y) in a linear form (see Figure 3(a)). Such a network is given by the following system:

ξ (c) x y η ξ (d)
ẋ = f (t, x, 0) -ηHx ẏ = f (t, y, C 1 ) + ηHx, ( 25
)
where x = (r 1 , c 1 , p 1 , q 1 ) T , y = (r 2 , c 2 , p 2 , q 2 ) T , C 1 > 0, H is defined by [START_REF] Langa | Stability, instability, and bifurcation phenomena in non-autonomous differential equations[END_REF], and η > 0 corresponds to the coupling strength.

Proposition 5. The system (25) admits a unique equilibrium O ∈ R 8 . Furthermore, it is locally asymptotically stable.

Proof. We first look for the equilibrium points of node (x), which can be seen as a perturbation of a PCR system. To that aim, we solve:

         γ(t)q 1 (1 -r 1 ) -(B 1 + B 2 )r 1 + F (r 1 , c 1 )r 1 c 1 + G(r 1 , p 1 )r 1 p 1 -ηr 1 = 0 B 1 r 1 -C 2 c 1 -F (r 1 , c 1 )r 1 c 1 + H(c 1 , p 1 )c 1 p 1 -ϕ(t)c 1 (r 1 + c 1 + p 1 + q 1 ) -ηc 1 = 0 B 2 r 1 + C 2 c 1 -G(r 1 , p 1 )r 1 p 1 -H(c 1 , p 1 )c 1 p 1 -ηp 1 = 0 -γ(t)q 1 (1 -r 1 ) = 0.
The sum of the 4 equations leads to:

-ϕ(t)c 1 (r 1 + c 1 + p 1 + q 1 ) -η(r 1 + c 1 + p 1 ) = 0,
thus r 1 = c 1 = p 1 = q 1 = 0, since the components of the solution are non negative. We then write the 4 equations of node (x) as:

ẋ = M x + N (t)x + (t, x),
where M and N (t) are two matrices of order 4 defined by:

M =     -B 1 -B 2 -η 0 0 1 B 1 -C 2 -η 0 0 B 2 C 2 -η 0 0 0 0 -1     , N (t) =     0 0 0 γ(t) -1 0 0 0 0 0 0 0 0 0 0 0 -γ(t) + 1     ,
and contains non linear terms:

(t, x) =     -γ(t)q 1 r 1 + F (r 1 , c 1 )r 1 c 1 + G(r 1 , p 1 ) -F (r 1 , c 1 )r 1 c 1 + H(c 1 , p 1 )c 1 p 1 -ϕ(t)c 1 (r 1 + c 1 + p 1 + q 1 ) -G(r 1 , p 1 )r 1 p 1 -H(c 1 , p 1 )c 1 p 1 γ(t)q 1 r 1     .
The eigenvalues of M are given by:

-B 1 -B 2 -η, -C 2 -η, -η, -1,
thus they are negative, since η > 0. Furthermore, some basic algebraic computations, and property ( 14), lead to:

lim t→+∞ N (t) = 0, lim x →0 (t, x) x = 0, uniformly in t.
The smoothness of functions γ, ϕ, F , G and H implies that of , which is consequently locally Lipschitz in its second argument. Thus the Poincaré-Lyapunov theorem (see [START_REF] Verhulst | Nonlinear differential equations and dynamical systems[END_REF] for instance) applies and guarantees that the equilibrium (0, 0, 0, 0) is locally asymptotically stable. Finally, we look for the equilibrium points of node (y), which are given by:

         γ(t)q 2 (1 -r 2 ) -(B 1 + B 2 )r 2 + F (r 2 , c 2 )r 2 c 2 + G(r 2 , p 2 )r 2 p 2 + ηr 1 = 0 B 1 r 2 + C 1 p 2 -C 2 c 2 -F (r 2 , c 2 )r 2 c 2 + H(c 2 , p 2 )c 2 p 2 -ϕ(t)c 2 (r 2 + c 2 + p 2 + q 2 ) + ηc 1 = 0 B 2 r 2 -C 1 p 2 + C 2 c 2 -G(r 2 , p 2 )r 2 p 2 -H(c 2 , p 2 )c 2 p 2 + ηp 1 = 0 -γ(t)q 2 (1 -r 2 ) = 0.
Since (r 1 , c 1 , p 1 , q 1 ) = (0, 0, 0, 0) is the only equilibrium point of node (x), we obtain the equations corresponding to the equilibrium points of a PCR system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] with C 1 > 0. Thus (r 2 , c 2 , p 2 , q 2 ) = (0, 0, 0, 0), and the only equilibrium point for the two-nodes network ( 25) is 0 ∈ R 8 . Its stability follows from the stability of the equilibrium point (0, 0, 0, 0) of node (x), combined with proposition (1).

Remark 5. This first proposition means that an evacuation of individuals in panic behavior from a panic node towards a control node, brings the whole network to a global return to daily behavior.

In other words, the linear coupling affects the bifurcation on the panic node (x), and makes the panic persistence vanish. Furthermore, it will appear in the next paragraph, as a consequence of the general case, that the trivial equilibrium is actually asymptotically globally stable. Thus the two nodes synchronize, and the control node drives the panic node to its dynamic.

We know from theory of ordinary differential equations [START_REF] Coddington | Theory of ordinary differential equations[END_REF], that any solution x(t) of the first equation in system [START_REF] Thieme | Mathematics in population biology[END_REF] satisfies the property:

lim sup t→+∞ ln x(t) t ≤ max 1≤k≤4 (λ k ) ,
where λ k denote the eigenvalues of matrix M . The latter proof shows that if η < 1, then:

lim sup t→+∞ ln x(t) t ≤ -η,
thus an exponential acceleration of the convergence of node (x) towards the trivial equilibrium under an increase of the coupling strength η (see Figure 4). We then look ahead to the inverse situation, when a control node (y) is connected towards a panic node (x) (see Figure 3 26) exhibits a persistence of panic on node (x).

Proof. We begin with the research of the equilibrium points of node (y), which is very similar to the previous proof, and leads to the uniqueness and local asymptotic stability of (r 2 , c 2 , p 2 , q 2 ) = (0, 0, 0, 0). It follows that the equilibrium points of node (x) correspond to the equilibrium points of a PCR system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] with C 1 = 0, thus the persistence of panic in node (x).

Remark 6. This second proposition suggests that a displacement of individuals of a control node towards a panic node should be avoided, at the risk to worsen the panic persistence level.

Next, we consider a quadratic coupling from a panic node towards a control node (see Figure 3(c)): ẋ = f (t, x, 0) -ξκ(x, y) ẏ = f (t, y, C 1 ) + ξκ(x, y), [START_REF] Verdière | Mathematical modeling of human behaviors during catastrophic events[END_REF] where the quadratic coupling κ(x, y) is defined by [START_REF] Pecora | Master stability functions for synchronized coupled systems[END_REF].

Proposition 7. System (27) presents a persistence of panic on node (x).

We skip the proof, since it is very similar to the latter. Figure 5 shows the shape of the panic component of node (x) for each of the latter situations. Proposition [START_REF] Chua | Local activity is the origin of complexity[END_REF] shows that a quadratic coupling is not efficient to empty the panic behavior in the panic nodes, and does not affect the bifurcation in the system.

Finally, we prove that a superposition of a linear and a quadratic couplings (see Figure 3(d)) can inhibit the panic persistence of node (x), by considering the following system:

ẋ = f (t, x, 0) + ηHy -ξκ(x, y) ẏ = f (t, y, C 1 ) -ηHy + ξκ(x, y). ( 28
)
Proposition 8. The solution of system [START_REF] Verhulst | Nonlinear differential equations and dynamical systems[END_REF] stemming from the initial condition (x 0 , y 0 ) = (0, 0, 0, 1, 0, 0, 0, 1)

presents a persistence of panic, which decreases if ξ increases and η is sufficiently small.

Proof. The demonstration is based on the Taylor expansion of the solution X ξ of system [START_REF] Verhulst | Nonlinear differential equations and dynamical systems[END_REF] according to parameter ξ. We write:

X ξ = X 0 + ξX 1 + ξ 2 X 2 + . . . ,
where the first term X 0 corresponds to the solution of system ( 28) with ξ = 0, and the dots indicate terms of order higher than 2 in ξ. After some basic but tedious computations, we obtain:

r (1) 1 (4) (0) = -3(B 1 + B 2 ), c (1) 1 (4) (0) = -3B 1 , p (1) 1 (4) (0) = -3B 2 ,
while the previous derivatives evaluated in 0 are null. This guarantees the negativeness of the components r

(1)

1 , c

(1)

1 and p

(1)

1 of node (1) on a small interval [t 0 , t 0 + τ [. Finally, we prove that τ = +∞ for η sufficiently small, by a similar reasoning as in the proof of the positiveness of the solution of the PCR system [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF]. Quadratic coupling from panic node to control node: the panic persistence survives, but is inhibited.

Condition for a global return to normal behavior and global stability of the trivial equilibrium

In this section, we present and prove our main result, which gives a necessary and sufficient condition for the synchronization of a PCR network. We next analyze the global stability of the trivial equilibrium. To that aim, we introduce, for each k ≥ 1, the set P k of all panic nodes which are connected to at least one control node, by an oriented chain of linear edges of length k, and not by a shorter chain (see Figure 6). Thus P k ∩ P l = ∅ if k = l. We recall that a panic node can be evacuated towards a control node by multiple paths (see Definition 3), and that the length of such a path is equal to the number of edges along it. A panic node is also likely to be isolated, which means the opposite of being evacuated.

Theorem 1. The PCR network (20) admits a unique equilibrium, which is the trivial equilibrium O ∈ R 4(n+m) , if and only if every panic node is evacuated by an oriented chain made of linear edges, towards one control node. In that case, the trivial equilibrium is locally stable.

Proof. The proof is divided into three steps. First, we look for the equilibrium points in the system [START_REF] Prigogine | Self organization in non-equilibrium systems[END_REF], and show that the only components which are not obviously null are the panic components of the panic nodes. Hence, all the quadratic coupling terms vanish in the subsequent equilibrium

x 1 x 2 x 3 x 4 x 5 x 6 x 7 y 1 y 2 L 1 2,1 L 1 2,7 L 1 3,2 L 3 1,2 L 1 5,4 L 1 1,5 L 1 6,3 L 3 2,3 P 1 P 2 P 3 P 4
Figure 6: PCR network with different subsets of panic nodes corresponding to the possible evacuation paths towards control nodes. The nodes x 2 and x 3 belong to the subset P 1 since they are evacuated to one control node by a chain of length 1. The panic node x 6 does not belong to any subset P k . equations. In the second step, we examine if each panic node is evacuated towards a control node by a chain made of linear edges, using an induction reasoning. Finally, we consider a Lyapunov function in order to analyze the local stability of the trivial equilibrium (the global stability will be studied in the next theorem).

First step. In order to find equilibrium points, we solve the following system:

                       f (t, x i , 0) -     n k=1 k =i L 1 k,i + m l=1 L 3 l,i     Hx i + n k=1 k =i L 1 i,k Hx k + m l=1 L 2 i,l Hy l + Q 1 i (X) = 0 f t, y j , C j 1 -     n k=1 L 2 k,j + m l=1 l =j L 4 l,j     Hy j + n k=1 L 3 j,k Hx k + m l=1 l =j L 4 j,l Hy l + Q 2 j (X) = 0, (29)
for all t ≥ t 0 , 1 ≤ i ≤ n and 1 ≤ j ≤ m. The sum of the 4(n + m) equations in (29) leads to:

-ϕ(t)   n i=1 c (1) i r (1) i + c (1) i + p (1)
i + q

(1) i + m j=1 c (2) j r (2) j + c (2) j + p (2) j + q (2) j   = 0.
Since ϕ(t) > 0 for all t > t 0 , and the components of the solution are non negative, we obtain:

c (1) i = 0, 1 ≤ i ≤ n, c (2)
j = 0, 1 ≤ j ≤ m.
Next, considering the second components equations, we obtain for nodes of type (1):

B 1 r (1) i = 0, 1 ≤ i ≤ n,
and for nodes of type (2):

B 1 r (2) j + C j 1 p (2) j = 0, 1 ≤ j ≤ m. This leads to r (1) i = 0, 1 ≤ i ≤ n, r (2)
j = 0, 1 ≤ j ≤ m, since we have B 1 > 0, and similarly:

p (2) j = 0, 1 ≤ j ≤ m, since C j
1 > 0 for all j ∈ {1, . . . , m}. Moreover, we recall that the components q

(1)

i , 1 ≤ i ≤ n and q

(2) j , 1 ≤ j ≤ m are not coupled (see [START_REF] Langa | Stability, instability, and bifurcation phenomena in non-autonomous differential equations[END_REF]). The corresponding equilibrium equations give:

q (1) i = 0, 1 ≤ i ≤ n, q (2) j = 0, 1 ≤ j ≤ m.
Second step. We examine if each panic node is evacuated towards a control node by a chain made of linear edges.

Considering that p

(2) j = 0 for all j ∈ {1, . . . , m}, we obtain for the panic components of the panic nodes:

-     n k=1 k =i L 1 k,i + m l=1 L 3 l,i     p (1) i + n k=1 k =i L 1 i,k p (1) k = 0, (30)
for all i ∈ {1, . . . , n}. The sum of those n equations gives:

m l=1 L 3 l,1 p (1) 1 + m l=1 L 3 l,2 p (1) 2 + • • • + m l=1 L 3 l,n p (1) n = 0,
which can be rewritten:

λ 1 p (1) 1 + λ 2 p (1) 2 + • • • + λ n p (1) n = 0, where the coefficient λ i = m l=1 L 3 l,i , 1 ≤ i ≤ n,
corresponds to the possible connections from the panic node x i towards the control nodes y l , 1 ≤ l ≤ m.

First case. Assume that P 1 = {x 1 , . . . , x n }. Then λ i > 0 for i ∈ {1, . . . , n}, which means that all panic nodes are directly connected to at least one control node by a linear oriented edge. We obtain: p

(1) i = 0, ∀i ∈ {1, . . . , n}, which guarantees that O ∈ R 4(n+m) is the only equilibrium for the PCR network [START_REF] Prigogine | Self organization in non-equilibrium systems[END_REF]. T Second case. Assume that P 1 = ∅. Consequently, P k = ∅ for all k ≥ 1, because a panic node which is evacuated by a chain of length k, is necessarily connected to a panic node which is evacuated by a chain of length k -1. Hence, λ i = 0 for all i ∈ {1, . . . , n}, and L 3 l,j = 0 for all l ∈ {1, . . . , m}, j ∈ {1, . . . , n}. Hence, system (30) becomes:

-     n k=1 k =i L 1 k,i     p (1) i + n k=1 k =i L 1 i,k p (1) k = 0, (31)
for all i ∈ {1, . . . , n}, which is a linear system of rank at most n -1, and admits non trivial solutions, thus an infinite number of equilibrium points of the form X = (0, 0, pi , 0) 1≤i≤n , (0, 0, 0, 0) 1≤j≤m , (

with arbitrary pi > 0, corresponding to a persistence of panic. Third case. Assume that P 2 = ∅. We can without loss of generality reorder the indexes and write

λ i = 0, 1 ≤ i ≤ r 1 , λ i > 0, r 1 + 1 ≤ i ≤ n,
with 1 ≤ r 1 ≤ n, so that P 2 ⊂ {x 1 , . . . , x r1 } and P 1 = {x r1+1 , . . . , x n }. Next, we obtain:

λ r1+1 p (1) r1+1 + • • • + λ n p (1) n = 0, which leads to: p (1) i = 0, ∀i ∈ {r 1 + 1, . . . , n}. (33)
System (30) becomes:

-     n k=1 k =i L 1 k,i     p (1) i + r1 k=1 k =i L 1 i,k p (1) k = 0, (34)
for all i ∈ {1, . . . , r 1 }. If we sum those r 1 equations, we get:

n k=r1+1 L 1 k,1 p (1) 1 + n k=r1+1 L 1 k,2 p (1) 2 + • • • + n k=r1+1 L 1 k,r1
p (1) r1 = 0, which can be rewritten:

δ 1 p (1) 1 + δ 2 p (1) 2 + • • • + δ r1 p (1) r1 = 0, (35)
where the coefficient

δ i = n k=r1+1 L 1 k,i , 1 ≤ i ≤ r 1 corresponds
to the linear connections from the panic nodes p (1) i , 1 ≤ i ≤ r 1 (which are not directly evacuated to a control node), towards the panic nodes p (1) i , r 1 + 1 ≤ i ≤ n, which do admit a direct linear connection to at least one control node.

Once again, we distinguish the following three cases:

1. P 2 = {x 1 , . . . , x r1 },
2. P 2 = ∅,

3. P 3 = ∅.
An induction reasoning enables us to build a finite sequence of indexes

1 ≤ r s < • • • < r 2 < r 1 < n, such that P s+1 ⊂ {x 1 , . . . , x rs }. If s+1 k=1 P k = {x 1 , . . . , x n }, then every panic node p (1) i
is evacuated towards at least one control node by a linear oriented chain, and simultaneously, we have:

p (1) i = 0, 1 ≤ i ≤ n.
It shows that the only equilibrium in the network corresponds to the global return to daily behavior.

Else, that is, if

s+1 k=1 P k {x 1 , . . . ,
x n }, it means that some of the panic nodes are not connected to the control nodes by a linear oriented chain, forming a cluster of panic, admitting equilibrium points of the form X = (0, 0, pi , 0) 1≤i≤rs , (0, 0, 0, 0) rs+1≤i≤n+m , (

with arbitrary pi > 0, 1 ≤ i ≤ r s . Third step. To achieve the proof, we analyze the stability of the equilibrium O ∈ R 4(n+m) . Indeed, this simply follows from the construction of the Lyapunov function V that we have already considered in order to build an invariant region (see Equation ( 23)), defined by:

V = 1 2 T 2 , ( 37
)
where T denotes the total population density involved in the disaster (see Equation ( 21)). More precisely, V is semi definite positive. Its orbital derivative is given by

V = T Ṫ . ( 38
)
But Ṫ ≤ 0 (see Proposition (3)), so V is semi definite negative, and this achieves the proof.

This theorem gives a necessary and sufficient condition for the solution of a PCR network to converge to the trivial equilibrium O ∈ R 4(n+m) , which corresponds to a global return of all individuals to a daily behavior. Its demonstration constructs an oriented chain made of linear edges, from each non isolated panic node to one control node. In practice, those evacuation chains can be determined by running a graph theory algorithm of path finding. Next we analyze the global stability of the trivial equilibrium.

Theorem 2. The trivial equilibrium is globally asymptotically stable if and only if every panic node is evacuated.

Proof. First, assume that the trivial equilibrium is globally asymptotically stable, and that some panic nodes are not evacuated. Then, as pointed in the latter proof, the system (20) admits an infinite number of panic persistence equilibria of the form (36), which are arbitrary close to the trivial equilibrium. Since it is supposed to be globally asymptotically stable, it must be isolated in its basin of attraction, thus a contradiction.

Conversely, assume that every panic node is evacuated. Let us consider the auxiliary system obtained by substituting 1 in place of γ(t) and ϕ(t) in the system [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF]:

Ẏ = Φ(t, Y, C) + L Ỹ + Q(Y ), (39)
where the vectors Y, Ỹ ∈ R 4(n+m) and C ∈ R n+m are given by

     Y = (
T , with f (t, x, C 1 ) =     q(1 -r) -(B 1 + B 2 )r + F (r, ĉ)rĉ + G(r, p)r p B 1 r + C 1 p -C 2 ĉ -F (r, ĉ)rĉ + H(ĉ, p)ĉp -ĉ(r + ĉ + p + q) B 2 r -C 1 p + C 2 ĉ -G(r, p)r p -H(ĉ, p)ĉp -q(1 -r)     , x = (r, ĉ, p, q),
the rest of the system being unmodified, in particular the couplings terms. Let V be the Lyapunov function defined by (37). It is easily seen that the set Ω introduced in Proposition 4 is also an invariant region for the modified system (39). Let define E = {Y ∈ Ω ; V (Y ) = 0}. Some basic computations show that Y ∈ E if and only if ĉ(1)

i = ĉ (2) j
= 0 for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}. The La Salle invariance theorem (see [START_REF] Lasalle | Some extensions of Liapunov's second method[END_REF][START_REF] Krasovskij | Stability of motion: applications of Lyapunov's second method to differential systems and equations with delay[END_REF][START_REF] Korobeinikov | Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models[END_REF]) guarantees that the orbits of system (39) approach the largest invariant set M included in E.

Next, we show that M is not more than the trivial equilibrium. So, let us consider an orbit Y (t, t 0 , Y 0 ) of system (39) starting from an initial condition Y 0 ∈ M at t = t 0 . Since M ⊂ E, we have ĉ( 1) i (t) = ĉ(2) j (t) = 0 for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m} and for all t. The corresponding equations in system (39) become:

ċ(1) i = B 1 r(1) i , 1 ≤ i ≤ n, ċ(2) j = B 1 r(2) j + C 1 p(2) j , 1 ≤ j ≤ m.
Necessarily, we have r(1) i (t) = 0 for all i ∈ {1, . . . , n}, t ≥ t 1 and r(2) j (t) = p(2) j (t) = 0 for all j ∈ {1, . . . , m}, t ≥ t 1 . Similarly, the equations of the reflex behavior component r become:

ṙ(1) i = q(1) i , 1 ≤ i ≤ n, ṙ(2) j = q(2) j , 1 ≤ j ≤ m.
which proves that q(1) i (t) = 0 and q(2) j (t) = 0 for all i, j, t. Then we consider xi ∈ P 1 . By assumption, there exists a control node ŷj such that xi is connected to ŷj . The equation of the panic component of node ŷj becomes:

ṗ(2) j = n k=1 L 3 j,k p(1) k ≥ L 3 j,i p(1)
i , thus p(1) i (t) = 0 for all t ≥ t 1 , since L 3 j,i > 0. Analogously, we show that p(1) i (t) = 0 for all t ≥ t 1 and i ∈ {1, . . . , n}, since by assumption every panic node is evacuated.

Finally, let us consider an orbit X(t, t 0 , X 0 ) of the original system [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF], starting from an initial condition X 0 ∈ Ω at t = t 0 . Because of Proposition (4), we have X(t, t 0 , X 0 ) ∈ Ω for all t ≥ t 0 . Furthermore, property [START_REF] Korobeinikov | Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models[END_REF] guarantees that γ(t) = ϕ(t) = 1 for all t ≥ t 1 , which implies

X(t, t 0 , X 0 ) = Y (t, t 1 , X 1 ), ∀t ≥ t 1 ,
where X 1 = X(t 1 , t 0 , X 0 ). The previous reasoning guarantees that Y (t, t 1 , X 1 ) converges to the trivial equilibrium, so the same is for X(t, t 0 , X 0 ), and this achieves the proof. This network is inspired by the geographical background of some epicenter type catastrophe. The position of a node on the affected region influences its nature: when the node is close to the heart of the disaster, the panic behavior is predominant. The first configuration is weakly densely connected, with 10 linear couplings and 10 quadratic couplings. It admits 13 isolated panic nodes. Not surprisingly, the network exhibits a high level persistence of panic (see Figures 7(a) and 8(a)). The second configuration is densely connected, with 50 linear couplings, but only 5 quadratic couplings. It still admits 3 isolated panic nodes, and thus a lesser persistence of panic (see Figures 7(b) and 8(b)). Finally, the third configuration is made with only 5 linear couplings, and 15 quadratic couplings which are suitably chosen from panic nodes towards control nodes. It admits 13 isolated panic nodes, but the level of panic persistence is attenuated by the quadratic connections, according to Proposition 8. Remark 7. Those numerical results clearly indicate which choices should be made in order to avoid the persistence of panic behavior in the context of a brutal catastrophic event. Connecting the nodes, which are susceptible to exhibit the highest level of panic, towards other zones which would be a favorable refuge for a return to a daily behavior, is the key of synchronizing the network and avoiding panic persistence. If physical displacements are not possible, one should favor other type interactions to limit the effect of panic. If a cluster of panic nodes is identified, the addition of refuge zones, considered as control nodes, with evacuation paths from the panic nodes, can be sufficient to reach a global return to the daily behavior. At the opposite, a broken connection can have a dramatic consequence, by letting in panic persistence some of the isolated nodes.

Effect of the length of evacuation chains

Theorem 1 guarantees that any panic node which is evacuated towards a control node presents a return to daily behavior. The coupling strength can accelerate that convergence, as shown in Figure 4. The length of the evacuation chain is another important parameter. Indeed, let us consider a PCR network built with n panic nodes (x i ), 1 ≤ i ≤ n, a unique control node y, and linear edges forming a directed chain of length n:

x 1 → x 2 → • • • → x n → y.
Introduce p y the population density in panic behavior in the control node y. Figure 9 shows that the convergence of p y towards 0 is as fast as the evacuation chain is short, in conformity with intuition.

Remark 8. This suggests that a preventive organization of high risk zones, with a disposition of refuges which could facilitate the evacuation of individuals of the panic nodes, would accelerate the return to daily behavior among the affected population.

Effect of quadratic couplings

We end our paper with the numerical results of a simulation of 200 randomly generated PCR networks. We compare the solutions of the networks built with only linear couplings, with the solutions of the networks built with the same nodes, the same linear couplings, and additional quadratic couplings chosen from panic nodes towards control nodes. Figure 10 shows the number the relationship between the number of isolated panic nodes and the number of linear edges in the network. More precisely, for each randomly generated network, we compute the number N i of isolated panic nodes by running a path-finding algorithm, and the number | E L | of linear edges; each red cross has coordinates (| E L |, N i ). Roughly speaking, the number of isolated panic nodes decreases when the number of linear edges increases. However, as mentioned previously, a densely connected network is not an absolute warranty for avoiding panic persistence. A clever disposal of a low number linear edges can be sufficient to evacuate n panic nodes. Theoretically, n linear edges should be sufficient. But the consideration of the geographical relief of the zone affected by the catastrophe can make that disposal become a complex problem.

Remark 9. The shape of the cloud of points in Figure 10 suggests an approximation by an inverse power law of the type N i = k1 |E l | ν -k 2 , with positive constants k 1 , k 2 and ν. The emergence of such heuristic laws occurs frequently in the study of complex systems arising in various domains of physics, biology or social sciences, and is linked to the self-organization feature of complex networks (see for instance [START_REF] Prigogine | Self organization in non-equilibrium systems[END_REF], [START_REF] Haken | Information and Self-organization: A Macroscopic Approach to Complex Systems[END_REF], [START_REF] Arena | Self-organization in nonrecurrent complex systems[END_REF], [START_REF] Chua | Local activity is the origin of complexity[END_REF] and the references therein). is lower than the density P L of individuals in panic behavior after the same time T , in the PCR network obtained by deleting the quadratic couplings. As mentioned before, this indicates the possible benefit of imitation phenomena, when evacuation is made difficult by the geographical relief of the impacted zone. In this work, we have studied non identical coupled networks of PCR systems, built with a superposition of linear and quadratic couplings. Patterns emerging from basic two-nodes networks show the decisive role of individuals displacements from panic nodes towards control nodes. A detailed analysis of general abstract networks reveals a necessary and sufficient condition for synchronization towards the trivial equilibrium, that is, evacuation of panic nodes towards control nodes by a directed chain of linear connections. In that case, the global stability of the trivial equilibrium is proved using an extension of Lyapunov's theorem, namely, the LaSalle invariance principle.

Those qualitative results are illustrated by numerical simulations, and represent a new step in the validation of the PCR system, as a mathematical model for understanding and predicting human behaviors during catastrophic events.

In a forthcoming paper, we will study another improvement of the modeling, taking into account some local diffusion phenomena, bringing us to the analysis of a semi-linear parabolic problem.

Figure 1 :

 1 Figure 1: A 3 colors directed coupled network, composed with 4 nodes, exhibiting 3 different internal dynamics, corresponding to 3 different parameter values in system ẋ = f (t, x, λ).

Figure 3 :

 3 Figure 3: Two-nodes PCR networks. (a) Linear connection from a panic node towards a control node. (b) Linear connection from a control node towards a panic node. (c) Quadratic connection from a panic node towards a control node. (d) Superposition of quadratic and linear connections of opposite directions between a panic node and a control node.

Figure 4 :Proposition 6 .

 46 Figure 4: An increase of the coupling strength η accelerates the convergence of the total population density P (t) of individuals in panic behavior towards the trivial equilibrium.

Figure 5 :

 5 Figure 5: Panic component of the panic node in a two-nodes PCR network. (a) Linear coupling from panic node to control node: the coupling makes the panic persistence vanish. (b) Linear coupling from control node to panic node: the coupling exacerbates the panic persistence. (c) Quadratic coupling from panic node to control node: the panic persistence survives, but is inhibited.

Figure 7 :

 7 Figure 7: Three couplings configurations for a particular network. Green models control nodes, while red represents panic nodes. (a) Weakly connected network, without favored orientation between panic and control nodes. (b) Densely connected network, still admitting isolated panic nodes, improved with quadratic couplings from panic nodes towards control nodes. (c) Weakly connected network, with quadratic connections from every panic node towards one control node.

Figure 8 :

 8 Figure 8: Numerical results for the three above coupling configurations. (a) The great number of isolated panic nodes accounts for the high level persistence of panic. (b) The low number of isolated panic reduces the panic persistence. (c) The systematic quadratic connection of panic nodes towards control nodes avoids a high level panic persistence.

Figure 9 :

 9 Figure 9: Effect of the length of evacuation chain. The convergence of p y towards 0 is as fast as the evacuation chain is short.

Figure 11 illustrates

 11 Figure 11 illustrates Proposition 8 by comparing the solutions of the PCR networks with or without the effect of quadratic couplings. The blue dots indicate that the density P Q of individuals in panic behavior after a given time T , in a PCR network with both linear and quadratic couplings,

Figure 10 :

 10 Figure 10: Numerical results for the random simulation of 200 PCR networks. The red crosses show that the number N i of isolated panic nodes roughly decreases as the density |E l | of linear connections increases. Indeed, a low number a linear edges can be sufficient to evacuate the panic nodes. The cloud of points is approximated by the blue dotted curve, given by an inverse power law of the type N i = k1 |E l | ν -k 2 , with positive constants k 1 , k 2 and ν.

Figure 11 :

 11 Figure 11: Numerical results for the random simulation of 200 PCR networks. The quadratic connections of panic nodes towards control nodes reduce the panic persistence. After a given time T , the density P Q of individuals in panic behavior is lower than the density P L of individuals in panic behavior in the PCR network obtained by deleting the quadratic couplings.

j

  Figure 2: Notations for the linear coupling terms in a PCR network. Red models panic nodes, while green represents control nodes. It is observed by geographers that the physical displacements are of two types. The first type is the result of a crowd motion in which the behavioral evolution is almost frozen, especially when the time of displacement is short. We have chosen to model this type of displacements by linear couplings. The second type occurs when a communication takes place between individuals

	Remark 3.

standing on different places. This communication corresponds to an interaction which can take hold through a vocal message, a mobile call, or just arms signs in the case of two nodes which are separated by a tiny distance. Thus, we have chosen to model the second type of displacements by quadratic couplings.

  x 1 , . . . , xn , ŷ1 , . . . , ŷm ) T Ỹ = (Hx 1 , . . . , Hx n , Hŷ 1 , . . . , Hŷ m ) T

	C = 0, . . . , 0, C 1 1 , . . . , C m 1	,
	and Φ corresponds to the modified dynamic
	Φ(t, Y, C) = f (t, x1 , 0), . . . , f (t, ĉn , 0), f t, ŷ1 , C 1 1 , . . . , f (t, ŷm , C m 1 )

Table 1 :

 1 Values of the parameters involved in the PCR network for the numerical simulations.

	Parameter Numerical value
	B 1	0.5
	B 2	0.5
	C 1	0.3 for control nodes, 0 else
	C 2	0.2
	α 1	0.1
	α 2	0.1
	δ 1	0.1
	δ 2	0.1
	µ 1	0.1
	µ 2	0.1
		(a)
		(b)
		(c)

§4 Numerical simulations

The aim of this section is to illustrate the previous qualitative results by numerical simulations, and to explore the quantitative aspect of the variation of the parameters involved in the PCR networks. The numerical integration has been performed using the scientific library scipy of the python language, on a personal computer equipped with the Debian-GNU/Linux operating system. The networks have been randomly generated by considering random integers for the numbers of linear edges and quadratic edges. The linear edges are randomly associated to the nodes of the networks, whereas the quadratic edges are oriented from the panic nodes towards the control nodes. The graph theory library networkx, which implements a path-finding algorithm [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF][START_REF] Gallo | Shortest path algorithms[END_REF], is used in order to detect whether the panic nodes are evacuated or not. The functions γ and ϕ are given by

where ζ is defined by

and the functions F , G, H are given by

where ν 0 = 0.01 and ξ is defined by

The values of the parameters are indicated in Table 1. The values of B 1 and B 2 are equal to each other, which means that the evolution processes from reflex to panic, or from reflex to control, are of similar intensity. The value of C 1 for control nodes is greater than the value of C 2 , which favors the evolution from panic to control. The values of the parameters α i , δ i and µ i , 1 ≤ i ≤ 2, involved in the functions F , G and H, are lesser than B i and C i , which means that the imitation phenomena are dominated by the evolution processes. Along each edge of the network, the linear and quadratic coupling strengths have been fixed at 0.1. Those numerical values can easily be varied.

The initial condition corresponds, as mentioned previously, to the situation when all individuals are in a daily behavior before the catastrophe, that is

(44)

Influence of the topology of the network

We first consider a particular network made with 15 panic nodes and 15 control nodes, and experiment three different couplings disposals (see Figure 7).


Magali Lagrange

Charlotte Boulade-Ladame

Laurent Mailly

Etienne Weiss

Georges Orfanoudakis

François Deryckere
email: francois.deryckere@esbs.u-strasbg.fr

Intracellular scFvs against the viral E6 oncoprotein provoke apoptosis in human papillomavirus-positive cancer cells

Keywords: Intrabodies, scFv, HPV, E6, adenovirus, apoptosis

The E6 protein of Human papillomavirus type 16 (16E6) is involved in the tumorigenesis of human cervical cells by targeting numerous cellular proteins. We have designed a strategy for neutralizing 16E6 based on the intracellular expression of single-chain Fv antibodies (scFvs) specific to 16E6.

Recombinant adenovirus vectors were constructed to allow expression of two 16E6-binding scFvs and one 16E6-non-binding scFv in HPV16-positive and -negative cells. Expression of the scFvs provoked two types of effects: i) inhibition of proliferation of all cell lines tested, this aspecific toxicity being likely due to the aggregation of unfolded scFvs. ii) apoptosis observed only in HPV16positive cervical cancer cell lines after expression of 16E6-binding scFvs, this specific effect being proportional to the intracellular solubility of the scFvs. These data demonstrate the feasibility of intracellular immunization with anti-16E6 scFvs and highlight the importance of the solubility of the intracellular antibodies.

INTRODUCTION

Cervical cancer is the second most common cause of cancer fatality in women worldwide. More than 99% of these cancers are linked to viral infection with high-risk human papillomaviruses (HPVs) [START_REF] Walboomers | Human papillomavirus is a necessary cause of invasive cervical cancer worldwide[END_REF].

Epidemiological studies have demonstrated that HPV16 is the most prevalent type corresponding to more than 50% of HPVs detected in cervical cancers [START_REF] Munoz | Against which human papillomavirus types shall we vaccinate and screen? The international perspective[END_REF]. The oncogenic effect of these viruses is principally due to the expression of two early viral proteins, E6 and E7, which play a crucial role in immortalization and transformation of the host cells by interfering with cellular proteins involved in the regulation of the cell cycle [START_REF] Hausen | Papillomavirus infections--a major cause of human cancers[END_REF]. The 16E6 is a small zinc-binding protein, the solution structure of which has revealed a novel zinc-binding fold [START_REF] Nomine | Structural and functional analysis of E6 oncoprotein: insights in the molecular pathways of human papillomavirus-mediated pathogenesis[END_REF]. The stimulation of the ubiquitin-mediated degradation of the p53 tumor suppressor protein via interaction with the cellular E6AP ubiquitinprotein ligase was the first described activity of the high-risk HPV E6 proteins associated with carcinogenesis [START_REF] Scheffner | The E6 oncoprotein encoded by human papillomavirus types 16 and 18 promotes the degradation of p53[END_REF]. However, in addition to E6AP and p53, many cellular targets of high-risk HPV E6 protein have been identified [START_REF] Longworth | Pathogenesis of human papillomaviruses in differentiating epithelia[END_REF][START_REF] Mantovani | The human papillomavirus E6 protein and its contribution to malignant progression[END_REF] and it appears that some 16E6-induced cell-transformations in tissue culture and in vivo are independent of p53 degradation [START_REF] Kiyono | Both Rb/p16INK4a inactivation and telomerase activity are required to immortalize human epithelial cells[END_REF][START_REF] Liu | Studies of the molecular mechanisms in the regulation of telomerase activity[END_REF][START_REF] Song | The human papillomavirus type 16 E6 gene alone is sufficient to induce carcinomas in transgenic animals[END_REF].

The 16E6 protein is sufficient to induce immortalization of human mammary epithelial cells [START_REF] Band | Loss of p53 protein in human papillomavirus type 16 E6-immortalized human mammary epithelial cells[END_REF] and continued expression of this protein is required for the maintenance of the transformed phenotype of HPV16-positive cervical cancer cells [START_REF] Wells | Papillomavirus E2 induces senescence in HPV-positive cells via pRB-and p21(CIP)-dependent pathways[END_REF]. Therefore, 16E6 represents an ideal target for therapeutic strategies of HPV16-positive cervical carcinoma and different approaches such as antisense RNA [START_REF] Alvarez-Salas | Growth inhibition of cervical tumor cells by antisense oligodeoxynucleotides directed to the human papillomavirus type 16 E6 gene[END_REF][START_REF] Cho | HPV E6 antisense induces apoptosis in CaSki cells via suppression of E6 splicing[END_REF][START_REF] Hamada | Adenovirus-mediated transfer of HPV 16 E6/E7 antisense RNA to human cervical cancer cells[END_REF][START_REF] Venturini | Kinetic selection of HPV 16 E6/E7-directed antisense nucleic acids: anti-proliferative effects on HPV 16-transformed cells[END_REF], ribozymes [START_REF] Alvarez-Salas | Inhibition of HPV-16 E6/E7 immortalization of normal keratinocytes by hairpin ribozymes[END_REF], siRNA [START_REF] Butz | siRNA targeting of the viral E6 oncogene efficiently kills human papillomavirus-positive cancer cells[END_REF][START_REF] Yamato | Induction of cell death in human papillomavirus 18-positive cervical cancer cells by E6 siRNA[END_REF][START_REF] Yoshinouchi | In vitro and in vivo growth suppression of human papillomavirus 16-positive cervical cancer cells by E6 siRNA[END_REF], peptide aptamers [START_REF] Butz | Induction of apoptosis in human papillomaviruspositive cancer cells by peptide aptamers targeting the viral E6 oncoprotein[END_REF] and intracellular antibodies [START_REF] Griffin | Inhibition of papillomavirus protein function in cervical cancer cells by intrabody targeting[END_REF] have been used to down regulate or inactivate E6 in HPV-positive cervical cancer cells. The decrease of E6 activity observed in HPV-transformed cells led to inhibition of cellular proliferation [START_REF] Alvarez-Salas | Growth inhibition of cervical tumor cells by antisense oligodeoxynucleotides directed to the human papillomavirus type 16 E6 gene[END_REF][START_REF] Hamada | Adenovirus-mediated transfer of HPV 16 E6/E7 antisense RNA to human cervical cancer cells[END_REF][START_REF] Venturini | Kinetic selection of HPV 16 E6/E7-directed antisense nucleic acids: anti-proliferative effects on HPV 16-transformed cells[END_REF][START_REF] Alvarez-Salas | Inhibition of HPV-16 E6/E7 immortalization of normal keratinocytes by hairpin ribozymes[END_REF] or apoptosis [START_REF] Cho | HPV E6 antisense induces apoptosis in CaSki cells via suppression of E6 splicing[END_REF][START_REF] Butz | siRNA targeting of the viral E6 oncogene efficiently kills human papillomavirus-positive cancer cells[END_REF][START_REF] Yamato | Induction of cell death in human papillomavirus 18-positive cervical cancer cells by E6 siRNA[END_REF][START_REF] Butz | Induction of apoptosis in human papillomaviruspositive cancer cells by peptide aptamers targeting the viral E6 oncoprotein[END_REF][START_REF] Griffin | Inhibition of papillomavirus protein function in cervical cancer cells by intrabody targeting[END_REF]. These findings confirmed the suitability of targeting the E6 protein as a possible therapeutic approach for carcinoma of the cervix.

Materials and methods

Cell lines. The following cell lines were used in this study: SiHa, CaSki (both derived from a HPV16 integrated human cervical carcinoma), HeLa (derived from a HPV18 integrated human cervical adenocarcinoma), A549 (derived from a human epithelial lung carcinoma). All the above cells were cultured in monolayer cultures in Dulbecco's modified Eagle's medium (DMEM) supplemented with 2 mM L-glutamine, 10% (v/v) foetal calf serum, 50 µg/ml penicillin and 50 µg/ml streptomycine (Invitrogen).

Expression of scFvs and dot-blot assays. The scFvs fused to the B10 tag were expressed in the periplasm of E. coli XL1-Blue as described [START_REF] Giovane | Targetting of the N-terminal domain of the human papillomavirus type 16 E6 oncoprotein with monomeric ScFvs blocks the E6-mediated degradation of cellular p53[END_REF]. Equivalent amounts (10 µL) of periplasmic extracts from bacteria transformed with scFv-expressing plasmids were loaded on SDS-PAGE and analyzed by Western blot using the B10 monoclonal antibody and peroxydase-labeled goat anti mouse IgG.

Nitrocellulose strips carrying dots (50 ng) of either the Maltose Binding Protein (MBP) or 16E6 fused to MBP (MBP-E6) were incubated with 2 mL of periplasmic extracts. Binding of scFv to 16E6 was revealed using the B10 monoclonal antibody and peroxydase-labeled goat anti mouse IgG.

Recombinant adenoviruses. Coding sequences of 1F4-L177P, 1F4 and 1F4-P41L scFvs fused to a B10 tag [START_REF] Orfanoudakis | Bacterially expressed Fabs of monoclonal antibodies neutralizing tumour necrosis factor alpha in vitro retain full binding and biological activity[END_REF] under the control of the CMV promoter were inserted in replacement of the E3 region into the adenovirus 5 genome using homologous recombination in E. coli as described [START_REF] Chartier | Efficient generation of recombinant adenovirus vectors by homologous recombination in Escherichia coli[END_REF], Fig. 2A). The recombinant adenoviruses were produced and purified after transfection of 293 cells [START_REF] Chartier | Efficient generation of recombinant adenovirus vectors by homologous recombination in Escherichia coli[END_REF].

Immunofluorescence analysis. Twenty-four hours after transduction with recombinant adenoviruses at a MOI of 100 plaque-forming-units (pfu)/cell, cells were fixed and analysed by immunofluorescence as previously described [START_REF] Lagrange | Binding of human papillomavirus 16 E6 to p53 and E6AP is impaired by monoclonal antibodies directed against the second zinc-binding domain of E6[END_REF]. The scFvs were revealed by the B10 monoclonal antibody and by a FITC-conjugated anti-mouse secondary antibody. The nuclei of the cells were stained with Hoechst 33258 (Molecular Probes).

Western blot analysis. Twenty-four hours transduction with recombinant adenoviruses at a MOI of 100 pfu/cell, cells were detached with a rubber policeman and incubated 30 min on ice in lysis buffer (150 mM NaCl, 20 mM Tris/HCl pH8.0, 2 mM DTT, 1% NP-40 containing protease inhibitor mix "Complete TM " from Roche Applied Science). After a 30-min centrifugation at 12 000 rpm, cellular extracts were fractionated into soluble (supernatant) and insoluble (pellet) forms.

Equivalent amounts of soluble or insoluble extracts were analyzed by SDS/PAGE 12% and Western blotting. The scFvs were revealed with the B10 monoclonal antibody and peroxydase-labeled goat anti mouse IgG. The percentages of soluble and insoluble scFvs were quantified using the Quantity-One software (Bio-Rad).

Methylthiazolyldiphenyl-tetrazolium bromide (MTT) assays. To measure proliferation, cells

were seeded into 96-well plates one day before transduction. At 50% confluency, cells were transduced with recombinant adenoviruses at a MOI of 100 pfu/cell. After 12 h, 24 h or 48 h of transduction, cells were washed with PBS and incubated at 37°C with 100 µL of a 500 µg/ml MTT solution in media. Two hours later, the blue formazan crystals were dissolved by adding 100µL of MTT stop solution (64% isopropanol, 32% butanol-2, 40 mM HCl) and the absorbance at 595nm was measured.

Apoptosis. Cells were stained with annexin V-FITC (Annexin V-FLUOS Staining kit, Roche Diagnostics) and propidium iodide according to the manufacturer instructions 24h or 48h after transduction by recombinant adenoviruses at a MOI of 100 pfu/cell. Cells were analysed on a FACScalibur (BD-Biosciences) using a dual filter for FITC and propidium iodide. In this test, live, apoptotic and necrotic cells are distinguished on the basis of annexin V-FITC binding and PI exclusion: live cells are not stained with any of the reagents, apoptotic cells are positive for annexin V binding and negative for PI whereas necrotic cells are stained with both reagents

RESULTS AND DISCUSSION

scFvs are small functional antibody fragments consisting of the heavy and light chain variable (V H and V L ) segments held together by a flexible linker that possess the binding and specificity of the whole antibody [START_REF] Huston | Protein engineering of antibody binding sites: recovery of specific activity in an anti-digoxin single-chain Fv analogue produced in Escherichia coli[END_REF]. During the antibody production, the formation of intra-and inter-chain disulfide bonds inside the endoplasmic reticulum allows the stabilization of each immunoglobulin domain and the assembly of heavy and light chains. The intracellular immunization strategy [START_REF] Biocca | Intracellular immunization with cytosolic recombinant antibodies[END_REF] aimed at neutralizing an antigen inside cells with intracellular scFvs (intrabodies) [START_REF] Lobato | Intracellular antibodies as specific reagents for functional ablation: future therapeutic molecules[END_REF] requires the expression of functional scFvs in the cytoplasm of the target cells. However, the reducing environment of this compartment does not allow the formation of intra-domain disulfide bonds within the V H and V L [START_REF] Biocca | Intracellular immunization: antibody targeting to subcellular compartments[END_REF][START_REF] Sibler | Extended half-life upon binding of destabilized intrabodies allows specific detection of antigen in mammalian cells[END_REF][START_REF] Worn | Different equilibrium stability behavior of ScFv fragments: identification, classification, and improvement by protein engineering[END_REF]. As a result the main challenge of intracellular immunization remains the correct folding and stability of scFvs expressed in the cytoplasm. In previous reports, we described monoclonal and single-chain antibodies against 16E6 [START_REF] Giovane | Targetting of the N-terminal domain of the human papillomavirus type 16 E6 oncoprotein with monomeric ScFvs blocks the E6-mediated degradation of cellular p53[END_REF][START_REF] Lagrange | Binding of human papillomavirus 16 E6 to p53 and E6AP is impaired by monoclonal antibodies directed against the second zinc-binding domain of E6[END_REF][START_REF] Masson | Preferential nuclear localization of the human papillomavirus type 16 E6 oncoprotein in cervical carcinoma cells[END_REF][START_REF] Schwalbach | Production of fluorescent single-chain antibody fragments in Escherichia coli[END_REF] and we evaluated here the potential of 16E6specific intrabodies for intracellular immunization of HPV-positive cancer cells using recombinant adenoviruses. We previously reported the construction of 1F4 scFv which binds to the N-terminus of 16E6 [START_REF] Giovane | Targetting of the N-terminal domain of the human papillomavirus type 16 E6 oncoprotein with monomeric ScFvs blocks the E6-mediated degradation of cellular p53[END_REF]. To test its capacity to function as an intrabody, we investigated its solubility and activity when expressed in the cytoplasm of E. coli [START_REF] Schwalbach | Production of fluorescent single-chain antibody fragments in Escherichia coli[END_REF] or COS cells [START_REF] Sibler | Nucleocytoplasmic shuttling of antigen in mammalian cells conferred by a soluble versus insoluble single-chain antibody fragment equipped with import/export signals[END_REF]. We found that a strong expression of 1F4 in the cytoplasm of E. coli or COS cells resulted in nearly 100% of non-folded, aggregated and insoluble scFv [START_REF] Sibler | Nucleocytoplasmic shuttling of antigen in mammalian cells conferred by a soluble versus insoluble single-chain antibody fragment equipped with import/export signals[END_REF]. This aggregation was toxic for the transfected COS cells and resulted in aspecific cell death. To overcome this problem, a random mutagenesis of 1F4 was performed in an attempt to isolate scFvs that would be more stable when expressed in the cytoplasm. This allowed us to isolate two scFvs each presenting one point mutation, i.e. 1F4-L177P [START_REF] Schwalbach | Production of fluorescent single-chain antibody fragments in Escherichia coli[END_REF] and 1F4-P41L (EW, unpublished results). The 1F4-L177P (L47P in V L ) had lost the capacity to bind to E6 and still aggregated when expressed in the cytoplasm whereas 1F4-P41L (P41L in V H ) still bound to E6 (Fig. 1B) and had an increased solubility after cytoplasmic expression (EW, unpublished results). The three scFvs were expressed in E. coli and we confirmed, using a dot-blot assay, that 1F4 and 1F4-P41L did bind to 16E6 whereas 1F4-L177P did not (Fig. 1B).

As plasmid transfection efficiency can be very variable from one human cell line to another, we chose to use a vector derived from adenovirus type 5, known for its very high gene transfer efficacy.

Coding sequences of 1F4-L177P, 1F4 and 1F4-P41L were inserted into the adenovirus type 5 genome to obtain three recombinant viruses, namely Ad1F4-L177P, Ad1F4 and Ad1F4-P41L. To assess the intracellular immunization potential of the 16E6-specific scFvs we employed two HPV16transformed cell lines, SiHa and CaSki, and two control cell lines, A549 and HeLa. HeLa cells are transformed by HPV18 and express 18E7 and 18E6, 18E6 is not recognized by the scFvs reported here [START_REF] Giovane | Targetting of the N-terminal domain of the human papillomavirus type 16 E6 oncoprotein with monomeric ScFvs blocks the E6-mediated degradation of cellular p53[END_REF]. The four different cell lines were transduced with the scFv-expressing vectors (Ad1F4-L177P, Ad1F4 and Ad1F4-P41L) and analyzed 24 hours later by immunofluorescence. The use of a MOI of 100 pfu/cell allowed expression of the scFv in nearly 100% of the cells (Fig. 2B). In these conditions, we observed a lower level of scFv-production per cell than in COS cells transfected with scFv-expressing plasmid [START_REF] Sibler | Nucleocytoplasmic shuttling of antigen in mammalian cells conferred by a soluble versus insoluble single-chain antibody fragment equipped with import/export signals[END_REF]. A higher MOI gave a stronger expression of the scFvs resulting in a high aspecific toxicity and a lower MOI resulted in a weaker percentage of cells transduced (data not shown).

After validating the high gene transfer efficacy of the scFv-encoding adenoviruses, we determined the solubility of each 16E6-intrabody by Western blotting of cellular extracts (Fig. 2C).

Although the majority of the scFvs were present into the insoluble fraction of cellular extracts, the relative proportion of soluble intrabodies was higher in this study (5-10%; Fig. 2C) than in previous results with transfected COS cells (<1%) [START_REF] Sibler | Nucleocytoplasmic shuttling of antigen in mammalian cells conferred by a soluble versus insoluble single-chain antibody fragment equipped with import/export signals[END_REF]. This global higher solubility of scFvs was due to a lower expression in the human cell lines after transduction with recombinant adenoviruses at a MOI of 100 pfu/cell than in COS cells after plasmid transfection (data not shown). The intracellular solubility of 1F4-P41L (9.9%) was nearly twice as high as the solubility of 1F4-L177P (5.2%) and 1F4 (5.6%).

We next investigated the effects of the expression of the anti-16E6 intrabodies on the proliferation of HPV16-positive cancer cell lines using a methylthiazolyldiphenyl-tetrazolium bromide (MTT) assay (Fig. 3A). In all cases, there was a strong inhibition of proliferation of the cell lines tested after expression of the scFvs, even with the negative control scFv (1F4-L177P) and negative control cell lines (HeLa and A549, Fig. 3A). This was very likely due to the toxic effect of the aggregation of unfolded intrabodies even though not as strong as was observed earlier with transfected COS cells [START_REF] Sibler | Nucleocytoplasmic shuttling of antigen in mammalian cells conferred by a soluble versus insoluble single-chain antibody fragment equipped with import/export signals[END_REF]. However in addition to this non-specific effect we observed a specific effect namely that the intracellular expression of E6-binding scFvs (1F4 and 1F4-P41L) provoked cell death in HPV-positive cells (SiHa and CaSki) whereas the 1F4-L177P did not. Interestingly, the increased cytosolic solubility of the 1F4-P41L intrabody correlated with a more pronounced specific cell death in HPV16-positive cervical cells.

To further analyze the observed cell death, cells were stained with annexin V-FITC/propidium iodide (PI) and examined by flow cytometry (Fig. 3A). HeLa, A549, SiHa and CaSki cells were analyzed 24h or 48h after transduction by the scFv-expressing adenoviruses. Intracellular expression of the three scFvs in HPV16-negative cell lines led to a very small increase of apoptotic cells when compared to non-transduced cells (1-3% at 48h, Fig. 3B). Similarly the negative control intrabody, 1F4-L177P, did not enhance apoptotic cell rates significantly. In contrast, transduction of 1F4 and 1F4-P41L increased apoptosis specifically in HPV16-transformed cells (up to 13.9% with 1F4-P41L in CaSki), this effect being proportional to the intracellular solubility of the intrabody. Of note, the proportion of apoptotic cells induced by the 1F4-P41L was 9.5 fold higher in the HPV16-positive CaSki cells than in HeLa cells. Our results corroborate previous reports showing that targeting E6 with either antisense RNA [START_REF] Cho | HPV E6 antisense induces apoptosis in CaSki cells via suppression of E6 splicing[END_REF], peptide aptamer [START_REF] Butz | Induction of apoptosis in human papillomaviruspositive cancer cells by peptide aptamers targeting the viral E6 oncoprotein[END_REF], siRNA [START_REF] Butz | siRNA targeting of the viral E6 oncogene efficiently kills human papillomavirus-positive cancer cells[END_REF] or transfection of intrabodies [START_REF] Griffin | Inhibition of papillomavirus protein function in cervical cancer cells by intrabody targeting[END_REF] provokes apoptosis in HPV16-transformed cells. However the apoptosis induced by the 1F4-P41L intrabody in SiHa or CaSki cells was not strictly correlated with an increase in p53 protein level as assayed by immunofluorescence or Western blot (data not shown), the precise pathway involved in the apoptosis that we observed remains to be investigated. This work demonstrates the feasibility of intracellular immunization with anti-16E6 scFvs for specific killing of cervical cancer cells. Our study was greatly facilitated by the use of adenovirus as a gene transfer and expression vector. This vector not only allowed the expression of the scFv uniformly in all the cells transduced in vitro (Fig. 2B) but also provides a useful vector for expression of scFv in vivo for instance for preclinical studies in animal models. The importance of intracellular solubility of intrabodies is shown by the fact that increased cytosolic solubility of the 1F4-P41L intrabody was correlated with increased specific cell death in HPV16-positive cervical cells. It may therefore be worthwhile to further improve the intracellular solubility of anti-E6 scFvs by using protein-engineering approaches [START_REF] Martineau | In vitro folding and thermodynamic stability of an antibody fragment selected in vivo for high expression levels in Escherichia coli cytoplasm[END_REF][START_REF] Proba | Antibody scFv fragments without disulfide bonds made by molecular evolution[END_REF][START_REF] Tanaka | Single domain intracellular antibodies: a minimal fragment for direct in vivo selection of antigen-specific intrabodies[END_REF][START_REF] Tse | Intracellular antibody capture technology: application to selection of intracellular antibodies recognising the BCR-ABL oncogenic protein[END_REF] since this could lead to the development of scFvs with improved therapeutic potential against cervical carcinoma.

Fig. 1 .

 1 Fig. 1. Expression of scFvs in E. coli and detection of 16E6. (A) Equivalent amounts (10 µL) of

Fig. 2 .

 2 Fig. 2. Expression of scFvs 1F4-L177P, 1F4 and 1F4-P41L. (A) Schematic representation of the

Fig. 3 :

 3 Fig. 3: Effects of the anti-16E6 scFv in HPV16-transformed cell lines. (A) The proliferation of HeLa,

Acknowledgements

We thank Claude Kedinger for his encouragement and support. We acknowledge Claudine Ebel for assistance with the FACS experiments. We thank Marc van Regenmortel, Philip Robinson,

Murielle

Masson and Gilles Travé for comments and critical reading of the manuscript. This work was supported by the Université Louis Pasteur de Strasbourg, the Centre National de la Recherche Scientifique, the Association pour la Recherche contre le Cancer, the Ligue Nationale Contre Le Cancer Comité du Haut Rhin and the Cancéropôle-Grand-EST. M.L. and C.B-L. were supported by scholarships of the Ministère de la Recherche (M.L. and C.B-L.) and the Ligue Régionale Contre le Cancer (ML).