Antoine Perasso
email: antoine.perasso@univ-fcomte.fr

Ulrich Razafison
email: ulrich.razafison@univ-fcomte.fr

ASYMPTOTIC BEHAVIOR AND NUMERICAL SIMULATIONS FOR AN INFECTION LOAD-STRUCTURED EPIDEMIOLOGICAL MODEL ; APPLICATION TO THE TRANSMISSION OF PRION PATHOLOGIES

Keywords: Structured population dynamics, epidemic models, transport equations, nonlinear PDE, stability analysis, finite volume method AMS subject classifications. 35B40, 65M08, 92D30, 92D40

In this article is studied an infection load-structured SI model with exponential growth of the infection, that incorporates a potential external source of contamination. We perform the analysis of the time asymptotic behavior of the solution by exhibiting epidemiological thresholds, such as the basic reproduction number, that ensure extinction or persistence of the disease in the contagion process. Moreover, a numerical scheme adapted to the model is developped and analyzed. This scheme is then used to illustrate the model with numerical simulations, applying this last to the transmission of prion pathologies.

1. Introduction. Epidemiological compartmental models of SIR type have been studied and extended in several directions since the classical Kermack and McKendrick [22,[START_REF] Kermack | Contributions to the Mathematical Theory of Epidemics: II[END_REF][START_REF] Kermack | Contributions to the Mathematical Theory of Epidemics: III[END_REF] model was built. The goal of such epidemiological models is to describe the propagation of an infectious disease in a population, by incorporating in the model a susceptible class S linked to an infective class I and potentially a recovered class R. Various SIR epidemics models are described either by ordinary differential equations (ODEs) or by partial differential equations (PDEs), these latter models being structured according to other variables than the time (age, space...). We refer to Bailey [START_REF] Bailey | The mathematical Theory of Epidemics[END_REF], Brauer and Castillo-Chavez [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology[END_REF], Capasso [START_REF] Capasso | Mathematical Structures of Epidemic Systems[END_REF], Diekmann and Heesterbeck [START_REF] Diekmann | Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and computational biology[END_REF], Murray [START_REF] Murray | Mathematical Biology I : An Introduction[END_REF], Perthame [START_REF] Perthame | Transport Equations in Biology[END_REF] and Thieme [START_REF] Thieme | Mathematics in Population Biology[END_REF] and references therein for a presentation and examples of SIR models described by ODEs or PDEs.

This article is devoted to the analysis of asymptotic properties and to the analysis of a numerical scheme for an infection load-structured epidemiological SI model that describes an infection process with an exponential growth of a fatal-issued disease. As illustrated at the end of the article, such a structuration allows to model the transmission of prion pathologies (BSE, scrapie...).

Denoting by i the infection load in the infective class, it is supposed that, identically to size-structured models (see Arino [START_REF] Arino | A survey of structured cell population dynamics[END_REF], Cushing [START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF], Webb [START_REF] Webb | Population models structured by age, size, and spatial position[END_REF] and references therein), the following evolution equation is satisfied with respect to time t ≥ 0, (1.1) di dt = νi, where the positive constant ν denotes the growth velocity of the infection. The model also incorporates, additionnaly to the contagion process, a potential external source of contamination that may affect the population, that is supposed to be constant and permanent. Taking into account a mass action law as in SI models, the infection process is then described by a nonlinear system of ODE and PDE of transport type given for t ≥ 0 and i ∈ J = (i -, +∞) ⊂ R + by (1.2)

                       dS(t) dt = γ -(µ 0 + α)S(t) -S(t)T (βI)(t), t ≥ 0, ∂I(t, i) ∂t = - ∂(νiI)(t, i) ∂i -µ(i)I(t, i) + Φ(i)S(t)T (βI)(t), t ≥ 0, i ∈ J, νi -I(t, i -) = αS(t), S(0) = S 0 ∈ R + , I(0, •) = I 0 ∈ L 1 + (J).
In Problem (1.2), T is the integral operator defined for some integrable function h on J by

T : h → J h(i) di.
As a consequence, S(t) + T (I)(t) denotes the total population at time t ≥ 0, with initial population S 0 + T (I 0 ).

Throughout the article the following hypotheses are made on the model: (i) µ 0 , ν > 0 and α, γ ≥ 0 ;

(ii) function Φ ∈ C ∞ (J) is a non negative function such that lim i→+∞ Φ(i) = 0 and J Φ(i)di = 1 ;

(iii) functions µ, β ∈ L ∞ (J) are such that µ(i) ≥ µ 0 and β(i) ≥ β 0 > 0 for almost every (f.a.e) i ∈ J.

As shown in the second equation, Problem (1.2) incorporates the infection load-structure i ∈ J = (i -, +∞) of the infected population, with i -as minimal infection load: the infimum i -is a threshold in the evolution of the infection process from which the individuals are considered to be infected and infectious, so appear in the infective class I.

The model also takes into account demographics of the studied population by assigning a constant mortality rate µ 0 and a constant entering flux γ into the susceptible class S. The mortality rate for the infective class is an infection-load dependent function i → µ(i). A consequence of the assumption (iii) is that function µ satisfies the following asymptotic property, (1.3) lim i→+∞ i i - µ(s)ds = +∞.

The limit in equation (1.3) models that all the infected individuals leave the stage I by dying of the disease with a finite infection load. The infectious process is due to an horizontal transmission of the disease, that is modeled with a mass action law of SI type, where the force of infection is given by T (βI) with β transmission rate. When infection occurs, individuals get variable initial infection loads taking values in J. This variability of the initial infected load at the contamination is therefore represented by the probability density function Φ, whose role is to attribute an initial infection load to susceptible individuals when they get infected. Such an approach has been developed for modelling the transmission of some prion pathologies [START_REF] Perasso | Well-posedness of an epidemiological model described by an evolution PDE[END_REF][START_REF] Stringer | A mathematical model of the dynamics of scrapie in a sheep flock[END_REF][START_REF] Touzeau | Modelling the spread of scrapie in a sheep flock: evidence for increased transmission during lambing seasons[END_REF].

In the model is also considered a potential external contamination source, supposed to be constant in time, that is modeled as an input into the system that affects susceptibles with a constant rate α by attributing the minimal initial infection load i -when the contamination occurs. This process of contamination is stated in Problem (1.2) by the loopback boundary condition νi -I(t, i -) = αS(t), where a zero value of α translates the case without external contamination. It can be seen that such a boundary condition models an intering flux of the disease using the conservation law ∂ t I(t, i) = -∂ i (νiI)(t, i). Indeed, one then gets for every t ≥ 0 and i ∈ J,

d dt i i - I(t, i) = νi -I(t, i -) entering flux in i - -νiI(t, i) entering flux in i .
When studying the evolution of a population over time, a fundamental question in ecology is the persistence of the considered population, which consists to know if this last is eternally present or not. In this article, we aim at studying the disease persistence otherwise its exitinction for the infinite dimensional system (1.2). In other words, we wonder if the infected population is persistent or not. This question has been studied in finite dimensional epidemiological systems, see for instance [START_REF] Burton | Disease persistence in epidemiological models: The interplay between vaccination and migration[END_REF][START_REF] Dhirasakdanon | A sharp threshold for disease persistence in host metapopulations[END_REF][START_REF] Hagenaars | Spatial heterogeneity and the persistence of infectious diseases[END_REF][START_REF] Ireland | The effect of seasonal host birth rates on disease persistence[END_REF][START_REF] Rosà | Thresholds for disease persistence in models for tick-borne infections including non-viraemic transmission, extended feeding and tick aggregation[END_REF][START_REF] Thieme | Mathematics in Population Biology[END_REF] and all references therein, and in some cases in infinite dimensional systems, see for instance [START_REF] Liu | Infectious disease models with time-varying parameters and general nonlinear incidence rate[END_REF][START_REF] Smith | Dynamical systems and population persistence[END_REF][START_REF] Thieme | An epidemic model with post-contact prophylaxis of distributed length I. Threshold for disease persistence and extinction[END_REF] and references therein, and [START_REF] Magal | Lyapunov functional and global asymptotic stability for an infection-age model[END_REF][START_REF] Magal | Two group infection age model: an application to nosocomial infection[END_REF][START_REF] Smith | Persistence of bacteria and phages in a chemostat[END_REF][START_REF] Thieme | How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS ?[END_REF] for persistence study in the specific case of infection-age models. In the present case, the study strongly depends on the value of the contamination parameter α. We prove that, as we could expect by considering a permanent input in the model for the disease, that this last is persistent when the parameter α is positive, and exhibit thresholds on parameters, linked to the basic reproductive number of the disease R 0 , that ensure persistence or extinction of the disease when α is zero. Moreover, drawing on existing results for the case of age-dependent infectivity SI models, we study This article is structured as follows: we firstly recall in Section 2 some mathematical backgrounds adapted to Problem (1.2): well-posedness, regularity and existence of a semiflow for the problem. In a second step, the Section 3 is dedicated to the mathematical analysis of the time-asymtotic behavior of Problem (1.2). We start this section by recalling some definitions of mathematical epidemiology, that are the notion of epidemic persistence and the concept of R 0 . Then Section 3 follows with the asymptotic analysis of the model, in both cases with external contamination and without. In the case with contamination, we prove by the use of fluctuation methods that the disease is uniformly stronglypersistent whatever the value of the positive parameter α and exhibit a concrete lower bound for the infective population. In the case without contamination, that is α = 0, we prove, according to threshold values linked to R 0 , some asymptotic properties of the problem, such as persistence or extinction of the disease by a study of stability of equilibria of the system. These results are obtained with fluctuation methods and also considering some spectral properties of the differential operator of Problem (1.2). Then Section 4 is devoted to the introduction of a numerical scheme for Problem (1.2) and its numerical analysis. Finally, we apply the results of the previous sections to the transmission of prion diseases in Section 5, where we perform some simulations in specific scenarios (persistence vs. extinction) with concrete parameter values.

2. Well-posedness, regularity and semiflow. In all that follows, (X, • X ) is the Banach space given by

X = R × L 1 (J),
equiped by the product norm, and X + is the non-negative cone of X, that is

X + = R + × L 1 + (J). For every constant R > 0, B R denotes the ball of X, B R = {x ∈ X, x X ≤ R}.
Linked to Problem (1.2), we consider the differential operator A : D(A) ⊂ X → X defined by

D(A) = {(x, ϕ) ∈ X, (iϕ) ∈ W 1,1 (J) and ϕ(i -) = αx}, A = -µ 0 -α 0 0 L , with Lϕ = - d di (νiϕ) -µϕ and the function f : X → X is given by (2.1) f (u, v) = γ -uT (βv) Φ uT (βv) .
Then Problem 1.2 rewrites as (2.2)

     d dt S(t) I(t) = A S(t) I(t) + f (S(t), I(t)), S(0) = S 0 ∈ R + , I(0, •) = I 0 ∈ L 1 + (J).
Note the following properties of function f : Lemma 2.1. The function f : X → X given in (2.1) satisfies:

1. ∀M > 0, ∀((u 1 , v 1 ), (u 2 , v 2 )) ∈ B 2 M , f (u 1 , v 1 ) -f (u 2 , v 2 ) X ≤ 2 β ∞ M (u 1 , v 1 ) -(u 2 , v 2 ) X 2.
f is continuously differentiable and its differential is a compact operator on X. Proof. The first point of the lemma is proved in [START_REF] Perasso | Infection Load Structured SI Model With Exponential Velocity And External Source of Contamination[END_REF]. From (2.1) one deduces that for (u 0 , v 0 ) ∈ X the differential D (u0,v0) f of f is given by

D (u0,v0) f = -T (βv 0 ) -u 0 T (β •) ΦT (βv 0 ) Φu 0 T (β •) .
From here, for (u, v) ∈ X one gets, denoting • op the operator norm,

D (u,v) f -D (u0,v0) f op ≤ 2 β ∞ (|T (v) -T (v 0 )| + |u -u 0 |) ≤ 2 β ∞ (u, v) -(u 0 , v 0 ) X ,
so f is continuously differentiable from X to X and for every (u, v) ∈ X, the operator D (u,v) f is bounded. Moreover, its range is clearly a finite dimensional space so this operator is compact.

The following theorem, stating the existence and the uniqueness of the solution of Problem (1.2), is proved in [START_REF] Perasso | Infection Load Structured SI Model With Exponential Velocity And External Source of Contamination[END_REF].

Theorem 2.2.

1. The differential operator (A, D(A)) is an infinitesimal generator of a strongly continuous positive semigroup {T A (t)} t≥0 on X that satisfies

(2.3) T A (t) ≤ 2 e (ν-µ0)t ∀t ≥ 0.
2. For every (S 0 , I 0 ) ∈ X + , Problem (1.2) has a unique mild solution (S, I) ∈ C(R + , X + ). Corollary 2.3. Problem (1.2) induces a continuous semiflow via

R + × X + → X + (t, x 0 ) → φ t (x 0 ) = (S(t), I(t)),
where (S(t), I(t)) is the unique solution that satisfies (S(0), I(0)) = x 0 . Proof. For convenience, let us denote x(t) = (S(t), I(t)) the unique mild solution of Problem (1.2) with initial condition x 0 ∈ X + . From the definition of a mild solution, the map φ t satisfies (2.4)

φ t (x 0 ) = T A (t)x 0 + t 0 T A (t -s)f (φ s (x 0 )) ds,
where {T A (t)} t≥0 is the semigroup of Theorem 2.2 and function f is defined in (2.1).

Then the semigroup property of {T A (t)} t≥0 clearly implies that for every τ ≥ 0,

φ t • φ τ (x 0 ) = φ t+τ (x 0 ),
and so (t, x 0 ) → φ t (x 0 ) is a semiflow on R + × X + . We now prove the continuity. Let M > 0 and (τ, x 0 ),(t, y 0 ) ∈ B M . Then, from Lemma 2.1 and equation (2.3), one gets

φ τ (x 0 ) -φ t (y 0 ) X ≤ φ τ (x 0 ) -φ t (x 0 ) X + 2e (ν-µ0)t (1 + 2M t β ∞ ) x 0 -y 0 X
A consequence of Theorem 2.2 is that t → φ t (x 0 ) ∈ C(R + , X + ) and so the latter inequality proves the continuity of the semiflow. Since from Lemma 2.1 function f is continuously differentiable, the following corollary is a consequence of the regularity theorem p.187 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] :

Corollary 2.4. Suppose that the initial condition satisfies (S 0 , I 0 ) ∈ D(A). Then the mild solution of Problem (1.2) is a classical solution.

3. Asymptotic properties. We assume in all that follows that the initial condition (S 0 , I 0 ) ∈ D(A) ∩ X + and is not zero.

3.1. Definitions. In this section, we express some standard notions from mathematical ecology by formulating in the context of Problem (1.2) the definitions of R 0 (see [START_REF] Anderson | Infectious Diseases of Humans: Dynamics and control[END_REF][START_REF] Diekmann | Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and computational biology[END_REF] for an introduction) and persistence (see [START_REF] Butler | Persistence in dynamical systems[END_REF][START_REF] Freedman | Persistence definitions and their connections[END_REF][START_REF] Thieme | Mathematics in Population Biology[END_REF]).

In epidemiology, a fundamental concept is the basic reproduction number of the disease, denoted R 0 , i.e. the number of secondary infections resulting from a single primary infection into an otherwise susceptible population. Mathematically, R 0 is looked as a threshold, depending on epidemiological parameters of the problem, that ensures or not the stability of an equilibrium point called disease-free equilibrium when this latter exists.

Another fundamental epidemiological concept linked to R 0 is the disease persistence. Let ρ : X → R + be a nonnegative uniformly continuous function on X and consider the composition σ ρ (t, x) = ρ(φ t (x)), where φ t is the semiflow defined in Corollary 2.3. Remark that this latter corollary implies that σ ρ is a continuous map from R + × X + to R + . Let us introduce the following notations, that will be used in all that follows :

σ + ρ (x) = lim sup t→+∞ σ ρ (t, x), σ - ρ (x) = lim inf t→+∞ σ ρ (t, x) Remark 1.
If no misunderstanding about the initial condition x ∈ X is possible, we may omit the x-dependence by writing σ ρ (t) rather than σ ρ (t, x) and σ ± ρ rather than σ ± ρ (x). Here is the definition of persistence for our system. Definition 3.1. The semiflow φ t is said

• strongly ρ-persistent if ∀x ∈ X, ρ(x) > 0 ⇒ σ - ρ (x) > 0 • uniformly strongly ρ-persistent if there exists ε > 0 such that ∀x ∈ X, ρ(x) > 0 ⇒ σ - ρ (x) ≥ ε
The semiflow φ t is said weakly ρ-persistent, respectively uniformly weakly ρ-persistent, if the considered limits above are σ + ρ (x) instead of σ - ρ (x). With the above definition, we say that:

• the total population in Problem (1.2) is persistent if the semiflow φ t is ρ-persistent for the mapping ρ :

(u, v) ∈ X → u + T (v); • the disease in Problem (1.2) is persistent if the semiflow φ t is ρ-persistent for the mapping ρ : (u, v) ∈ X → T (v).
One can now give a necessary condition to get persistence of the disease. Lemma 3.2. The total population in Problem (1.2) is a non negative bounded function which is uniformly strongly persistent if γ > 0 and converges to 0 if γ = 0. Moreover, for γ > 0 the set X 0 = {x ∈ X + , ρ(x) ≤ γ µ0 } is an invariant set for the system (1.2). Proof. The quantity σ ρ(t, (S 0 , I 0 )) clearly represents the total population in Problem (1.2). The positivity of the solution stated by Theorem 2.2 directly yields the positivity of the total population for initial densities such that ρ(S 0 , I 0 ) ≥ 0. Applying the operator T to the transport equation in (1.2) one deduces, using the boundary condition, assumption (ii) on Φ and equation (3.5), the equality

(3.1) σ ′ ρ(t) = γ -µ 0 S(t) -T (µI)(t).
Since µ 0 ≤ µ(i) ≤ µ ∞ for a.e. i ∈ J, the latter equality implies that the total population σ ρ(t) satisfies the differential inequalities

γ -µ ∞ σ ρ(t) ≤ σ ′ ρ(t) ≤ γ -µ 0 σ ρ(t).
Then using standard Gronwall arguments one deduces that

(3.2) f ∞ (t) ≤ σ ρ(t) ≤ f 0 (t),
where f 0 is the following function,

f 0 (t) = e -µ0t ρ(S 0 , I 0 ) + γ µ 0 1 -e -µ0t ,
and f ∞ (t) is the same function as f 0 but with µ ∞ instead of µ 0 . It follows from (3.2) that the total population is a bounded function. Moreover, letting t → +∞ in (3.2) one gets

(3.3) γ µ ∞ ≤ σ - ρ ≤ σ + ρ ≤ γ µ 0 ,
which proves the persistence result. Finally, from (3.2) one deduces that for γ > 0, σ ρ(t, x) ≤ γ µ0 for every x ∈ X 0 , that is equivalent to φ t (X 0 ) ⊂ X 0 , which ends the proof of the lemma.

As a consequence of Lemma 3.2, we focus on the study in the case where the total population does not extinct by assuming in all that follows that γ > 0.

Study in a particular case.

We now look at the particular case where µ and β are non negative constant parameters. We therefore suppose in all that follows the following assumption, (H) µ(i) := µ 1 ≥ µ 0 , and β(i) := β 0 for a.e. i ∈ J.

Then denoting Y (t) = T (I)(t) one gets the following proposition. Proposition 3.3. Suppose that assumption (H) is satisfied. Then (S(t), Y (t)) satisfies for every t ≥ 0 the following differential equations system,

(3.4)            dS(t) dt = γ -(µ 0 + α)S(t) -β 0 S(t)Y (t), Y (t) dt = αS(t) + β 0 S(t)Y (t) -µ 1 Y (t), S(0) = S 0 , Y (0) = T (I 0 ).
Proof. Let us start by proving that the infected stage satisfies the following asymptotic property,

(3.5) lim i→+∞ iI(t, i) = 0.
Indeed from Corollary 2.4 one gets iI(t, i) ∈ W 1,1 (J) for every t ≥ 0. Then from the equality

iI(t, i) = i i - ∂ u (uI(t, u)) du + α ν S(t),
one deduces that for every t ≥ 0 there exists l(t) ≥ 0 such that lim i→+∞ iI(t, i) = l(t). Finally, since iI(t, i) ∈ W 1,1 (J) ⊂ L 1 (J), then necessarily l(t) = 0 for every t ≥ 0 and (3.5) is then proved. Integrating w.r.t. i ∈ J the transport equation of Problem (1.2) yields, as a consequence of the expression of the boundary condition and of the limit stated by (3.5),

dY (t) dt = νi -I(t, i -) + β 0 S(t)Y (t) -µ 1 Y (t) = αS(t) + β 0 S(t)Y (t) -µ 1 Y (t),
and so the proposition is proved. Remark 2. Observe that from (3.4), we obtain the following balance equation

(3.6) d dt S(t) + Y (t) = γ -µ 0 S(t) + µ 1 Y (t),
which states that the total population can change only due to the entering flux and death. Note that, using the mapping ρ and ρ previously defined, this equation can also be written as

d dt σ ρ(t) = γ -µ 0 σ ρ(t) + (µ 0 + µ 1 )σ ρ (t),
where the initial condition (S 0 , I 0 ) is deliberately avoided.

In the particular case where assumption (H) is satisfied, we consider the R 0 value given by

R 0 = γβ 0 µ 0 µ 1 . Proposition 3.4. Suppose that assumption (H) is satisfied. Then • If α > 0, Problem (3.4
) has a unique equilibrium point ( S, Ȳ ) given by

(3.7) Ȳ = δ + δ 2 + 4αγβ 0 µ 1 2βµ 1 , S = µ 1 Ȳ α + β 0 Ȳ with δ = γβ 0 -(µ 0 + α). • If α = 0, then - R 0 ≤ 1 ⇒ there exists a unique equilibrium E 0 = γ µ0 , 0 , - R 0 > 1 ⇒
there are two equilibria, E 0 and E * = µ1 β0 , µ0 β0 (R 0 -1) Proof. Suppose that α > 0. Looking for a steady state ( S, Ȳ ) in (3.4), the second equation gives the equality that links S to Ȳ in (3.7). From here, substituting the S value in the first equation one gets

β 0 µ 1 Ȳ 2 -δ Ȳ -αγ = 0.
The latter equation has only one admissible point that is the non negative value Ȳ stated in (3.7). Suppose now that α = 0. Then for Ȳ = 0 one gets S = γ µ0 from the first equation leading to the equilibrium E 0 . Moreover, taking α = 0 in (3.7) yields S = µ1 β0 and then the first equation in (3.4) implies that γ -µ0µ1 β0 -µ 1 Ȳ = 0 and so one deduces E * as given in the proposition. In the case where assumption (H) is satisfied and α > 0, the latter proposition implies that the disease may not extinct. The following theorem is a particular result of a more general result that will proved without the assumption (H) (see Theorem 3.11 and 3.12).

Theorem 3.5. Suppose that assumption (H) is satisfied and

I 0 > 0. If α > 0 then the disease is uniformly strongly persistent. If α = 0 then R 0 < 1 ⇒ E 0 is locally asymptotically stable, R 0 > 1 ⇒ E * is
locally asymptotically stable and the disease is uniformly strongly persistent.

3.3. Mathematical study in the general case. We now suppose that assumption (H) is not satisfied, translating that the mortality rate and the transmission rate are infection-load dependent functional parameters satisfying hypothesis (iii) .

3.3.1.

Disease with external contamination. This case is translated by the assumption α > 0. Theorem 3.6. Suppose that I 0 > 0 and α > 0. Then the disease is uniformly strongly persistent, and we have the following lower bound,

σ - ρ (S 0 , I 0 ) ≥ αγµ 0 µ ∞ (µ 2 0 + αµ 0 + β ∞ γ) .
Proof. To prove this theorem, we make several uses of fluctuation method (see [START_REF] Hirsch | Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior[END_REF][START_REF] Thieme | Mathematics in Population Biology[END_REF]). Let us denote in all the proof the following limit, By fluctuation method, there exists a sequence t n → +∞ such that

lim n→+∞ S(t n ) = S -, lim n→+∞ S ′ (t n ) = 0.
Since γ > 0, Lemma 3.2 implies that t → ρ(t) is bounded and so σ + ρ < +∞ and same for S + . Let ε > 0. By definition of σ + ρ one gets lim sup n→+∞ σ ρ (t n ) ≤ σ + ρ < σ + ρ + ε and so σ ρ (t n ) ≤ σ + ρ + ε for every n big enough. The differential equation of susceptibles in Problem (1.2) then gives for every n big enough,

S ′ (t n ) ≥ γ -(µ 0 + α)S(t n ) -β ∞ S(t n )(σ + ρ + ε).
From the latter equation, for n → ∞ one gets

(3.9) S -≥ γ µ 0 + α + β ∞ (σ + ρ + ε) > 0.
Furthermore, the fluctuation method also implies that there exists a sequence s n → +∞ such that

lim n→+∞ σ ρ (s n ) = σ - ρ lim n→+∞ σ ′ ρ (s n ) = 0.
Since S -> 0, one can consider now 0 < ε ′ < S -. Then one gets S(s n ) > S --ε ′ for every n big enough. The inequality µ(i) ≥ µ ∞ f.a.e. i ∈ J and equation (3.1) imply that for every n big enough,

σ ′ ρ (s n ) ≥ α(S --ε ′ ) -µ ∞ σ ρ (s n ).
For n → ∞ one finally gets, taking (3.9) into account,

σ - ρ ≥ α µ ∞ γ µ 0 + α + β ∞ (σ + ρ + ε) -ε ′ . Now, letting ε, ε ′ → 0, σ - ρ ≥ αγ µ ∞ (µ 0 + α + β ∞ σ + ρ )
.

Moreover, equation (3.3) induces that σ + ρ ≤ γ µ0 so the latter bound is independent of the initial condition, and the disease is uniformly persistent and the lower bound in the theorem is satisfied.

Disease without external contamination.

We now study the case without external contamination, that is α = 0. To perform the persistence analysis, we exhibit some spectral properties of the differential operator A and of the semigroup {T A (t)} t≥0 of Theorem 2.2. Before proceeding, we introduce some notations and recall some definitions of spectral theory. Denoting ρ(A) the resolvent set of A : D(A) ⊂ X → X, we define the spectrum σ(A), the point spectrum σ p (A) and the spectral bound s(A) by

σ(A) = C\ρ(A), σ p (A) = {λ ∈ C, λI -A is not injective}, s(A) = sup{Re λ, λ ∈ σ(A)}.
For an isolated λ 0 ∈ σ(A), the associated spectral projection P λ0 is

P λ0 = 1 2πi γ λ 0 R λ dλ,
where R λ is the resolvent operator of (A, D(A)) and γ λ0 is a Jordan path in the complement of σ(A)\{λ 0 } enclosing {λ 0 }.

The algebraic multiplicity m a (λ 0 ) ∈ N ∪ {+∞} of λ 0 is the dimension of the range of P λ0 . We recall the following fundamental result of spectral theory (see Yosida [START_REF] Yosida | Functional Analysis[END_REF]) :

Theorem 3.7. If m a (λ 0 ) < ∞ then λ 0 ∈ σ p (A).
Denoting L(X) the set of bounded linear operators on X and K(X) the subset of compact operators on X, we then define the essential norm L ess of L ∈ L(X) by

L ess = inf K∈K(X) L -K X .
We recall that the quotient L(X)/ K(X) is called the Calkin algebra which, when providing the norm L = L ess where L = L + K(X), is a Banach algebra with unit (see [START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF] and references cited in for details). The growth bound ω 0 (A) ∈ [-∞, +∞) of A is defined by

ω 0 (A) = lim t→+∞ 1 t ln ( T A (t) X ) ,
and the essential growth bound

ω ess (A) ∈ [-∞, +∞) of A is ω ess (A) = lim t→+∞ 1 t ln ( T A (t) ess ) .
The following theorem that gives a characterization of the growth bound using the spectrum of A was proved by Engel and Nagel [START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF]. Theorem 3.8. The growth bound of A satisfies ω 0 (A) = max{ω ess (A), s(A)}, and for every ω > ω ess (A) the set σ ω = {λ ∈ σ(A), Re (λ) > ω} is finite and composed of finite algebraic multiplicity elements.

A consequence of Theorems 3.7 and 3.8 is that

(3.10) ω > ω ess (A) ⇒ σ ω ⊂ σ p (A).
Remark 3. Another definition of the essential norm can be stated by the use of the Kuratovski measure of non-compactness ( [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF][START_REF] Webb | Dynamics of populations structured by internal variables[END_REF]). However, this definition is equivalent to the one we use (see [START_REF] Arino | Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics[END_REF]). Due to the quotient defined previously, the use of the Calkin algebra shows well why the compact operators do not affect the growth bound values. More specifically, one gets ω ess (A + K) = ω ess (A) for every K ∈ K(X).

In all that follows, let us consider for every µ ∈ L ∞ (J) the function π µ ∈ L 1 (J), and the values R -, R 0 and R + given by

π µ (i) = 1 νi i i - Φ(s)e -i s µ(l) νl dl ds R -= γβ 0 µ 0 µ ∞ , R 0 = γ µ 0 T (βπ µ ), R + = γ β ∞ µ 2 0 .
Remark 4. One can check that the following equalities are satisfied,

R -= γ µ 0 T (β 0 π µ ∞ ), R + = γ µ 0 T ( β ∞ π µ0 ),
and so the three values R -, R 0 and R + are related by the inequalities

R -≤ R 0 ≤ R + . As a consequence, one gets R -> 1 ⇒ R 0 > 1 and R + < 1 ⇒ R 0 < 1. Proposition 3.9. • If R 0 ≤ 1 then Problem (1.
2) has a unique equilibrium E 0 ∈ X + , that is the disease free equilibrium given by

E 0 = (S F , 0) = γ µ 0 , 0 • If R 0 > 1 then Problem (1.
2) has two equilibria in X + , that are the disease free equilibrium E 0 and an endemic equilibrium E * given by

E * = (S * , I * ) = S F R 0 , γ(R 0 -1) R 0 π µ Proof. Looking for ( S, Ī) ∈ X + constant solutions of Problem (1.2) one gets        0 = γ -µ 0 S -ST (β Ī), (νi Ī) ′ = -µ(i) Ī(i) + Φ(i) ST (β Ī), i ∈ J, νi -I(i -) = 0.
The first equation and the integration of the second equation give

S = γ µ 0 + T (β Ī) , (3.11) Ī(i) = ST (β Ī)π µ (i). (3.12)
Applying the T (β •)-operator to (3.12) yields

T (β Ī) 1 - SR 0 S F = 0.
From here, either T ( Ī) = 0, and since T ( Ī) ≤ T (β Ī) β0 so Ī = 0 in L 1 (J) and (3.11) implies that E 0 is the desired disease free equilibrium, or T ( Ī) > 0. In that case Ī is not zero in L 1 (J). Then necessarily S = S * and from (3.11) follows

T (β Ī) = γ S * -µ 0 = µ 0 (R 0 -1) .
This latter is positive if and only if R 0 > 1, and then (3.12) finally implies

Ī = S * T (β Ī)π µ = I * .
Since the semilinear part f defined in (2.1) is continuously differentiable, a result from the theory of semilinear evolution equations proves that, under some conditions on the growth bound ω 0 (A) and on the essential growth bound ω ess (A), it is possible to approach the stability of some equilibrium points for the infinite-dimensional system (1.2) in the same way as for a finite dimensional system. More precisely, the following proposition hods: Proposition 3.10. For every equilibrium point E of the Problem (2.2), we have 1. ω 0 (A + D E f ) < 0 implies that E is locally asymptotically stable.

2. ω 0 (A + D E f ) > 0 and ω ess (A + D E f ) ≤ 0 implies instability of E. We refer to [START_REF] Prüß | Equilibrium solutions of Age-Specific Populations Dynaics of several Species[END_REF] or Proposition 4.19 in [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF] for the proof of this result. Using this result one gets the following theorem proving that the disease may extinct when R 0 < 1 whereas the disease-free equilibrium is unstable when R 0 > 1. Moreover, we exhibit stronger assumptions on R -and R + that provide either the global stability of the disease-free equilibrium, either the uniform strong persistence of the disease.

Theorem 3.11. Suppose that I 0 > 0 and R 0 < 1. Then the disease free equilibrium E 0 is locally asymptotically stable. Moreover, under the stronger assumption R + < 1, the disease free equilibrium E 0 is globally asymptotically stable.

Proof. The linearized system (1.2) about the disease-free equilibrium E 0 is

(3.13) d dt S(t) I(t) = (A + D E0 f ) S(t) I(t) ,
where D E0 f is the differential at the disease free equilibrium. We first prove that ω 0 (A + D E0 f ) < 0. The estimation (2.3) shows that ω 0 (A) ≤ ν -µ 0 , so does not guarantee a negative value of the growth bound when the exponential velocity ν is too big. However, one can prove that ω ess (A) < 0. Indeed the change of variables using the mapping (t, ξ) → (t, i) = (t, i -e ν(t-ξ) ) and standard integrative computations imply that the semigroup T A (t) satisfies for every t ≥ 0 and every (u, v) ∈ X,

(3.14) T A (t) u v = v(ie -νt ) 0 e -t 0 (µ(ie -νs )+ν)ds χ {θ(i)≥t} + u e -µ0t 0 ,
where θ(i) = 1 ν ln i i -. Moreover, a change of variables clearly gives {θ≤t} v(ie -νt )ds ≤ e νt v L 1 (J) , and since the u-part in (3.14) has rank 1, then it is compact and one gets with assumption (iii) on µ,

T A (t) ess ≤ e -µ0t .
From here, ω ess (A) ≤ -µ 0 , and since from Lemma 2.1.2 the differential D E0 f is a compact operator, one deduces that ω ess (A + D E0 f ) < 0. As a consequence of Theorems 3.7 and 3.8 we only have to focus on the point spectrum of A + D E0 f , by proving that sup{Re λ, λ ∈ σ p (A + D E0 f )} < 0 when R 0 < 1.

For λ ∈ C such that Re λ > -µ 0 and looking for a solution I(t, i) = z(i)e λt of the linearized problem (3.13), function z satisfies

λz(i) + (iνz(i)) ′ = -µ(i)z(i) + γ µ 0 Φ(i)T (βz).
Since the boundary is homogeneous, an integration then gives where ξ(λ) = γ µ0 T (βπ µ+λ ). Suppose now that R 0 < 1. On can check that ξ(0) = R 0 < 1. Moreover, for λ ∈ R the Lebesgue dominated convergence theorem implies that λ → ξ(λ) is a strictly decreasing function on (-µ 0 , ∞) and lim λ→+∞ ξ(λ) = 0.

z(i) = γ µ 0 T (
Then necessarily (3.15) implies that λ ∈ (-µ 0 , 0) and sup{σ p (A + D E0 f ) ∩ R} < 0. Now, for λ ∈ C such that Re λ > -µ 0 one gets, considering the real part in (3.15),

1 = γ µ 0 T β(i) νi i i - Φ(s) cos(Im λ ln(i/s))e -i s
Re λ+µ(l) νl dl ds ≤ ξ(Re λ).

From here, R 0 < 1 ⇒ Re λ < 0. To conclude, A + D E0 f has a dominant eigenvalue with negative real part for R 0 < 1 so Theorem 3.8 implies that ω 0 (A + D E0 f ) < 0. Proposition 4.19 in [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF] finally gives the stability of the disease-free equilibrium E 0 .

We now suppose that R + < 1 and prove the global stability of the disease free equilibrium using the fluctuation method. The goal is to prove that σ + ρ = 0, where ρ : (u, v) ∈ X → T (v). Applying the operator T to the infected equation in Problem (1.2), one deduces from assumption (iii) on parameter µ that

dT (I) dt ≤ αS + β ∞ ST (I) -µ 0 T (I).
Then the fluctuation method implies that

σ + ρ ( β ∞ S + -µ 0 ) ≥ 0,
where

S + = lim sup t→+∞ S(t).
From (3.2) one deduces that γ µ 0 ≥ σ + ρ ≥ S + , so one gets

σ + ρ (R + -1) ≥ 0,
which proves that, unformly in (S 0 , I 0 ), σ + ρ = 0 when R + < 1. Then the global stability of E 0 yields. Theorem 3.12. Suppose that I 0 > 0 and R 0 > 1. Then the disease free equilibrium E 0 is unstable and the endemic equilibrium E * is locally asymptotically stable. Moreover, under the stronger assumption R -> 1, the disease is uniformly strongly persistent.

Proof. Same calculus than in the proof of Theorem 3.11 proves that for R 0 > 1 the disease free equilibrium E 0 is unstable. Indeed, since ξ(0) = R 0 > 1, continuity and decreasing of λ → ξ(λ) on R imply that there exists an unique λ 0 > 0 such that ξ(λ 0 ) = 1. So ω 0 (A + D E0 f ) > 0 and the unstability yields from Proposition 3.10. We now prove on the stability of the endemic equilibrium E * . Since ω ess (A + D E * f ) < 0, we only have to focus on the point spectrum of A + D E * f . The linearized system (1.2) about E * rewrites, using the expression of equilibria E 0 , E * and R 0 ,

(3.16)      dS(t) dt = -µ 0 R 0 S(t) -SF R0 T (βI)(t), t ≥ 0, ∂I(t, i) ∂t = - ∂(νiI)(t, i) ∂i -µ(i)I(t, i) + Φ(i) SF R0 T (βI)(t) + Φ(i)µ 0 (R 0 -1)S(t)
For λ ∈ C such that Re λ > -µ 0 and looking for S(t) = e λt s and I(t, i) = z(i)e λt of the linearized problem (3.16), one gets

λs = -µ 0 R 0 s -SF R0 T (βz) λz(i) + (iνz(i)) ′ = -µ(i)z(i) + Φ(i) γ µ0 T (βz) + µ 0 (R 0 -1)s .
The latter equation may be integrated and then applying the T (β •) operator one deduces that (s, T (z)) satisfies the linear system

(λ + µ 0 R 0 ) γ µ0R0 T (βπ µ+λ )µ 0 (R 0 -1) γT (βπ µ+λ ) µ0R0 -1 . s T (βz) = 0.
Since R 0 > 1, several computations lead to the characteristic equation for Re λ > -µ 0 ,

γT (βπ µ+λ )(λ + µ 0 ) µ 0 R 0 (λ + µ 0 R 0 ) = 1.
Let us denote by Γ(λ) the left hand side of the previous equality. Considering the real and imaginary parts of λ then identifying the real part of the equation Γ(λ) = 1 gives

µ 0 R 0 [(Re λ + µ 0 R 0 )(Re λ + µ 0 ) + (Im λ) 2 ] (Re λ + µ 0 ) 2 + (Im λ) 2 = γ T β(i) νi i i - Φ(s) cos(Im λ ln(i/s))e -i s
Re λ+µ(l) νl dl ds .

Using the equality S * = γ µ0R0 , this latter equality can be also written as

µ 0 (Re λ + µ 0 )(R 0 -1) = [(Re λ + µ 0 ) 2 + (Im λ) 2 ]× S * T β(i) νi i i - Φ(s) cos(Im λ ln(i/s))e -i s
Re λ+µ(l) νl dl ds -1 . (3.17) Suppose now that Re λ ≥ 0. Then one gets

S * T β(i) νi i i - Φ(s) cos(Im λ ln(i/s))e -i s Re λ+µ(l) νl dl ds ≤ S * T (βπ µ ) = 1.
Since R 0 > 1, a contradiction yields from the last equation and (3.17). As a consequence, A + D E * f always has a dominant eigenvalue with negative real part. Finally one deduces, as in the proof of Theorem 3.11, that the endemic equilibrium E * is locally stable. We now prove that the disease is uniformly strongly persistent under the assumption R -> 1. We start by proving the uniform weak persistence of the disease. Suppose by contradiction that the disease is not uniformly-weakly persistent: then for a fixed 0 < ε < µ0(R--1) β ∞ , there exists initial conditions (that may depend on ε) such that σ + ρ < ε. The same arguments than ones developed to prove (3.9) then give S -≥ γ µ0+ β ∞ε , where S -is defined in (3.8). Moreover, Applying the operator T to the transport equation in (1.2) one deduces, using the boundary condition, assumption (ii) on Φ and equation (3.5), that there exists τ > 0 such that for every t > τ , σ ′ ρ (t) ≥ (-µ ∞ + β 0 S -)σ ρ (t). After an integration, one gets for every t ≥ τ ,

σ ρ (t) ≥ e µ 0 µ ∞ µ 0 + β ∞ε R--1-β ∞ε µ 0 (t-τ ) σ ρ (τ ). Since ε < µ0(R--1) β ∞
, one finally gets σ + ρ = +∞, which contradicts σ + ρ < ε. Consequently, the disease is uniformly weakly persistent. To prove that the disease persistence is uniformly strong, we now refer to a compactness argument that ensures that uniform weak persistence implies uniform strong persistence when there exists a compact attractor under the semiflow φ t (see Thieme [START_REF] Thieme | Mathematics in Population Biology[END_REF], Theorem A.32). Using the equality (2.4) for the semiflow φ t = (φ S t , φ I t ) and the expression (3.14) of the semigroup, one deduces that the second component T A (t-s)[f (φ s (x))]

I satisfies for every x = (x S , x I ) ∈ X + the equality

T A (t -s)[f (φ s (x))] I (i) = φ S s (x)T (βφ I s (x))Φ(ie ν(s-t) )e -t-s 0 (µ(ie -νξ )+ν)dξ χ {θ(i)≥t-s} .
Then an obvious change of variable implies that the component φ I t of the semiflow φ t rewrites as

φ I t (x) = φ 1 (t, x, •) + φ 2 (t, x, •) with φ 1 (t, x, i) = x I (ie -νt )e -t 0 (µ(ie -νs )+ν)ds χ {θ(i)≥t} , φ 2 (t, x, i) = R+ φ S t-s (x)T (βφ I t-s (x))Φ(ie -νs )e -s 0 (µ(ie -νξ )+ν)dξ χ {0≤s≤min(θ(i),t)} ds
From here, consider the subset B ⊂ X + defined by B = {(φ S t (x), φ 2 (t, x, •)), t ≥ 0, x ∈ X 0 } where X 0 is defined in Lemma 3.2. We now prove that the adherence of B in X is a compact attractor in the following sense:

1. B X is a compact subset of X ; 2. ∀x ∈ X 0 , lim t→+∞ d(φ t (x), B) = 0 -1. We start to prove the relative compactness of B. Since φ S t (x) is a real variable, we only need to consider the family {φ 2 (t, x, •), t ≥ 0, x ∈ X 0 } of L 1 (J). We identify L 1 (R) with the set of those functions that are zero on (-∞, i -) and we use the well-known Riesz-Frechet-Kolmogorov criterion (see for instance Yosida [START_REF] Yosida | Functional Analysis[END_REF]). We leave to the reader easier conditions needed for the criterion, and concentrate the proof to check that the family {φ 2 (t, x, •), (t,

x) ∈ R + × X 0 } satisfies (3.18) lim h→0 sup (t,x)∈R+×X0 τ h φ 2 (t, x, •) -φ 2 (t, x, •) L 1 = 0,
where τ h is the translation operator in L 1 . Note that the limit above is then satisfied uniformly for t ≥ 0 and for all solutions of the model. since x ∈ X 0 which is an invariant set for the semiflow as stated in Lemma 3.2 and θ is an increasing function, one gets by spliting the integral the following majoration,

(3.19) τ h φ 2 (t, x, •) -φ 2 (t, x, •) L 1 ≤ β ∞ γ µ 0 2 (I 1 (t, h) + I 2 (t, h)),
with

I 1 (t, h) = J×R+ Φ((i + h)e -νs )e -s 0 µ((i+h)e νξ )dξ -Φ(ie -νs )e -s 0 µ(ie νξ )dξ χ {s≤min(t,θ(i)} ds di and I 2 (t, h) = J×R+ Φ((i + h)e -νs )e -s 0 µ((i+h)e νξ )dξ χ {min(t,θ(i))≤s≤min(t,θ(i+h))} ds di Since θ is an increasing function one gets {min(t, θ(i)) ≤ s ≤ min(t, θ(i + h))} ⊂ {θ(i) ≤ s ≤ θ(i + h)},
and the assumption (iii) on µ then implies that (3.20) I 2 (t, h) ≤ J×R+ Φ((i + h)e -νs )e -µ0s χ {θ(i)≤s≤θ(i+h)} ds di.

Moreover, standard computations and majorations give

I 1 (t, h) ≤ J×R+ Φ((i + h)e -νs ) -Φ(ie -νs ) e -µ0s ds di + J×R+ Φ(ie -νs ) e -s 0 (µ((i+h)e νξ )+ν)dξ -e -s 0 (µ(ie νξ )+ν)dξ ds di. (3.21)
From here, since assumptions (ii) and (iii) on Φ and µ imply that that the following functions are in Antoine PERASSO and Ulrich RAZAFISON -2. To prove this limit, we check that for x ∈ X 0 ,

L 1 (J × R + ), (i, s) → Φ((i + h)e -νs )e -µ0s χ {θ(i)≤s≤θ(i+h)} , (i, s) → Φ((i + h)e -νs ) -Φ(ie -νs ) e -µ0s , (i, s) → Φ(ie -νs ) e -
φ 1 (t, x, •) L 1 = θ(i)≥t x I (ie -νt )e -t 0 (µ(ie -νs )+ν)ds di ≤ γ µ 0 e -(µ0+ν)t ,
which ends the proof.

4. The numerical method. In this section we describe and study the discretization of (1.2). The scheme is then used in the next section to perform numerical simulations on the model applied to the transmission of prion pathologies.

4.1. The numerical scheme. We introduce an infection load-time grids where the infection load and the time steps are ∆i and ∆t respectively. We use the following discretizations i j+1/2 = i -+ j∆i, t n = n∆t where j, n ∈ N. We define the cells

K j =]i j-1/2 , i j+1/2 [ centered at i j = 1 2 (i j-1/2 + i j+1/2
), for j ≥ 1. We denote by I n j the approximation of the average of I(t n , i) over the cell K j , namely

I n j ≃ 1 ∆i Kj I(t n , i)di.
We use an implicit upwind finite volume scheme in order to compute I n j . The general scheme is as follows: • We compute the initial states and the data:

S 0 = S(0), I 0 j = 1 ∆i Kj I 0 (i)di, β j = 1 ∆i Kj β(i)di, µ j = 1 ∆i Kj µ(i)di and Φ j = 1 ∆i Kj Φ(i)di. (4.1)
• Assume now S n and I n = (I n j ) j≥1 are computed, ⊲ we define

T (βI n ) = ∆i j≥1 β j I n j , ⊲ we compute (4.2) S n+1 = γ∆t + S n 1 + ∆t(µ 0 + α + T (βI n )) ,
⊲ we compute I n+1 by solving the following linear system:

(4.3) -ν ∆t ∆i i j-1/2 I n+1 j-1 + 1 + ν ∆t ∆i i j+1/2 + ∆tµ j I n+1 j = I n j + ∆tΦ j S n+1 T (βI n ).
Let us now prove some basic properties of the scheme, namely the stability and that it preserves the positivity of the unknowns and it satisfies a balance equation. (i) For any n ≥ 0, j ≥ 1, we have S n ≥ 0 and I n j ≥ 0. (ii) Under the assumption (H), S n and I n satisfy

(4.4) S n+1 -S n ∆t + ∆i ∆t j≥1 I n+1 j -I n j = γ -µ 0 S n+1 -µ 1 T (I n+1 ),
which is a discrete version of the balance equation (3.6).

(iii) Let T > 0. Then for any t n ≤ T , we have the following stability results:

S n ≤ γ T + S 0 , (4.5)

T (I n ) ≤ e T β ∞(γT +S 0 ) (T (I 0 ) + αT (γ T + S 0 )). (4.6)
Proof. (i) Since S 0 ≥ 0 and I 0 ∈ L 1 + (J), then from (4.1), we have that S 0 ≥ 0 and for any j ≥ 1, I 0 j ≥ 0. Next an easy induction argument shows, from (4.2)-(4.3), that S n ≥ 0 and I n j ≥ 0 for any n ∈ N and j ≥ 1.

(ii) Observe that one can rewrite (4.2)-( 4.3) into the form

S n+1 -S n ∆t = γ -(µ 0 + α)S n+1 -S n+1 T (βI n ), I n+1 j -I n j ∆t + 1 ∆i ν i j+1/2 I n+1 j -ν i j-1/2 I n+1 j-1 = -µ j I n+1 j + Φ j S n+1 T (βI n ). (4.7)
This leads to

S n+1 -S n ∆t + ∆i ∆t j≥1 I n+1 j -I n j = γ -(µ 0 + α)S n+1 -S n+1 T (βI n ) + νi 1/2 I n+1 0 -∆i j≥1 µ j I n+1 j + T (βI n )S n+1 ∆i j≥1 Φ j .
Then the discrete balance equation (4.4) follows from the boundary condition of (1.2) which implies that νi 1/2 I n+1 0 = αS n+1 , from the fact that ∆i j≥1 Φ j = J Φ(i)di = 1 and from assumption (H) that implies µ j = µ 1 , for any j ≥ 1.

(iii) From (4.2), we clearly have S n ≤ (γ∆t+S n-1 ), for any n ≥ 1. Then we can write S n ≤ 2γ∆t+S n-2 and by induction, for any n ≥ 1, such that n∆t ≤ T we obtain (4.5). Now, summing the second equation of (4.7) over j ≥ 1, we get

j≥1 I n+1 j -I n j ∆t - ν ∆i i 1/2 I n+1 0 = - j≥1 µ j I n+1 j + S n+1 T (βI n ) j≥1 Φ j .
Using the same arguments as in (ii), we arrive at

j≥1 (1 + ∆tµ j )I n+1 j = 1 ∆i T (I n ) + ∆t ∆i S n+1 T (βI n ) + ∆t ∆i αS n+1 .
Therefore, we have

T (I n+1 ) ≤ (1 + ∆t β ∞ S n+1 )T (I n ) + ∆tαS n+1 .
It follows from (4.5) that

T (I n+1 ) ≤ (1 + ∆t β ∞ (γT + S 0 ))T (I n ) + ∆tα(γT + S 0 ).
Using discrete version of Gronwall's lemma we obtain (4.6).

Comparison with explicit solutions.

A possibility to validate the scheme is to compare the numerical solutions with explicit available solutions. Since no explicit solutions are available for the complete model (1.2), we consider the particular case of system (3.4) where we assume β(i) = 0 f.a.e. i in J. Hence (3.4) is reduced to the following system of ODE (4.8)

               dS(t) dt = γ -(µ 0 + α)S(t), dT (I)(t) dt = αS(t) -µ 1 T (I)(t), S(0) = S 0 , T (I)(0) = T (I 0 ).
For convenience, we assume µ 0 +α-µ 1 = 0. Then system (4.8) has a unique pair of solution (S(t), T (I)(t)) defined by

S(t) = S 0 - γ µ 0 + α e -(µ0+α)t + γ µ 0 + α and T (I)(t) = T (I 0 )e -µ1t + α µ 0 + α -µ 1 S 0 - γ µ 0 + α e -µ1t -e -(µ0+α)t + αγ µ 1 (µ 0 + α) 1 -e -µ1t .
We now consider the following arbitrary parameters: µ 0 = 0.1, µ 1 = 0.15, ν = 0.1, α = 0.02, γ = 1. The initial conditions are assumed to be S 0 = 100 and T (I)(0) = 0.1. In order to reduce the computation, we truncate the domain by assuming that i -≤ i ≤ i + and 0 ≤ t ≤ T , where we set i -= 0.1, i + = 100 and T = 50. The infected load grid ∆i is fixed to ∆i = 0.05. We define the following relative L 1 -discrete error norms

e S = 0≤n≤N |S(t n ) -S n | 0≤n≤N |S(t n )| , e T = 0≤n≤N |T (I)(t n ) -T (I n )| 0≤n≤N |T (I)(t n )| ,
where N is an integer such that T = N ∆t. In Figure 4.1, we report the L 1 -discrete error norms e S and e T using logarithmic scale. We deduce the orders of convergence for each variables, which are respectively 1 for S and 0.68 for T (I). Note that these orders can be improved using more accurate schemes (see e.g. [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF][START_REF] Shu | Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws[END_REF]). On Figure 4.2, both explicit and numerical solutions are plotted. For the numerical solutions we use the time step ∆t = 0.025. We observe good agreements.

5. Application: transmission of prion pathologies. Prion pathologies, such as Scrapie or BSE, are fatal-issued diseases characterized by variable incubation periods during which the disease cannot be detected. At the end of the incubation period, infected individuals usually develop distinctive clinical signs which are rapidly followed by death. As a consequence, no recovery is possible. Modeling the same way as in [START_REF] Hagenaars | The transmission dynamics of the aetiological agent of scrapie in a sheep flock[END_REF][START_REF] Stringer | A mathematical model of the dynamics of scrapie in a sheep flock[END_REF], incubation time heterogeneity may be incorporated in the model through the infection load variable i. Indeed, Equation (1.3) on µ implies that the quantity Π : J → (0, 1] given by Π(i) = e -i i -µ(l)dl denotes the probability to survive for an infected individual with infection load i ∈ J. Consider now a given threshold Π d ∈ (0, 1) such that an individual with an infection load that satisfies Π(i) ≤ Π d is considered to be dead or about to die, meaning that the incubation time of the disease is gone. Since Π is a strictly decreasing function, there is a unique i d > i -such that Π(i d ) = Π d . Then for an initial infection load at contamination i 0 ∈ (i -, i d ) the incubation time τ is given by

τ = 1 ν ln i d i 0 .
In other words, the incubation time τ represents the time for the infectious process to reach the value i d from the initial value i 0 . Then the map i 0 → τ (i 0 ) defines a natural change of variable between incubation time of the disease and initial infection loads at contamination and enables us to incorporate in the model the variable incubation times using the distribution Φ.

The goal of this section is to perform simulations that either allow to strengthen our theoretical results or to show that these latter could be improved. To achieve that goal, we focus on a specific prion pathology that is scrapie: indeed, Problem (1.2) is a variation of scrapie models that describe the propagation of the disease in flocks [START_REF] Hagenaars | The transmission dynamics of the aetiological agent of scrapie in a sheep flock[END_REF][START_REF] Perasso | Well-posedness of an epidemiological model described by an evolution PDE[END_REF][START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF][START_REF] Stringer | A mathematical model of the dynamics of scrapie in a sheep flock[END_REF][START_REF] Touzeau | Modelling the spread of scrapie in a sheep flock: evidence for increased transmission during lambing seasons[END_REF], where the age-structure and the genotype specificity are avoided. More precisely, the following table sums up the specific characteristics of scrapie and why they are well-translated by the model: The distribution of initial infection load Φ is choosed, according to [START_REF] Stringer | A mathematical model of the dynamics of scrapie in a sheep flock[END_REF], as a Γ distribution of parameters (g, c), that is

Characteristics Model

Φ(i) = i c-1 e -i g Γ(c)g c .
The mean and standard deviation, that caracterize the distribution Φ, are given by m Φ = gc and σ Φ = g 2 c.

Parameters g and c in Table 5.1 are computed to ensure that m Φ = 0.35 and σ Φ = 0.05, as in [START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF].

In [START_REF] Stringer | A mathematical model of the dynamics of scrapie in a sheep flock[END_REF], the natural mortality rate is an age-dependent function µ(a) given by a Weibull distribution: µ(a) = 0.22 2 a, with the average natural lifespan that is ā = 4 years. The value of µ 0 in Table 5.1 is then computed from the weibull distribution and the value ā. From here, the infection mortality rate is taken as µ

(i) = µ 0 2 -1+i -2 1+i 2
, so that assumption (iii) is satisfied. The entering flux γ, choosed in order to simulate different scenarios, allows to compute some values of R 0 , R + or R -that are greater or smaller that 1. In both following scenarios the model is considered without external source of contamination so that α = 0.

Scenario 1. This scenario illustrates the case of disease persistence. Corresponding to the value γ 1 = 10 we have R -≈ 1.31. According to Theorem 3.12, one can check on Figures 5.1 that, even if the initial data of infected is small (T (I 0 ) 1 = 1), the disease is persistent. Moreover, another value of the entering flux is choosed in this scenario to test if the disease may persist when R -< 1 < R 0 . Taking the value γ1 = 7 for the enterring flux, one gets R -≈ 0.92 whereas R 0 ≈ 1.17. In that case, one can see that, although keeping a small intial data for the infected population, the persistence of the disease remains. Scenario 2. This scenario illustrates the case of disease extinction. The parameter value γ 2 , resp. γ2 , of the entering flux corresponds to the case where R + < 1, resp. R 0 < 1 < R + (see Figures 5.2). For these simulations, a big initial data for the infected population is choosed in proportion of the overall population (≈ 60%), confirming the global stability of the disease free equilibrium. Indeed, the infected curve (red color) converges to 0, while the susceptible curve (green color) tends to stabilize around the equilibrium point. Moreover, the value γ2 yields R + ≈ 1.31 and R 0 ≈ 0.84. In that particular case, Figure 5.2 (right) also suggests that the global stability of the disease free equilibrium is satisfied when R 0 < 1. 6. Final remarks. In this paper, we have studied the asymptotic behavior of the solution of an infection load-structured SI model. When the model incorporates a constant external contamination, we proved by the use of fluctuation methods that the disease is uniformly strongly persistent, and exhibit a lower uniform bound for the infected population. In the case whithout external contamination, we proved that the model exhibits the traditional threshold behavior where the disease-free equilibrium is locally asymptotically stable if the basic reproduction number is less than one, and the endemic equilibrium is locally asymptotically stable if the basic reproduction number is greater than one. These stability results are proved using semigroup theory. Moreover, linked to the basic reproduction number R 0 , the value R -, resp. R + , ensure that the disease is uniformly strongly persistent when R -is greater than one, resp. the disease-free equilibrium is globally stable if R + is less than one. These consistency results are proved using fluctuation methods coupled with compactness arguments. A numerical scheme adapted to the model is built and analyzed. We then show that such an infection loadstructured model can be applied to the transmition of prion pathologies and use the numerical scheme to perform simulations. The different scenario we proposed illustrate well the theoretical results that are proved in the paper. Furthermore, some simulations make us think that the consistency theoretical results that are proved with values R -> 1 and R + < 1 should remain by substituting the thresholds R -and R + by R 0 . Future works will be dedicated to the analyzis of the global stability of the disease-free equilibrium when R 0 < 1 and of the endemic equilibrium when R 0 > 1, on the basis of similar results proved in the case of SI models structured according to age of infectivity [START_REF] Magal | Lyapunov functional and global asymptotic stability for an infection-age model[END_REF][START_REF] Magal | Two group infection age model: an application to nosocomial infection[END_REF]. Another future work will be the study of a model incorporating a more general external source of contamination. For instance, in the particular case of prion pathologies, the food intake would lead to a linear but non-autonomous and periodic boundary condition. Some other cases, such as an environmental spatially localized source of contamination, would be modeled by a non-linear function of the density S(t) with a finite limit translating a saturation process of the number of susceptibles in contact with the source. This would lead to formulate a PDE problem with a non-linear boundary condition that may take into consideration more realistic cases than with a linear external source of contamination.

Fig. 1 . 1 .

 11 Fig. 1.1. Diagram of population dynamics fluxes

  βz)π µ+λ and applying the operator T (β •) leads to the following characteristic equation for λ ∈ C, Re λ > -µ 0 , (3.15) ξ(λ) = 1

I 1

 1 s 0 µ((i+h)e νξ )dξ -e -s 0 µ(ie νξ )dξ , one can apply the Lebesgue dominated convergence theorem in equations (3.20) and (3.21), proving that lim h→0 sup t≥0 (t, h) + I 2 (t, h) = 0. Equation (3.19) then proves that (3.18) is satisfied, and the relative compactness of B yields.

Proposition 4 . 1 .

 41 Assume that for any n ≥ 0, S n and I n are computed by (4.2)-(4.3).

Fig. 4 . 1 .

 41 Fig. 4.1. L 1 relative errors norms e S and e T in log/log scale

20 AntoineFig. 4 . 2 .

 2042 Fig. 4.2. Comparison between explicit solutions and computed solutions

Fig. 5 . 1 . 1 -

 511 Fig. 5.1. Scenario 1-Total infected curve for R -> 1 (left) and R -< 1 < R 0 (right)

Fig. 5 . 2 . 2 -

 522 Fig. 5.2. Scenario 2-Total infected curve for R + < 1 (a) and R 0 < 1 < R + (b)

Table 5 . 1

 51 Parameter values used for the simulations.

	Parameter definition	symbol	value
	initial susceptible population size	S 0	10 indiv.
	initial infected load range	[i -, i + ]	[10 -2 , 10 2 ] years
	basic mortality rate	µ 0	0.39 year -1
	horizontal transmission rate	β	4.10 -2 (indiv. • year) -1
	infection load velocity	ν	1.15 year -1
	contamination rate	α	0 year -1
	first infection load distribution Φ	(g; c)	(7.10 -3 ; 49)
		Scenario 1
	initial infected population size	T (I 0 ) 1	1 indiv.
	entering flux	(γ 1 ; γ1 )	(10 ; 7)indiv.year -1
		Scenario 2
	initial infected population size	T (I 0 ) 2	50 indiv.
	entering flux	(γ 2 ; γ2 )	(3 ; 5) indiv.year -1


Fanny Ademe

LIST Cea

Anastassova Margarita
email: margarita.anastassova@cea.fr

Pierre Falzon
email: pierre.falzon@cnam.fr

Cnam

The Environment: a Source of Capabilities for Older Adults? Le Morellec

Keywords:

This article presents an exploratory study on older adults' mobility. The study was based on both observations in a complex built environment (a university hospital), and on an online questionnaire distributed to people aged 50 and more. The main objective of the two studies presented in this paper was to determine the difficulties encountered by older adults when moving outdoors and indoors. A secondary objective was to investigate the resources used in order to cope with the difficulties.The results shows that the main mobility obstaclesfor older adults are related, firstly,to the salience of landmarks and the spatial organization of theenvironment and, secondly, to the age-related declinein physical, sensory and physiologicalabilities.Our studies show that the main resourcesto overcome these obstacles are landmarks and personnel support in the hospital and Internet, GPS, plans and mapsoutdoors.

INTRODUCTION

The aging of the population involves changes in different activities such as mobility. Studies on aging suggest that there are mainly changes in physical and cognitive capacities (e.g. walking speed [START_REF] Holland | The effect of age, gender and driver status on pedestrians' intentions to cross the road in risky situations[END_REF], cardiorespiratory capacity [START_REF] Maguire | Physiology of ageing[END_REF], memory and attention [START_REF] Rose | Age and individual differences in prospective memory during a" Virtual Week": The roles of working memory, vigilance, task regularity, and cue focality[END_REF]). On the average, these capacities tend to deteriorate starting from the age of 50. However, some authors [START_REF] Dunbar | Older pedestrians: a critical review of the literature[END_REF][START_REF] Li | Walking while memorizing: Age-related differences in compensatory behavior[END_REF]show that older adults use compensatory strategies to cope with this decline and that not all cognitive and physical capabilities decline with age [START_REF] Fisk | Designing for older adults. Principles and Creative Human Factors Approches[END_REF]. An example of a compensationstrategy used byolder people isavoiding drivingin challenging circumstances [START_REF] Gwyther | The effect of age, gender and attitudes on self-regulation in driving[END_REF]. Thus, withaging, individuals continue developing new skills. This development requires resources for learning.

In mobility, these resources are landmarks (e.g. signage), route knowledge (i.e. actions associated with landmarks) and survey knowledge (i.e.distance and directional relationships betweenlandmarks)."These kinds of knowledge help guide people's actions in adaptive ways, in other words, so that their behavior is coordinated not only to the environment as perceived but also to the environment as conceived and remembered" [START_REF] Ishikawa | Spatial knowledge acquisition from direct experience in the environment: Individual differences in the development of metric knowledge and the integration of separately learned places[END_REF].Landmarks are fundamental to the efficient navigation [START_REF] Brunner-Friedrich | Active Landmarks in Indoor Environments[END_REF]. They are preferentially selected by people for their content, form, color, and prominence [START_REF] Nothegger | Selection of Salient Features for Route Directions[END_REF].

The main objective of the two studies presented in this paper was to determine the difficulties encountered byolder adults when moving outdoors and indoors. A secondary objective was to investigate the resources usedin order to cope with the difficulties. A special focus was put on environmental resources and their role for helping older adults to compensateage-related decline in physical and cognitive abilities.

The paper is organized as follows.Thefirst part presents themethodology of the two studies. The second part presents the results of the studiesand a related discussion. The last part reflects ona number of limitations of the studies and the perspectives for our future work.

METHODOLOGY

The two studies are based on the use of two different methods, which are 1) open preliminary observations of older adults moving indoors, in a University hospital, and 2) an online questionnaire on older adults' mobility patterns and habits when moving outdoors. We were interested in these two aspects of older adults' mobility, since our studies were conducted as a part of a European project, called ENTRANCE,targeting the development of interfaces for facilitating older adults' mobility indoors and outdoors (http://www.entrance.fr/).

As for the choice of the University hospital, it was selected because of the important number of older adults visiting it and because of its complex spatial organization.In fact, it was a 14-floor building comprising 9 regular floors, a ground floor and two basement floors.Another reason for choosing this hospital was the demographic characteristics of the region in which it is located (i.e. the Limousin). With 21% of its inhabitants aged 65 and over, and its population having started to decline between the 1982 and 1990 censuses, the Limousin foreshadows the France of the 2020s (for more details see http://www.insee.fr/fr/ffc/docs_ffc /cs74a.pdf).

The methodologies of the two studies are presented in more details below.

Preliminary observations

There were 32.5 hours of open observations done over 5 days, during the regular working hours. No pre-established observation guide was used.The objective of the observations was to collectolder adults' difficulties in terms of orientation, navigation, as well as the use of environmental resources to cope with these difficulties. Thus, we focused on the effects of the spatial organization of the hospital, of time and physical constraints and ofinformational resourceson older adults' mobility. We also observed and registered older patients' and visitors' interactions with the hospital staff.

Subjects

We observed patients and visitors aged 50 or more. This choice can be explained by the fact thatthese are the elderly of tomorrow who could be the potential users of the technologies developed in the ENTRANCE project.We observed older adults moving without any difficulty, but also older adults who seemed lost and or searching for orientation information (e.g.signage or landmarks).

Chosen locations

When observing, we were standing at strategic locations such as lobbies, waiting rooms, reception desks, elevators accesses, parking lots, corridors. These locations were chosenon the basis of a literature analysis on problematic points for older adults' mobility [START_REF] Hölscher | Finding the Way Inside: Linking Architectural Design Analysis and Cognitive Processes[END_REF][START_REF] Rooke | Wayfinding: embedding knowledge in hospital environments[END_REF]. The choice of locations was also discussed and approved by the hospital staff.

Coding scheme

The notes taken during the observations were transcribed. A content analysis was performed on the data. Based on the content of the transcripts, the relevant literature on older adults' mobility and on discussions with the hospital staff, we defined the following coding categories:

1. Spatial organization: it is the physicalaspectof the hospital aswell as the distribution and organization of different services and levels. 2. Time and flow effects: these are mainly the peaks in hospital attendance. 3. Landmarks: signage, information points. 4. Staffresponsesto questions frompatients and visitors. 5. Personal characteristics such as anxiety and stress. This coding scheme was then applied to the transcripts of the observations. A complementary coding ofall the units in the 5abovementioned categories was then done. The objective of this second coding was to define, within each of the 5categories, the obstacles, the resources and the neutral elements used by older adults when moving in the hospital. We define as obstacles the elements disturbing fluid mobility, e.g. the complex spatial organization of thehematologyunit. We define as resources the elements with specific utility for mobility and navigation. A typical example is the presence ofmedical assistants near the elevators. Because elevators are a strategic location, such a presence facilitates orientation of lost visitors and patients.The neutral elements do not satisfy either of these definitions.

The units within the category « Landmarks » were further recoded. The objective was to determine whether the difficulties and the resources from this category were related to the content (i.e. the meaning conveyed by an informational element provided by the environment) or the form of this informational element. The form of the elemnts concerns the media on which it is presented (e.g. paper, traffic sign, etc..). Also, a complementary coding of the category "Spatial Organization" was done to determine whether the obstacles and resources within this category are related to a (mis)undertanding or an interaction between the visitor/patient and the environment.

Online questionnaire

The online questionnaire comprised 20 questions. Its objective was to collect information on the daily mobility patterns of older adults (aged 50 and more). In this paper, we only focus on the data related to mobility difficultiesand the navigation support used by older adults. We had 234 volunteer respondents (i.e. 140 persons aged 50 to 64; 76 aged 65 to 74; 18 aged 75 and more, all of them living in France).

RESULTS AND DISCUSSION

Preliminary observations

Figure1 -Environmental resources and obstacles

Figure 1 shows a clear tendency noted during the preliminary observations.The environment seems to cause more difficulties (47%) to older adults than it provides mobility resources (28%). However, this observation should be put in perspective, since an observer usually focuses on difficulties in human activity.

Our results suggest that the difficulties are mainly related to the spatial organization, to the landmarks available in the 0% 10% 20% 30% 40% 50% environment and to time and flow constraints. For example, in busy periods, the receptionis not visible, or there are long lines. In this case, it could be identified as an obstacle for some hospital users. In contrast, when the flow is limited, the reception is easy to locate and may be serve as a useful resource for orientation. Personal characteristics (Figure 1) do not seem to play an important role in mobility. Again, this result should be put in perspective, as it is related to the method used (i.e. open observations rather than observations of individual activity). The evaluation of the effects of personal characteristics on older adults' mobility requires a detailed analysis of individual activity (e.g. using interviews).

To understand why environmental elementscould be either an obstacle or a resource, a deeper data analysis was done.

The results of this analysis showed that the difficulties rooted in the spatial organization of the building were mainly due to the understanding that the users had of this organization (Figure 2). In the same time, severalstudies show that comprehension difficulties result in difficulties inconstructing a meaningful representation of space [START_REF] Hölscher | Finding the Way Inside: Linking Architectural Design Analysis and Cognitive Processes[END_REF]. In our study, the older adults' problems with spatial organization were mainly due to the disposition of a hospital service on several floors. The category "Landmarks" (Figure 3) includes information provided by traffic signs, maps, as well as by oral route descriptions. For this category, the form rather than the content of information provokes orientation and mobility difficulties. In our study, the main problems are due to the graphic layout of the informational support, their localization, as well as their large variety. According to part of the hospital personnel, these elements might provoke informational overload in users. Though in our study, the environment is often perceived as an obstacle, it may be a valuable resource for older adults' mobility. As shown in Figure 1, the medical staff is the main resource used to find one's way inthe hospital. Users also uselandmarks, route descriptions given by receptionists and audio information in lifts. For all these supports, the most problematic aspect is their form (Figure 3).

Online questionnaire

Figure 4 -Mobility difficulties

Figure 4 shows the main difficulties encountered by older adults in their mobility. Firstly, it is clear that with aging, the number of individuals experiencing difficulties increases. Thus, after 75 years, more than 40% of the respondents report difficulties when moving around. These difficulties are mainly due to physical problems (e.g.difficulties to carry heavy bags when walking) and to sensory and physiological limitations (e.g. cardiac, respiratory, visual, auditory problems). This may mean that the environment is not adapted to the mobility needs of older adults or that its physical characteristics provoke difficulties to their navigation and orientation. Physiological and sensory difficulties, though important, do not seem the main obstacle for older adults' mobility.

Figure 5 shows the frequency of travel outside one's home according to the age group. Two phenomenaseem to appear. The first one is thatthefrequencyof traveldecreases withadvancing age. The secondone is thatthe respondents aged 75 and moremovemore often than65-74. Howeverwe supposethat if they move more frequently, they cover shorter distances compared to the group aged 65-74. We canthus assume thatthe people aged 75 and morediffer fromother age groupsin terms ofmobility practices.

Figure 5 -Weekly frequency travel

Figure 6 shows that with the advance in age, people tend to use more navigation aids (e.g. technologies, route descriptions, classic maps). Moreover, with cohorts' effects we can imagine that older people of tomorrow will use more technology. However, there is an exception in the group of seniors aged 75 or more, who tend to use technology less often. This tendency may be explained by a generational effect or by the limited need of such technology if moving on short distances.

Figure 6-Use of technologies and other navigation aids

Figure 7 shows the resources used by older adultswhen moving around. Technologies (GPS andInternet) seem to be used very often as navigation support. It is probably because older adults have better performance in navigation with these technologies [START_REF] Goodman | Using Landmarks to Support Older People in Navigation[END_REF]. However,mobile interface (e.g. phones and tablets) appear to beless widely used. It is possible thatthis type of resourcesis lessused because they are too complex or becausetheir main functionare not adapted to older adults' needs.Also, older adults use very littleenvironmental information (e.g. billboards, traffic signs, audio information). If people rely on aid other than environmental information, it may mean that these resources are not sufficient for older adults to find their way. This could also mean that the built environment is not affordantor missing resources are not directly accessible and / or usable.

Figure 7 -Types of environmental resources

The results from our preliminary study show that environmental resources are crucial for the success or failure of older adults'navigationand orientation activity but they are not sufficient. Furthermore, we can suppose that resources or obstacles are differentaccording to age (Figure 4) and practice (e.g. Figure 5).

LIMITATIONS, IMPLICATIONS FOR DESIGN AND FUTURE WORK

The exploratory study presented in this paper shows an activity analysis focused on the relationship between individuals and the environment in which they move. It highlights the role of the environment in mobility (and activity, in a more general perspective). It also questions the potential of this environment as a learning resource and a support for the compensation mechanisms used by older adults in their mobility. We think that taking into account the changing environment influencing directly and indirectly older adultsmobility, is a valuable perspective to explore.

Of course, as a preliminary study, our study has a number of limitations summarized below.

Limitations of the studies

Thus, the interaction between older adults and the environment in which they movehas been evaluated on the basis of observations and a questionnaire on daily mobility patterns. However, as noted above, this analysis does not sufficiently integrate the individual user's activity. For this reason, interviews and more detailed observations will be donein the future. Furthermore, the questionnaire was conducted online, which limits the respondents to people already using information technologies in their daily life. In the future, we will complement this data by data based on interviews and observations of non-users of information technologies.

Implications for accessible design

Firstly, our preliminary observations indicate that older adults'difficulties when moving in complex built environments are mainly due to the media and the form in which landmarks is provided.Thus, graphical layouts may provoke difficulties in spotting in reading information. Another difficulty concerns complex spatial organization of indoor environments. The complex spatial organization of the building in which the preliminary observations were done lead to comprehension difficulties, which, in their turn, impeded the intuitive perception and use of environmental resources. In this sense, in our study, the architecture of the hospital does not seem intuitive. On the contrary, intuitivecomplex built environmentswould provide informational clues which could naturally and intuitively guide the older adult to his/her destination. However, as it is not enough to dispose of the necessary resources, the design should also provide the conditions for using these resources (e.g. learning technologies, variety of ways for achieving the same goal, etc...).Designers should also work on solutions that help users more easily integrate the logic of space, make landmarks more salient and construct a mental map of the environment.

Also, the online questionnaire suggests that with aging people travel less. This tendency is probably due to theage- related decline in walking or cardiorespiratory capacities.Thus, a good design should also allow older adults to compensate age-related sensory and physical limitations. A potentially useful design orientation can be the capability approach [START_REF] Sen | Development as freedom[END_REF][START_REF] Falzon | Enabling environments and reflective practices[END_REF], supporting the design of environments which allow older adults to develop new skills and knowledge, broaden their activity space and their control on the situation, as well as their autonomy. This approach could support the design of technologies enablingpersonalization of information and embedding learning elements.

Methodological perspective

This study questions the role of the environment as a mobility resource that helps older adults to compensate declines. When a person is engaged in mobility, his/her productive activity is focused on his/her interaction with the material and symbolic elements of the environment, as well as on their transformation according to task (e.g. go to a medical visit). The person then controls his actions throughout the activity to adapt to the context. Thus, he/she is involved in a constructive activity. This constructive nature of activity may provoke a change in the environmental resources used by the person, especially when the goal of the activity changes [START_REF] Quéré | L'environnement comme partenaire[END_REF].When this happens, the environment which was not earlier in the main focusof attention -i.e. the peripheral environment -can become a useful resource.

In ergonomics, the analysis of the role of the environment for human activity has mainly been focused on the very close environment with which the person interacts (Fig. 8).

The environment, in a larger perspective (i.e. not directly supporting actions or activities), is often neglected.We propose to consider the peripheral environment in the analysis of the activity because it participates in the regulation thereof (Fig. 7).For example, an analysis of the activity of walking would rarely take into account the possibilities offered by the environment such as other types of transportation (e.g. public transportation or taxi).What we propose is to take into account the potential offered bythe environment when designing for mobility. The potential of the environment should be considered as a new "open avenue" that enables people to cope withthe variability of situations and thus to compensate age-related decline.

Figure 2 -Figure 3 -

 23 Figure 2 -Spatial Organisation

Figure 8 -

 8 Figure 8 -Activity pillars (Translated and adapted from Leplat, 2006,[17]) However, this "peripheral" environment may become important when the goal of the activity changes dynamically, or whenpeople encounter obstacles creating situations of disability.

ACKNOWLEDGEMENTS

This study has been partially funded by French Environment and Energy Management Agency (ADEME), by the French Atomic Commission (CEA, LIST) and by the AAL ENTRANCE project.