Clément Foucart
email: s.clement.foucart@etu.upmc.fr

Olivier Hénard
email: henardo@cermics.enpc.fr

Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration

Keywords: Measure-valued processes, Continuous-state branching processes, Fleming-Viot processes, Immigration, Beta-Coalescent, Generators, Random time change Mathematics Subject classification (2010): 60J25 60G09 92D25

 which respectively connect the Feller diffusion with the classical Fleming-Viot process and the α-stable continuous state branching process with the Beta(2α, α)-generalized Fleming-Viot process. In a recent work, a new class of probabilitymeasure valued processes, called M -generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called M -coalescents. The purpose of this article is to investigate the links between this new class of processes and the continuous-state branching processes with immigration. In the specific case of the α-stable branching process conditioned to be never extinct, we get that its genealogy is given, up to a random time change, by a Beta(2α, α -1)coalescent.

Introduction

The connections between the Fleming-Viot processes and the continuous-state branching processes have been intensively studied. Shiga established in 1990 that a Fleming-Viot process may be recovered from the ratio process associated with a Feller diffusion up to a random time change, see [START_REF] Shiga | A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes[END_REF]. This result has been generalized in 2005 by Birkner et al in [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF] in the setting of Λ-generalized Fleming-Viot processes and continuous-state branching processes (CBs for short). In that paper they proved that the ratio process associated with an α-stable branching process is a time-changed Beta(2α, α)-Fleming-Viot process for α ∈ (0, 2). The main goal of this article is to study such connections when immigration is incorporated in the underlying population. The continuous-state branching processes with immigration (CBIs for short) are a class of time-homogeneous Markov processes with values in R + . They have been introduced by Kawazu and Watanabe in 1971, see [START_REF] Kawazu | Branching processes with immigration and related limit theorems[END_REF], as limits of rescaled Galton-Watson processes with immigration. These processes are characterized by two functions Φ and Ψ respectively called the immigration mechanism and the branching mechanism. A new class of measure-valued processes with immigration has been recently set up in Foucart [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. These processes, called M-generalized Fleming-Viot processes with immigration (M-GFVIs for short) are valued in the space of probability measures on [0, 1]. The notation M stands for a couple of finite measures (Λ 0 , Λ 1 ) encoding respectively the rates of immigration and of reproduction. The genealogies of the M-GFVIs are given by the so-called M-coalescents. These processes are valued in the space of the partitions of Z + , denoted by P 0 ∞ .

In the same manner as Birkner et al. in [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF], Perkins in [START_REF] Perkins | Conditional Dawson-Watanabe processes and Fleming-Viot processes[END_REF] and Shiga in [START_REF] Shiga | A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes[END_REF], we shall establish some relations between continuous-state branching processes with immigration and M-GFVIs. A notion of continuous population with immigration may be defined using a flow of CBIs in the same spirit as Bertoin and Le Gall in [START_REF] Bertoin | The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes[END_REF]. This allows us to compare the two notions of continuous populations provided respectively by the CBIs and by the M-GFVIs. Using calculations of generators, we show in Theorem 3 that the following self-similar CBIs admit time-changed M-GFVIs for ratio processes:

• the Feller branching diffusion with branching rate σ 2 and immigration rate β (namely the CBI with Φ(q) = βq and Ψ(q) = 1 2 σ 2 q 2 ) which has for ratio process a time-changed M-Fleming-Viot process with immigration where M = (βδ 0 , σ 2 δ 0 ),

• the CBI process with Φ(q) = d ′ αq α-1 and Ψ(q) = dq α for some d, d ′ ≥ 0, α ∈ (1, 2) which has for ratio process a time-changed M-generalized Fleming-Viot process with immigration where M = (c ′ Beta(2α, α -1), cBeta(2α, α)), c ′ = α(α-1) Γ(2-α) d ′ and c = α(α-1) Γ(2-α) d. We stress that the CBIs may reach 0, see Proposition 1, in which case the M-GFVIs involved describe the ratio process up to this hitting time only. When d = d ′ or β = σ 2 , the corresponding CBIs are respectively the α-stable branching process and the Feller branching diffusion conditioned to be never extinct. In that case, the M-coalescents are genuine Λ-coalescent viewed on P 0 ∞ . We get respectively a Beta(2-α, α-1)-coalescent when α ∈ (1, 2) and a Kingman's coalescent for α = 2, see Theorem 11. This differs from the α-stable branching process without immigration (already studied in [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF]) for which the coalescent involved is a Beta(2α, α)-coalescent. Last, ideas provided to establish our main theorem have been used by Handa [START_REF] Handa | Stationary distributions for a class of generalized Fleming-Viot processes[END_REF] to study stationary distributions for another class of generalized Fleming-Viot processes.

Outline. The paper is organized as follows. In Section 2, we recall the definition of a continuous-state branching process with immigration and of an M-generalized Fleming-Viot process with immigration. We describe briefly how to define from a flow of CBIs a continuous population represented by a measure-valued process. We state in Section 3 the connections between the CBIs and M-GFVIs, mentioned in the Introduction, and study the random time change. Recalling the definition of an Mcoalescent, we focus in Section 4 on the genealogy of the M-GFVIs involved. We establish that, when the CBIs correspond with CB-processes conditioned to be never extinct, the M-coalescents involved are actually classical Λ-coalescents. We identify them and, as mentioned, the Beta(2α, α -1)coalescent arises. In Section 5, we compare the generators of the M-GFVI and CBI processes and prove the main result.

2 A continuous population embedded in a flow of CBIs and the M -generalized Fleming-Viot with immigration

Background on continuous state branching processes with immigration

We will focus on critical continuous-state branching processes with immigration characterized by two functions of the variable q ≥ 0:

Ψ(q) = 1 2 σ 2 q 2 + ∞ 0 (e -qu -1 + qu) ν1 (du) Φ(q) = βq + ∞ 0 (1 -e -qu ) ν0 (du)
where σ 2 , β ≥ 0 and ν0 , ν1 are two Lévy measures such that

∞ 0 (1 ∧ u) ν0 (du) < ∞ and ∞ 0 (u ∧ u 2 ) ν1 (du) < ∞.
The measure ν1 is the Lévy measure of a spectrally positive Lévy process which characterizes the reproduction. The measure ν0 characterizes the jumps of the subordinator that describes the arrival of immigrants in the population. The non-negative constants σ 2 and β correspond respectively to the continuous reproduction and the continuous immigration. Let P x be the law of a CBI (Y t , t ≥ 0) started at x, and denote by E x the associated expectation. The law of the Markov process (Y t , t ≥ 0) can then be characterized by the Laplace transform of its marginal as follows: for every q > 0 and x ∈ R + ,

E x [e -qYt ] = exp -xv t (q) - t 0 Φ(v s (q))ds
where v is the unique non-negative solution of ∂ ∂t v t (q) = -Ψ(v t (q)), v 0 (q) = q.

The pair (Ψ, Φ) is known as the branching-immigration mechanism. A CBI process (Y t , t ≥ 0) is said to be conservative if for every t > 0 and

x ∈ [0, ∞[, P x [Y t < ∞] = 1.
A result of Kawazu and Watanabe [START_REF] Kawazu | Branching processes with immigration and related limit theorems[END_REF] states that (Y t , t ≥ 0) is conservative if and only if for every ǫ > 0

ǫ 0 1 |Ψ(q)| dq = ∞.
Moreover, we shall say that the CBI process is critical when Ψ ′ (0) = 0: in that case, the CBI process is necessarily conservative. We follow the seminal idea of Bertoin and Le Gall in [START_REF] Bertoin | The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes[END_REF] to define a genuine continuous population model with immigration on [0, 1] associated with a CBI. Emphasizing the rôle of the initial value, we denote by (Y t (x), t ≥ 0) a CBI started at x ∈ R + . The branching property ensures that (Y t (x + y), t ≥ 0) law = (Y t (x) + X t (y), t ≥ 0) where (X t (y), t ≥ 0) is a CBI(Ψ, 0) starting from y (that is a CB-process without immigration and with branching mechanism Ψ) independent of (Y t (x), t ≥ 0). The Kolmogorov's extension theorem allows one to construct a flow (Y t (x), t ≥ 0, x ≥ 0) such that for every y ≥ 0, (Y t (x + y) -Y t (x), t ≥ 0) has the same law as (X t (y), t ≥ 0) a CB-process started from y. We denote by (M t , t ≥ 0) the Stieltjes-measure associated with the increasing process x ∈ [0, 1] → Y t (x). Namely, define

M t (]x, y]) := Y t (y) -Y t (x), 0 ≤ x ≤ y ≤ 1. M t ({0}) := Y t (0).
The process (Y t (1), t ≥ 0) is assumed to be conservative, therefore the process (M t , t ≥ 0) is valued in the space M f of finite measures on [0, 1]. By a slight abuse of notation, we denote by (Y t , t ≥ 0) the process (Y t (1), t ≥ 0). The framework of measure-valued processes allows us to consider an infinitely many types model. Namely each individual has initially its own type (which lies in [0, 1]) and transmits it to its progeny. People issued from the immigration have a distinguished type fixed at 0. Since the types do not evolve in time, they allow us to track the ancestors at time 0. This model can be viewed as a superprocess without spatial motion (or without mutation in population genetics vocable). Let C be the class of functions on M f of the form [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF] (see Theorem 9.18 p. 218) ensures that the following operator acting on the space M f is an extended generator of (M t , t ≥ 0). For any η ∈ M f ,

F (η) := G ( f 1 , η , ..., f n , η ) , where f, η := [0,1] f (x)η(dx), G ∈ C 2 (R n ) and f 1 , ..., f n are bounded measurable functions on [0, 1]. Section 9.3 of Li's book
LF (η) := σ 2 /2 1 0 1 0 η(da)δ a (db)F ′′ (η; a, b) (1)
+ βF ′ (η; 0) (2)
+ 1 0 η(da) ∞ 0 ν1 (dh)[F (η + hδ a ) -F (η) -hF ′ (η, a)] (3)
+ ∞ 0 ν0 (dh)[F (η + hδ 0 ) -F (η)] (4)
where

F ′ (η; a) := lim ǫ→0 1 ǫ [F (η + ǫδ a ) -F (η)]
is the Gateaux derivative of F at η in direction δ a , and F ′′ (η; a, b) := G ′ (η; b) with G(η) = F ′ (η; a). The terms (1) and (3) correspond to the reproduction, see for instance Section 6.1 p. 106 of Dawson [START_REF] Dawson | Measure-valued Markov processes[END_REF]. The terms (2) and (4) correspond to the immigration. We stress that in our model the immigration is concentrated on 0, contrary to other works which consider infinitely many types for the immigrants. For the interested reader, the operator L corresponds with that given in equation (9.25) of Section 9 of Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF] by setting

H(dµ) = ∞ 0 ν0 (dh)δ hδ 0 (dµ) and η = βδ 0 .
For all η ∈ M f , we denote by |η| the total mass |η| := η([0, 1]). If (M t , t ≥ 0) is a Markov process with the above operator for generator, the process (|M t |, t ≥ 0) is by construction a CBI. This is also plain from the form of the generator L: let ψ be a twice differentiable function on R + and define F : η → ψ(|η|), we find LF (η) = zG B ψ(z) + G I ψ(z) for z = |η|, where

G B ψ(z) = σ 2 2 ψ ′′ (z) + ∞ 0 [ψ(z + h) -ψ(z) -hψ ′ (z)] ν1 (dh) (5)
G I ψ(z) = βψ ′ (z) + ∞ 0 [ψ(z + h) -ψ(z)] ν0 (dh). (6)
2.2 Background on M-generalized Fleming-Viot processes with immigration

We denote by M 1 the space of probability measures on [0, 1]. Let c 0 , c 1 be two non-negative real numbers and ν 0 , ν 1 be two measures on [0, 1] such that 1 0 xν 0 (dx) < ∞ and 1 0 x 2 ν 1 (dx) < ∞. Following the notation of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF], we define the couple of finite measures M = (Λ 0 , Λ 1 ) such that

Λ 0 (dx) = c 0 δ 0 (dx) + xν 0 (dx), Λ 1 (dx) = c 1 δ 0 (dx) + x 2 ν 1 (dx).
The M-generalized Fleming-Viot process with immigration describes a population with constant size which evolves by resampling. Let (ρ t , t ≥ 0) be an M-generalized Fleming-Viot process with immigration. The evolution of this process is a superposition of a continuous evolution, and a discontinuous one. The continuous evolution can be described as follows: every couple of individuals is sampled at constant rate c 1 , in which case one of the two individuals gives its type to the other: this is a reproduction event. Furthermore, any individual is picked at constant rate c 0 , and its type replaced by the distinguished type 0 (the immigrant type): this is an immigration event. The discontinuous evolution is prescribed by two independent Poisson point measures N 0 and N 1 on R + × [0, 1] with respective intensity dt ⊗ ν 0 (dx) and dt ⊗ ν 1 (dx). More precisely, if (t, x) is an atom of N 0 + N 1 then t is a jump time of the process (ρ t , t ≥ 0) and the conditional law of ρ t given ρ t-is:

• (1 -x)ρ t-+ xδ U , if (t, x) is an atom of N 1 , where U is distributed according to ρ t- • (1 -x)ρ t-+ xδ 0 , if (t, x) is an atom of N 0 . If (t, x
) is an atom of N 1 , an individual is picked at random in the population at generation t-and generates a proportion x of the population at time t: this is a reproduction event, as for the genuine generalized Fleming-Viot process (see [START_REF] Bertoin | Stochastic flows associated to coalescent processes[END_REF] p278). If (t, x) is an atom of N 0 , the individual 0 at time t-generates a proportion x of the population at time t: this is an immigration event. In both cases, the population at time t-is reduced by a factor 1x so that, at time t, the total size is still 1. The genealogy of this population (which is identified as a probability measure on [0, 1]) is given by an M-coalescent (see Section 4 below). This description is purely heuristic (we stress for instance that the atoms of N 0 + N 1 may form an infinite dense set), to make a rigorous construction of such processes, we refer to the Section 5.2 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] (or alternatively Section 3.2 of [START_REF] Foucart | Generalized Fleming-Viot processes with immigration via stochastic flows of partitions[END_REF]). For any p ∈ N and any continuous function f on [0, 1] p , we denote by G f the map

ρ ∈ M 1 → f, ρ ⊗p := [0,1] p f (x)ρ ⊗p (dx) = [0,1] p f (x 1 , ..., x p )ρ(dx 1 )...ρ(dx p ).
Let (F , D) denote the generator of (ρ t , t ≥ 0) and its domain. The vector space generated by the functionals of the type G f forms a core of (F , D) and we have (see Lemma 5.2 in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]):

F G f (ρ) = c 1 1≤i<j≤p [0,1] p [f (x i,j ) -f (x)]ρ ⊗p (dx) (1') + c 0 1≤j≤p [0,1] p [f (x 0,j ) -f (x)]ρ ⊗p (dx) (2'
)

+ 1 0 ν 1 (dr) ρ(da)[G f ((1 -r)ρ + rδ a ) -G f (ρ)] (3'
)

+ 1 0 ν 0 (dr)[G f ((1 -r)ρ + rδ 0 ) -G f (ρ)]. ( 4

')

where x denotes the vector (x 1 , ..., x p ) and

• the vector x 0,j is defined by x 0,j k = x k , for all k = j and x 0,j j = 0,

• the vector x i,j is defined by x i,j k = x k , for all k = j and x i,j j = x i .

3 Relations between CBIs and M -GFVIs

Forward results

The expressions of the generators of (M t , t ≥ 0) and (ρ t , t ≥ 0) lead us to specify the connections between CBIs and GFVIs. We add a cemetery point ∆ to the space M 1 and define (R t , t ≥ 0) := ( Mt |Mt| , t ≥ 0), the ratio process with lifetime τ := inf{t ≥ 0; |M t | = 0}. By convention, for all t ≥ τ , we set R t = ∆. As mentioned in the Introduction, we shall focus our study on the two following critical CBIs:

(i) (Y t , t ≥ 0) is CBI with parameters σ 2 , β ≥ 0 and ν0 = ν1 = 0, so that Ψ(q) = σ 2 2 q 2 and Φ(q) = βq. (ii) (Y t , t ≥ 0) is a CBI with σ 2 = β = 0, ν0 (dh) = c ′ h -α 1 h>0 dh and ν1 (dh) = ch -1-α 1 h>0 dh for 1 < α < 2, so that Ψ(q) = dq α and Φ(q) = d ′ αq α-1 with d ′ = Γ(2-α) α(α-1) c ′ and d = Γ(2-α) α(α-1) c
Notice that the CBI in (i) may be seen as a limit case of the CBIs in (ii) for α = 2. We first establish in the following proposition a dichotomy for the finiteness of the lifetime, depending on the ratio immigration over reproduction.

Proposition 1. Recall the notation τ = inf{t ≥ 0, Y t = 0}. • If β σ 2 ≥ 1 2 in case (i) or c ′ c ≥ α-1 α in case (ii), then P[τ = ∞] = 1. • If β σ 2 < 1 2 in case (i) or c ′ c < α-1 α in case (ii), then P[τ < ∞] = 1.
We then deal with the random change of time. In the case of a CB-process (that is a CBI process without immigration), Birkner et al. used the Lamperti representation and worked on the embedded stable spectrally positive Lévy process. We shall work directly on the CBI process instead. For 0 ≤ t ≤ τ , we define:

C(t) = t 0 Y 1-α s ds,
in case (ii) and set α = 2 in case (i).

Proposition 2. In both cases (i) and (ii), we have:

P (C(τ ) = ∞) = 1.
In other words, the additive functional

C maps [0, τ [ to [0, ∞[. By convention, if τ is almost surely finite we set C(t) = C(τ ) = ∞ for all t ≥ τ . Denote by C -1 the right continuous inverse of the functional C. This maps [0, ∞[ to [0, τ [,
a.s. We stress that in most cases, (R t , t ≥ 0) is not a Markov process. Nevertheless, in some cases, through a change of time, the process (R t , t ≥ 0) may be changed into a Markov process. This shall be stated in the following Theorem where the functional C is central. For every x, y > 0, denote by Beta(x, y)(dr) the finite measure with density r x-1 (1r) y-1 1 (0,1) (r)dr, and recall that its total mass is given by the Beta function B(x, y). Theorem 3. Let (M t , t ≥ 0) be the measure-valued process associated to a process (Y t (x), x ∈ [0, 1], t ≥ 0).

-In case (i), the process (R C -1 (t) ) t≥0 is a M-Fleming-Viot process with immigration with Λ 0 (dr) = βδ 0 (dr) and Λ 1 (dr) = σ 2 δ 0 (dr).

-In case (ii), the process (R C -1 (t) ) t≥0 is a M-generalized Fleming-Viot process with immigration with

Λ 0 (dr) = c ′ Beta(2 -α, α -1)(dr) and Λ 1 (dr) = cBeta(2 -α, α)(dr).
The proof requires rather technical arguments on the generators and is given in Section 5.

Remark 4.

• The CBIs in the statement of Theorem 3 with σ 2 = β in case (i) or c = c ′ in case (ii), are also CBs conditioned on non extinction and are studied further in Section 4.

• Contrary to the case without immigration, see Theorem 1.1 in [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF], we have to restrict ourselves to α ∈ (1, 2].

So far, we state that the ratio process (R t , t ≥ 0) associated to (M t , t ≥ 0), once time changed by C -1 , is a M-GFVI process. Conversely, starting from a M-GFVI process, we could wonder how to recover the measure-valued CBI process (M t , t ≥ 0). This lead us to investigate the relation between the time changed ratio process (R C -1 (t) , t ≥ 0) and the process (Y t , t ≥ 0).

Proposition 5. In case (i) of Theorem 3, the additive functional (C(t), t ≥ 0) and (R C -1 (t) , 0 ≤ t < τ ) are independent.
This proves that in case (i) we need additional randomness to reconstruct M from the M-GFVI process. On the contrary, in case (ii), the process (Y t , t ≥ 0) is clearly not independent of the ratio process (R t , t ≥ 0), since both processes jump at the same time. The proof of Propositions 1, 2 are given in the next Subsection. Some rather technical arguments are needed to prove Proposition 5. We postpone its proof to the end of Section 5.

Proofs of Propositions 1, 2

Proof of Proposition 1. Let (X t (x), t ≥ 0) denote an α-stable branching process started at x (with α ∈ (1, 2]). Denote ζ its absorption time, ζ := inf{t ≥ 0; X t (x) = 0}. The following construction of the process (Y t (0), t ≥ 0) may be deduced from the expression of the Laplace transform of the CBI process. We shall need the canonical measure N which is a sigma-finite measure on càdlàg paths and represents informally the "law" of the population generated by one single individual in a CB(Ψ), see Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]. We write:

(Y t (0), t ≥ 0) = i∈I X i (t-t i ) + , t ≥ 0 (7)
with i δ (t i ,X i ) a Poisson random measure on R + ×D(R + , R + ) with intensity dt⊗µ, where D(R + , R + ) denotes the space of càdlàg functions, and µ is defined as follows:

• in case (ii), µ(dX) = ν0 (dx)P x (dX), where P x is the law of a CB(Ψ) with Ψ(q) = dq α . Formula (7) may be understood as follows: at the jump times t i of a pure jump stable subordinator with Lévy measure ν0 , a new arrival of immigrants, of size X i 0 , occurs in the population. Each of these "packs", labelled by i ∈ I, generates its own descendance (X i t , t ≥ 0), which is a CB(Ψ) process.

• in case (i), µ(dX) = β N(dX), where N is the canonical measure associated to the CB(Ψ) with Ψ(q) = σ 2 2 q 2 . The canonical measure may be thought of as the "law" of the population generated by one single individual. The link with case (ii) is the following: the pure jump subordinator degenerates into a continuous subordinator equal to (t → βt). The immigrants no more arrive by packs, but appear continuously.

Actually, the canonical measure N is defined in both cases (i) and (ii), and we may always write µ(dX) = Φ(N(dX)). The process (Y t (0), t ≥ 0) is a CBI(Ψ, Φ) started at 0. We call R the set of zeros of (Y t (0), t > 0): R := {t > 0; Y t (0) = 0}.

Denote ζ i = inf {t > 0, X i t =
R := R ⋆ + \ i∈I ]t i , t i + ζ i [.
we have R ⊂ R with equality in case (i) only.

The lengths ζ i have law µ(ζ ∈ dt) thanks to the Poisson construction of Y (0). We now distinguish the two cases:

• Feller case: this corresponds to α = 2. We have Ψ(q) := σ 2 2 q and Φ(q) := βq, and thus

µ[ζ > t] = β N[ζ > t] = 2β σ 2 1 t
see Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF] p. 62. Using Example 1 p. 180 of Fitzsimmons et al. [START_REF] Fitzsimmons | The set of real numbers left uncovered by random covering intervals[END_REF], we deduce that R = ∅ a.s. if and only if

2β σ 2 ≥ 1. (8)
• Stable case: this corresponds to α ∈ (1, 2). Recall Ψ(q) := dq α , Φ(q) := d ′ αq α-1 . In that case, we have,

N(ζ > t) = d -1 α-1 [(α -1)t] -1 α-1 . Thus, µ[ζ > t] = Φ(N(ζ > t)) = α α-1 d ′ d 1 t . Recall that d ′ d = c ′ c
. Therefore, using reference [START_REF] Fitzsimmons | The set of real numbers left uncovered by random covering intervals[END_REF], we deduce that

R = ∅ a.s. if and only if c ′ c ≥ α -1 α . (9)
This allows us to establish the first point of Proposition 1: we get R ⊂ R = ∅, and the inequality Y t (1) ≥ Y t (0) for all t ensures that τ = ∞.

We deal now with the second point of Proposition 1. Assume that c ′ c < α-1 α or β σ 2 < 1 2 . By assertions (8) and ( 9), we already know that R = ∅. However, what we really need is that R is a.s. not bounded. To that aim, observe that, in both cases (i) and (ii),

µ[ζ > s] = Φ(N(ζ > s)) = κ s with κ = α α-1 d ′ d = α α-1 c ′ c < 1 if 1 < α < 2 and κ = 2β σ 2 < 1 if α = 2. Thus u 1 µ[ζ > s]ds = κ ln(u) and we obtain exp - u 1 µ[ζ > s]ds = 1 u κ . Therefore, since κ < 1, ∞ 1 exp - u 1 µ[ζ > s]ds du = ∞,
which implies thanks to Corollary 4 (Equation 17 p 183) of [START_REF] Fitzsimmons | The set of real numbers left uncovered by random covering intervals[END_REF] that R is a.s. not bounded.

Since R = R in case (i), the set R is a.s. not bounded in that case. Now, we prove that R is a.s. not bounded in case (ii). The set R is almost surely not empty and not bounded. Moreover this is a perfect set (Corollary 1 of [START_REF] Fitzsimmons | The set of real numbers left uncovered by random covering intervals[END_REF]). Since there are only countable points (t i , i ∈ I), the set R = R \ i∈I {t i } is also uncountable and not bounded.

Last, recall from Subsection 2.1 that we may write Y t (1) = Y t (0) + X t (1) for all t ≥ 0 with (X t (1), t ≥ 0) a CB-process independent of (Y t (0), t ≥ 0). Let ξ := inf{t ≥ 0, X t (1) = 0} be the extinction time of (X t (1), t ≥ 0). Since R is a.s. not bounded in both cases (i) and (ii), R ∩ (ξ, ∞) = ∅, and τ < ∞ almost surely.

Proof of Proposition 2. Recall that Y t (x) is the value of the CBI started at x at time t. We will denote by τ x (0) := inf {t > 0, Y t (x) = 0}. With this notation, τ 1 (0) = τ introduced in Section 3.1. In both cases (i) and (ii), the processes are self-similar, see Kyprianou and Pardo [START_REF] Kyprianou | Continuous-state branching processes and self-similarity[END_REF]. Namely, we have

(xY x 1-α t (1), t ≥ 0) law = (Y t (x), t ≥ 0) ,
where we take α = 2 in case (i). Performing the change of variable s = x 1-α t, we obtain

τ x (0) 0 dt Y t (x) 1-α law = τ 1 (0) 0 ds Y s (1) 1-α . ( 10
)
According to Proposition 1, depending on the values of the parameters:

• Either P(τ x (0) < ∞) = 1 for every x. Let x > 1. Denote τ x (1) = inf {t > 0, Y t (x) ≤ 1}. We have P(τ x (1) < ∞) = 1. We have:

τ x (0) 0 dt Y t (x) 1-α = τ x (1) 0 dt Y t (x) 1-α + τ x (0) τ x (1) dt Y t (x) 1-α
By the strong Markov property applied at the stopping time τ x (1), since Y has no negative jumps:

τ x (0) τ x (1) dt Y t (x) 1-α law = τ 1 (0) 0 dt Ỹt (1) 1-α ,
with ( Ỹt (1), t ≥ 0) an independent copy started from 1. Since

τ x (1) 0 dt Y t (x) 1-α > 0, a.s.,
the equality ( 10) is impossible unless both sides of the equality are infinite almost surely. We thus get that C(τ ) = ∞ almost surely in that case.

• Either P(τ x (0) = ∞) = 1 for every x, on which case we may rewrite [START_REF] Etheridge | An introduction to superprocesses[END_REF] as follows:

∞ 0 dt Y t (x) 1-α law = ∞ 0 ds Y s (1) 1-α .
Since, for x > 1, the difference (Y t (x) -Y t (1), t ≥ 0) is an α-stable CB-process started at x -1 > 0, we deduce that C(τ ) = ∞ almost surely again.

This proves the statement.

Remark 6. The situation is quite different when the CBI process starts at 0, in which case the time change also diverges in the neighbourhood of 0. The same change of variables as in [START_REF] Etheridge | An introduction to superprocesses[END_REF] yields, for all 0 < x < k,

ι x (k) 0 dt Y t (x) 1-α law = ι 1 (k/x) 0 dt Y t (1) 1-α , with ι x (k) = inf{t > 0, Y t (x) ≥ k} ∈ [0, ∞].
Letting x tend to 0, we get ι 1 (k/x) -→ ∞ and the right hand side diverges to infinity. Thus, the left hand side also diverges, which implies that:

P ι 0 (k) 0 dt Y t (0) 1-α = ∞ = 1.

Genealogy of the Beta-Fleming-Viot processes with immigration

To describe the genealogy associated with stable CBs, Bertoin and Le Gall [START_REF] Bertoin | Stochastic flows associated to coalescent processes. III. Limit theorems[END_REF] and Birkner et al. [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF] used partition-valued processes called Beta-coalescents. These processes form a subclass of Λcoalescents, introduced independently by Pitman and Sagitov in 1999. A Λ-coalescent is an exchangeable process in the sense that its law is invariant under the action of any permutation. In words, there is no distinction between the individuals. Although these processes arise as models of genealogy for a wide range of stochastic populations, they are not in general adapted to describe the genealogy of a population with immigration. Recently, a larger class of processes called M-coalescents has been defined in [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] (see Section 5). These processes are precisely those describing the genealogy of M-GFVIs. Remark 7. We mention that the use of the lookdown construction in Birkner et al. [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF] may be easily adapted to our framework and yields a genealogy for any conservative CBI. Moreover, other genealogies, based on continuous trees, have been investigated by Lambert [START_REF] Lambert | The genealogy of continuous-state branching processes with immigration[END_REF] and Duquesne [START_REF] Duquesne | Continuum random trees and branching processes with immigration[END_REF].

Background on M-coalescents

Before focusing on the M-coalescents involved in the context of Theorem 3, we recall their general definition and the duality with the M-GFVIs. Contrary to the Λ-coalescents, the M-coalescents are only invariant by permutations letting 0 fixed. The individual 0 represents the immigrant lineage and is distinguished from the others. We denote by P 0 ∞ the space of partitions of Z + := {0} N. Let π ∈ P 0 ∞ . By convention, we identify π with the sequence (π 0 , π 1 , ...) of the blocks of π enumerated in increasing order of their smallest element: for every i ≤ j, min π i ≤ min π j . Let [ n ] denote the set {0, ..., n} and P 0 n the space of partitions of [ n ]. The partition of [ n ] into singletons is denoted by 0 [ n ] . As in Section 2.2, the notation M stands for a pair of finite measures (Λ 0 , Λ 1 ) such that:

Λ 0 (dx) = c 0 δ 0 (dx) + xν 0 (dx), Λ 1 (dx) = c 1 δ 0 (dx) + x 2 ν 1 (dx),
where c 0 , c 1 are two non-negative real numbers and ν 0 , ν 1 are two measures on [0, 1] subject to the same conditions as in Section 2.2. Let N 0 and N 1 be two Poisson point measures with intensity respectively dt ⊗ ν 0 and dt ⊗ ν 1 . An M-coalescent is a Feller process (Π(t), t ≥ 0) valued in P 0 ∞ with the following dynamics.

• At an atom (t, x) of N 1 , flip a coin with probability of "heads" x for each block not containing 0. All blocks flipping "heads" are merged immediately in one block. At time t, a proportion x share a common parent in the population.

• At an atom (t, x) of N 0 , flip a coin with probability of "heads" x for each block not containing 0. All blocks flipping "heads" coagulate immediately with the distinguished block. At time t, a proportion x of the population is children of immigrant.

In order to take into account the parameters c 0 and c 1 , imagine that at constant rate c 1 , two blocks (not containing 0) merge continuously in time, and at constant rate c 0 , one block (not containing 0) merged with the distinguished one. We refer to Section 4.2 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF] for a rigorous definition. Let π ∈ P 0 n . The jump rate of an M-coalescent from 0 [ n ] to π, denoted by q π , is given as follows:

• If π has one block not containing 0 with k elements and 2 ≤ k ≤ n, then

q π = λ n,k := 1 0 x k-2 (1 -x) n-k Λ 1 (dx).
• If the distinguished block of π has k + 1 elements (counting 0) and 1 ≤ k ≤ n then

q π = r n,k := 1 0 x k-1 (1 -x) n-k Λ 0 (dx).
The next duality property is a key result and links the M-GFVIs to the M-coalescents. For any π in P 0 ∞ , define α π : k → the index of the block of π containing k.

We have the duality relation (see Lemma 4 in [START_REF] Foucart | Generalized Fleming-Viot processes with immigration via stochastic flows of partitions[END_REF]): for any p ≥ 1 and f ∈ C([0, 1] p ),

E [0,1] p+1 f (x α Π(t) (1) , ..., x α Π(t) (p) )δ 0 (dx 0 )dx 1 ...dx p = E [0,1] p f (x 1 , ..., x p )ρ t (dx 1 )...ρ t (dx p ) ,
where (ρ t , t ≥ 0) is a M-GFVI started from the Lebesgue measure on [0, 1]. We establish a useful lemma relating genuine Λ-coalescents and M-coalescents. Consider a Λ-coalescent taking values in the set P 0 ∞ ; this differs from the usual convention, according to which they are valued in the set P ∞ of the partitions of N (see Chapters 1 and 3 of [START_REF] Berestycki | Recent progress in coalescent theory, volume 16 of Ensaios Matemáticos [Mathematical Surveys[END_REF] for a complete introduction to these processes). In that framework, Λ-coalescents appear as a subclass of M-coalescents and the integer 0 may be viewed as a typical individual. The proof is postponed in Section 4.3.

Lemma 8. A M-coalescent, with M = (Λ 0 , Λ 1 ) is also a Λ-coalescent on P 0 ∞ if and only if (1 -x)Λ 0 (dx) = Λ 1 (dx).
In that case Λ = Λ 0 .

4.2

The Beta(2α, α -1)-coalescent

The aim of this Section is to show how a Beta(2α, α -1)-coalescent is embedded in the genealogy of an α-stable CB-process conditioned to be never extinct. Along the way, we also derive the fixed time genealogy of the Feller CBI.

We first state the following straightforward Corollary of Theorem 3, which gives the genealogy of the ratio process at the random time C -1 (t): Corollary 9. Let (R t , t ≥ 0) be the ratio process of a CBI in case (i) or (ii). We have for all t ≥ 0:

E [0,1] p+1 f (x α Π(t) (1) , ..., x α Π(t) (p) )δ 0 (dx 0 )dx 1 ...dx p = E [0,1] p f (x 1 , ..., x p )R C -1 (t) (dx 1 )...R C -1 (t) (dx p ) ,
where:

• In case (i), (Π(t), t ≥ 0) is a M-coalescent with M = (βδ 0 , σ 2 δ 0 ), • In case (ii), (Π(t), t ≥ 0) is a M-coalescent with M = (c ′ Beta(2 -α, α -1), cBeta(2 -α, α)).
In general, we cannot set the random quantity C(t) instead of t in the equation of Corollary 9. Nevertheless, using the independence property proved in Proposition 5, we get the following Corollary, whose proof may be found in Section 4.3.. Corollary 10. In case (i), assume β σ 2 ≥ 1 2 , then for all t ≥ 0,

E [0,1] p+1 f (x α Π(C(t)) (1) , ..., x α Π(C(t)) (p) )δ 0 (dx 0 )dx 1 ...dx p = E [0,1] p f (x 1 , ..., x p )R t (dx 1 )...R t (dx p ) ,
where (Π(t), t ≥ 0) is a M-coalescent with M = (βδ 0 , σ 2 δ 0 ), (Y t , t ≥ 0) is a CBI in case (i) independent of (Π(t), t ≥ 0) and (C(t), t ≥ 0) = t 0 1 Ys ds, t ≥ 0 .

We stress on a fundamental difference between Corollaries 9 and 10. Whereas the first gives the genealogy of the ratio process R at the random time C -1 (t), the second gives the genealogy of the ratio process R at a fixed time t. Notice that we impose the additional assumption that β σ 2 ≥ 1 2 in Corollary 10 for ensuring that the lifetime is infinite. Therefore, R t = ∆ for all t ≥ 0, and we may consider its genealogy.

We easily check that the M-coalescents for which M = (σ 2 δ 0 , σ 2 δ 0 ) and M = (cBeta(2α, α -1), cBeta(2α, α)) fulfill the conditions of Lemma 8. Recall from Section 3.1 the definitions of the CBIs in case (i) and (ii) .

Theorem 11. (i) If the process (Y t , t ≥ 0) is a CBI such that σ 2 = β > 0, ν1 = ν0 = 0, then the process (Π(t/σ 2 ), t ≥ 0) defined in Corollary 9 is a Kingman's coalescent valued in P 0 ∞ . (ii) If the process (Y t , t ≥ 0) is a CBI such that σ 2 = β = 0 and ν0 (dh) = ch -α dh, ν1 (dh) =
ch -α-1 dh for some constant c > 0 then the process (Π(t/c), t ≥ 0) defined in Corollary 9 is a Beta(2α, α -1)-coalescent valued in P 0 ∞ . In both cases, the process (Y t , t ≥ 0) involved in that Theorem may be interpreted as a CB-process (X t , t ≥ 0) without immigration (β = 0 or c ′ = 0) conditioned on non-extinction, see Lambert [START_REF] Lambert | Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct[END_REF]. We then notice that both the genealogies of the time changed Feller diffusion and of the time changed Feller diffusion conditioned on non extinction are given by the same Kingman's coalescent. On the contrary, the genealogy of the time changed α-stable CB-process is a Beta(2α, α)-coalescent, whereas the genealogy of the time changed α-stable CB-process conditioned on non-extinction is a Beta(2α, α -1)-coalescent. We stress that for any α ∈ (1, 2) and any borelian B of [0, 1], we have Beta(2α, α -1)(B) ≥ Beta(2α, α)(B). This may be interpreted as the additional reproduction events needed for the process to be never extinct.

Proofs.

Proof of Lemma 8. Let (Π ′ (t), t ≥ 0) be a Λ-coalescent on P 0 ∞ . Let n ≥ 1, we may express the jump rate of (Π

′ | [ n ] (t), t ≥ 0) from 0 [ n ] to π by q ′ π =    0 if π has more than one non-trivial block [0,1] x k (1 -x) n+1-k x -2 Λ(dx)
if the non trivial block has k elements. Consider now a M-coalescent, denoting by q π the jump rate from 0 [ n ] to π, we have

q π =            0 if π has more than one non-trivial block [0,1] x k (1 -x) n-k x -2 Λ 1 (dx) if π 0 = {0} and the non trivial block has k elements [0,1] x k-1 (1 -x) n+1-k x -1 Λ 0 (dx) if #π 0 = k.
Since the law of a Λ-coalescent is entirely described by the family of the jump rates of its restriction on [ n ] from 0 [ n ] to π for π belonging to P 0 n (see Section 4.2 of [START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF]), the processes Π and Π ′ have the same law if and only if for all n ≥ 0 and π ∈ P 0 n , we have q π = q ′ π , that is if and only if

(1 -x)Λ 0 (dx) = Λ 1 (dx).
Proof of Corollary 10. Since C -1 (C(t)) = t,

E [0,1] p f (x 1 , ..., x p )R t (dx 1 )...R t (dx p ) = E [0,1] p f (x 1 , ..., x p )R C -1 (C(t)) (dx 1 )...R C -1 (C(t)) (dx p ) .
Then, using the independence between R C -1 and C, the right hand side above is also equal to:

P(C(t) ∈ ds) E [0,1] p f (x 1 , ..., x p )R C -1 (s) (dx 1 )...R C -1 (s) (dx p ) .
Using Corollary 9 and choosing (Π(t), t ≥ 0) independent of (C(t), t ≥ 0), we find:

P(C(t) ∈ ds) E [0,1] p f (x 1 , ..., x p )R C -1 (s) (dx 1 )...R C -1 (s) (dx p ) = P(C(t) ∈ ds) E [0,1] p+1 f (x α Π(s) (1) , ..., x α Π(s) (p) )δ 0 (dx 0 )dx 1 ...dx p = E [0,1] p+1 f (x α Π(C(t)) (1) , ..., x α Π(C(t)) (p) )δ 0 (dx 0 )dx 1 ...dx p .
Remark 12. Notice the crucial rôle of the independence in order to establish Corollary 10. When this property fails, as in the case (ii), the question of describing the fixed time genealogy of the α-stable CB or CBI remains open. We refer to the discussion in Section 2.2 of Berestycki et. al [START_REF] Berestycki | Kingman's coalescent and Brownian motion[END_REF].

Proof of Theorem 3 and Proposition 5

We first deal with Theorem 3. The proof of Proposition 5 is rather technical ans is postponed at the end of this Section. In order to get the connection between the two measure-valued processes (R t , t ≥ 0) and (M t , t ≥ 0), we may follow the ideas of Birkner et al. [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF] and rewrite the generator of the process (M t , t ≥ 0) using the "polar coordinates": for any η ∈ M f , we define z := |η| and ρ := η |η| .

The proof relies on five lemmas. Lemma 13 establishes that the law of a generalized Fleming-Viot process with immigration is entirely determined by the generator F on the test functions of the form ρ → φ, ρ m with φ a measurable non-negative bounded map and m ∈ N. Lemmas 14, 15 and 17 allow us to study the generator L on the class of functions of the type F : η → 1 |η| m φ, η m . Lemma 16 (lifted from Lemma 3.5 of [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF]) relates stable Lévy-measures and Beta-measures. We end the proof using results on time change by the inverse of an additive functional. We conclude thanks to a result due to Volkonskiȋ in [START_REF] Volkonskiȋ | Random substitution of time in strong Markov processes[END_REF] about the generator of a time-changed process.

Lemma 13. The following martingale problem is well-posed: for any function f of the form:

(x 1 , ..., x p ) → p i=1 φ(x i )
with φ a non-negative measurable bounded map and p ≥ 1, the process

G f (ρ t ) - t 0 F G f (ρ s )ds is a martingale.
Proof. Only the uniqueness has to be checked. We shall establish that the martingale problem of the statement is equivalent to the following martingale problem: for any continuous function f on [0, 1] p , the process

G f (ρ t ) - t 0 F G f (ρ s )ds
is a martingale. This martingale problem is well posed, see Proposition 5.2 of [START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]. Notice that we can focus on continuous and symmetric functions since for any continuous f , G f = G f with f the symmetrized version of f . Moreover, by the Stone-Weierstrass theorem, any symmetric continuous function f from [0, 1] p to R can be uniformly approximated by linear combination of functions of the form (x 1 , ..., x p ) → p i=1 φ(x i ) for some function φ continuous on [0, 1]. We now take f symmetric and continuous, and let f k be an approximating sequence. Plainly, we have

|G f k (ρ) -G f (ρ)| ≤ ||f k -f || ∞
Assume that (ρ t , t ≥ 0) is a solution of the martingale problem stated in the lemma. Since the map h → G h is linear, the process

G f k (ρ t ) - t 0 F G f k (ρ s )ds
is a martingale for each k ≥ 1. We want to prove that the process

G f (ρ t ) - t 0 F G f (ρ s )ds
is a martingale, knowing it holds for each f k . We will show the following convergence

F G f k (ρ) -→ k→∞ F G f (ρ) uniformly in ρ.
Recall expressions (1') and (2') in Subsection 2.2, one can check that the following limits are uniform in the variable ρ

1≤i<j≤p [0,1] p [f k (x i,j ) -f k (x)]ρ ⊗p (dx) -→ k→∞ 1≤i<j≤p [0,1] p [f (x i,j ) -f (x)]ρ ⊗p (dx) and 1≤i≤m [0,1] p [f k (x 0,i ) -f k (x)]ρ ⊗p (dx) -→ k→∞ 1≤i≤p [0,1] p [f (x 0,i ) -f (x)]ρ ⊗p (dx).
We have now to deal with the terms (3') and (4'). In order to get that the quantity

1 0 ν(dr) 1 0 [G f k ((1 -r)ρ + rδ a ) -G f k (ρ)]ρ(da) converges toward 1 0 ν(dr) 1 0 [G f ((1 -r)ρ + rδ a ) -G f (ρ)]ρ(da), we compute f k -f, ((1 -r)ρ + rδ a ) ⊗p -f k -f, ρ ⊗p .
Since the function f k -f is symmetric, we may expand the p-fold product f k -f, ((1r)ρ + rδ a ) ⊗p , this yields

f k -f, ((1 -r)ρ + rδ a ) ⊗p -f k -f, ρ ⊗p = p i=0 p i r i (1 -r) p-i f k -f, ρ ⊗p-i ⊗ δ ⊗i a -f k -f, ρ ⊗p = pr(1 -r) p-1 f k -f, ρ ⊗p-1 ⊗ δ a -f k -f, ρ ⊗p + p i=2 p i r i (1 -r) p-i f k -f, ρ ⊗p-i ⊗ δ ⊗i a -f k -f, ρ ⊗p .
We use here the notation g, µ ⊗m-i ⊗ δ ⊗i a := g(x 1 , ..., x m-i , a, ..., a i terms

)µ(dx 1 )...µ(dx m-i ).

Therefore, integrating with respect to ρ, the first term in the last equality vanishes and we get

1 0 ρ(da) (G f -f k ((1 -r)ρ + rδ a ) -G f -f k (ρ)) ≤ 2 p+1 ||f -f k || ∞ r 2 where ||f k -f || ∞ denotes the supremum of the function |f k -f |. Recall that the measure ν 1 verifies 1 0 r 2 ν 1 (dr) < ∞, moreover the quantity ||f k -f || ∞ is bounded.
Thus appealing to the Lebesgue Theorem, we get the sought-after convergence. Same arguments hold for the immigration part (4') of the operator F . Namely we have

|G f -f k ((1 -r)ρ + rδ 0 ) -G f -f k (ρ)| ≤ 2 p+1 r||f k -f || ∞
and the measure ν 0 satisfies 1 0 rν 0 (dr) < ∞. Combining our results, we obtain

|F G f k (ρ) -F G f (ρ)| ≤ C||f -f k || ∞ for a positive constant C independent of ρ. Therefore the sequence of martingales G f k (ρ t ) - t 0 F G f k (ρ s )ds converges toward G f (ρ t ) - t 0 F G f (ρ s )ds,
which is then a martingale.

Lemma 14. Assume that ν0 = ν1 = 0 the generator L of (M t , t ≥ 0) is reduced to the expressions (1) and ( 2):

LF (η) = σ 2 /2 1 0 1 0 η(da)δ a (db)F ′′ (η; a, b) + βF ′ (η; 0)
Let φ be a measurable bounded function on [0, 1] and F be the map η → G f (ρ) := f, ρ ⊗m with f (x 1 , ..., x p ) = p i=1 φ(x i ). We have the following identity

|η|LF (η) = F G f (ρ),
for η = 0, where F is the generator of a Fleming-Viot process with immigration with reproduction rate c 1 = σ 2 and immigration rate c 0 = β, see expressions (1') and (2').

Proof. By the calculations in Section 4.3 of Etheridge [START_REF] Etheridge | An introduction to superprocesses[END_REF] (but in a non-spatial setting, see also the proof of Theorem 2.1 p. 249 of Shiga [START_REF] Shiga | A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes[END_REF]), we get: [f (x i,j )f (x)]ρ ⊗m (dx).

We focus now on the immigration part. We take f a function of the form f : (x 1 , ..., x m ) → m i=1 φ(x i ) for some function φ, and consider F (η) := G f (ρ) = f, ρ ⊗m . We may compute:

F (η + hδ a ) -F (η) = φ, η + hδ a z + h m -φ, ρ m = m j=2 m j z z + h m-j h z + h j [ φ, ρ m-j φ(a) j -φ, ρ m ] (11)
+ m z z + h m-1 h z + h [ φ, ρ m-1 φ(a) -φ, ρ m ] . (12)
We get that: coincides with (Y C -1 (t) , R C -1 (t) ) t≥0 . We establish now that (R C -1 (t) , t ≥ 0) is a Markov process with the same generator as the Fleming-Viot processes involved in Theorem 3. Let G(z, ρ) = G f (ρ) = φ, ρ m (taking f : (x 1 , ..., x m ) → m i=1 φ(x i )). In both cases (i) and (ii) of Theorem 3, we have: z α-1 UG(z, ρ) = z α-1 LF (η) with F : η → G f (ρ) = F G f (ρ).

First equality holds since we took ψ ≡ 1 and the second uses Lemma 14 and Lemma 17. Since it does not depend on z, the process (R C -1 (t) , t ≥ 0) is a Markov process, moreover it is a generalized Fleming-Viot process with immigration with parameters as stated.

Proof of Proposition 5. Let (Y t ) t≥0 be a Feller branching diffusion with continuous immigration with parameters (σ 2 , β). Consider an independent M-Fleming-Viot (ρ t , t ≥ 0) with M = (βδ 0 , σ 2 δ 0 ). We first establish that (Y t ρ C(t) , 0 ≤ t < τ ) has the same law as the measure-valued branching process (M t , 0 ≤ t < τ ). Recall that L denote the generator of (M t , t ≥ 0) (here only the terms (1) and ( 2) are considered). Consider F (η) := ψ(z) φ, ρ m with z = |η|, ψ a twice differentiable map valued in R + and φ a non-negative bounded measurable function. Note that the generator acting on such functions F characterizes the law of (M t∧τ , t ≥ 0). First we easily obtain that We recognize in the first line the generator of (Y t , t ≥ 0) and in the second, 1 z F G f (ρ) with f (x 1 , ..., x m ) = m i=1 φ(x i ) and c 0 = β, c 1 = σ 2 . We easily get that this is the generator of the Markov process (Y t ρ C(t) , t ≥ 0) with lifetime τ . We conclude that it has the same law as (M t∧τ , t ≥ 0). We rewrite this equality in law as follows:

(Y t ρ C(t) , 0 ≤ t < τ ) law = (|M t | R C -1 (C(t)) , 0 ≤ t < τ ), (16)
with C defined by C(t) = t 0 |M s | -1 ds for 0 ≤ t < τ on the right hand side. Since (C(t), t ≥ 0) and (ρ t , t ≥ 0) are independent on the left hand side and the decomposition in ( 16) is unique, we have also (C(t), 0 ≤ t < τ ) and (R C -1 (t) , 0 ≤ t < τ ) independent on the right hand side.

Concerning the case (ii) of Theorem 3, we easily observe that the presence of jumps implies that such a decomposition of the generator cannot hold. See for instance Equation (2.7) of [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF] p344. The processes (R C -1 (t) , t ≥ 0) and (Y t , ≥ 0) are not independent.

1 0 1 0∂ 2

 112 )δ a (db)F ′′ (η; a, b) = |η| -1 σ 2 2 G f ∂ρ(a)∂ρ(b) (ρ)[δ a (db)ρ(db)]ρ(da) = |η| -1 σ 2 1≤i<j≤m [0,1] p

F

  ′ (η; a) = m z φ(a) φ, ρ m-1φ, ρ m .Thus,F ′ (η; 0) = |η| -1 1≤i≤m [0,1] p [f (x 0,i )f (x)]ρ ⊗m (dx)and F ′ (η; a)η(da) = 0[START_REF] Foucart | Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration[END_REF]

F 2 ψ

 2 ′ (η; 0) = ψ ′ (z) φ, ρ m + m ψ(z) z [φ(0) φ, ρ m-1φ, ρ m ], F ′′ (η; a, b) = ψ ′′ (z) φ, ρ m + m ψ ′ (z) z (φ(b) + φ(a)) φ, ρ m-1 -2 φ, ρ m + m ψ(z) z 2 (m -1)φ(a)φ(b) φ, ρ m-2m(φ(a) + φ(b)) φ, ρ m-1 + (m + 1) φ, ρ m . Simple calculations yield, LF (η) = z σ 2 ′′ (z) + βψ ′ (z) φ, ρ m + ψ(z) z σ 2 m(m -1) 2 φ 2, ρ φ, ρ m-2φ, ρ m + βm φ(0) φ, ρ m-1φ, ρ m .

  0} the lifetime of the branching process X i . The intervals ]t i , t i + ζ i [ and [t i , t i + ζ i [ represent respectively the time where X i is alive in case (i) and in case (ii) (in this case, we have X i t i > 0.) Therefore, if we define R as the set of the positive real numbers left uncovered by the random intervals ]t i , t i + ζ i [, that is:

Acknowledgments. The authors would like to thank Jean Bertoin and Jean-François Delmas for their helpful comments and advice. C.F thanks the Statistical Laboratory of Cambridge where part of this work was done with the support of the Foundation Sciences Mathématiques de Paris. O.H. thanks Goethe Universität of Frankfurt for hospitality, and École Doctorale MSTIC for support. This work is partially supported by the "Agence Nationale de la Recherche", ANR-08-BLAN-0190.

for such function f . This proves the Lemma. This first lemma will allow us to prove the case (i) of Theorem 3. We now focus on the case (ii). Assuming that σ 2 = β = 0, the generator of (M t , t ≥ 0) reduces to

where, as in equations ( 3) and (4) of Subsection 2.1,

The following lemma is a first step to understand the infinitesimal evolution of the non-markovian process (R t , t ≥ 0) in the purely discontinuous case.

Lemma 15. Let f be a continuous function on [0, 1] p of the form f (x 1 , ..., x p ) = p i=1 φ(x i ) and F be the map η → G f (ρ) = φ, ρ p . Recall the notation ρ := η/|η| and z = |η|. We have the identities:

Proof. The identity for L 0 is plain, we thus focus on L 1 . Combining Equation ( 13) and the term (12) we get

We easily check from the terms of ( 11) that the map h →

] is integrable with respect to the measure ν1 . This allows us to interchange the integrals and yields:

The previous lemma leads us to study the images of the measures ν0 and ν1 by the map φ z :

The following lemma is lifted from Lemma 3.5 of [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF].

Lemma 16. There exist two measures ν 0 , ν 1 such that λ 0 z (dr) = s 0 (z)ν 0 (dr) and λ 1 z (dr) = s 1 (z)ν 1 (dr) for some maps s 0 , s 1 from R + to R if and only if for some α ∈ (0, 2), α ′ ∈ (0, 1) and c, c ′ > 0:

In this case:

Proof. The necessary part is given by the same arguments as in Lemma 3.5 of [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF]. We focus on the sufficient part. Assuming that ν0 , ν1 are as above, we have

and thus s 0 (z) = z -α ′ . The next lemma allows us to deal with the second statement of Theorem 3.

Lemma 17. Assume that σ 2 = β = 0, ν0 (dh) = ch -α 1 h>0 dh and ν1 (dh) = ch -1-α 1 h>0 dh. Let f be a function on [0, 1] p of the form f (x 1 , ..., x p ) = p i=1 φ(x i ) , and F be the map η → G f (ρ). We have

for η = 0, where F is the generator of a M-Fleming-Viot process with immigration, with

Proof. Recall Equation ( 14):

Recall from Equation ( 13) that we have

Applying Lemma 15 and Lemma 16, we get that in the case σ 2 = β = 0 and ν1 (dx

Recalling the expressions (3'), (4'), the factorization h(z)LF (η) = F G(ρ) holds for some function h if s 0 (z) = zs 1 (z),

We are now ready to prove Theorem 3. To treat the case (i), replace α by 2 in the sequel. The process (Y t , R t ) t≥0 with lifetime τ has the Markov property. The additive functional

From Theorem 65.9 of [START_REF] Sharpe | General theory of Markov processes[END_REF] and Proposition 2, the process (Y C -1 (t) , R C -1 (t) ) t≥0 is a strong Markov process with infinite lifetime. Denote by U the generator of (Y t , R t ) t≥0 . As explained in Birkner et al. [START_REF] Birkner | Alpha-stable branching and beta-coalescents[END_REF] (Equation (2.6) p314), the law of (Y t , R t ) t≥0 is characterized by U acting on the following class of test functions:

(z, ρ) ∈ R + × M 1 → F (z, ρ) := ψ(z) φ, ρ m for φ a non-negative measurable bounded function on [0, 1], m ≥ 1 and ψ a twice differentiable non-negative map. Theorem 3 of Volkonskiȋ, see [START_REF] Volkonskiȋ | Random substitution of time in strong Markov processes[END_REF] (or Theorem 1.4 Chapter 6 of [START_REF] Ethier | Markov processes[END_REF]) states that the Markov process with generator ŨF (z, ρ) := z α-1 UF (z, ρ)


M Atef Yekta

E Cox

B M Goddeeris

D Vanrompay

Maryam Atef Yekta
email: maryam.atefyekta@ugent.be

Reduction of Escherichia coli O157:H7 excretion in sheep by oral lactoferrin administration

Keywords: H7 Lactoferrin, E. coli O157:H7, sheep

Ruminants are an important reservoir of E. coli O157:H7, therefore reducing E. coli O157:H7 excretion by these animals could play a key role in reducing human infections.

The present study investigates the potential of bovine lactoferrin, a natural antimicrobialimmunomodulatory protein of milk, to prevent colonization and excretion of E. coli O157:H7 in sheep. The effect of two different doses of lactoferrin (1.5 g or 0.15 g per 12 hours) was evaluated on colonization of sheep intestine and faecal excretion of the NCTC12900 strain. Hereto, lactoferrin was orally administered to sheep during 30 consecutive days and sheep were experimentally infected with E. coli O157:H7 on the second day of the lactoferrin administration. Interestingly, both lactoferrin dosages significantly reduced the number of E. coli O157:H7 in faeces as well as the duration of faecal excretion. The high dose group showed a significantly higher antibody response against EspA and EspB, two structural proteins of the bacterial type III secretion system (TTSS), than the colonization control group. The results suggest that oral lactoferrin administration could be used to prevent persistent colonization of sheep with E. coli

Enterohemorrhagic Escherichia coli (EHEC) are zoonotic pathogens associated with haemorrhagic colitis (HC) and haemolytic uremic syndrome (HUS) in humans [START_REF] Mead | Escherichia coli O157: H7[END_REF]. The main EHEC serotype responsible for these clinical signs is E. coli O157:H7 (Nataro and Kaper, 1998). Ruminants are the main reservoir of this microorganism which normally resides in their gut without causing apparent illness [START_REF] Besser | Escherichia coli O157: H7 gastroenteritis and the hemolytic uremic syndrome: An emerging infectious disease[END_REF]. Many human E. coli O157:H7 infections originate, either directly or indirectly via contaminated food or water, from exposure to ruminant's feces. Therefore a key step towards protecting humans from E. coli O157:H7 infection could be the control and/or prevention of E. coli O157:H7 colonization of the ruminant's intestine. Several approaches have been suggested with variable success including vaccination, probiotic and antibiotic treatment and even diet management [START_REF] Molbak | Antimicrobial therapy in patients with Escherichia coli O157: H7 infection[END_REF]Potter et al., 2004;[START_REF] Callaway | Colicin concentrations inhibit growth of Escherichia coli O157 : H7 in vitro[END_REF][START_REF] Callaway | Diet, Escherichia coli O157:H7, and cattle: a review after 10 years[END_REF]. Although some of these approaches seem promising, none of them stop bacterial excretion for a 100 %.

Milk feeding protects young mammals from intestinal infections. This protection is attributed to the multiple anti-microbial, anti-inflammatory and immunomodulatory factors present in milk of which lactoferrin (LF) is one of the most important [START_REF] Gallois | Maternal milk contains antimicrobial factors that protect young rabbits from enteropathogenic Escherichia coli infection[END_REF][START_REF] Walker | Breast Milk as the Gold Standard for Protective Nutrients[END_REF]. This molecule which is also present in mucosal secretions like tears, saliva and vaginal and airway surface fluid (Travis et al., 1999;[START_REF] Gonzalez-Chavez | Lactoferrin: structure, function and applications[END_REF], is involved in host defense against pathogenic bacteria, fungi and protozoa, both directly and through regulation of the inflammatory response [START_REF] Vorland | Lactoferrin: A multifunctional glycoprotein[END_REF]Tian et al., 2010). Previously we showed that lactoferrin reduces E. coli O157:H7 growth and its attachment to epithelial cells in vitro [START_REF] Atef Yekta | Lactoferrin inhibits E. coli O157:H7 growth and attachment to intestinal epithelial cells[END_REF]. This reduction is at A c c e p t e d M a n u s c r i p t 4 least partly due to a proteolytic effect of lactoferrin on the bacterial type III secretion system (TTSS) proteins such as EspA and EspB. This effect has also been shown on other bacteria (EPEP) [START_REF] Ochoa | Lactoferrin impairs type III secretory system function in enteropathogenic Escherichia coli[END_REF][START_REF] Ashida | Cellular internalization of lactoferrin in intestinal epithelial cells[END_REF]). However we could not show proteolytic effect of lactoferrin on bacterial protein intimin. The potential for using oral lactoferrin in animals has been validated with studies that showed no toxic effect attributed to oral delivery of lactoferrin in rats [START_REF] Appel | Sub-chronic (13-week) oral toxicity study in rats with recombinant human lactoferrin produced in the milk of transgenic cows[END_REF].

Many studies on the control of E. coli O157:H7 focus on cattle, but sheep are also an important well-established model [START_REF] Vande Walle | Oral infection with a Shiga toxin-negative Escherichia coli O157:H7 strain elicits humoral and cellular responses but does not protect sheep from colonisation with the homologous strain[END_REF], a b, Woodward et al., 2003).

The aim of the present study was to investigate the potential of lactoferrin to prevent E. coli O157:H7 excretion in a sheep model.

Materials and methods

Bacterial inoculum

NCTC12900, a well characterized Shiga toxin (Stx) negative, nalidixic acid (Nal) resistant E. coli O157:H7 strain was kindly provided by Prof. M. Woodward [START_REF] Woodward | Non-toxigenic Escherichia coli O157:H7 strain NCTC12900 causes attaching-effacing lesions and eae-dependent persistence in weaned sheep[END_REF]. A Stx negative strain was used since this allowed us to perform experiments in A2 isolation units. Bacteria were grown overnight in Luria Bertani broth (LB) at 37°C while shaking (200 rpm), centrifuged (550 g, 10 min, 4°C) and subsequently resuspended in sterile phosphate-buffered saline (PBS) to a concentration of 10 10 CFU.

Lactoferrin

Non-iron-saturated bovine lactoferrin (Ingredia nutritional, France) with 90% purity was used.

A c c e p t e d M a n u s c r i p t 5

Animals

Seventeen 3-month-old male sheep (Belgian cross-breed, Zootechnical Centre, Leuven, Belgium) were used in this study. The faeces of these sheep were free of E. coli O157:H7 as demonstrated by immunomagnetic separation (IMS) and culturing (procedure described further). Animals were seronegative for antibodies against EspA, EspB and intimin, as determined by ELISA (described further). Selected animals were allowed to acclimatise for one week after arrival in our animal facility. Sheep were housed in groups of 8, 7 and 2 animals per pen and received grain-based pellets and water ad libitum.

Experimental procedures

Five animals received lactoferrin at high dose (1.5 g per 12 hours) (high LF group)

and three animals at a 10 times lower dose (0.15 g per 12 hours) (low LF group) for 30 days. Animals were housed in the same pen as group 1 (n = 8). Lactoferrin was given orally to group 1 in a volume of 10 ml sodium bicarbonate buffer (10%) via a syringe allowing the sheep to drink the solution. The colonization control group 2 (n = 7) and the LF control group 3 (n = 2) received the sodium bicarbonate buffer or the high lactoferrin dose, respectively (Table 1). The LF control group allowed monitoring for visual side effects of the lactoferrin administration. After 1 day of lactoferrin administration, groups 1 and 2 received 10 10 E. coli O157:H7 in 10 ml PBS orally for 2 consecutive days, whereas the LF control group 3 received PBS only.

Excretion of E. coli O157:H7 was monitored twice a week. Blood was collected weekly from the vena jugularis to test for serum antibodies against intimin, EspA and

Monitoring of E. coli O157:H7 excretion

Faecal excretion of E. coli O157:H7 was monitored as described by Vande Walle et al., (2010 a), who formerly demonstrated the reproducibility of the E. coli O157:H7 excretion pattern in this sheep model. Briefly, 10 g faeces were diluted in modified tryptone soy broth (Oxoid Ltd, Hanst, UK) supplemented with 20 mg/ml novobiocin and subsequently homogenized using a stomacher. Ten-fold serial dilutions were spreadplated onto MacConkey agar plates supplemented with sorbitol, cefixime and tellurite and Nal (NalCT-SMAC) (MERCK, Darmstadt, Germany). Remaining broth was enriched for 6h at 42°C and subjected to immuno-magnetic separation (IMS) with O157 Dynabeads ® (Invitrogen, Merelbeke, Belgium) according to the manufacturer's instructions. Finally, 100 µl was plated onto NalCT-SMAC. Colonies were confirmed to be E. coli O157 by a latex agglutination test (Oxoid Ltd, Basingstoke, UK). Colony counts were log10 transformed for data analysis. If E. coli O157 was not detected by direct plating, but only detected by enrichment, a concentration of 10 CFU/g was assigned [START_REF] Vande Walle | Oral infection with a Shiga toxin-negative Escherichia coli O157:H7 strain elicits humoral and cellular responses but does not protect sheep from colonisation with the homologous strain[END_REF]. Excretion results were considered negative after 2 successive negative IMS results.

Serum antibody response against virulence factors of E. coli O157:H7

As described by Vande Walle et al., (2010 a), sera were tested for the presence of antibodies against the following E. coli O157:H7 virulence factors: intimin, EspA and

Effect of lactoferrin on faecal E. coli O157:H7 excretion in sheep

In the present study, eight sheep receiving lactoferrin (five the high dose and three the low dose) in sodium bicarbonate buffer during 30 days and seven sheep receiving only the buffer, were inoculated on the second and third day of the lactoferrin administration with E. coli O157:H7. Four days after the first inoculation, animals in the colonization control group shed between 10 5 and 10 8 CFU E. coli O157:H7/g faeces with an average of 7x10 7 CFU/g (Fig 1). Subsequently, the number of excreted bacteria declined gradually and five of seven animals stopped shedding between 18 and 28 days post inoculation (PI). The two remaining animals excreted E. coli O157:H7 till the end of the experiment (day 28 PI).

In the high LF group, reduced faecal shedding was observed from day four onwards.

On day seven, two animals already ceased excreting E. coli O157:H7. One week later, the faeces of two additional animals became negative (day 14 PI), while bacterial excretion in the single remaining animal ceased at day 21 PI. The shedding period for the high LF group (12.6 ± 1.17 days (mean ± SEM)) was significantly shorter than for the colonization control group (24.71 ± 0.68 days (mean ± SEM)) (p < 0.05).

Reduced faecal shedding was observed in the LF low group from day four PI onwards, however this reduction was only significant on day 4 and 7 PI. Day 14, one of the animals stopped excreting and day 17 the two remaining animals became negative.

Nevertheless, during the second week of the experiment the number of bacteria excreted by the low LF group was higher than for the high LF group. Differences between the high LF and low LF groups were not significant (Fig 1). However, again the duration of A c c e p t e d M a n u s c r i p t 9 excretion (16.66 ± 0.76 days (mean ± SEM)) was significantly shortened in comparison with the colonization control group (p < 0.05), but not in comparison with the high LF group. The LF control group remained negative throughout the experiment.

Effect of lactoferrin on the IgG response against E. coli O157:H7 antigens

Since lactoferrin has been described to modulate the adaptive immunity [START_REF] Legrand | A critical review of the roles of host lactoferrin in immunity[END_REF], the effect of oral lactoferrin treatment on the serum antibody response against virulence factors of E. coli O157:H7 including EspA, EspB and intimin was determined (Fig. 2). Two weeks after the experimental inoculation, the colonization control group, receiving E. coli O157:H7 only, showed a very low serum IgG response against intimin, EspA and EspB with maximal log 2 titres of 4.54, 4.05 and 4.62, respectively. A similar response was observed in the low LF group. However, for the high LF group, the IgG response against EspA and EspB significantly (p < 0.05) increased with a peak at 2 and 3 weeks post infection for EspB and EspA, respectively.

The IgG response against intimin did not significantly raised, as compared to the LF control and the colonization control group.

Discussion

Escherichia coli O157:H7 colonizes the intestinal tract of ruminants, making them the main reservoir for this human pathogen [START_REF] Dean-Nystrom | Pathogenesis of Escherichia coli O157: H7 in weaned calves[END_REF][START_REF] La Ragione | Escherichia coli O157:H7 colonization in small domestic ruminants[END_REF]. Therefore, reducing or eliminating E. coli O157:H7 excretion by ruminants is important to decrease the rate of human infection.

A c c e p t e d M a n u s c r i p t 10 Several strategies have been described to reduce excretion such as isolating animals which excrete more than 10 4 CFU/g E. coli O157:H7, the so called "super shedders" (Stephens et al., 2008;Stephens et al., 2009), vaccination (Potter et al., 2004;[START_REF] Peterson | Effect of a A c c e p t e d M a n u s c r i p t 18[END_REF], antibiotic treatment [START_REF] Molbak | Antimicrobial therapy in patients with Escherichia coli O157: H7 infection[END_REF], probiotics [START_REF] Callaway | Colicin concentrations inhibit growth of Escherichia coli O157 : H7 in vitro[END_REF], bacteriophages [START_REF] Callaway | Colicin concentrations inhibit growth of Escherichia coli O157 : H7 in vitro[END_REF] and dietary changes [START_REF] Callaway | Diet, Escherichia coli O157:H7, and cattle: a review after 10 years[END_REF].

However, continued research for effective pre-harvest interventions is needed.

Lactoferrin has a direct antimicrobial effect on E. coli O157:H7 [START_REF] Atef Yekta | Lactoferrin inhibits E. coli O157:H7 growth and attachment to intestinal epithelial cells[END_REF] and is responsible for the proteolytic degradation of EspA and EspB (Atef Yekta et al., 2010). In addition, lactoferrin is an immunomodulatory protein [START_REF] Legrand | A critical review of the roles of host lactoferrin in immunity[END_REF]. These findings resulted in the following hypothesis: lactoferrin might be used to reduce E. coli O157:H7 colonization of the intestinal tract and consequently, faecal shedding by sheep.

Previously, we developed an oral infection model in sheep creating persistent E. coli O157:H7 shedders [START_REF] Vande Walle | Oral infection with a Shiga toxin-negative Escherichia coli O157:H7 strain elicits humoral and cellular responses but does not protect sheep from colonisation with the homologous strain[END_REF]. At present, the model was used to study the effect of the natural antimicrobial-immunomodulatory milk protein lactoferrin on E. coli O157:H7 excretion. Moreover, the sheep infection model was used to examine the antibody responses against intimin and the E. coli O157:H7 type III secretion proteins EspA and EspB.

In our study, lactoferrin was administered in bicarbonate buffer. The buffer closes the esophageal groove, so that lactoferrin passes rumen, reticulum and omasum and directly reaches the abomasum (Rosenberger, 1979). Delivering lactoferrin in the abomasum will prevent its bacterial degradation in rumen and reticulum and will enhance its degradation by pepsin. Pepsin cuts lactoferrin in different peptides of which one peptide, lactoferricin, A c c e p t e d M a n u s c r i p t 11 has been demonstrated to exert a stronger bactericidal, immunomodulatory and inflammatory effect than lactoferrin [START_REF] Gifford | Lactoferricin: a lactoferrin-derived peptide with antimicrobial, antiviral, antitumor and immunological properties[END_REF]. Once lactoferrin reaches the small intestine it probably is only slowly degraded, since it is strongly resistant to trypsin and trypsin-like enzymes [START_REF] Brines | The effect of trypsin and chymotrypsin on the in vitro antimicrobial and iron binding properties of lactoferrin in human milk and bovine colostrum, unusual resistance of human apo-lactoferrin to proteolytic digestion[END_REF]. Whether lactoferricin shows a similar resistance, has not been tested yet.

Both lactoferrin dosages were capable to significantly reduce the concentration of E. coli O157:H7 in the faeces and to shorten the duration of faecal excretion, indicating that sufficient functional active lactoferrin reaches the intestine to interfere with bacterial colonization. Interestingly, sheep, which received the high dose of lactoferrin, showed a higher antibody response against EspA and EspB in comparison to the control group. The peak of the antibody response was 2 to 3 weeks post inoculation, when the excretion of E. coli O157:H7 had completely ceased. A similar increase in antibody response was observed in chickens orally fed lactoferrin from birth on and orally vaccinated with an infectious bursal disease (Gumboro) vaccine at 1 and 3 weeks of age [START_REF] Hung | Porcine lactoferrin as feedstuff additive elevates avian immunity and potentiates vaccination[END_REF].

As in our study, this effect was observed with the highest dose. Furthermore, a significant increase in total serum IgA and IgG and an increased mitogen-induced proliferation of peripheral blood lymphocytes was demonstrated. Studies in mice demonstrated increases in total antibody, B-, T-and NK-cells amounts (Teraguchi et al., 2004;[START_REF] Varadhachary | Oral lactoferrin inhibits growth of established tumors and potentiates conventional chemotherapy[END_REF]Wolf et al., 2007). In humans, oral supplementation resulted in an enhanced T cell activation, but no effects on B-cells or antibody production have been described [START_REF] Mulder | Bovine lactoferrin supplementation supports immune and antioxidant status in healthy human males[END_REF].

Lactoferrin has important immunomodulatory activities such as increased maturation of B-and T-lymphocytes and increased recruitment and maturation of antigen-presenting A c c e p t e d M a n u s c r i p t 12 cells which occur by influencing pattern recognition receptor-mediated cell signaling [START_REF] Curran | Lactoferrin activates macrophages via TLR4-dependent and -independent signaling pathways[END_REF][START_REF] De La Rosa | Lactoferrin acts as an alarmin to promote the recruitment and activation of APCs and antigen-specific immune responses[END_REF][START_REF] Legrand | A critical review of the roles of host lactoferrin in immunity[END_REF]. Whether such effects are responsible for the increased antibody response in our study still has to be determined. In this study, we used an Stx-negative strain and there is evidence that infections with STEC can suppress the development of specific cellular immune responses in cattle [START_REF] Hoffman | Bovine immune response to shiga-toxigenic Escherichia coli O157:H7[END_REF]. Stx could counteract the immuno-modulatory effects of lactoferrin. This will be examined in a future experiment in biosafety level 3 large animal facilities.

Conclusion

This is the first study demonstrating the reduction of E. coli O157:H7 excretion by oral administration of the natural antimicrobial protein, lactoferrin. Moreover, the results suggest that lactoferrin could become an important tool to decrease colonization pressure on farms and to prevent contamination of food by E. coli O157:H7 and, consequently to decrease E. coli O157:H7 associated illness in humans. However, more convenient lactoferrin administration methods need to be evaluated.

  animal experiments were approved by the ethical committee of the Faculty of Veterinary Medicine (approval 2009/074).

  A c c e p t e d M a n u s c r i p t 7 EspB. Briefly, sera were heat-inactivated (30 min. at 56°C) and kaolin-treated. Polysorb 96-well plates (NUNC, Polysorb Immuno Plates, Roskilde, Denmark) were coated with 200 ng/well of recombinant intimin, EspA or EspB in PBS for 2h at 37°C and subsequently blocked overnight at 4°C using PBS + 0.2% Tween ® 80. After washing with PBS + 0.2% Tween ® 20, plates were incubated with two-fold serial dilutions of serum in PBS + 0.05% Tween ® 20 and with HRP-conjugated anti-sheep IgG-specific donkey antibodies (AbD Serotec, UK). Sera of sheep intramuscularly immunized with intimin, EspA and EspB during a former study (Vande Walle et al., 2010 a), served as positive control. The cut-off values were calculated as the mean OD 405 -values of all sera(dilution 1/10) obtained at day 0 increased with three times the standard deviation (cutoff values for intimin, EspA and EspB were 0.387, 0.412 and 0.312, respectively).2.7. Statistical analysisStatistical analysis was performed using SAS software version 1999. The Proc MIXED test was used to analyze bacterial excretion. The statistical model included the fixed effect of LF dose (low, high), the random effect of sheep nested within treatment and the residual errors. The duration of the excretion was compared in three groups of infected animals by using the t-test. Serum antibody titres were statistically analyzed by use of a General linear Model (repeated measures analysis of variance). Differences were considered statistically significant at p <0.05.

  D. Karpman and Dr. D. O'Brien for providing the plasmids. K. Vande Walle and A. Pezeshki are acknowledged for support during data analyses. The authors wish to thank B. Driessen from the Zoological Centre in Leuven for assistance during the screening of the sheep and Rudy Cooman as animal caretaker.

Figure 1 .

 1 Figure 1. Effect of lactoferrin administration on faecal excretion of E. coli O157:H7 by

Figure 2 .

 2 Figure 2. Mean serum antibody responses in the groups infected with E. coli O157:H7.

  Figure 1

Table 1 :

 1 Experimental set-up

					Lactoferrin (LF)
	Group number	N	Manipulation	E. coli O157:H7
					for 30 days
	1	5	High LF	+	1.5 g/per 12 hours (high dose)
	1	3	Low LF	+	0.15 g/per 12 hours (low dose)
	2	7	Colonization control	+	-
	3	2	LF control	-	1.5 g/per 12 hours (high dose)

Table Page

 Page

21 of 22

Acknowledgments

The authors wish to thank Gent University (BOF) for providing a PhD grant to Maryam Atef Yekta. This study was funded by the Federal Public Service of Health, Food Chain Safety and Environment. The Research Foundation -Flanders (FWO -Vlaanderen) is also acknowledged for financial support. The authors gratefully acknowledge Dr. M. J. Woodward for providing E. coli O157:H7 strain NCTC12900 and

Travis, S. M., Conway, B. A. D. Zabner, J. Smith, J. J. ,Anderson, N. N., Singh, P. K., Greenberg, E. P., Welsh., M. J. 1999. Activity of abundant antimicrobials of the human airway. Am J Respir Cell Mol Biol 20(5):872-879.


Elisabeth Chéreau
email: elisabeth.chereau@gmail.com

MD M Ballester

MD R Rouzier

Ovarian cancer in the elderly: impact of surgery on morbidity and survival

Keywords: ovarian cancer, elderly, optimal surgery, morbidity, outcome

Background: Elderly ovarian cancer patients often undergo non-optimal surgery due to their age despite of the high risk of recurrence. The aim of this study was to determine if more postoperative complications occurred in patients over 70 years and to compare extent of surgery with younger patients.

Materials and Methods

: Between 2001 and 2009, 172 patients with ovarian cancer were included. We compared patient characteristics, surgical course, postoperative complications and outcome for patients under and over 70 years. Results: 143 patients were under 70 years and 29 over. There were no difference between the two groups for tumors carateristics, time of surgery, FIGO stage, standard surgical procedures and rate of optimal resection. Patients over 70 years had less peritoneal surgery (p<0.001) especially diaphragmatic surgery (p=0.006), pelvic (p=0.02) and para aortic (p=0.003) lymphadenectomy. There was no difference in the occurence of per-or post operative complications and patients over 70 years had shorter duration of hospitalisation (p=0.04).

There was no difference between the two groups for disease free survival (DFS) (p=0.08) but overall survival (OS) was better in patients under 70 years (p=0.002).

Conclusion:

Elderly ovarian cancer patients undergo less extensive surgery and have lower OS despite similar postoperative morbidity, optimal resection and DFS. OS decrease could be explained by difference in the management of recurrences.

Introduction

Cumulative data support that complete cytoreduction with adjuvant or neo-adjuvant chemotherapy is the cornerstone treatment for advanced epithelial ovarian cancer [START_REF] Bristow Re | Survival effect of maximal cytoreductive surgery for advanced ovarian carcinoma during the platinum era: a meta-analysis[END_REF][START_REF] Ds | What is the optimal goal of primary cytoreductive surgery for bulky stage IIIC epithelial ovarian carcinoma (EOC)?[END_REF] .

However, as the peak of incidence of ovarian cancer is 61 years a high proportion of patients are elderly and this raises specific concerns about the choice of therapeutic strategy. Therapeutic management of these patients constitutes a challenge between achieving the goal of complete cytoreduction to improve survival, and the occurrence of postoperative complications, which can lead to severe outcome or death. Recent studies have suggested that elderly patients are less likely to be optimally debulked [START_REF] Ng | Management of ovarian cancer in patients older than 80 years of Age[END_REF] , receive standard therapy less often [START_REF] Sundararajan | Variations in the use of chemotherapy for elderly patients with advanced ovarian cancer: a population-based study[END_REF][START_REF] Uyar D | Treatment patterns by decade of life in elderly women (> or =70 years of age) with ovarian cancer[END_REF][START_REF] Markman | Epithelial ovarian cancer in the elderly. The Memorial Sloan-Kettering Cancer Center experience[END_REF] and are less able to tolerate medical treatments [START_REF] Kn | Ovarian cancer in the octogenarian: does the paradigm of aggressive cytoreductive surgery and chemotherapy still apply?[END_REF] . Furthermore, patients over 65 years are rarely included in clinical trials. Overall then, management of these patients remains a matter of debate [START_REF] Lewis | Participation of patients 65 years of age or older in cancer clinical trials[END_REF] .

Aletti et al [START_REF] Aletti Gd | Relationship among surgical complexity, short-term morbidity, and overall survival in primary surgery for advanced ovarian cancer[END_REF] demonstrated that the main independent predictors of morbidity were age, American Society of Anesthesia (ASA) status, and surgical complexity score. Cloven et al reported up to 20% of mortality in patients over 80 years [START_REF] Ng | Management of ovarian cancer in patients older than 80 years of Age[END_REF] . Nevertheless, geriatric practitioners believe that the physiological age should be taken into account when evaluating patients so as to be able to adopt the most effective treatment approach 10 . And while Chi et al [START_REF] Ds | Nomogram for survival after primary surgery for bulky stage IIIC ovarian carcinoma[END_REF] have proposed a nomogram to evaluate survival in this group of patients, few data are available on management of ovarian cancer in elderly [START_REF] Ng | Management of ovarian cancer in patients older than 80 years of Age[END_REF][START_REF] Sundararajan | Variations in the use of chemotherapy for elderly patients with advanced ovarian cancer: a population-based study[END_REF][START_REF] Uyar D | Treatment patterns by decade of life in elderly women (> or =70 years of age) with ovarian cancer[END_REF][START_REF] Markman | Epithelial ovarian cancer in the elderly. The Memorial Sloan-Kettering Cancer Center experience[END_REF][START_REF] Marchetti Dl | Treatment of advanced ovarian carcinoma in the elderly[END_REF][START_REF] Tp | Surgical care of elderly women with ovarian cancer: a population-based perspective[END_REF][START_REF] Mclean | Ovarian cancer in the elderly: outcomes with neoadjuvant chemotherapy or primary cytoreduction[END_REF] .

Therefore, the purpose of this retrospective study was to determine whether patients over 70 years were more likely to undergo incomplete or less aggressive surgery and experience more postoperative complications and worse outcome.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Materials and methods

From 2001 to 2009, 172 patients underwent surgery for ovarian cancer in our department of gynaecologic surgery. Individual records of all the patients were reviewed and analyzed. Patient and tumor characteristics, intraoperative findings, postoperative courses and outcomes were analyzed.

Patients were divided into two groups according to their age: under or over 70 years. Surgical procedures were separated into standard procedures and upper abdominal surgery including splenectomy, cholecystectomy, distal pancreatectomy, liver resection and lesser omentum resection. Residual disease was reported as complete (no residual disease, CC0), optimal (CC1 i.e. residual disease less than 2.5 mm) or sub-optimal (CC2 i.e. residual disease between 2.5 and 25 mm) according to the Sugarbaker score of cytoreduction [START_REF] Sugarbaker Ph | Management of peritoneal-surface malignancy: the surgeon's role[END_REF] .

Intraoperative estimated blood loss, operative time, and length of hospitalization were recorded. Postoperative complications such as digestive fistula, lymphocyst and infection were also recorded. Disease-free survival (DFS) and overall survival (OS) were calculated and compared between the two groups. All the women gave informed written consent to therapeutic procedures and to the analysis of data related to their malignancy in accordance with institutional guidelines and the Declaration of Helsinki.

Statistical Analysis

Data were analyzed using the chi-square test, t-test and Mann Whithney test. Differences were considered significant at a level of p < 0.05. Overall survival time was calculated in months from the date of surgery to death, or the date of last follow-up for surviving patients and DFS time from the date of surgery to recurrence. The Kaplan-Meier method was used to estimate the survival distribution and comparisons of survival were made by the use of the log-rank test.

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Results

Epidemiological and surgical characteristics of the population

Patients and surgical characteristics are reported in Table 1. A hundred and forty-three patients were under 70 years (group < 70y) and 29 patients were over 70 years (group ≥ 70y).

Mean age was 54.3 years in the group < 70y and 75.8 years in the group ≥ 70y. The comorbidity rate in the group < 70y and the group ≥ 70y was 49% and 69%, respectively (p=0.06). There were no differences in the mean preoperative CA 125 serum level, proportion of patients undergoing initial or interval surgery after neoadjuvant chemotherapy, and proportion of FIGO stage between groups. The mean number of chemotherapy cycles during neoadjuvant chemotherapy was also not different (5 cycles +/-2 for >70y, 6 cycles +/-4 for <70y). Among the patients of >70y group, 6 could not acchieved at least six cycles of chemotherapy (neoadjuvant and adjuvant chemotherapy). One patient died in the post operative course, the five others were not able to afford the whole treatment. According to our local guidelines, all patients received association of platinum and taxane during the first line of chemotherapy.

The mean operative time was shorter for the group ≥ 70y (p=0.04). No difference was found between the groups in the mean estimated blood loss or in the type of resection at the end of surgery; optimal resection was obtained in 86% of patients in the group < 70y and 90% in the group ≥ 70y. In the group ≥ 70y there was less peritoneal surgery (p<0.001), and pelvic (p=0.02) and para-aortic (p=0.003) lymphadenectomy. For upper abdominal surgery, the only statistical difference was for diaphragmatic surgery that was less often performed in the group ≥ 70y (p=0.006). Using Aletti's surgical complexity score (SCS) system based upon complexity and the number of surgical procedures performed [START_REF] Aletti Gd | Relationship among surgical complexity, short-term morbidity, and overall survival in primary surgery for advanced ovarian cancer[END_REF] , the rate of patients with low, intermediate or high

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SCS were respectively 36%, 36% and 28% in the group < 70y and 55%, 41% and 4% in the group ≥ 70y. The proportion of patients with a low SCS was higher in the group ≥ 70y (p=0.01).

Seven intraoperative complications occurred in the group < 70y compared with only one in the group ≥ 70y without reaching a statistical significance (Table 2). Patients ≥ 70 years had a shorter duration of hospitalisation both in the intensive care unit (p=0.04) and overall (p=0.03). No difference in the rate of postoperative complications was observed (28% in the group < 70y and 10% in the group ≥ 70y) whatever in terms of digestive fistula, lymphocyst, pneumothorax, pulmonary embolism,fever, infection, pleural effusion and iterative surgery. In each group one patient died in the post operative time. In group <70y from pulmonary embolism and in group >70y from septic shock.

Survival

The mean follow-up time for survival patients was 42.1 +/-25 months. The two-year DFS was 57% for the group < 70y and 35% for the group ≥ 70y. The five-year DFS was 40% for the group < 70y and 23% for the group ≥ 70y. There was no significant difference between the two groups (p (log Rank)=0.08) (Figure 1). The five-year OS was lower in the group ≥ 70y than in the group < 70y: 55% and 70% respectively (p(log Rank)=0.002) (Figure 2). All deceased patients died of their cancer.

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Discussion

The present study has demonstrated that five-year overall survival (OS) in patients with ovarian cancer is lower in patients aged 70 years or over in spite of similar complete cytoreduction and surgical complications rates to patients under 70 years. Two-year DFS is similar in the two groups. This difference cannot be explained by differences in the epidemiological characteristics of patients as no difference in comorbidity and FIGO stages were observed between the groups. Hence, in addition to the age, other parameters such as variations in surgical or adjuvant therapies could play a role. The absence of difference for surgical outcome despite a lower proportion of peritoneal and diaphragmatic surgery between the groups with same rate of advanced stage could be explained by a higher rate of patients with 3C FIGO stage related to positive paraaortic lymph nodes in the >70y group (4% in the <70y group and 16% in the >70y group (p=0.02)).

Impact of lymphadenectomy on survival

A closer look at the type of surgery performed in both groups shows that, although there was no difference in the rate of complete and optimal cytoreduction, there were lower rates of pelvic and paraaortic lymphadenectomy and peritoneal surgery in the older group. This is of particular importance especially for patients with advanced stages of ovarian cancer, which was the case for most of our patients. When the residual disease was taken into account, Beneditti Panici et al [START_REF] Benedetti Panici | Bulky lymph node resection in patients with recurrent epithelial ovarian cancer: impact of surgery[END_REF] showed that the hazard ratios for occurrence of the first event and mortality from any cause were almost unchanged. However, analyzing the surveillance, epidemiology, and end results (SEER) database for the period from 1988 to 2001 when taxanes were little used, Chan et al [START_REF] Jk | The potential therapeutic role of lymph node resection in epithelial ovarian cancer: a study of 13918 patients[END_REF] underlined the detrimental impact of not performing lymphadenectomy. A similar recent study using the SEER database on the period up to 2004 when taxanes were widely used as in our population, Rouzier et al [START_REF] Rouzier R | The role of lymph node resection in ovarian cancer: analysis of the surveillance, epidemiology, and end results (SEER) database[END_REF]

also confirmed the

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positive impact of performing lymphadenectomy independently of the extent of cytoreduction. This advantage was proved for women with stage-II ovarian cancer in whom the risk of incomplete cytoreductive surgery is relatively low and the incidence of lymph node involvement is only about 20% as well as for patients with stage IIIC and IV. All these data reinforce the fact that surgical effort even in elderly women should take into account not only peritoneal disease but also retroperitoneal spread.

Morbidity according to the extent of surgery

The risk ratio of extensive surgery including diaphragmatic removal and lymphadenectomy in elderly women deserves discussion due to the risk of postoperative complications. Although no difference in the overall complication rate was found between the two groups in our study, three patients in the group < 70y developed digestive fistula compared with none in the group ≥ 70y. Moreover, lymphocyst, pneumothorax or pulmonary embolism were only observed in the group of patients under 70 years. In accordance with Chi et al [START_REF] Ds | Improved optimal cytoreduction rates for stages IIIC and IV epithelial ovarian, fallopian tube, and primary peritoneal cancer: a change in surgical approach[END_REF] , although cytoreductive surgery and chemotherapy offer the best chance of long term survival, this strategy in elderly patients is associated with a significant morbidity and mortality, hence patients and their families should be counseled on these potential risks. Using Aletti's surgical complexity score (SCS) system [START_REF] Aletti Gd | Relationship among surgical complexity, short-term morbidity, and overall survival in primary surgery for advanced ovarian cancer[END_REF] , the association of pelvic and paraaortic lymphadenectomy to other surgical procedures shifts one third of the patients ≥ 70 years from low to intermediate or high SCS groups. Aletti et al underlined that the relative risk of postoperative complication compared to low SCS were 1.35 and 4.59 for intermediate and high SCS groups respectively. A recent study has confirmed the relevance of the Aletti scoring system to predict postoperative complications with an area under the ROC curve superior to Fagotti, Fagotti modified, Sugarbaker's Peritoneal Cancer Index, Eisenkop and FIGO classification 20 . However, the relevance of the SCS system to evaluate OS and DFS

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was not confirmed [START_REF] Aletti Gd | Relationship among surgical complexity, short-term morbidity, and overall survival in primary surgery for advanced ovarian cancer[END_REF] .

Criteria to opt for extensive surgery

In a previous study 21 using a multivariable analysis, the decision to perform lymph node assessment appeared dependent on the surgeon's choice, low residual disease, an ASA of 1 or 2 and the absence of carcinomatosis. Once lymphadenectomy had been decided on, the main independent criterion to perform complete lymphadenectomy versus lymph node sampling was a patient age of under 65 years. However, as there is a lack of clear recommendations for patients over 65 years linked to them being underrepresented in clinical trials [START_REF] Lewis | Participation of patients 65 years of age or older in cancer clinical trials[END_REF] , these criteria would seem to be subjective. There is therefore a need for objective criteria to opt for extensive surgery including pelvic and paraaortic lymphadenctomy. Chi et al [START_REF] Ds | Nomogram for survival after primary surgery for bulky stage IIIC ovarian carcinoma[END_REF] advocated the use of a nomogramme to predict survival taking into account age at surgery, tumor grade, histology, preoperative platelet count, presence of ascites and residual disease. Except for tumor grade, which is sometimes difficult to assess on intraoperative histology but which has little impact on the nomogram score, these parameters are evaluable during surgery and could be used to opt for lymphadenectomy in patients with a high probability of survival.

Impact of chemotherapy regimen on survival

In addition to surgical effort, neo-adjuvant or adjuvant platinum/taxane based chemotherapy is the cornerstone for treating patients with advanced stages of ovarian cancer.

In our study, no difference in the rate of neo-adjuvant chemotherapy or in the chemotherapy regimen was noted between the groups. Therefore, the difference in survival between the groups cannot be explained by this parameter. Brun et al [START_REF] Brun | Neoadjuvant chemotherapy or primary surgery for stage III/IV ovarian cancer: contribution of diagnostic laparoscopy[END_REF] showed that neo-adjuvant chemotherapy leads to a high rate of complete cytoreduction in nearly three quarters of

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patients and offers similar survival to initial surgery. However, a recent meta-analysis on interval debulking surgery found no advantage on survival of this option compared to initial surgery followed by adjuvant chemotherapy [START_REF] Tangjitgamol | Interval debulking surgery for advanced epithelial ovarian cancer[END_REF] . Despite its proven efficacy in treating ovarian cancer, chemotherapy seems to be used less in elderly patients. In a multiple logistic regression model with patients aged 65 to 69 years as the reference, Sundararajan et al [START_REF] Sundararajan | Variations in the use of chemotherapy for elderly patients with advanced ovarian cancer: a population-based study[END_REF] found that the odds ratios of receiving chemotherapy were 0.96 for ages 70 to 74, 0.65 for 75 to 79, 0.24 for 80 to 84, and 0.12 for patients over 85 years. Moreover, among patients over 80 years who underwent complete cytoreductive surgery, 13% were unable to receive adjuvant therapy.

This contrasts with the data of Eisenhauer et al [START_REF] El | Response and outcomes in elderly patients with stages IIIC-IV ovarian cancer receiving platinum-taxane chemotherapy[END_REF] showing that patients over 65 years had similar rates of initial response, platinum resistance, DFS and OS to younger patients.

This also contrasts with the potential benefit of chemotherapy, especially neo-adjuvant. Indeed, McLean et al 14 demonstrated that neo-adjuvant chemotherapy was associated with odds ratios of 0.80 (95% CI 0.37-1.75) for surgical complications and 0.79 (95% CI 0.33-1.90) for chemotherapeutic complications in elderly women. Further studies are required to evaluate whether neo-adjuvant chemotherapy should be the first line therapy in elderly women.

Limitations of study on elderly patients

Some limitations of the present study have to be underlined. First, the retrospective nature cannot exclude the risk of potential bias such as the exclusion of patients from first surgical effort linked to age. Second, there is a lack of consensus about what constitutes an elderly patient. Gardner reported that the term "elderly" has been variably defined in the ovarian cancer debulking literature as being older than 60, 70, or 80+ years of age. However, most studies determine a cut-off for elderly women as patients aged 70 years or more [START_REF] Sundararajan | Variations in the use of chemotherapy for elderly patients with advanced ovarian cancer: a population-based study[END_REF][START_REF] Uyar D | Treatment patterns by decade of life in elderly women (> or =70 years of age) with ovarian cancer[END_REF][START_REF] Markman | Epithelial ovarian cancer in the elderly. The Memorial Sloan-Kettering Cancer Center experience[END_REF][START_REF] Kn | Ovarian cancer in the octogenarian: does the paradigm of aggressive cytoreductive surgery and chemotherapy still apply?[END_REF]

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which is what we opted to use here. Third, no attempt was made to replace the birth age by criteria evaluating life expectancy. Further studies are required to evaluate the contribution of geriatric evaluation to define patients with high life expectancy that could benefit from initial surgical effort including lymphadenectomy. Finaly, the decrease in OS for elderly patients observed in our study could be explained by difference in the management at the time of recurrence. Indeed, patients over 70 years are usually not included in medical trials and so couldn't benefit from new therapy.

Conclusion

Our results support that as many elderly patients with ovarian cancer benefit from complete cytoreductive surgery as younger patients. However, extensive surgery including lymphadenectomy was less frequently performed in patients ≥ 70 years impacting on overall survival. There is therefore a need to establish objective criteria to identify elderly candidates who could benefit from lymphadenectomy and extensive peritoneal resection such as diaphragmatic surgery.

Conflict of interest statement

None of the authors had any financial and personal relationships with other people or organisations that could inappropriately influence this work.

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