C Bourdarias
email: bourdarias@univ-savoie.fr

M Gisclon
email: gisclon@univ-savoie.fr

S Junca
email: junca@unice.fr

EXISTENCE OF WEAK ENTROPY SOLUTIONS FOR GAS CHROMATOGRAPHY SYSTEM WITH ONE OR TWO ACTIVE SPECIES AND NON CONVEX ISOTHERMS *

Keywords: boundary conditions, systems of conservation laws, Godunov scheme, entropies, composite waves, Liu entropy-condition. 35L65, 35L67, 35Q35

Existence of weak entropy solutions for gas chromatography system with one or two active species and non convex isotherms

Christian Bourdarias, Marguerite Gisclon, Stéphane Junca

Introduction

Heatless adsorption is a cyclic process for the separation of a gaseous mixture, called "Pressure Swing Adsorption" cycle. During this process, each of the d species (d ≥ 2) simultaneously exists under two phases, a gaseous and movable one with concentration c i (t,x) and velocity u(t,x), or a solid (adsorbed) other with concentration q i (t,x), 1 ≤ i ≤ d. Following Ruthwen (see [START_REF] Ruthwen | Principles of adsorption and adsorption processes[END_REF] for a precise description of the process) we can describe the evolution of u, c i , q i according to the following system:

∂ t c i + ∂ x (uc i ) = A i (q i -q * i (c 1 ,••• ,c d )), (1.1)
∂ t q i + A i q i = A i q * i (c 1 ,••• ,c d ) t ≥ 0, x∈ (0,1), (1.2)
with suitable initial and boundary data. In (1.1)-(1.2) the velocity u(t,x) of the mixture has to be found in order to achieve a given pressure (or density in this isothermal model)

d X i=1 c i = ρ(t), (1.3)
where ρ represents the given total density of the mixture. The experimental device is realized so that it is a given function depending only upon time. The function q * i is defined on (R + ) d , depends upon the assumed model and represents the equilibrium concentrations. Its precise form is usually unknown but is experimentally obtained. Simple examples of such a function are for instance the linear isotherm

q * i = K i c i (1.4)
with K i ≥ 0 and the Langmuir isotherm

q * i = Q i K i c i 1+ d X j=1 K j c j (1.5)
with K i ≥ 0, Q i > 0 (see for instance [START_REF] Bourdarias | Approximation of the solution to a system modeling heatless adsorption of gases[END_REF], [START_REF] James | Sur la modélisation mathématique des équilibres diphasiques et des colonnes de chromatographie[END_REF], [START_REF] Liu | The entropy condition and the admissibility of shocks[END_REF]). The right hand side of (1.1)-(1.2) rules the matter exchange between the two phases and quantifies the attraction of the system to the equilibrium state: it is a pulling back force and A i is the "velocity" of exchange for the species i. A component with concentration c

i is said to be inert if A i = 0 and q i = 0. A theoretical study of the system (1.1)-(1.2)-(1.3) was presented in [START_REF] Bourdarias | Sur un système d'edp modélisant un processus d'adsorption isotherme d'un mélange gazeux. (french) [on a system of p.d.e. modelling heatless adsorption of a gaseous mixture[END_REF] and a numerical approach was developed in [START_REF] Bourdarias | Approximation of the solution to a system modeling heatless adsorption of gases[END_REF]. Let us point out that one of the mathematical interests of the above model is its analogies and differences compared to various other classical equations of physics or chemistry. First, when d = 1 (and eventually with A i = 0) this model shares a similar structure with conservation laws under the form

∂ t ρ + ∂ x (ρu(ρ)) = 0, ∂ x u(ρ) = F (ρ)
where u(ρ) has an integral dependence upon ρ, while in scalar conservation laws u depends upon ρ. In [START_REF] Bourdarias | Sur un système d'edp modélisant un processus d'adsorption isotherme d'un mélange gazeux. (french) [on a system of p.d.e. modelling heatless adsorption of a gaseous mixture[END_REF] both BV and L ∞ theory are developed for this model, but oscillations can propagate thus differing from Burger's example (see Tartar [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF], Lions, Perthame, Tadmor [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related questions[END_REF]). Secondly, when the coefficients A i tend to infinity (instantaneous equilibrium) we get formally

q i -q * i = - 1 A i ∂ t q i → 0
and the equations (1.1)-(1.2) reduce to

∂ t (c i + q * i (c 1 ,••• ,c d )) + ∂ x (uc i ) = 0, i= 1,••• ,d. (1.6)
Joined to the algebraic constraint (1.3), the system of conservation laws (1.6) generalizes the system of chromatography which has been intensively studied (see [START_REF] James | Sur la modélisation mathématique des équilibres diphasiques et des colonnes de chromatographie[END_REF][START_REF] Rhee | On the theory of multicomponent chromatography[END_REF] for the Langmuir isotherm) whereas the system (1.1)-(1.2) enters more in the field of relaxation systems (see for instance Jin and Xin [START_REF] Jin | The relaxing schemes for systems of conservation laws in arbitrary space dimensions[END_REF], Katsoulakis and Tzavaras [START_REF] Katsoulakis | Contractive relaxation systems and the scalar multidimensional conservation law[END_REF]). Actually the system of chromatography corresponds, like in (1.6), to instantaneous adsorption, but the fluid speed is a constant u(t,x) = u. One may consult James [START_REF] James | Sur la modélisation mathématique des équilibres diphasiques et des colonnes de chromatographie[END_REF] for a numerical analysis and the relationships with thermodynamics, Canon and James [START_REF] Canon | Resolution of the Cauchy problem for several hyperbolic systems arising in chemical engineering[END_REF] in the case of the Langmuir isotherm. In [START_REF] James | Convergence results for some conservation laws with a reflux boundary condition and a relaxation term arising in chemical engineering[END_REF], James studied a system closely related to (1.1)-(1.2) in which the speed is constant and the coefficients A i are equal to 1/ε, where ε is a small parameter. Using compensated compactness, he proved, under some assumptions on the flux, that the solution of this system converges, as ε → 0, to a solution of a system of quasilinear equations similar to (1.6) satisfying a set of entropy inequalities. The extension of his method to (1.6) with constraint (1.3) seems not straightforward and is still an open problem.

In a previous work [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF], we focused on the system of equations (1.6)-(1.3) with two components (d = 2), one adsorbable with concentration c 1 and one inert with concentration c 2 . Moreover in (1.3) we assumed that ρ ≡ 1, which is not really restrictive from a theoretical point of view. In that case, setting c := c 2 and h(c) = -q * 1 (c 1 ,c 2 ) = -q * 1 (1 -c,c), we studied the system (1.6)-(1.3) under the form:

Ω ∂ t c + ∂ x (uc) = 0, ∂ t h(c) -∂ x u = 0, x>0, t > 0, (1.7)
supplemented by initial and boundary values (assuming an influx boundary condition). Single-component adsorption is of course of a poor physical meaning, but must be understood as a preliminary theoretical study. In this context and under some restrictive conditions on the function h, we obtained an existence theorem for a weak solution. The uniqueness was obtained in the class on piecewise C 1 functions.

In this paper we look at the system (1.6)-(1.3) with d = 2 and ρ ≡ 1 with one or two active components under more general and physically relevant assumptions on the so-called isotherms q * i and the related functions. Moreover, using the hyperbolicity property of this system with respect to the (x,t) variables (see Rouchon and al. [14]) we find all the entropy-flux pairs, which is a significant improvement. The system writes:

∂ t (c 1 + q * 1 (c 1 ,c 2 )) + ∂ x (uc 1 ) = 0, (1.8) ∂ t (c 2 + q * 2 (c 1 ,c 2 )) + ∂ x (uc 2 ) = 0, (1.9) c 1 + c 2 = ρ ≡ 1.
(1.10)

Notice that we seek positive solutions (c 1 ,c 2 ), thus, in view of (1.10), c 1 , c 2 must satisfy

0 ≤ c 1 , c 2 ≤ 1.
We make use of the following notations: we set c = c 1 (as in [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF], instead of c 2 in [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF]) and

q i (c) = q * i (c,1 -c), i= 1,2 h(c) = q 1 (c)+q 2 (c) I(c) = c + q 1 (c)
(notice the change of sign in h with respect to [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF]). We assume that

dq * i dc i ≥ 0
(general property of all isotherms) and that dq * i dc j ≤ 0 for j = i (general property in the case of two species : see [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF]), thus :

dq 1 dc = q 1 ≥ 0, dq 2 dc = q 2 ≤ 0. (1.11)
Adding (1.8) and (1.9) we get, thanks to (1.10):

∂ t (q 1 (c)+ q 2 (c)) + ∂
x u = 0 thus our purpose is to study the following system:

( ∂ t I(c)+ ∂ x (uc) = 0, ∂ t h(c)+ ∂ x u = 0, (1.12)
supplemented by initial and boundary values:

8 > > < > > : c(0,x) = c 0 (x) ∈ [0,1], x>0, c(t,0) = c b (t) ∈ [0,1], t>0, u(t,0) = u b (t) > 0, t>0. (1.13)
Notice that we assume in (1.13) an incoming flux at the boundary, i.e. ∀t > 0, u b (t) > 0. In the case where the the first species is inert, that is q * 1 = 0, the I function reduces to identity. In the sequel, in order to solve the Riemann problem, we will make use of the fundamental function

f = q 1 c 2 -q 2 c 1 (1.14)
introduced by Douglas and al. in [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF], written here under the form

f (c) = q 1 (c) -ch(c), (1.15)
and of the function H defined by

H(c) = 1+(1-c)q 1 -cq 2 = 1+q 1 (c) -ch (c).
The function f satisfies

d 2 f dc 2 = f = H -h (1.16)
and H, in view of (1.11):

H(c) ≥ 1.
(1.17)

In order to investigate some properties of the function h we look at some common used isotherm (see [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF] for instance).

For linear isotherm and two adsorbable components we have q 1 (c) = K 1 c and

q 2 (c) = K 2 (1 -c) with K 1 , K 2 > 0, then f (c) = 2(K 2 -K 1 ) (1.18)
have a constant sign.

For the binary Langmuir isotherm (i. e. the Langmuir isotherm for two species) we have

q 1 (c) = Q 1 K 1 c 1+K 1 c + K 2 (1 -c) , q 2 (c) = Q 2 K 2 (1 -c) 1+ K 1 c + K 2 (1 -c) with K 1 , K 2 , Q 1 , Q 2 > 0 and we find f (c) = 2 (Q 2 K 2 -Q 1 K 1 )(1 + K 1 )(1 + K 2 ) (1 + K 1 c + K 2 (1 -c)) 3 (1.19)
which have also a constant sign. Notice that it is not alway the case : for one adsorbable component (number one for instance) in an inert gas with the so called BET isotherm defined by

q * 1 (c 1 ,c 2 ) = QKc 1 (1 + Kc 1 -(c 1 /c s ))(1 -(c 1 /c s )) , Q>0, K > 0, c s > 0,
we have q 2 (c) = 0 and f (c) = (1-c)q 1 (c) is no longer concave or convex. Except some important cases: Langmuir, ammonia, water vapor, isotherms are generally non convex.

Indeed non convex isotherm has a chemical meaning: at each inflexion point gas adsorbed cover the mixed bed and a new level of adsorbed gas begins on the previous.

The paper is organized as follows: analyzing the system (1.12)-(1.13) as an hyperbolic system with respect to the (x,t) variables, that is with x as the evolution variable, we find a set of two Riemann invariants. Then we investigate the existence of smooth convex entropies. Next we solve the Riemann problem and build a Godunovtype scheme in the "physical variables" (t,x). This scheme enjoys some BV estimates which allows us to get an existence result of entropy weak solutions.

The system of fixed-bed adsorption for two species

As pointed out by Rouchon and al. ([14]), it is possible to analyse the system (1.12) in terms of hyperbolic system of P.D.E. provided we exchange the time and space variables. In this framework the vector state will be U =

µ u m ∂ where m = uc
is the flow rate of the first species. The first component u of this vector must be understood as uρ, that is the total flow rate. Actually it appears that this choice of set of conservative variables is better than the choice (u,c) for the analysis of convexity of the entropies of the system, nevertheless notice that the corresponding systems have the same weak solutions and the same convex entropies since (u,c) Dafermos [5] for instance). Thus wee seek for solutions satisfying u > 0.

→ (u,m) is a diffeomorphism between R * + ×]0,1[ and R * + × R * + (see

Hyperbolicity

The system (1.12) takes the form

∂ x U + ∂ t Φ(U ) = 0 with U = µ u m ∂ and Φ(U ) = 0 @ h(m/u) I(m/u) 1 A . (2.1)
The Jacobian matrix, written in terms of the variables (u,c) for the sake of simplicity, is:

D U Φ = 1 u 0 @ -ch (c) h (c) -c(1 + q 1 (c)) 1 + q 1 (c) 1 A (2.
2)

The eigenvalues are 0 and λ = H(c) u , thus in view of (1.17) the system is strictly hyperbolic. The associated eigenvectors are

µ 1 c ∂ and r = µ h (c) 1+ q 1 (c) ∂ respectively.
The zero eigenvalue is of course "strongly degenerated", moreover

dλ • r = H(c) u 2 f (c), (2.3)
thus λ is genuinely nonlinear in each domain where f " = 0. In most cases (a finite number of inflexion points), λ is piecewise nonlinear.

Riemann invariants

Proposition 2.1. The system (1.12) admits the two Riemann invariants: c and W = ue g(c) , where g satisfies g (c) = -h (c) H(c) .

Proof : for smooth solutions, (1.12) writes:

∂ x u = -h (c)∂ t c u∂ x c + c∂ x u +(1+ q 1 (c))∂ t c = 0 and we get immediately u∂ x c +(1+ q 1 (c) -ch (c))∂ t c = 0 which writes ∂ x c + H(c) u ∂ t c = 0. (2.4)
Multiplying the second equation of the previous system by h (c) 1+q

1 (c) and using the first one we get also

∂ x u - h (c)u H(c) ∂ x c = 0. (2.5)
Let us define g and G such that:

g = - h H , G(c) = exp(g(c)). (2.6)
The function G satisfies G = -Gh H . Thus, multiplying (2.5) by G(c) we get

∂ x (uG(c)) = 0 (2.7) (read ∂ x (uG(c)) + 0 • ∂ t (uG(c)) = 0). § Remark 2.1.
The so called "concentration velocity" (velocity of a given concentration) defined in [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF] is nothing else than the inverse of the eigenvalue λ, i.e. u H(c) associated to the Riemann invariant c (recall that we have exchanged the t and x variables).

Remark 2.2. In the sequel we will make use of the λ-Riemann invariant w = lnW = lnu + g(c).

Particular important cases

• Case of an inert carrier gas: if the gas number 1 (for instance) is inert, i.e. q 1 = 0, we have

I(c) = c, h = q 2 , h < 0, g > 0 and H = 1-ch (c).
A study of this case, assuming that f never vanish, has be done in [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF]. • Case where a gas is most active than the other: assume for instance that q * 1 < q * 2 , then h < 0 and g > 0. The previous case is a particular case of this one. Notice that h is monotonous if and only if there is one gas more active than the other. The same holds for g and G.

Entropies

Denote S = S(c,u) any smooth entropy and Q = Q(c,u) any associated entropy flux. Then, for smooth solutions, ∂

x S + ∂ t Q = 0. Moreover: Lemma 3.1. The entropy flux Q does not depend on u.

Proof : on one hand we have

S u ∂ x u + S c ∂ x c + Q c ∂ t c + Q u ∂ t u = 0.
On the other hand, multiplying the second equation of (1.12) by a function α(u,c) and the first one by a function β(u,c) we get, for smooth solutions:

(α + cβ)∂ x u + β u∂ x c +(αh + β (c + q 1 ) )∂ t c = 0 thus S u = α + cβ, S c = βu, Q c = αh + β(c + q 1 ) and Q u = 0. §
We are now able to describe all the smooth entropy-entropy flux pairs:

Proposition 3.2. The smooth entropy functions for the system (1.12) are given by

S(c,u) = φ(w)+ uψ(c) (3.1)
where φ and ψ are any smooth real functions and w = lnu + g(c). The corresponding entropy flux satisfies

Q (c) = h (c)ψ(c)+ H(c)ψ (c).
Proof : using the preceding lemma and writing S as a function of the Riemann invariants (c,w) we are looking for smooth functions S(c,w) and

Q = Q(c) satisfying ∂ x S(c,w)+ ∂ t Q(c) = 0.
From equations (2.4) and (2.7) we get

S w ∂ x w + S c ∂ x c + S c H(c) u ∂ t c = 0 that is ∂ x S(c,w)+ S c H(c) u ∂ t c = 0. Thus we have Q (c) = S c H(c) u and then S c (c,w) u is a function χ(c) = Q (c) H(c)
. Now we have:

S(c,w) = S(0,w)+ Z c 0 uχ(s)ds = φ(w)+W Z c 0 χ(s) G(s) ds = φ(w)+ W F (c) = φ(w)+ uψ(c)
where

ψ(c) = G(C)F (C) = G(c) Z c 0 χ(s) G(s) ds.
The entropy flux Q satisfies:

Q H = S c (c,w) u = G(c)S c (c,w) w = G(c) µ ψ(c) G(c) ∂ c = ψ (c) -ψ(c) G (c) G(c)
which gives Q = H ψ + h ψ thanks to (2.6). §

Convex entropies

We are looking for convex entropies for the system (2.1) (i.e. the system (1.12) written in the (u,m) variables). A straightforward computation gives

u 2 D 2 S = 0 @ (1 -cg ) 2 φ +(c 2 g +2cg -1) + c 2 uψ g (1 -cg )φ -(g + cg )φ -cuψ g (1 -cg )φ -(g + cg )φ -cuψ g 2 φ + φ g 1 A
with determinant and trace such that:

u 4 det(D 2 S) = (φ -φ )(φ (g + g 2 )+ uψ ), (3.2)
u 2 tr(D 2 S) = (g 2 +(1-cg ) 2 )φ +((1+c 2 )g + 2cg -1)φ +u(1+c 2 )ψ . (3.3)
In the simple case where S = uψ(c), i.e. φ = 0, these conditions reduce to ψ ≥ 0, but such entropies are never strictly convex since they are linear with respect to u. We get: Proposition 3.3 (Existence of degenerate convex entropies). For each convex or degenerate convex smooth function ψ (i.e. ψ ≥ 0) the corresponding entropy S = uψ(c) is degenerate convex.

In order to study the existence of strictly convex entropies, notice that for a symmetric matrix A = (a ij ) 1≤i,j≤2 , A is definite positive if and only if detA > 0 and a 22 > 0. Thus an entropy S = φ(w)+ uψ(c) is stricly convex if and only if:

(φ -φ )((g + g 2 )φ + uψ ) > 0, (3.4) φ g + φ g 2 > 0. (3.5)
Rewriting (3.5) under the form:

φ (g + g 2 )+(φ -φ )g 2 > 0 (3.6)
we see, with (3.4) and (3.6), that the strict convexity of S is somewhat related to the sign of g + g 2 = G G , i.e. to the sign of G (indeed G > 0, see (2.6)).

Remark 3.1 (Case of an inert gas). A straightforward computation gives:

H 2 (g + g 2 ) = -q 1 (1 -q 2 ) -q 2 (1 + q 1 ),
moreover we notice, thanks to (1.11), that 1 -q 2 > 0 and 1+q 1 > 0. Thus G have a constant sign as soon as q 1 and q 2 have the same constant sign. This condition is always (and trivially) satisfied when one of the gases is inert and is associated with a convex (or concave) isotherm, and the following proposition ensures the existence of λ-Riemann invariants which also are strictly convex entropies. So, for such cases, W is monotonous with respect to x for any entropy solution. Actually there are some few cases where the isotherm is convex, for instance those of water vapor or ammonia. Proof : When S is also a λ-Riemann invariant, i.e. ψ ≡ 0, we can rewrite the previous inequalities as:

φ (φ -φ )(g + g 2 ) > 0, (3.7) φ (g + g 2 )+(φ -φ )g 2 > 0, (3.8)
and we conclude easily , taking φ(W ) = e αW . §

We investigate now a general negative result in the case when G has an inflexion point.

Proposition 3.5 (Non existence of a strictly convex entropy).

If the sign of G changes then the system (1.12) does not admit any convex smooth entropy.

Proof

(φ -φ )(e -w φ (c)G (c)+ψ (c)) > 0, (3.9) φ G (c)+(φ -φ )g 2 G(c) > 0.
(3.10)

From (3.9) we deduce that φφ has a constant sign on R w . Assume for instance that φφ > 0, then setting φφ = rhs(w) > 0 we get: We study, as an example, the case of one inert gas with active gas equipped with the so-called B.E.T. isotherm.

Working with adimensional concentration c/c s instead of c we can write the BET isotherm as follows:

q(c) = αc (1 + δc)(1 -c) , α>0, 0 ≥ c < 1.
The increasing property of q implies δ > -1. Furthermore q > 0 for δ < 1, and q have an inflexion point for δ > 1. So, the chemical relevant case with one inflexion point is δ > 1. Thus, according to the remark 3.1 and the proposition 3.5 we conclude that there is no strictly convex entropy as soon as δ > 1.

Definition of weak entropy solution

We have seen that there are two families of entropies: uψ(c) and Φ(uG(c)). The First family is degenerate convex (in variables (u,uc)) provided ψ ≥ 0. So, we seek after weak entropy solutions which satisfy: ∂

x (uψ(c)) + ∂ t Q(c) ≤ 0 in the distribution sense. The second family is not always convex. There are only two interesting cases, namely ±G (c) > 0 for all c ∈ [0,1]. When G > 0 and α > 1, we expect to have ∂

x (uG(c)) α ≤ 0 from proposition 3.4. But, the mapping W → W α is increasing on R + . So, the last inequality reduces to ∂

x (uG(c)) ≤ 0. In the same way, if G < 0, we get ∂

x (uG(c)) ≥ 0. Now, we can state a mathematical definition of weak entropy solutions. Definition 3.6. Let be T > 0, X > 0, u ∈ L ∞ ((0,T ) × (0,X),R + ), 0 ≤ c(t,x) ≤ ρ ≡ 1 for almost all (t,x) ∈ (0,T ) × (0,X). Then (c,u) is a weak entropy solution of (1.12) if, for all convex (or degenerate convex) ψ:

∂ ∂x (uψ(c)) + ∂ ∂t Q(c) ≤ 0, (3.14)
in the distribution sense, and where Q = Hψ + h ψ. Moreover, if G keeps a constant sign on [0,1], (c,u) has to satisfy:

± ∂ ∂x (uG(c)) ≤ 0, if ± G ≥ 0 on [0,1]. (3.15)
Notice that the entropies uψ(c) and entropy uG(c) are linear with respect to the velocity u. Furthermore, the entropy uG(c) is contained in the family of entropies (uψ(c)). Indeed, if ±G ≥ 0, ψ = ±G satisfies ψ ≥ 0 with entropy-flux Q ≡ 0. But, inequality (3.15) is so important (when it occurs) and so unusual that we prefer to emphasize it in previous definition.

Riemann problem

We are looking for a weak solution of the following Riemann problem:

( ∂ x u + ∂ t h(c) = 0, ∂ x (uc)+∂ t I(c) = 0, (4.1)
c(0,x) = c -∈ [0,1], x>0, Ω c(t,0) = c + ∈ [0,1], u(t,0) = u + > 0, t > 0. (4.2)
and we search a selfsimilar solution, i.

dC dz = H(C) z f (C) , (4.3)
U (z) = H(C) z . (4.4)
In particular, dC dz has the same sign as f (C).

Proof : setting C (z) = dC dz and U (z) = dU dz , we get from (1.12)

z U = h (C)C , (4.5)
-z (U C + C U )+ C + q 1 (C)C = 0. ( 4
C(z) = c -, 0 < z < z -, dC dz = H(C) z f (C) , z -< z < z + C(z) = c + , z + < z, , (4.7)
where

z + = H(c + ) u + , z -= z + e -Φ(c + ) with Φ(c) = Z c c - f (ξ) H(ξ) dξ. Moreover u -= H(c -)
z - and U is given by:

8 > < > : U (z) = u -, 0 < z < z -, U (z) = H(C(z)) z , z -< z < z + , U (z) = u + z + < z.
(4.8)

Proof : let be z + = H(c + ) u + and Φ(c) = Z c c - f (ξ) H(ξ) dξ. Equation (4.3) writes Φ (C)C = 1 z
thus we can connect c -and c + through a rarefaction wave defined on ]z -,z + [ provided that z + = z -e Φ(C + ) . This gives z -and, according to (4.4),

u -= H(c -) z -. § Remark 4.1.
It appears that c is always monotone along a rarefaction wave but no longer u because the sign of h may change. Indeed the Riemann invariant w = lnu + g(c) is constant along such a wave and g , h have opposite signs. However notice that in the case where one gas is inert, u is monotone (see also [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF]).

Shock waves

We are looking now for admissible shocks in the sense of Liu [START_REF] Liu | The entropy condition and the admissibility of shocks[END_REF]. Let us first recall the Liu entropy criterion: let U -be a given state. Assume λ i (U -) is a (simple) eigenvalue of (2.2). Let W i (s) be the i-shock curve through U -. Any i-shock of speed σ that joins U -= W i (0), on the left, to U + = W i (1), on the right, satisfies the Liu entropy-condition if

∀τ ∈ [0,1] σ ≤ σ i (τ ) (4.9)
where σ i (τ ) is the speed of the shock (U -,W i (τ )). Proposition 4.3. Let (U -,U + ,σ) any weak shock with c -= c + , then

[f ] u -[c] + 1+ h - u -= σ = [f ] u + [c] + 1+h + u + (4.10)
where

[c] = c + -c -, [f ] = f + -f -= f (c + ) -f (c -), h + = h(c + ), h -= h(c -).
Proof : the Rankine-Hugoniot conditions for the system (1.12) are:

σ [u] = [h] (4.11) σ [uc] = [I(c)] = [c + q 1 (c)] (4.12) thus, writing [uc] = u + [c]+[u]c -in the equality [h][uc] = [u][c + q 1 (c)] we get [u] = u + [c][h] [c + q 1 (c)] -[h]c -. If c -= c + (4.11) implies that [u] = 0, thus: σ = [h] [u] = [c + q 1 (c)] -[h]c - u + [c] = 1 u + + [q 1 (c)] -[h]c - u + [c] . From f (c) = q 1 (c) -ch(c) we get [f ] = [q 1 ] -c + h + + c -h -= [q 1 ] -[h]c --h + [
c] and we get the second equality in (4.10). The first one is obtained in a similar way. § Corollary 4.4. Along a shock curve starting at U -, σ has the same sense of variation with respect to c + as

[f ] [c] = f (c + ) -f (c -) c + -c - .
Proof : this is a straightforward consequence of the first equality in (4.10) joined to u -> 0. § As a consequence of this corollary and the Liu entropy-condition we get immediately: Corollary 4.5. A shock connecting two states U -and U + such that c -= c + satisfies the Liu entropy-condition if and only if: for all c between c -and c + , f (c

+ ) -f (c -) c + -c - ≤ f (c) -f (c -) c -c - . ( 4
C(z) = Ω c -if 0 < z < s, c + if s < z , U(z) = Ω u -if 0 < z < s, u + if s < z , (4.14)
where u -and the speed s of the shock are obtained through (4.10). Proof : the first case is corresponding to the linearly degenerate eigenvalue 0, the second case comes directly from the equation (2.3). §

Contact discontinuities

Application to the Riemann problem

We make use of the following wave fan admissibility criterion (see [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum physics[END_REF] for instance): the fan is admissible if each one of its shocks, individually, satisfies the Liu shock admissibility criterion. Then, in view of proposition 4.1, corollaries 4.2 and 4.5, we get the solution of the Riemann problem (4.1)-(4.2) in a very simple way, similar to the scalar case with flux f . Case c -< c + : we consider the lower convex envelope f c of the function f (see Fig. 4.2, left). On the subintervals where f is strictly convex (then f = f c ) we get a rarefaction wave according to corollary 4.2. Elsewhere we get admissible shock waves (or contact discontinuities if f is affine). Case c -> c + : we use the upper convex envelope f c (see Fig. On the left c -is connected to c + via a shock (S), a rarefaction wave (R) and a shock. On the right, c -is connected to c + via a shock and a rarefaction wave. Now, we can prove the following useful lemma which will allow us to control the BV

x norm of ln(u) with the total variation of c. Notice that this estimate is only valid for λ-waves. For 0-waves, we have no control. Lemma 4.8 (BV estimate for ln(u) through a λ-wave). Let c -, c + , u + be given as in (4.2), U the solution of the associated Riemann problem, U -= (u -,c -) t and U + = (u + ,c + ) t . U + is connected to an intermediate state U 0 by a λ-composite wave for z 0 < z < z + and U 0 is connected to U -by a 0-contact discontinuity (c 0 = c -). Then, there exists a true constant γ depending only on q 1 , q 2 and its derivatives such that:

T V [ln(u(z)),(z 0 ,z + )] ≤ γ |c + -c 0 |. (4.15)
Proof : recall that c 0 = c -through a 0-contact discontinuity. We study three cases:

U + is connected to U 0 by a shock wave: from Proposition 4.3, we see that

u + u 0 = S(c + ,c 0 ) = S(c + ,c -) = [f ]/[c]+1+ h + [f ]/[c]+1+
h -where S is smooth and S ≡ 1 when c + = c 0 . Taking the logarithm, we deduce the existence of a constant γ 1 such that |ln(u

+ ) - ln(u 0 )| ≤ γ 1 |c + -c 0 |.
U + is connected to U 0 by a rarefaction wave: the result is an easy consequence of the constance of w = ln(u)+g(c) through a rarefaction wave. So, we have

T V [ln(u(z)),(z 0 ,z + )] ≤ γ 2 |c + -c 0 | with γ 2 := sup [0,1]
|g (c)|. Notice that c is ever monotone on (z 0 ,z + ) but u is monotone on (z 0 ,z + ) if and only if g is monotone on (c + ,c 0 ). U + is connected to U 0 by a composite wave: then there exists k intermediary states

U k ,U k-1 ,••• ,U 1 such that the sequence (c j ) k+1
j=0 is monotone and U j+1 is connected to U j by a shock wave or a rarefaction wave, with U k+1 = U + . Setting γ := max(γ 1 ,γ 2 ) we conclude the proof:

T V [ln(u(z)),(z 0 ,z + )] ≤ γ X j |c j+1 -c j | = γ|c + -c 0 |.
If the number of intermediary states is not finite, the previous inequality remains valid since X j |c j+1 -c j | is convergent. §

Godunov scheme

There is two different approaches to build a Godunov scheme for the system (1.12)-(1.13): the first one uses the "natural" space-time framework (the evolution variable is t), the second one uses the hyperbolicity of the system where the roles of time and space are exchanged, as performed by Rouchon and al. and following the preceding theoretical study (x viewed as the evolution variable). This last option, because of the hyperbolic structure, is the classical one. The corresponding (CFL) condition requires a lower bound for u which is easily obtained thanks to the BV control of ln(u) given in Lemma 4.8. Furthermore, a control of the total variation of u with respect to time is required to show that a subsequence of approximate solutions converges toward a weak (entropy) solution. This cannot be a priori expected: fix c a constant in [0,1] and u b only in L ∞ , then (c(t,x),u(t,x)) ≡ (c,u b (t)) is always a weak entropy solution of (1.12) with c b (t) ≡ c and c 0 (t) ≡ c. However, in the context of the Godunov scheme, we have just to ensure a BV control when connecting U 0 and U + through a λ-composite wave. This is done thanks to Lemma 4.8 again. Finally, we chose the first option because it seemed interesting to show that it is possible (and perhaps clearer) to proceed numerically while keeping the physical structure (t as the evolution variable) despite the loss of hyperbolicity. In this context we have to state an upper bound for u, which is again achieved thanks to the fundamental lemma 4.8 (see Lemma 5.1 below). Furthermore, it is simpler to prove that scheme gives us a weak entropy solution. Indeed, the projection step in this Godunov scheme involves only the concentration c and not the velocity u.

We now present more precisely our scheme and investigate its main properties. Notice that in the following, according with our option, the superscriptsand + for both c and u unknowns are exchanged compared with the preceding section (see Fig. , where M is an integer depending upon N and will be chosen later to satisfy a CFL-type condition. We are going to build an approximate solution (c N ,u N ) of (1.12) on (0,T ) × (0,X). For i = 0,••• ,N and j = 0,•

c N (0,x) = c N (0,x i+1/2 ) := 1 ∆x Z x i+1 xi c 0 (x)dx,x i < x < x i+1 , c N (t,0) = c N (t j+1/2 ,0) := 1 ∆t Z tj+1 tj c b (t)dt, t j < t < t j+1 , u N (t,0) = u N (t j+1/2 ,0) := 1 ∆t Z t j+1 t j u b (t)dt, t j < t < t j+1 ,
where 0 ≤ i ≤ N and 0 ≤ j ≤ M . For the Godunov scheme we need a (CFL) condition: solving a Riemann problem on the box B = [0,∆t[×[0,∆x[ with the initial value c + and the boundary values c -, u -on {x = 0} (see Fig. 5.1), we want that the wave leaves the box B by its upper side {∆t}×[0,∆x[, i.e. z + ∆t < ∆x for a rarefaction wave and s∆t < ∆x for a shock. Since z + < max(u -,u + ) or s < max(u -,u + ), this is clearly satisfied under the following (CFL) condition sup

[0,∆t[×[0,∆x[ u = max(u -,u + ) < ∆x ∆t (5.1)
If this (CFL) condition is always satisfied, we can compute (c N ,u N ) row by row (i.e. for each fixed j) solving the Riemann problem on each box B i,j , i = 1,••• ,N , according to the following procedure.

Assume that, for a given i, we have given c Notice that we could proceed as well by columns before rows, (i before j), which is equivalent to exchange the time and space variable. To ensure the (CFL) condition (5.1), we need to control supu. Therefore, by lemma 4.8, we have to control the total variation in space of c for all time. Recall that, for any function v defined on (a,b):

N (t j ,x) = c + on [x i ,x i+1 [, c N (t,x i ) = c -and u N (t,x i ) = u -on [t j ,
T V [v,(a,b)] = sup ( n X k=0 |v(z k+1 ) -v(z k )|; n ∈ N, a < z 0 < ••• < z n+1 < b ) = sup (Ø Ø Ø Ø Ø Z b a v(z)φ (z)dz Ø Ø Ø Ø Ø ; φ ∈ C ∞ c (a,b), |φ| ≤
Let be λ = u b ∞ × exp(γ T V (c I )) > 0.
We have u ≤ λ, thus if λ∆t < ∆x then the (CFL) condition is fulfilled.

Proof : by induction, using Lemma 4.8. See [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF] for details. §

From lemma 4.8 we easily get the following results. We refer the reader to [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF] for detailed proofs when f is strictly convex (or concave) on [0,1]. Lemma 5.2. Let be γ the constant defined in lemma 4.8. If the (CFL) condition is fulfilled, then, for all t ∈ (0,T ):

T V [ln(u(t,.)),(0,X)] ≤ γ T V [c(t,.),(0,X)].

Existence and BV regularity of an entropy solution

We state now our main result of existence of a weak entropy solution, see definition 3.6. Theorem 6.1 (Global large weak entropy solution). Let be X > 0, T > 0. Assume c 0 ∈ BV (0,X), c b ∈ BV (0,T ), u b ∈ L ∞ (0,T ), satisfying 0 ≤ c 0 , c b ≤ 1 and inf 0<t<T u b (t) > 0. Then the system (1.12)-(1.13) admits a weak entropy solution given by Godunov scheme. Furthermore, c and u satisfy: c ∈ L ∞ ((0,T ) × (0,X)) ∩ L ∞ ((0,T );BV (0,X)) ∩ L ∞ ((0,X);BV (0,T )), (6.1) c ∈ Lip(0,T ; L 1 (0,X)) ∩ Lip(0,X ; L 1 (0,T )), (6.2) ln(u) ∈ L ∞ ((0,T ) × (0,X)) ∩ L ∞ ((0,T );BV (0,X)). (6.3) Furthermore for all convex (or degenerate convex) entropy S with flux Q we have in the sense of distribution on (0,T ) × (0,X):

∂

x S(c,u)+ ∂ t Q(c) ≤ 0. (6.4)

We also have the following bounds: 0 ≤ min(inf c b ,inf c 0 ) ≤ c ≤ max(supc b ,supc 0 ) ≤ 1, (6.5) max °c L ∞ ((0,T ),BV (0,X)) ,c

L ∞ ((0,X),BV (0,T )) (γ is the constant defined in lemma 4.8 and depending only on the q 1 , q 2 functions)

¢ ≤ T V (c I ), (6.6)
Proof : In a classical way, see [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF] for instance, with previous bounds obtained from Godunov scheme, we get BV bounds in space and in time for c, and BV bounds for ln(u) only in space. So, as in [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF], by strong and weak compactness, we get the existence of a weak solution. Furthermore, c N is an weak entropy solution inside each box B ij since c N satisfies Liu-entropy condition in the open set B ij . That means ∂

x (u N ψ(c N )) + ∂ t Q(c N ) ≤ 0 in D (B ij ) for all (i,j). Recall, from definition 3.6, that ψ = ±G and Q ≡ 0 occurs if ±G ≥ 0. Since, (up to a subsequence), (u N ) converges weakly towards u and (c N ) converges strongly towards c and entropies are linear with respect to u we obtain: ∂

x (uψ(c)) + ∂ t Q(c) ≤ 0 in D (B ij ) for all (i,j). Now, to satisfy previous inequality in D ((0,T ) × (0,X)) we have to control error E N coming from projection steps in Godunov scheme. Indeed, for any nonnegative smooth test function ϕ we have, as in [START_REF] Bourdarias | Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases[END_REF],

E N = M X j=1 Z X 0 °[Q(c N )]∂

Proposition 3 . 4 (

 34 When λ-Riemann invariant is a strictly convex entropy). There are strictly convex entropies of the form S = φ(w) if and only if G does not vanish for c ∈ [0,1]. More precisely S α (c,u) := u α G α (c) is a increasing entropy with respect to W , strictly convex for α > 1 if G > 0 and for α < 1 if G < 0.Notice that ±uG(c) is a degenerate convex entropy provided ±G ≥ 0 on [0,1] thanks to proposition 3.3.

  : the assumption on G means that there exists c + , c -∈ [0,1] such that G (c + ) > 0 and G (c -) < 0. Assume the existence of a strictly convex entropy S written under the form S(c,w) = φ(w)+ e w G(c) ψ(c) defined on [0,1] c × R w . The inequalities (3.4)-(3.6) become:

φ

  (w) = p(w)e w , with p (w) > 0(3.11) where p(w) = k + Z w 0 rhs(τ )e -τ dτ for some real constant k. Then, with(3.11), the preceding inequalities write:p(w)G (c)+ψ (c) > 0, (3.12) p(w)G (c)+ p (w)g 2 G(c) > 0. (3.13) With c = c -in (3.12) we get p(w) < -ψ (c -) G (c -) and thus the increasing function p has a finite limit in +∞. The same holds in -∞ using c = c + . Hence there exists a sequence (w n ) n≥0 such that lim n→∞ w n = +∞ and lim n→∞ p (w n ) = 0. Using this sequence in (3.13) with c = c -(resp. c = c + ) we get lim +∞ p ≤ 0 (resp. lim +∞ p ≥ 0 ), thus p ≡ 0, which is absurd (recall that p is increasing). The same holds assuming that φφ < 0. §

Fig. 4 . 1 .

 41 Fig. 4.1. data for the Riemann problem

. 6 )Corollary 4 . 2 .

 642 Using (4.5), equation(4.6) gives z U = H(C) as long as C = 0. Differentiating this last relation with respect to z we get, using (4Assume for instance that 0 ≤ a < c -< c + < b ≤ 1 and f > 0 in ]a,b[. Then the only smooth self-similar solution of (4.1)-(4.2) is such that :

Proposition 4 . 7 .

 47 Two states U -and U + are connected by a contact discontinuity if and only if c -= c + (with of course u -= u + ), or c -= c + and f affine between c - and c + .

Fig. 4 . 2 .

 42 Fig. 4.2. shocks chords are shown as dashed lines. On the left c -is connected to c + via a shock (S), a rarefaction wave (R) and a shock. On the right, c -is connected to c + via a shock and a rarefaction wave.

Fig. 5 . 1 .

 51 Fig. 5.1. Riemann problem in a box B ij

1 )( 5 . 2 ) 5 . 1 (

 15251 and v ∈ BV (a,b) if and only if T V [v,(a,b)] < +∞.Let us define the total variation of initial-boundary concentration by:T V (c I ) := T V (c I ,(-T,X)) where c I (s) := Ω c 0 (s) if 0< s c b (-s) if s < 0Lemma Upper bound for u and (CFL) condition).

Lemma 5 . 3 .

 53 If the (CFL) condition is fulfilled, then, for all N ≥ 0:sup 0<t<T T V [c N (t,.),(0,X)] ≤ T V (c I ), sup 0<x<X T V [c N (.,x),(0,T )] ≤ T V (c I )

u

  L ∞ ((0,T )×(0,X)) ≤ u b ∞ exp(γ T V (c I

  x)dx = O(∆x) since (c N ) is uniformly bounded in L ∞ t ((0,T ),BV x ((0,X))). So the previous solution (c,u) is an entropy solution.§Notice that, with such kind of regularity this solution satisfies initial-boundary value data strongly. Uniqueness is still an open problem.

  t j+1 [, then: 1. if the solution is a shock-wave, we compute s and u + according to Prop. [ as the mean value of the solution of the Riemann problem. 3. If the solution consists in a composite wave, we entirely solve the Riemann problem, see subsection 4.4. Then we take the c mean value on the upper side of the box B

	4.3. Thanks to the (CFL) condition (5.1) and the formula (4.14) we get
	c N (t,x i+1 ) = c + ,
	u N (t,x i+1 ) = u + on [t	j ,t	j+1 [ and we define c N (t	j+1 ,x) on [x i ,x i+1 [ as the
	mean value of the solution of the Riemann problem, that is:
	c N (t j+1 ,x) = c N (t j+1 ,x i+1/2 ) := λsc

-+(1 -λs)c + with λ = ∆t ∆x 2. if the solution is a rarefaction wave, we compute z -by Proposition (4.1). Then, z + is computed as the unique solution of C(z + ) = c + with C defined through(4.7). U is defined by (4.8) with u + = z + H(z + ). As in the preceding case we have c N (t,x i+1 ) = c + , u N (t,x i+1 ) = u + on [t j ,t j+1 [ and we define c N (t j+1 ,x) on [x i ,x i+1 ij .


Youssef Bouanan
email: youssef.bouanan@ims-bordeaux.fr

Gregory Zacharewicz
email: gregory.zacharewicz@ims-bordeaux.fr

Bruno Vallespir
email: bruno.vallespir@ims-bordeaux.fr

DEVS Modelling and Simulation of Human Social Interaction and Influence

Keywords: Modeling and Simulation, DEVS Formalism, Agent-Based Model, Information Impact, Diffusion Network Model

. It provides operational semantics applicable to this domain. The contributions of this work are dynamic models of individuals using low-level language to simulate the propagation of information among a group of individuals and its influence on their behaviour. In more details, we define a set of models of individuals characterized by a set of state variables and the mesh between the individuals within a social network. Then, we introduce the information diffusion based on epidemic spreading algorithms and we transpose them into the case of the message propagation in a social network. Finally, a basic scenario is used to give a beginning of validation to our models using a platform based on DEVS formalism.

Introduction and Motivation

In the field of social influence, a fundamental problem is to develop a model for finding an efficient way to spread information within an informal network. It seems natural that many people are often influenced by their friends' opinions and recommendations. In industrial engineering, even if the enterprise has reached the age of the electronic communication, we assume here that the human direct communication is not sufficiently considered although it remains one critical communication vector to transmit information. This study is focussing on informal social networks that remain a critical communication vector such as coffee break or others informal meetings. The idea is to evaluate the informal communication existing in the enterprise or other organisations that takes other ways than hierarchical or process orders.

These commutations can influence and change the members' opinion in the field of organizational management (for instance the implementation of a new information system).

Therefore, the behaviours of individuals and user groups have to be modelled and integrated into simulators to get more realistic results. The research in social science domain gives significant results but the impact of information on individuals within a social network is, mostly, statically modelled where the dynamic and sequential aspects are not frequently tackled.

Social interaction plays an important role in studying the propagation of information, innovation, ideas, and influence among its members. When an event appears -for example, the use of cell phones among students, the adoption of a new information system within the enterprise, or the rise of a political movement in an unstable society -it can either die out quickly or makes significant inroads into a population. Network diffusion process allows us to understand the dynamics and the propagation of information in social network. These networks can consist of individuals, group of person or organizations. The interactions could be done by physical contact, remote collaboration, any types of social meetings or some forms of verbal or written communication depending on the situations. The diffusion studies are numerous because diffusion phenomena are discussed in several disciplines: computer science (computer virus, information diffusion in a social network) [START_REF] Girvan | Simple model of epidemics with pathogen mutation[END_REF], biology (epidemics) [START_REF] Cauchemez | Role of social networks in shaping disease transmission during a community outbreak of 2009 H1N1 pandemic influenza[END_REF], physics, etc. In our case we diffuse a message enclosed in a data packet. This packet contains several pieces of information which influence the reaction of the individual and time life of the packet. The intention is to go further than most of current models which generally reduce the individuals as simple obstacles or information transmitters without enough nuances in their behaviour and the influence they can have on a message.

The choice of an appropriate modelling technique to study the behaviour of complex systems is very important [START_REF] Robinson | Simulation: The practice of model development and use[END_REF]. To inform the choice of modelling technique, the relevant literature spanning the fields of Economics, Social Science, Psychology, Retail, Marketing, Operational Research, Artificial Intelligence, and Computer Science has been reviewed.

Agent-based Modelling and Simulation (ABMS) play an important role in the field of computational social science. It is a method of computational representation of individual interactions from which social patterns emerge; the latter is a method that affords (dynamic) structural analysis of (socio) structural patterns. Agent-Based Modelling (ABM) shows microlevel processes that affect macro level outcomes. The macro level behaviour is not explicitly modelled; it emerges from the micro-decisions made by the individual entities [START_REF] Pourdehnad | System dynamics and intelligent agent-based simulation: where is the synergy[END_REF]. Today, Agent-Based Models (ABMs) are mostly implemented as object-oriented computer programs. They consist of autonomous agents that can be perceived as computer programs themselves. Agents have in principal three features: they behave and interact according to a given set of rules, possess cognitive capabilities to process information, and constitute their own environment [START_REF] Ferber | Multi-agent systems: an introduction to distributed Artificial Intelligence[END_REF]. In this study we use the Discrete EVent Systems specifications (DEVS) formalism to represent agents. DEVS has the interest to be timed, highly modular and hierarchical for the description of reactive systems. This low-level language permits a full control of the agents' models and it promotes modularity and reusability.

A cellular automaton is a very simple abstraction invented in the 1940s by John von Neumann to study self-reproducing systems [START_REF] Von Neumann | Theory of self-reproducing automata[END_REF]. Since then, Cellular Automata (CA) have been widely used to reproduce and better understand the dynamics of complex phenomena found in the real world, especially when these phenomena only require very simple programming to be reproduced digitally. Despite their high ability to describe complex phenomena, however, CA still suffer from a low ability in supporting structural rules at various levels. In this article, we recall an extension of the CA that compensates this missing structure by means of another simple but equally powerful mathematical abstraction called DEVS. First, we give a comprehensive description of the resulting Cell-DEVS formalism. Then, we demonstrate the ability of this formalism to express the phenomena of social influence. In more detail, this paper will participate in the definition of a set of models that addresses the entities, the structure of a population and the social interaction. We begin by recalling the DEVS formalism and the CD++ toolkit. In addition, we provide a model of individual characterized by a set of attributes using DEVS. Finally, we present a basic scenario of message diffusion over a population to validate our models using a platform based on DEVS formalism.

Background and related work

In this section we introduce the concepts used along the article and we analyse some related works from the literature. We first introduce the DEVS formalism and its extension (focusing on Cell-DEVS). Then we describe some DEVS simulation tools and we focus on CD++, the DEVS simulation environment in which the actual implementation of our work is based on.

Finally, we analyse some related work on the diffusion model.

DEVS formalism

The DEVS formalism for modelling and simulation [START_REF] Zeigler | Theory of Modelling and Simulation[END_REF] is based on discrete events modelling. It provides a framework with mathematical concepts based on the sets theory and systems theory to describe the structure and the behaviour of a system. With DEVS, there is an explicit separation between a model and its simulator. Once a model is defined, it is used to build a simulator (i.e. a device able to execute the model's instructions).

DEVS proposes two kinds of models: the atomic models, which describe behaviour, and the coupled models which describe a structure and hierarchy. DEVS has been used as a simulation paradigm for various systems (Sarjoughian et al., 2001). In our work, we use it to model human behaviour.

The DEVS formalism has numerous advantages. We can classify them into two aspects: theoretical and applicable perspectives (Figure 1). In the theoretical perspective, the DEVS formalism allows the building of a very complex model by connecting different DEVS models, either atomic or coupled models, in a hierarchical manner (Figure 1 Left Circle).

Besides, the formalism provides simple but clear semantics for the basic model behavior

Formal specification of an atomic DEVS model

An atomic DEVS model is defined as: AM = < X, Y, S, ta, δ int , δ ext , λ >  X: is the set of input events;  Y: is the set of output events;  S: is the set of sequential states (or also called the set of partial states);  Ta: is the time advance function which is used to determine the lifespan of a state;  δ ext : is the external transition function which defines how an input event changes a state of the system;  δ int: is the internal transition function which defines how a state of the system changes internally (when the elapsed time reaches to the lifetime of the state);  λ: is the output function. It defines how a state of the system generates an output event (when the elapsed time reaches to the lifetime of the state).

The semantics for this definition is given as follows. At any time, a DEVS atomic model is in a state s  S. In the absence of external events, the model stays in this state for the duration specified by ta(s). When the elapsed time e= ta (s), the state duration expires and the atomic model sends the output λ(s) and performs an internal transition to a new state specified by δ int (s). Transitions that occur due to the expiration of ta(s) are called internal transitions.

However, state transition can also happen due to the arrival of an external event which will place the model into a new state specified by δ ext (s,e,x); where s is the current state, e is the elapsed time, and x is the input value. The time advance function ta(s) can take any real value from 0 to +. A state with ta(s) value of zero is called transient state, and on the other hand, if ta(s) is equal to + the state is said to be passive, in which the system will remain in this state until receiving an external event.

Formal specification of a coupled DEVS model

The classic DEVS coupled model can be described as: N = < X, Y, D, {Mi}, {Ii}, {Zi,j}, select >

DEVS extensions

The DEVS formalism supports an open approach to formalism extension, allowing the researcher to explore new extended or specialized formalism [START_REF] Zeigler | DEVS formalism and methodology: unity of conception/diversity of application[END_REF].

These extensions facilitate the development of models for various applications in many different domains such as biology, engineering, and sociology. For example, [START_REF] Barros | The dynamic structure discrete event system specification formalism[END_REF] proposed the Dynamic Structure DEVS (DSDEVS) formalism which allows changes in model structure during execution. [START_REF] Chow | Parallel DEVS: a parallel, hierarchical, modular modeling formalism and its distributed simulator[END_REF] proposed the Parallel DEVS (P-DEVS) for parallel execution benefits. [START_REF] Hong | A real-time discrete event system specification formalism for seamless real-time software development[END_REF] proposed the Real Time DEVS (RT-DEVS) for executing DEVS models within a real-world environment. From a network modelling perspective, [START_REF] Uhrmacher | Combining micro and macro-modeling in devs for computational biology[END_REF] proposed Multi-Level-DEVS (ml-DEVS) which supports an explicit description of macro and micro level and Wainer and Giambiasi (2002) proposed the Cell-DEVS formalism which is a combination of cellular automata and DEVS that allows the implementation of cellular models with timing delays. Among them, we will use Cell-DEVS in this study to model and simulate the dissemination and the impact of information. Wainer andGiambiasi (2001, 2002) proposed the Cell-DEVS formalism to describe cell spaces as discrete-event models where each cell is represented by a DEVS basic model component to combine the advantages of CA and DEVS methodologies in a systematic way.

Using a modular interface, each DEVS basic model can communicate with its neighboring cells in the cell space, as well as other models outside of the cell space. Cell-DEVS is a combination of CA and DEVS that allows the implementation of cellular models with timing delays. A Cell-DEVS model is defined as a lattice of cells holding a state variable and a computing apparatus to update the cell state. This is done using the present cell state and a set of inputs coming from cells in the neighborhood. Cell-DEVS improves the execution performance of cellular models by using a discrete-event approach. It also enhances the cell's timing definition by making it more expressive. Each cell is defined as an atomic model using timing delays. It can be later integrated to a coupled model by putting together a number of cells interconnected by a neighborhood relationship as showed in Figure 2.

DEVS tools

The DEVS formalism presents an explicit separation between model specification and its implementation (or simulation development). In other words, implementing DEVS models is easily achievable by using an implementation framework supporting the DEVS formalism, such as CD++ [START_REF] Wainer | CD++: a toolkit to define discrete event models[END_REF], a framework which uses C++, JDEVS [START_REF] Filippi | JDEVS: an implementation of a DEVS based formal framework for environmental modelling[END_REF] and DEVSJAVA (ACIMS, 2012) which both (JDEVS and DEVSJAVA) use JAVA, PythonDEVS [START_REF] Bolduc | A modeling and simulation package for classic hierarchical DEVS[END_REF] which is written in Python and VLE (virtual laboratory environment) which is a multi-modelling and simulation platform developed in C++ [START_REF] Quesnel | The Virtual Laboratory Environment-An operational framework for multi-modelling, simulation and analysis of complex dynamical systems[END_REF].

Several simulators for DEVS have been implemented. They are built with various Objectoriented languages (OOL) and even for those which use the same language, the simulation algorithms are different. The proposed DEVS and Cell-DEVS models in this study will be implemented in the CD++ simulator. CD++ is a modelling and simulation toolkit that implements DEVS and Cell-DEVS theory [START_REF] Wainer | CD++: a toolkit to define discrete event models[END_REF]. It has been used to solve a variety of complex applications [START_REF] Wainer | Applying cellular automata and DEVS methodologies to digital games: a survey[END_REF][START_REF] Wainer | Applying Cell-DEVS methodology for modeling the environment[END_REF]. The tool allows models to be defined according to the specifications introduced in the previous section. DEVS atomic models can be incorporated into a class hierarchy in C++, while coupled models are defined using a built-in specification language. The tool also includes an interpreter for a specification language that allows describing Cell-DEVS models. The model specification includes the definition of the size and dimension of the cell space, the shape of the neighbourhood and the borders. The cell's local computing function is defined using a set of rules with the form:

{POSTCONDITION} {DELAY} {PRECONDITION}.
These indicate that when the PRECONDITION is satisfied, the state of the cell changes to the designated POSTCONDITION, whose computed value will be transmitted to other components after consuming the DELAY.

Network diffusion process

Social interaction plays an important role in studying the propagation of information, innovation, ideas, and influence among its members. An event will appear -for example, the use of cell phones among students, the adoption of a new information system within the enterprise -and it can either die out quickly or make significant inroads into the population.

The network diffusion processes have a long history in social sciences [START_REF] Rogers | Diffusion of innovativeness[END_REF]. With the advent of sufficient storage and computational power, this network diffusion process became an emerging research area in computer science [START_REF] Domingos | Mining social networks for viral marketing[END_REF]. Propagation models are designed to reproduce the phenomena that can be observed in social networks with applications in viral marketing, spread of disease and diffusion of ideas and innovations. Most models proposed recently are extensions from the Independent Cascade (IC) [START_REF] Goldenberg | Talk of the network: A complex systems look at the underlying process of word-of-mouth[END_REF] and the Linear Threshold models (LT) [START_REF] Granovetter | Threshold models of collective behavior[END_REF]. The two models characterize two different aspects of social interaction. The IC model focuses on individual (and independent) interaction and influence among friends in a social network. The IC models can also be identified with the so-called Susceptible/Infective/Recovered (SIR) model for the spread of disease in a network [START_REF] Bailey | The mathematical theory of infectious diseases and its applications[END_REF]. The LT model focuses on the threshold behaviour in influence propagation, which we can frequently relate to; when enough of our friends bought a new phone, played a new computer game, or used new online social networks, we may be converted to follow the same action.

Modelling information flow through various social networks is an active research area, with work on diffusion of innovation and technology [START_REF] Bass | A new product growth for model consumer durables[END_REF][START_REF] Brown | Social ties and word-of-mouth referral behaviour[END_REF]Rogers, 1995), viral marketing [START_REF] Leskovec | Patterns of Influence in a Recommendation Network[END_REF], the spread of computer viruses [START_REF] Albert | Error and attack tolerance of complex networks[END_REF], and the spread of diseases [START_REF] Newman | Spread of epidemic disease on networks[END_REF][START_REF] Morris | Contagion[END_REF]. Diffusion models are used to study the adoption of products and spread of innovation influence by viewing them as a process of social interactions. The diffusion of innovation theory, introduced by Everett Rogers in 1962, defines stages of product adoption process which includes knowledge, persuasion, decision, implementation, and confirmation (Rogers, 1995). With respect to marketing, ABM has often been applied to model the diffusion of innovations. For instance, several researchers [START_REF] Goldenberg | The role of hubs in the adoption process[END_REF][START_REF] Rahmandad | Heterogeneity and network structure in the dynamics of diffusion: Comparing agent-based and differential equation models[END_REF][START_REF] Stephen | A comparison of the effects of transmitter activity and connectivity on the diffusion of information over online social networks[END_REF] have used an agent-based approach integrated with network science to model the role of influential in diffusion. [START_REF] Goldenberg | The chilling effect of network externalities[END_REF] have used ABM to explore network effects in product adoption. [START_REF] Garcia | Uses of agent-based modeling in innovation/new product development research[END_REF] provides a good review of how ABM had been used in innovation research. A restricted form of ABM, known as cellular automata (CA), has also proven useful. For instance, [START_REF] Garber | From density to destiny: Using spatial dimension of sales data for early prediction of new product success[END_REF] and [START_REF] Moldovan | Cellular automata modeling of resistance to innovations: Effects and solutions[END_REF] use CA to model the diffusion of innovations. [START_REF] Dabbaghian | Social interactions of eating behaviour among high school students: a cellular automata approach[END_REF] use CA to explore how social interactions among high school adolescents can affect their eating behavior and food choice.

Materials and methods

Human behaviour modelling

General approach

Human Behaviour Modelling is an important area of computational science with implications not only for social sciences, but also for economics, epidemiology and other fields. Scientific literature abounds in heterogeneous and highly specialized, theoretically founded concepts of human cognition, emotion and other behaviour aspects. There are many lines of research on such models, which span several disciplines, have different goals, and often use different terminologies and various approaches. Human behaviour modelling or Human Behaviour Representation (HBR) refer to computer-based models which imitate either the behaviour of a single person or the collective actions of a team [START_REF] Pew | Modeling Human and Organizational Behavior: Application to Military Simulations[END_REF]. Nowadays, research in human behaviour modelling is discussed in a variety of application areas. The important fields of study are military service research [START_REF] Fei | Verification of Human Decision Models in Military Simulations[END_REF], robotics (Kubota and Nichida, 2006), human machine interface (HMI) [START_REF] Okuda | Modeling of human behavior. in man-machine cooperative system based on hybrid system framework[END_REF] and some specially oriented anthropology studies [START_REF] Ding | The effects of time delay of internet on characteristics of human behaviors[END_REF]. Human behaviour models are often represented by finite state machines, rules, fuzzy rules [START_REF] Dorsey | Mathematical Modeling of Decision making: A Soft and Fuzzy Approach to Capturing Hard Decisions[END_REF] or multi-agent based modelling [START_REF] Sun | Multi-Agent Based Modeling: Methods and Techniques for Investigating Human Behaviors[END_REF]. [START_REF] Silverman | Toward a Human Behavior Modeling Anthology for Developing Synthetic Agents[END_REF] proposes a framework for integrating Performance Moderator Functions (PMF) in behavior simulation models. His framework named PMFserv is a model of an agent's cognitive-affective state and reasoning abilities that is applied to profile the traits, cognitions, and reasoning of individual leaders, followers, and others. It is composed of various behavior moderators in different categories (physiology, stress, emotions and decision) and allows the agent to adopt different coping modes depending on its internal state.

We note that human behavior modelling in various fields is evolving and tending to be more realistic. However, these models remain specified in the shape of graphs or mathematical formulas and then directly implemented and coded using classical programming languages.

The disadvantage of such an approach is that there cannot be a clear distinction between modeling and simulation. It is why the applications are specific and difficult to reuse in a different context. The DEVS Formalism allows the development of robust model representation based on the concept of atomic models and on the concept of higher-level models coupling. DEVS allows a hierarchical modelling and it is totally independent of the implementation of simulators that use such system modelling.

Human behaviour modelling using DEVS

In the agent-based model, individuals or group of individuals are represented as agents. Each agent is described by a set of attributes:

 Static attributes: gender, social status, religion, age class, ethnicity, leadership and language, etc.

 Dynamic attributes (variables): opinion, interest, un/satisfied-needs, etc.

Static attributes are intrinsic or unchanged parameters, i.e., time has no effect on them.

Dynamic attributes evolve with time or events. For example, individuals can be reached or not by the information depending on its opinion and the social network configuration.

We use the graphical DEVS notations [START_REF] Song | The DEVS framework for discrete event systems control[END_REF] to describe the human behavior. Figure 1 is an example that describes the message influence on the individual behavior and potentially its dissemination using the DEVS specifications. The first phase is used to configure and initialize the agent' attributes. Then, when the agent is in the "IDLE" phase and if it receives an external event from another agent on port "In_info" (In_info? Packet), it moves to "phase_0". Here, the agent calculates the trust factor α(i,j) between him and the sender depending on the sender's religion, language and age class. If the trust factor is higher than a predefined threshold, the receiver moves to "phase_1" else it returns to "IDLE". Then, if the message is still strong enough, the receiver moves to "phase_2" else it returns to "IDLE". This message creates an impact on the individual, and eventually its behavior depending on the agent's opinion and the relationship between him and the sender. After a period, processing_time, the receiver transmits the message on its ego-network. After the contact between receiver and sender, the receiver's variables (opinion, interest, satisfaction) change according to the message content, the sender and cultural factors. There are clear relations between epidemic disease and information diffusion through social networks. In epidemiological modelling, the spread of diseases is often represented as the disease state of individuals, such as susceptible, infectious, and recovered, as well as possible transition and time constants for transition between states. The state and transition framework is then applied to a resolution level of the population that is appropriate for application. This may be the whole population, subpopulations, communities, or individuals. Although there are many epidemiological models available, most applications have their own specific requirements. In epidemiology, the classical disease spreading models are based upon the cycle of disease in a host: a person is first susceptible (S) to the disease. If then exposed to the disease by an infectious contact, the person becomes infected (I) (and infectious) with some probability. The disease then runs its course in that host, who is subsequently recovered (R) (or removed, depending on the virulence of the disease).

Various epidemic models have been proposed and studied over many years. Recently, researchers (Bailey, 1975 ;[START_REF] Mccormack | An application of epidemiological modeling to information Diffusion[END_REF][START_REF] Jin | Epidemiological modeling of news and rumors on twitter[END_REF] have also applied epidemic models to the diffusion of information and influence in social networks. Both diseases and information can spread from person to person, across similar kinds of networks that connect people, and in this respect, they exhibit very similar structural mechanisms. We propose a basic analogy (Table 1) between the dissemination of information among a group of individuals and the transmission of infectious disease between the individuals themselves. In both cases, something (e.g. a message, feeling, etc.) is communicated through a given contact. 4) it has been a long time (higher than a given threshold) since the scenario occurred.

Influence and opinion change modelling

The challenge in creating a social simulation addresses how groups or individuals influence each other, how they update their opinion and alter behaviour based on messages diffused from others. Many psychological results and theories address the social influence phenomenon [START_REF] French | A formal theory of social power[END_REF]De Groot, 1974;[START_REF] Friedkin | Choice Shift and Group Polarization[END_REF][START_REF] Cialdini | Influence: Science and practice[END_REF]. The mechanisms of opinion change in our work are based on social influence and bounded confidence models (Hegselmann and Krause, 2002). Social influence occurs when a message emitter and a message receiver get in contact. The message emitter attempts to communicate position about an opinion to the receiver. As result of this conversation, the receiver of the message may shift her/his opinion some distance towards or away from that of the emitter's opinion. This opinion is characterized by the opinion number and the opinion confidence bounds (Figure 6). This opinion is held with a certain degree of confidence. This confidence is usually represented as either a symmetrical or asymmetrical set thresholds. This confidence impacts the ability of the receiver's opinion to change. If the emitter's opinion falls beyond the thresholds of the receiver's opinion, then the receivers will not change her/his opinion. This kind of study can be exploited to study the information diffusion in various domains such as the collaboration between several enterprises, in an organization or in the military area.

Simulation experiment

In this section, we present a simple model of information impact (evolution of agent's opinion) using the formalism Cell-DEVS in order to validate the interest to use DEVS and Cell-DEVS in social influence modelling. These models implement limited number of data in order to facilitate its understanding in a pedagogical approach. We find encouragingly that such simple model with a modest amount of readily available data produces credible potential useful predictions of trends.

Conceptual model description

The model involves the propagation of information within a group of individuals (from infosources to info-targets) and it evaluates the affects generated by the information. Using Cell-DEVS, we constructed 441 agents (individuals) that represent an entire population. In Cell-DEVS model the population is represented by two-dimensional square grids where each cell represents an individual in a population. We assume that they are residing near the same area.

Each individual, residing in (i,j) cell, is equipped with variables indicating different parameters and states of the individual; Static attributes: gender, social status, religion, age class, ethnicity, leadership, language and dynamic attributes (variables): opinion, interest, un/satisfied-needs. The state of each cell can vary depending on pre-determined rules. These rules are derived from human behaviour model described in section 2.1. Our assumption is that individuals belong to any of three states; Active (A-agent), Uninformed (B-agent) and passive (C-agent).

The protocol of opinion change is based on social influence and bounded confidence models.

Social influence occurs when there is contact between an info-source (message emitter) and an info-target (message receiver). The message sender communicates a position about an opinion to the sender. As a result of this discussion, the receiver of the message may change its opinion; towards or away from that of the sender's opinion.

Formal specification

We now present the formal definition for the Cell-DEVS model defined in the previous paragraph. DEVS and Cell-DEVS (as CA as well) provide the advantage of a formal approach. It allows formalizing and validating the conceptual model and improves the error detection process, while reducing testing time (thus improving the quality and development costs of a simulation). DEVS permits formal specifications to be translated into executable models, facilitating the verification of the simulator and validation against as well as the controlled investigation of performance through alternative models and simulators [START_REF] Shiginah | Multi-layer cellular DEVS formalism for faster model development and simulator efficiency[END_REF].

Atomic cell-DEVS models

The Atomic cell-DEVS specification of the agent is:

CD = < X, Y, I, S, θ, N, d, δint, δext, τ, λ, D >
Where  X is input port to receive the package or information  Y is output port to send the package or information  S is the set of possible states for a given cell.

 N represents the von Neumann neighborhood {(-1, 0), (0, -1), (0, 1), (1, 0), (0, 0),}  d = transport delay, 100 ms  τ: NS, introduced in figure 2, is defined by following rules.

 Agent state change: After a delay, uninformed agent will become Active (from 0 to 1) if at least one neighbour (Von Neumann's neighbourhood) is an active agent (1).  Opinion change: Each agent updates its opinion when it receives a message from its neighbours according to bounded Confidence (BC) mechanism.



Where

Is the current opinion of the agent (x) and is degree of influence of agent y.

The model we are going to exhibit portrays bounded confidence among the agents in the following sense. An agent i takes only those agents j into account whose opinions differ from his own not more than a certain confidence level (Section 2.3).

 Message adoption: After a delay, when the agent changes its opinion it can adopt the message depending on its age class and its current opinion (opinion > 6).

 I, θ, δint, δext, λ, D will be defined according to coupled Cell-DEVS definitions.

Coupled cell-DEVS models

Once the behavior of one cell is defined, they can be combined into a coupled model: M=<I, X, Y, Xlist, Ylist, η, N,{m,n}, C, B, Z, select> Where  Xlist = Φ  Ylist = Φ  η = 5  I = < P X ,P Y >, with P X ={Φ},P Y ={Φ}  N = { (-1,0), (1,0), (0,0), (0,-1), (0,1) }  m = 40; n = 40  Select = {(0, -1), (0,1), (0,0), (-1,0), (1,0)}

 B = {Φ}  C = {Cij/i∈[0,
The coupled model rules P is built as an array of atomic cells, each connected to its neighborhood. The space borders are provided with different behavior, or their cells are connected with those in the opposite border.

An agent changes its opinion (encourage or not the implementation of an information system) depending on the sender's opinion and its own current status. An agent's opinion is characterized by the opinion number, and the opinion's confidence bounds. When the agent receives an opinion, from another agent within his social network, the receiver modifies its opinion partially, in the direction of the sender's opinion. When the sender's opinion is outside the receiver's confidence bounds, the receiver ignores the sender's opinion. This model and the opinion scaling equations follow largely [START_REF] Friedkin | Choice Shift and Group Polarization[END_REF]Hegselmann and Krause, 2002) introduced in section 2.2.

We suppose that the dimension of for/against the innovation is from 0 to 10, with 5 as indifference towards the change. Initially, the entities may be uninformed individuals or information sources with an opinion about the change affected randomly. They also have a set of thresholds around these opinions. Source nodes initiate the original message. The infosources are chosen randomly.

Implementation

Overview

The CD++ environment includes facilities to build DEVS and Cell-DEVS models. Cellular models are defined using a built-in language based on the formal specifications of Cell-DEVS, which includes the definition of the size and dimension of the cell space, the shape of the neighbourhood, and the type of the cells that represent the borders of the bouncing area.

The cell's local computing function is defined using a set of rules in the following format.

POSTCONDITION DELAY { PRECONDITION }

That is, when the PRECONDITION is met, the state of the cell will change to the designated POSTCONDITION after the duration specified by DELAY. If the precondition is not met, then the next rule is evaluated until a rule is satisfied or no more rules are available. As discussed in [START_REF] Wainer | Applying Cell-DEVS methodology for modeling the environment[END_REF][START_REF] Wainer | Applying cellular automata and DEVS methodologies to digital games: a survey[END_REF], this formalism allows security to be improved and development cost to be reduced thanks to the simulations.

In this section, we show the basic definitions of Cell-DEVS models by defining the behaviour of diffusion process within a group of persons. The model is specified by a 21 × 21 cell space using the C++ specification language. The model defines a set of rules to control the dynamic of agent's opinion and the spread of the message, including four sets of rules for the corner and border cells each as well as a set of rules for the rest of the cell space.

Settings

Our Model of the message dissemination consists of four properties:

1) We use a 2-D regular uniform lattice (21 x 21) to represent the area where innovation diffuses.

2) Each cell has a neighbourhood that includes the surrounding cells at the four possible 2D directions. Each cell in the model has a neighbourhood definition (using the relative positions to the central cell itself) that includes the central cell (0, 0), northwest (NW) cell (-1, -1), northeast (NE) cell (-1, 1), southwest (SW) cell (1, -1), and southeast (SE) cell (1, 1). Each one representing an agent, which can accept discrete values representing various states. Each agent can be in one of two states: "0", agent did not adopt the innovation, "1", agent adopted the innovation.

3) The neighbourhood is a Von Neumann neighbourhood.

4) About transition rules, the variables at each site are updated simultaneously, based on the values of the variables in their neighbors at the preceding time step, and according to the definite set of "local rules". As in this model, the agent updates its opinion when it receives a message of his neighbour. Based on the sender' opinion and its attributes (age and position in the organization) it can change its opinion converging to the sender' opinion and decide to adopt or no the innovation. In the case of this model, the irreversibility of transition is assumed, so that customers cannot "un-adopt" after adoption. Each set of rules is applied to its corresponding zone defined in the cell space.

Figure 5 represents the specification of the initialization part of this model in CD++.

Simulation Results

The figure 6 shows the simulation set and results of a message propagation from info-sources to info-targets. This model represents the opinion of enterprise human resources regarding a change within the organization. Some people are pro, others are not aware about the innovation. The simulation of this model details the evolution of the individual's opinion about supporting opponents' ideas. It can be seen as a way to measure some informal information that can circulate within an enterprise. First, we generate a population of 441 individuals with some attributes; gender, position and opinion. Then, we diffuse a message from 10 sources (info-sources) represented by blue cells. In the beginning of the simulation (Figure 6A), the agents are initialized with the some degree of opinion (randomly) concerning the implementation of a new system (Red cells opinion is between 6 and 10. Orange cells represent individuals with an opinion between 3 and 6. Yellow cell contains individual who are opinion between 0 and 3). When they receive the message they change their opinion based on the sender's opinion and message content. In the end (Figure 6B), we observe that the agents represented by white cells did not receive the message because they were not interest with message or because they do not share the same social networks with their neighborhood and others are changed their opinion depending on criteria presented previously. formalism can be appropriately used to model a complex phenomenon of social influence.

The results of this experiment can be exploited to reveal a communication path and to identify potential exchange of information between partners or collaborators. These informal communications can have negative or positive impacts on official orders and process actions.

These estimations done through the simulation can permit to have a more global vision on the potential impact of information in enterprise, organizations or more generally on population.

This study can be used to resolve the problem of influence maximization (IM); i.e. find a set of β initially activated nodes (info-sources) with the maximum number of activated nodes after the time step t by launching several simulations with different nodes. Then, we compare the final results to select the nodes that maximum influence or spread the information. As perspective, we are integrating progressively more variables to describe more accurately A B human behavior including social and psychological feelings and health. The inner idea is to obtain, thanks to simulation, a better understanding of the influence chain and how information can affect behavior of people and population.

At this level, our illustration model is based on fabricated data to test and simulate our design.

We present a computer check of the rules and the diffusion process implemented in CD ++.

Because it is based for a great part on random variable values assignment, we are aware that we must replicate the model simulation to obtain trustable tendencies. Then, the next step will be the analysis of these data to check different hypotheses.

CONCLUSION AND PERSPECTIVES

This paper introduced the formal modelling and simulation of the impact of information on individuals in a group. We have expressed how DEVS and Cell-DEVS can be appropriate and As a first perspective, we are integrating progressively more variables to describe more accurately human behavior including social and psychological feelings and health. The inner idea is to obtain, thanks to simulation, a better understanding of the influence chain and how information can affect behavior of people and population. This kind of simulation is promising for identifying and measuring the effects of informal information in society. Also, it opens up an interesting perspective to model human behaviour as actor of production systems. In a more general context it can serve to observe the effect of information through defined media on population.

The main perspective remains the definition of a multi-level social network. In detail the real social network is complex, it can refer to several networks spheres from primary networks to less important networks, for e.g. to the family environment, work, , geography or any other connecting networks to transport the information from one individual to another. The final result will be the combination of several networks graphs.

One future step will consist in acquiring and using realistic data in social media for this simulation, for instance to observe the psychological effects of influence information on the population. To go further, we will also consider other experimental terrains, including domain dedicated electronic social networks.

(Figure 1

 1 Top Circle). It can specify a specific state at any point of time as well as connect with other models with I/O events which are caused by state transitions. In addition it permits structural coupling to assembly DEVS and DEVS extension so it encourages interoperability (Figure 1 Right Circle). It is well knows and several implementations exist (Figure 1 Down Circle).

Figure 1 :

 1 Figure 1 : Advantages of DEVS formalism from two perspectives

Figure 2 :

 2 Figure 2: Informal Description of Cell-DEVS Each cell uses N inputs to compute its next state. These inputs, which are received through the model's interface, activate a local computing function (t). A delay (d) can be associated with each cell. The state (s) changes can be transmitted to other models, but only after the consumption of this delay. Two kinds of delays can be defined: transport delays model a variable commuting time (every state change is transmitted), and inertial delays, which have pre-emptive semantics (scheduled events can be discarded).

Figure 3 :

 3 Figure 3 : Specification of individual model 2.2 Message dissemination propagation modelling

x

  Contacts per unit of time and per individual Contacts per unit of time and per individual ß Probability of transmission in a contact between an infective and a susceptible Probability of transmission in a contact between an information holder and a susceptible Force of infection: probability per unit of time for a susceptible to become infective Transmission Strength: probability per unit of time for a node to receive the information The diffusion model presented in this paper is motivated by concepts from existing diffusion models. The key concepts found in the SIR models used in epidemiology and the standard threshold and cascade models are reflected in the models. The SIR model divides the population into three possible categories (susceptible, infected, and removed) that reflect the status of the individuals. Susceptible are individuals who are not infected but may become infected when they get in contact with an infected individual. Infected are individuals who are carrying the disease and have the potential to spread it. Removed are individuals who have either recovered from the disease or died, and cannot spread the disease. The model assigns a disease transmission probability based on a given average rate of contact, and assumes that all individuals are equally likely to become infected. An agent is assumed to transmit messages to another agent based on the SIR model, where agents are classified into three types: A-, Band C-type. A-agent (active agent) is an agent who knows the information and is willing to make others known because the agent is itself interested by the information. B-agent (uninformed agent) has not received any message, but can be so through interaction with Aagent. C-agent (passive agent) has received a message but does not spread it. Each A-agent tries to transmit information to other uninformed agents. Four conditions can cause the end of the simulation: (1) The individual who receives the information is a type B; (2) The strength of the message to be propagated falls below a given threshold; (3) All nodes are type C; or (

Figure 4 :

 4 Figure 4: The mechanism of opinion change

Figure 5 :

 5 Figure 5: Diffusion model specification in CD++

Figure 6 :

 6 Figure 6: Simulation of human behaviour: (A) Initialization; (B) End of simulationThe individual can be opposed to the information and/or person that is transmitting the message and/or favourable to the emitter and message content. Some individuals can be reached by the information where some others not. It depends on the individuals' opinion and the configuration of the social network. This test is basic; it is based on the abstract geographical situation of individuals and it takes one dimension of a social network into account. Nevertheless, the approach already shows how hierarchical DEVS and Cell-DEVS

  formal techniques for modelling and simulating of social influence. At the individual level, DEVS models integrate the set of attributes (e.g. gender, age and languages), variables (e.g. opinion, satisfaction degrees and interests) and behavioral functions. The individual models can transmit the message after being affected itself by the message and having potentially changed the message current strength according to several criteria. The Cell-DEVS structures use these DEVS models of human behaviour and start building on top of it connected modules to form Cell-DEVS network. In addition, the separation between the model and simulator followed by DEVS and CD++, has enabled the modeller to concentrate on building the behavioural model on one side and preparing the spreading logic on the other side using the CD++ toolkit. The model shows a correct, even simple, human behaviour impact regarding information perception and treatment.

Table 1 : Analytical model parameters in epidemiology versus diffusion Epidemiology Information diffusion N

 1

		Number of individuals	Number of individuals
	S(t)	Number of susceptible individuals at time t	Number of susceptible individuals at time t
	I(t)	Number of infectious individuals at time t	Number of individuals at time t that carry the information

ACKNOWLEDGMENTS

This work has been partially supported by the SICOMORES Project N° 132936073 funded by French DGA (Direction Générale de l'Armement). It involves the following partners: IMS university of Bordeaux, LSIS university of Marseille and MASA Group.

BIOGRAPHY

Bruno VALLESPIR is full professor at University of Bordeaux, IMS laboratory. His research specialty is enterprise engineering (enterprise modeling, performance evaluation, interoperability). He is the former responsible of the Production systems group of IMS. He is member of several international working groups (IFIP, IFAC), he participated to 5 European projects, has directed more than 20 PhD students and written more than 120 papers in journals and conferences.


Jean-Baptiste Charlier

Bernard Ladouche

Yvan Caballero

Julie Lions

Enora Lucas

What drives the long-term evolution of water quality in the Jura Mountains? A combined analysis of climate and anthropic impacts

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What drives the long-term evolution of water quality in the Jura Mountains? A combined analysis of climate and anthropic impacts

Jean-Baptiste Charlier 1,* , Bernard Ladouche 1 , Yvan Caballero 1 , Julie Lions 2 , Enora Lucas 1 1 BRGM, 1039 rue de Pinville, F-34000 Montpellier, France 2 BRGM, 3 avenue Claude Guillemin, BP 6009, F-45060 Orléans, France * corresponding author: e-mail: j.charlier@brgm.fr; Tel: +33 (0)4 67 15 79 77; Fax: +33 (0)4 67 64 58 51.

The aim of this study is to explore the long term decadal response of the chemical composition of surface and groundwaters to the global environmental changes. Focusing on karstic basins, we ask how climate and anthropic forcing could influence trends and/or large-scale oscillations of the physico-chemistry of springs and rivers.

In this setting, we present in this study an analysis of more than 45 yrs of hydrological and physicochemical data recorded in springs and rivers of the north of the French Jura Mountains (Doubs and Loue basins, i.e. about 5000 km²). We used climate data (air temperature, rainfall, snowfall) from METEO FRANCE, hydrological data (discharge) from Banque Hydro. Water quality (air temperature, electrical conductivity, NO 3 ) data were collected from public agencies (ADES, EauFrance) at a monthly or a quarterly time step since 50's.

A first analysis is carried out on the Loue river where continuous long-term time series are available. Increase of mean annual water temperature (2°C over 45 yr), daily evapotranspiration (0.4 mm over 45 yr), electrical conductivity (200 µS/cm over 35 yr), and NO 3 (2 mg/l over 45 yrs) were recorded. We investigated relationships between these evolutions in the water, and climatic and hydrological fluctuations. Statistical methods and wavelet analysis were applied to characterize trends and largescale oscillations, respectively.

Finally, our analysis on about 20 sites (springs and rivers) over the study area in the Jura Mountains showed that the physico-chemical variations observed in the Loue river is generalized at the regional scale. It highlights the regional influence of the environmental changes on the water quality evolution.