Luis Almeida
email: almeida@ljll.math.upmc.fr

Pierre-Alexandre Bliman
email: pierre-alexandre.bliman@inria.fr

Grégoire Nadin
email: gregoire.nadin@ljll.math.upmc.fr

Benoît Perthame
email: benoit.perthame@ljll.math.upmc.fr

Nicolas Vauchelet
email: vauchelet@math.univ-paris13.fr

Final size and convergence rate for an epidemic in heterogeneous population

Keywords: SIR model, Epidemic spread, structured population equations, next generation operator, long term asymptotics. 2010 Mathematics Subject Classification.: 35Q92, 35R09, 45C05, 45G15, 92D30

We formulate a general SEIR epidemic model in a heterogenous population characterized by some trait in a discrete or continuous subset of a space R d . The incubation and recovery rates governing the evolution of each homogenous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines the probability for an individual of a given trait to be infected by an individual with another trait. Our goal is to derive and study the final size equation fulfilled by the limit distribution of the population. We show that this limit exists and satisfies the final size equation. The main contribution is to prove the uniqueness of this solution among the distributions smaller than the initial condition. We also establish that the dominant eigenvalue of the next-generation operator (whose initial value is equal to the basic reproduction number) decreases along every trajectory until a limit smaller than 1. The results are shown to remain valid in presence of diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrix (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another).

Introduction

Observing the complex behaviour of the COVID-19, see Refs. [START_REF] Aguas | Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics[END_REF][START_REF] Britton | A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2[END_REF][START_REF] Britton | The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level[END_REF][START_REF] Di Lauro | The impact of network properties and mixing on control measures and disease-induced herd immunity in epidemic models: a mean-field model perspective[END_REF][START_REF] Dolbeault | Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model[END_REF][START_REF] Gomes | Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold[END_REF], reveals the importance of the population heterogeneity in the dynamics of an outbreak. Inspired by this question, but in contrast with recent papers aiming at reproducing the dynamics of the pandemic observed through various data, we adopt here an abstract point of view and investigate issues related to the concepts of epidemic final size and herd immunity in an ample setting.

Before explaining in more details the context and contribution, we present the system under study. We consider here the general SEIR epidemic model [START_REF] Aguas | Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics[END_REF], which describes the spread of a disease in a heterogeneous population characterized by a "trait" x ∈ Ω, for a given open subset Ω ⊂ R d . The underlying structure described by this trait can be very varied, typical examples being one or several of the following characteristics: propensity to have social contacts and hazardousness of the latter (in the sense of the transmission disease), susceptibility, infectivity, characteristics of the immunological response, age, spatial location, etc. ∂ t S(t, x) = -S(t, x) Ω β(x, y)I(t, y) dy, S(0, x) = S 0 (x) (1a)

∂ t E(t, x) = S(t, x) Ω β(x, y)I(t, y) dy -α(x)E(t, x), E(0, x) = E 0 (x),

∂ t I(t, x) = α(x)E(t, x) -γ(x)I(t, x), I(0, x) = I 0 (x), (1c)

∂ t R(t, x) = γ(x)I(t, x), R(0, x) = R 0 (x). ( 1d
)
Classical notations are used : S denotes the proportion of susceptible individuals, E the proportion of those who have been exposed to the disease, I the proportion of infected individuals, R the proportion of removed individuals. The component β(x, y) of the so-called (infinite-dimensional) contact matrix β is the product of the average contact rate between individuals of traits x and y and average transmission probability from an individual with trait y to individual with trait x. The force of infection Ω β(x, y)I(t, y) dy that applies to the individuals of trait x thus obeys the law of mass-action. The average incubation rate is denoted α(x) and the average recovery rate is denoted γ(x): both quantities may depend upon the trait x.

As can be seen, the dynamics of the outbreak is assumed to be sufficiently fast to neglect the effects of the demography in the population. A particular case of system (1) is when the parameters α, β, γ are independent of the trait: the classical ("homogeneous") SEIR model, which reads Ṡ = -βSI, Ė = βSI -αE, İ = αE -γI,

Ṙ = γI (2)
is then recovered after integration over Ω. A variant of (1) with diffusion operator in the infected compartment capable to account for possible mutations or spatial displacement of the infected population is also considered, see equation [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF] below.

For an epidemic model where the effects of demography are neglected, the number of exposed and infected individuals is expected to go to zero after reaching an epidemic peak (if the initial proportion of susceptible is sufficient). An important question is then to determine what will be the so-called epidemic final size, that is the cumulative number (or proportion) of individuals infected, and thus ultimately removed, during the outbreak. For the homogeneous SEIR model ( 2), the answer to this question is well-known. Indeed, due to the invariance of the function S(t) + E(t) + I(t)γ β ln S(t) along the trajectory, the asymptotic value S ∞ of S(t) is necessarily a root of the equation S ∞ -γ β ln S ∞ = S 0 + E 0 + I 0 -γ β ln S 0 . Indeed, one may show that it is the unique root of this equation smaller than the so-called herd immunity threshold γ β -the value of S below which the number E + I of exposed and infected necessarily decreases.

The study of the final size of an epidemic has been addressed in several works since its appearance in early contributions [START_REF] Norman | The mathematical theory of infectious diseases[END_REF][START_REF] Kermack | Contributions to the mathematical theory of epidemics -I[END_REF]. Ref. [START_REF] Ma | Generality of the final size formula for an epidemic of a newly invading infectious disease[END_REF] was among the first papers to unveil the generality of this concept, extending it to models with several distinct infectious stages, or with arbitrarily distributed mean contact rate, or again having spatially heterogeneous contact structures. Multiple susceptible classes were considered in Ref. [START_REF] Arino | A final size relation for epidemic models[END_REF]. Ref. [START_REF] Brauer | Age-of-infection and the final size relation[END_REF] established final size equation for age-of-infection model. Distributed susceptibility or distributed infectivity has been studied in Ref. [START_REF] Artem | On the spread of epidemics in a closed heterogeneous population[END_REF], and Ref. [START_REF] Andreasen | The final size of an epidemic and its relation to the basic reproduction number[END_REF] studied an epidemic model within a population of n different groups. Analysis in a general framework was presented in Ref. [START_REF] Miller | A Note on the derivation of epidemic final sizes[END_REF], establishing sufficient conditions under which epidemic final size equation exists. Sharp estimates and bounds are obtained in Ref. [START_REF] Katriel | The size of epidemics in populations with heterogeneous susceptibility[END_REF] for general model with heterogeneous susceptibility. Also, the case of a two-group SIR model has been studied in Ref. [START_REF] Magal | Final size of an epidemic for a two-group SIR model[END_REF]. Finally, we mention that, in a different biological context, related questions have been addressed on models of population dynamics with heterogeneity modeled by an internal variables [START_REF] Desvillettes | On selection dynamics for continuous populations[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF].

An important quantity to describe the dynamics of an epidemic is the basic reproduction number R 0 . From a biological point of view, the latter represents the expected number of secondary cases directly generated by an individual in a completely susceptible population during its entire period of infectiousness. A famous threshold property states that the disease can invade (in the sense that it will spread and reach an epidemic peak) if R 0 > 1 and the initial number of susceptible is sufficient, whereas it cannot if R 0 < 1. Mathematically, this number has been defined in full generality in the seminal paper [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF] as the dominant eigenvalue of the so-called next-generation operator.

Our aim in the present paper is to extend some of these previous results and completely characterize the final size of an epidemic in a heterogeneous population whose dynamics is governed by the general systems (1) or [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF]. On this occasion, we revisit and present in a unified setting the notions of herd immunity and basic reproductive number, and demonstrate the centrality and powerfulness of the concept of next-generation operator.

The outline of this work is the following. We present in Section 2 the assumptions used in the sequel, and provide the key results, Theorems 2.1 and 2.2. They correspond respectively to epidemic spreading without and with diffusion. Their proof is given in Sections 3 and 4. Section 5 shows how the present framework includes several examples previously considered in the literature.

Assumptions and main results

Heterogeneous populations without diffusion. To begin with, we investigate the final size of an epidemic in a heterogeneous population whose dynamics is modelled by system [START_REF] Aguas | Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics[END_REF]. Denote L ∞ + (Ω) the set of measurable functions which are bounded and nonnegative a.e. on Ω, and assume that

       α ∈ L ∞ + (Ω) and α := ess inf x∈Ω α(x) > 0, β ∈ L ∞ + (Ω × Ω) ∩ L 2 (Ω × Ω) and β > 0, γ ∈ L ∞ + (Ω) and γ := ess inf x∈Ω γ(x) > 0. (3)
In order for the disease to start, we assume also that the initial proportions of susceptible individuals, and of infected and exposed individuals, are nonzero :

S 0 , E 0 , I 0 , R 0 ∈ L ∞ + (Ω), S 0 > 0, E 0 + I 0 ≡ 0, Ω (S 0 + E 0 + I 0 + R 0 ) dx = 1. ( 4
)
The normalization assumption contained in the last identity materializes the fact that the variables constitute proportions of a given population whose overall value is conserved along time.

The first result concerns system (1).

Theorem 2.1 Assume that assumptions (3) and (4) hold, then for the solution of problem (1), t → S(t, •) is a decreasing function and there exist S ∞ , R ∞ ∈ L ∞ + (Ω) ∩ L 1 (Ω) such that, for a.e.

x ∈ Ω,

lim t→+∞ S(t, x) = S ∞ (x), lim t→+∞ R(t, x) = R ∞ (x), (5a) lim t→+∞ I(t, x) = lim t→+∞ E(t, x) = 0. (5b)
In addition, the convergence is uniform, S ∞ > 0, S ∞ is the unique solution such that S ∞ ≤ S 0 of the final size equation

ln S ∞ (x) - Ω β(x, y) γ(y) S ∞ (y) dy = ln S 0 (x) - Ω β(x, y) γ(y) S 0 (y) + E 0 (y) + I 0 (y) dy, (6)
and

R ∞ = S 0 + E 0 + I 0 + R 0 -S ∞ . Moreover, if ess inf Ω S 0 > 0 and β := ess inf (x,y)∈Ω×Ω β(x, y) > 0, (7)
then ess inf Ω S ∞ > 0 and the convergence in (5) is exponential.

An alternative form of ( 6), useful in the sequel, may be expressed as follows:

S ∞ (x) = S 0 (x) exp Ω β(x, y) γ(y) (S ∞ (y) -(S 0 (y) + E 0 (y) + I 0 (y))) dy . (8)
The proof of Theorem 2.1 is the subject of Section 3. The main difficulty in this demonstration is to prove uniqueness of the solution of the final size equation [START_REF] Britton | A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2[END_REF], in the set of those S at most equal to the initial condition S 0 . It uses as central tool an irreducibility property related to the assumption β > 0. It is possible to relax this hypothesis by only assuming that the next-generation operator (defined in (15) below) is strongly positive -this is indeed the case if β > 0. On the contrary, β ≥ 0 is not sufficient to deduce uniqueness, see a counterexample at the end of the present Section.

Heterogeneous populations with diffusion. Diffusion is introduced in the infected compartment for the second result, allowing to consider possible additional natural phenomenon as mutations or spatial diffusion of infected individuals. There is a long history on this problem, in particular to study the effect of diffusion on the basic reproduction number, the spread or extinction of the epidemic and the existence of propagating waves, see Refs. [START_REF] Ducasse | Threshold phenomenon and traveling waves for heterogeneous integral equations and epidemic models[END_REF][START_REF] Ducasse | Qualitative properties of spatial epidemiological models[END_REF][START_REF] Kendall | Discussion of "measles periodicity and community size[END_REF][START_REF] Horst | A model for the spatial spread of an epidemic[END_REF] and the references therein. Here, we are interested in characterizing the final size of the epidemic in a bounded domain.

The corresponding system reads, for t ≥ 0, x ∈ Ω,

∂ t S(t, x) = -S(t, x) Ω β(x, y)I(t, y) dy, S(0, x) = S 0 (x), (9a)
∂ t E(t, x) = S(t, x) Ω β(x, y)I(t, y) dy -α(x)E(t, x), E(0, x) = E 0 (x), (9b)
∂ t I(t, x) = α(x)E(t, x) -γ(x)I(t, x) + ∆I(t, x), I(0, x) = I 0 (x), ∂ n I(t, x) = 0 on (0, ∞) × ∂Ω, (9c)
∂ t R(t, x) = γ(x)I(t, x), R(0, x) = R 0 (x). ( 9d
)
Essentially the same assumptions than Theorem 2.1 allow to obtain the similar result.

Theorem 2.2 Assume that (3) and (4) hold. Assume moreover that Ω is a smooth, bounded open set. Then, for the solution of (9), t → S(t, •) is a decreasing function and there exist S ∞ , R ∞ ∈ L ∞ + (Ω) ∩ L 1 (Ω) such that (5) holds for a.e. x ∈ Ω, uniformly.

Moreover, denoting Φ 0 the solution of

-∆Φ 0 + γΦ 0 = S 0 + E 0 + I 0 , ∂ n Φ 0 = 0 on (0, ∞) × ∂Ω, (10)
S ∞ is characterized by

S ∞ = S 0 exp Ω β(x, y)(Φ ∞ (y) -Φ 0 (y)) dy , (11a)
where Φ ∞ is the unique solution such that Φ ∞ ≤ Φ 0 (with Φ 0 defined in (10)) of

-∆Φ ∞ + γ(x)Φ ∞ = S 0 exp Ω β(x, y)(Φ ∞ (y) -Φ 0 (y)) dy , ∂ n Φ ∞ = 0 on (0, ∞) × ∂Ω. (11b)
Last, if (7) holds, then the convergence is exponential.

Equation ( 11) constitutes, together with [START_REF] Di Lauro | The impact of network properties and mixing on control measures and disease-induced herd immunity in epidemic models: a mean-field model perspective[END_REF], the final size equation for equation [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF]. It appears analogous to [START_REF] Britton | A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2[END_REF] when ignoring the diffusion term. It may also be written as

-∆Φ ∞ + γΦ ∞ = S ∞ .
Again, we can relax the hypothesis β > 0 by only assuming that the modified next-generation operator (defined in (28) below) is strongly positive, which is of course true when β > 0. Another possible set of assumption guaranteeing this strong positivity, and thus the availability of Theorem 2.2, is β ≥ 0 and Ω connected. In that case, even if β = 0 in some parts of Ω, the diffusion ensures that the infection reaches such parts. If β is not positive and Ω is not connected, it is easy to construct counter-examples where a connected part of Ω is invaded whereas another is not.

Finally, we mention that these results may be adapted straightforwardly when the exposed compartment is neglected, that is for the heterogeneous SIR model. Also they may be extended to other operators than the diffusion, as the fractional Laplacian or integral operators, as long as a standard spectral theory is available. Last, diffusion could be included on the Exposed compartment with the same results and methods.

Nonuniqueness and positivity of β In order to illustrate the necessity of assumption β > 0 in (3), we provide a simple example for the discrete case with piecewise constant coefficient. More precisely, assume that for some disjoint sets Ω 1 , Ω 2 for which Ω = Ω 1 ∪ Ω 2 , and some positive constants α k , β k , γ k , k = 1, 2 and β 12 , the coefficients verify:

α(x) = α 1 1 Ω 1 (x) + α 2 1 Ω 2 (x), γ(x) = γ 1 1 Ω 1 (x) + γ 2 1 Ω 2 (x), β(x, y) =    β k , if (x, y) ∈ Ω k × Ω k , k = 1, 2 β 12 , if (x, y) ∈ Ω 1 × Ω 2 0 , if (x, y) ∈ Ω 2 × Ω 1 .
Then, denoting

S k (t) := Ω k S(t, x) dx, E k (t) := Ω k E(t, x) dx, I k (t) := Ω k I(t, x) dx
for k = 1, 2, system (1) implies (after integration in space) that Ṡ1 = -S 1 (β

1 I 1 + β 12 I 2 ), Ṡ2 = -β 2 S 2 I 2 , Ė1 = S 1 (β 1 I 1 + β 12 I 2 ) -α 1 E 1 , Ė2 = β 2 S 2 I 2 -α 2 E 2 , İ1 = α 1 E 1 -γ 1 I 1 , İ2 = α 2 E 2 -γ 2 I 2 ,
complemented with initial data S 0 k , E 0 k , I 0 k , for k = 1, 2 (with the notations S 0 k = Ω k S 0 (x) dx, and so on). Denoting S k∞ the limit of S k (t) when t → +∞, k = 1, 2, the final state is characterized, in this particular setting, by the system of two scalar identities ln S 1∞ -

β 1 γ 1 S 1∞ - β 12 γ 2 S 2∞ = ln S 0 1 - β 1 γ 1 (S 0 1 + E 0 1 + I 0 1 ) - β 12 γ 2 (S 0 2 + E 0 2 + I 0 2 ), ( 12a
) ln S 2∞ - β 2 γ 2 S 2∞ = ln S 0 2 - β 2 γ 2 (S 0 2 + E 0 2 + I 0 2 ). ( 12b
)
If E 0 2 + I 0 2 = 0, then (12b) admits two solutions, namely S 0 2 (corresponding to absence of disease in Ω 2 ), and another solution, denoted S 2 , which is strictly smaller than S 0 2 (corresponding to a spread of the epidemic in Ω 2 ). If now E 0 1 + I 0 1 > 0, for each of these solutions, solving (12a) for S 1∞ ≤ S 0 1 yields a unique solution. We thus obtain two distinct solutions of equation ( 12), smaller or equal to (S 1 0 , S 2 0 ). Notice that, for any initial condition of ( 6) constant on each of the two sets Ω 1 , Ω 2 , the solution itself retains this property. The previous considerations thus allows to exhibit for such a choice of initial condition, two distinct solutions of the final size equation ( 6). As a conclusion, the uniqueness property stated in Theorem 2.1 may not hold if β > 0.

General contact matrices in absence of diffusion

We investigate here system (1). After introducing some adequate notions and results, Theorem 2.1 is proved. Some conserved quantities are first studied in Section 3.1. The notions of basic reproduction number and herd immunity domain are then introduced in this general setting in Section 3.2, based on the next-generation operator. The limit behaviour is established in Section 3.3, together with the properties of the solution of the final size equation, formally achieving the proof of Theorem 2.1. Section 3.4 then exploits the tools introduced during this section to disentangle the link between herd immunity and equilibrium stability.

Conserved quantities and long time limit

A central role in the dynamics is played by the integral quantity

Φ(t, x) = Ω β(x, y) γ(y) [S(t, y) + E(t, y) + I(t, y)] dy, (13)
extending a similar and standard quantity for the SIR model.

Lemma 3.1 Under assumption (3), the solution of (1) verifies that, for a.e. x ∈ Ω, the quantities S(t, x) + E(t, x) + I(t, x) + R(t, x) and ln S(t, x) -Φ(t, x) are constant with respect to time.

Proof. Firstly, adding the equations in (1) provides directly the conservation of S + E + I + R with respect to time. Secondly, we compute

∂ ∂t [ln S(t, x) -Φ(t, x)] = Ṡ(t, x) S(t, x) - Ω β(x, y) γ(y) ( Ṡ(t, y) + Ė(t, y) + İ(t, y)) dy = - Ω β(x, y)I(t, y) dy + Ω β(x, y)I(t, y) = 0.
We thus deduce the conservation relation

∂ ∂t [ln S(t, x) -Φ(t, x)] = 0, (14)
and Lemma 3.1 is proved.

We now demonstrate the basic convergence properties stated in Theorem 2.1. The exponential convergence, which necessitates more advanced tools, will be treated later, see Section 3.3. Lemma 3.2 Assume that (3) and (4) hold. Then the solution of (1) verifies : there exists

S ∞ ∈ L ∞ + (Ω), 0 < S ∞ ≤ S 0 , and R ∞ = S 0 + E 0 + I 0 + R 0 -S ∞
, such that the convergence properties stated in (5) hold uniformly, for a.e. x ∈ Ω, as well as [START_REF] Britton | A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2[END_REF].

Proof. We deduce easily from (1) that for a.e. x ∈ Ω, t → S(t, x) is decreasing. Being bounded from below by 0, it admits a limit as t goes to +∞. By the same token, S(•, x) + E(•, x) and S(•, x) + E(•, x) + I(•, x) admit similarly a limit. As S(•, x) admits a limit, we deduce that

E(•, x) = (S(•, x) + E(•, x)) -S(•, x) and I(•, x) = (S(•, x) + E(•, x) + I(•, x)) -(S(•, x) + E(•, x
)) converge as well when t → +∞. The convergence of R then comes as a consequence of the first conservation property in Lemma 3.1.

Equation ( 6) and its equivalent form (8) follow from Lemma 3.1, allowing to deduce that S ∞ > 0. From the fact that lim t→∞ S(t, x) = S ∞ (x) for a.e. x ∈ Ω, one deduces, by the dominated convergence theorem, that this convergence is indeed L 1 (Ω). Then it follows from equation (1a) by application of Arzela-Ascoli's theorem that the convergence of S(t, x) towards S ∞ (x) is uniform a.e. over Ω.

Let us now study the limits of E and I. Adding (1a) and (1c) yields

∂ t (S(t, x) + E(t, x)) + α(x)(S(t, x) + E(t, x)) = α(x)S(t, x),
with S(t, x) → S ∞ (x) a.e. in Ω uniformly. Due to the fact that α(x) ≥ α > 0, see (3), one obtains by integration that S(t, x) + E(t, x) → S ∞ (x) when t → +∞, a.e. and uniformly on Ω. Therefore, E(t, x) converges to 0. Similarly,

∂ t (S(t, x) + E(t, x) + I(t, x)) + γ(x)(S(t, x) + E(t, x) + I(t, x)) = γ(x)(S(t, x) + E(t, x)).
Appealing to the hypothesis (made in (3)) that γ(x) ≥ γ > 0, permits to show that I(t, x) → 0 when t → +∞, a.e. and uniformly on Ω. This demonstrates [START_REF] Brauer | Age-of-infection and the final size relation[END_REF] and achieves the proof of Lemma 3.2.

Herd immunity and basic reproduction number

Recall that (3) and ( 4) have been assumed. Following the idea in Ref. [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF], we define now a key notion.

Definition 1 For any S(•) ∈ L 1 (Ω) ∩ L ∞ + (Ω)
, we call next-generation operator K S(•) associated to (1), the operator defined over L 2 (Ω) by:

K S(•) [φ](x) = S(x) Ω β(x, y) γ(y) φ(y) dy. (15)
Under assumptions (3) and ( 4), we know that, for any t ≥ 0, the solution S(t, •) of (1) belongs to L ∞ + (Ω) ∩ L 1 (Ω). The operator K S(t,•) is thus well-defined and it is a positive and compact operator over L 2 (Ω). This permits to define its norm and its spectral radius in the usual way recalled now.

Definition 2 The spectral radius r(K S(t,•) ) of K S(t,•) is defined by r(K S(t,•) ) = lim n→+∞ K S(t,•) n 1/n .
As β is irreducible thanks to (3) and using Krein-Rutman theory [START_REF] Grigor'evich Kreȋn | Linear operators which leave a cone in Banach space invariant[END_REF][START_REF] Zeidler | Nonlinear Functional Analysis and its Applications, I: Fixed Point Theorems[END_REF], we can identify the spectral radius r(K S(t,•) ) of K S(t,•) with its principal eigenvalue. The latter is simple, and defined as the unique eigenvalue associated to a positive eigenfunction. In particular, under the assumptions of Theorem 2.1, r(K S(t,•) ) > 0 for any t ≥ 0. We denote ϕ S(t,•) this eigenfunction, so that by definition,

K S(t,•) [ϕ S(t,•) ](x) = S(t, x) Ω β(x, y) γ(y) ϕ S(t,•) (y) dy = r(K S(t,•) )ϕ S(t,•) (x).
We also know that r(K S(t,•) ) = r(K ⋆ S(t,•) ), where K ⋆ S(t,•) is the adjoint operator, defined by

K ⋆ S(t,•) [ψ], φ = ψ, K S(t,•) [φ] = Ω ψ(x)S(t, x) Ω β(x, y) γ(y) φ(y) dy dx that is K ⋆ S(t,•) [ψ](y) := Ω S(t, x) β(x, y) γ(y) ψ(x) dx. ( 16
)
We let ψ S(t,•) be the associated normalized adjoint eigenfunction, defined by

K ⋆ S(t,•) [ψ S(t,•) ] = r(K S(t,•) )ψ S(t,•) ,
and

Ω ψ S(t,•) (x)S 0 (x)dx = 1.
We are now in position to define for (1) the key notions of basic reproduction number and herd immunity, based on the definition of next-generation operator in [START_REF] Jabin | On selection dynamics for competitive interactions[END_REF]. Definition 3 Under assumptions (3) and (4), we define:

• basic reproduction number R 0 the spectral radius of the operator K S 0 (•) , i.e., R 0 = r(K S 0 (•) );

• herd immunity domain the set of those elements S of L ∞ + (Ω) ∩ L 1 (Ω) such that r(K S(•) ) ≤ 1.
In contrast to the homogeneous SEIR system where herd immunity is reached for a uniquely defined proportion of the variable S, here the boundary of the herd immunity domain is a set of functions corresponding to various susceptible population distributions. It is hard to determine in which point of the boundary the trajectory will enter this domain, but this necessarily takes place after a certain time, as stated now.

Lemma 3.3 Assume that (3) and (4) hold, and that r(

K S 0 (•) ) > 1. Then, t → r(K S(t,•) ) is decreas- ing and continuous with lim t→+∞ r(K S(t,•) ) < 1, (17)
and there exists a unique T 0 > 0 such that r(K S(T 0 ,•) ) = 1. As a consequence S(t, •) belongs to the herd immunity domain iff t ≥ T 0 .

Proof. We first establish that t → r(K S(t,•) ) is non-increasing and continuous. This is a consequence of the fact that for a.e. x ∈ Ω, t → S(t, x) is decreasing and continuous. Indeed, by definition [START_REF] Jabin | On selection dynamics for competitive interactions[END_REF], for any φ ∈ L 2 (Ω), and any n ∈ N * , t → K S(t,•) n [φ] is decreasing and continuous. Hence, t → K S(t,•) n 1/n is decreasing and continuous for any n ∈ N * . The result follows by passing to the limit n → +∞.

Secondly, since r(K S 0 (•) ) > 1, we are left to prove that (17) holds. From Lemma 3.2, we may use the fact that S ∞ (x) is the limit of S(t, x) when t → +∞, for a.e. x ∈ Ω. Denote simply K ∞ := K S∞(•) the associated next-generation operator, and by ψ ∞ the principal eigenfunction associated to the adjoint operator

K ⋆ ∞ , that is, K ⋆ ∞ [ψ ∞ ] = r(K ⋆ ∞ )ψ ∞ .
It follows from this identity that r(K ⋆ ∞ ) > 0. Moreover one has ψ ∞ > 0. In addition, if [START_REF] Britton | The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level[END_REF] holds, then

ψ ∞ (x) ≥ β ess inf Ω S ∞ r(K ⋆ ∞ )ess sup Ω γ Ω ψ ∞ (y)dy > 0,
and in this second case, ess inf Ω ψ ∞ > 0.

Adding the second and third equations in [START_REF] Aguas | Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics[END_REF], multiplying the result by ψ ∞ and integrating, one obtains from (16) taken at the limit,

d dt Ω ψ ∞ (x) E(t, x) + I(t, x) dx = Ω×Ω ψ ∞ (x)S(t, x)β(x, y)I(t, y) dydx - Ω γ(x)ψ ∞ (x)I(t, x) dx ≥ Ω×Ω ψ ∞ (x)S ∞ (x)β(x, y)I(t, y)dydx - Ω γ(x)ψ ∞ (x)I(t, x) dx = K ⋆ ∞ [ψ ∞ ], γ(•)I(t, •) -ψ ∞ , γ(•)I(t, •) = r(K ⋆ ∞ ) -1 Ω ψ ∞ (y)γ(y)I(t, y) dy.
Hence, arguing by contradiction, if r(K ⋆ ∞ ) ≥ 1, then, on the one hand t → x ψ ∞ (x) E(t, x) + I(t, x) dx is non-decreasing. On the other hand, we have seen in Lemma 3.2 that lim t→+∞ E(t, x)+ I(t, x) = 0. It is a contradiction. Thus, r(K ⋆ ∞ ) < 1 and by continuity there exists T 0 such that r(K S(T 0 ,•) ) = 1. This achieves the proof of Lemma 3.3.

As is the case for the homogenous SEIR model, there exists a tight relation between herd immunity and stability of the equilibrium points of (1). This link is elucidated in Section 3.4 below.

Long time behaviour and final size equation -Proof of Theorem 2.1

We are now in position to achieve the proof of Theorem 2.1.

Step 1. Preliminaries. In Lemma 3.2 we already proved the decay of t → S(t, x) and the existence of limits for all variables S, E, I and R. Basically we have still to achieve two tasks: identifying the limit S ∞ , and showing the exponential convergence in the case where [START_REF] Britton | The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level[END_REF] holds. Integrating [START_REF] Gomes | Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold[END_REF] between t and +∞, we obtain that the latter satisfies necessarily

       ln S ∞ (x) -Φ ∞ (x) = ln S(t, x) - Ω β(x, y) γ(y) [S(t, y) + E(t, y) + I(t, y)] dy, Φ ∞ (x) = Ω β(x, y) γ(y) S ∞ (y) dy, (18)
for any t ≥ 0. Taking t = 0, yields an alternative form of ( 6)/(8) that will be useful in the sequel.

Step 2. Exponential convergence properties. We assume in this paragraph that (7) holds. From [START_REF] Desvillettes | On selection dynamics for continuous populations[END_REF], it is deduced directly that ess inf Ω S ∞ > 0.

From the uniform convergence of S(t, •) towards S ∞ (•) previously ascertained in Lemma 3.2, one deduces that, for all ε > 0, there is a T ε such that S(t, •) ≤ (1 + ε)S ∞ for t ≥ T ε . We also introduce θ ∈ (0, 1), to be chosen later, and compute

d dt [E(t, x) + θI(t, x)] ≤ (1 + ε)S ∞ (x) Ω β(x, y)I(t, y) dy -(1 -θ)αE(t, x) -θγ(x)I(t, x), (19)
where α > 0 has been defined in [START_REF] Arino | A final size relation for epidemic models[END_REF]. Multiplying again by ψ ∞ , the principal eigenfunction associated to the adjoint operator K ⋆ ∞ = K ⋆ S∞(•) introduced in the proof of Lemma 3.3, and using ( 16), we find

for t ≥ T ε d dt Ω ψ ∞ (x)[E(t, x) + θI(t, x)]dx ≤ (1 + ε) Ω×Ω ψ ∞ (x)S ∞ (x)β(x, y)I(t, y) dydx - Ω ψ ∞ (x)[(1 -θ)αE(t, x) + θγ(x)I(t, x)]dx ≤ (1 + ε)r(K ∞ ) Ω ψ ∞ (y)γ(y)I(t, y) dy - Ω ψ ∞ (x)[(1 -θ)αE(t, x) + θγ(x)I(t, x)] dx.
Since we know from Lemma 3.3 that r(K ∞ ) < 1, one may choose θ sufficiently close to 1 and ε small enough such that 1 > θ > (1 + ε)r(K ∞ ), in such a way that

λ := min (1 -θ)α, 1 - (1 + ε)r(K ∞ ) θ γ > 0.
We then obtain

d dt Ω ψ ∞ (x)[E(t, x) + θI(t, x)]dx ≤ -λ Ω ψ ∞ (x)[E(t, x) + θI(t, x)]dx,
and, applying Gronwall lemma, we find exponential decay to 0 with rate λ for both positive integrals Ω ψ ∞ (x)E(t, x)dx and Ω ψ ∞ (x)I(t, x)dx.

Now, as ess inf Ω ψ ∞ > 0 (see the proof of Lemma 3.3), one deduces that both Ω E(t, x)dx and Ω I(t, x)dx converge exponentially to 0. It follows from (1b) that E(t, x) converges exponentially to 0 as t → +∞ for a.e. x ∈ Ω. Similarly, one gets from (1c) the exponential convergence of I(t, x).

Last, using the fact that S(t, x) decreases along time, equation (1a) tells us that

0 ≤ S(t, x) -S ∞ (x) = ∞ t S(s, x) Ω β(x, y)I(s, y) dyds ≤ S 0 (x) β L ∞ inf ψ ∞ ∞ t Ω ψ ∞ (y)I(s, y) dyds = O(e -λt ).
Step 3. Contraction property for t large enough and local uniqueness of the solution of (6). From Lemma 3.3, there exists T > 0 such that r(K S(T,•) ) < 1. From the first line of equation ( 18), we deduce the expression for S ∞

S ∞ (x) = S(T, x) exp(Φ ∞ (x) -Φ(T, x)), (20)
where we use the notation [START_REF] Ducasse | Qualitative properties of spatial epidemiological models[END_REF]. Injecting this expression into the second line of [START_REF] Kendall | Discussion of "measles periodicity and community size[END_REF], yields the following fixed point problem for the function Φ ∞ :

Φ ∞ (x) = Ω β(x, y) γ(y) S(T, y) exp(Φ ∞ (y) -Φ(T, y)) dy. (21)
Notice that since t → S(t, x) is decreasing from (1a), we deduce that S ∞ ≤ S(T, •) and Φ ∞ ≤ Φ(T, •). Introduce

K := {u ∈ L 1 (Ω), such that 0 ≤ u ≤ Φ(T, •) a.e.}.
Clearly, K is stable by F (using the definition of Φ in ( 13)), and we may prove that F is a contraction on the set K. Indeed, we compute, for any u, v in K,

|F(u) -F(v)|(x) ≤ Ω β(x, y) γ(y) S(T, y) |exp(u(y) -Φ(T, y)) -exp(v(y) -Φ(T, y))| dy ≤ Ω β(x, y) γ(y) S(T, y)|u(y) -v(y)| dy. (22)
To deduce the last inequality, we used the fact that u(y) -Φ(T, y) ≤ 0, v(y) -Φ(T, y) ≤ 0 for almost every y ∈ Ω (as a consequence of the fact that u, v ∈ K and by definition of this set) and applied the mean value theorem. At this stage, we introduce on the set K the norm

u T := Ω ψ S(T,•) (x)S(T, x)|u(x)| dx,
where, as defined in ( 16), ψ S(T,•) is the principal eigenfunction of the adjoint K ⋆ S(T,•) . (Notice that, when ess inf Ω ψ S(T,•) > 0 by the same arguments as in Lemma 3.3, the norm • T is equivalent to L ∞ norm, and (K, • T ) is a Banach space; but this property will not be necessary in the sequel.) Multiplying by ψ S(T,•) and S(T, •), we get with the help of [START_REF] Magal | Final size of an epidemic for a two-group SIR model[END_REF], that

F(u) -F(v) T = Ω ψ S(T,•) (x)S(T, x)|F(u)(x) -F(v)(x)| dx ≤ Ω×Ω ψ S(T,•) (x) β(x, y) γ(y) S(T, x)S(T, y)|u(y) -v(y)| dydx = r(K S(T,•) ) Ω ψ S(T,•) (y)S(T, y)|u -v|(y)dy = r(K S(t,•) ) u -v T ,
by definition of the norm • T . As r(K S(T,•) ) < 1, we deduce that F is a contraction on K. The existence of the limit S ∞ has been established previously (see Lemma 3.2), together with the fact that it belongs to the set K. The contraction inequality now shows that Φ ∞ is the unique solution to [START_REF] Ma | Generality of the final size formula for an epidemic of a newly invading infectious disease[END_REF] in the set K, from which S ∞ is deduced, via [START_REF] Grigor'evich Kreȋn | Linear operators which leave a cone in Banach space invariant[END_REF].

Step 4. Uniqueness below S 0 . It remains to extend the uniqueness result to all function smaller or equal to S 0 . We make use of the following result.

Lemma 3.4

The solution S ∞ is larger than all sub-solutions of (6).

Indeed, we deduce from Lemma 3.4 that any other solution S of (6) satisfies S ≤ S ∞ . Therefore it also satisfies S ≤ S(T ) for T large enough. Thus by the contraction property of Step 3 it is equal to S ∞ . This achieves the demonstration of Theorem 2.1, with the exception of the proof of Lemma 3.4, which we now complete.

Proof. [Proof of Lemma 3.4] We first show that S ∞ is the limit of the sequence S k (x) departing from the initial condition S 0 and constructed, for k ≥ 0, by the following iteration:

ln S k+1 (x) = Ω β(x, y) γ(y) S k (y) dy + A(x), A(x) = ln S 0 (x) - Ω β(x, y) γ(y) [S 0 + E 0 + I 0 ](y) dy.
Obviously, we have

ln S 1 (x) = Ω β(x, y) γ(y) S 0 (y) dy + A(x) = ln S 0 (x) - Ω β(x, y) γ(y) [E 0 + I 0 ](y) dy < ln S 0 (x).
And iterating, if we know that S k < S k-1 , we find

ln S k+1 (x) = Ω β(x, y) γ(y) S k (y) dy + A(x) < Ω β(x, y) γ(y) S k-1 (y) dy + A(x) = ln S k (x),
which gives S k+1 < S k . As a consequence S k is a decreasing sequence which converges to a solution S of (6). Secondly, let S ≤ S 0 be a sub-solution of (6). We have

ln S(x) ≤ Ω β(x, y) γ(y) S(y) dy + A(x) ≤ Ω β(x, y) γ(y) S 0 (y) dy + A(x) = ln S 1 (x),
therefore S ≤ S 1 , and iterating the argument S ≤ S k for all k ∈ N and thus S ≤ S. In particular we have S ∞ ≤ S. Thirdly, we prove that S ≤ S ∞ . We know that S < S 0 and we are going to prove that S < S(t) for all t ≥ 0 from which we conclude, by definition of S ∞ , that S ≤ S ∞ and thus that S ∞ = S. To prove this, we consider, by contradiction, the first time t 0 when S(t 0 ) touches S, i.e., S ≤ S(t 0 ) and for some x 0 ∈ Ω, S(x 0 ) = S(t 0 , x 0 ). Using the fact, proved in Lemma 3.1, that the quantity ln S(t, x) -Φ(t, x) is constant with respect to time, we conclude that 0 = ln S(x 0 ) -ln S(t 0 , x 0 ) = Ω β(x, y) γ(y) [S(y) -S(t 0 , y) -E(t 0 , y) -I(t 0 , y)] dy < 0, a contradiction. This achieves the demonstration of Lemma 3.4.

Herd immunity and stability of the equilibrium points

Before closing Section 3 and the study of system (1), we analyze in the following result the link between herd immunity and the stability of the equilibria of this system.

Lemma 3.5 Assume (3) and (4) hold. The equilibrium points of system (1) are exactly the configurations (S * , 0, 0, R * ) that fulfil (4). Moreover each of them is stable if

r(K S * (•) ) < 1, unstable if r(K S * (•) ) > 1,
Proof. The equilibria for (1) are any configuration (S * , E * , I * , R * ) such that, for a.e. x ∈ Ω,

-S * (x) Ω β(x, y)I * (y) dy = 0, S * (x) Ω β(x, y)I * (y) dy -α(x)E * (x) = 0, α(x)E * (x) -γ(x)I * (x) = 0, γ(x)I * (x) = 0.
We deduce from the last two equations that (E * , I * ) = (0, 0). This condition is also sufficient to have an equilibrium, provided that it fulfils (4).

The stability of such an equilibrium is related to the stability of the linearized system

∂s ∂t = -S * (x) Ω β(x,
∂i ∂t = α(x)e(t, x) -γ(x)i(t, x), ∂r ∂t = γ(x)i(t, x). ( 23b
)
Notice that ( 23) admits a block-triangular structure: the evolution of (e, i) is independent of the other two components s and r; while the evolution of s and r only depends upon i.

Assume first that r(K S * (•) ) < 1. Let ψ S * (•) be the principal adjoint eigenvector of the nextgeneration operator K S * (•) associated to S * . Mimicking the computation achieved in Step 2 of Section 3.3, one finds that, for any θ ∈ (0, 1) and a.e. x ∈ Ω,

d dt Ω ψ S * (•) (x)[e(t, x) + θi(t, x)] dx = Ω×Ω ψ S * (•) (x)S * (x)β(x, y)i(t, y) dy - Ω ψ S * (•) (x)[(1 -θ)α(x)e(t, x) + θγ(x)i(t, x)] dx ≤ r(K S * (•) ) Ω ψ S * (•) (y)γ(y)i(t, y) dy - Ω ψ S * (•) (x)[(1 -θ)αe(t, x) + θγ(x)i(t, x)] dx.
Taking θ ∈ (r(K S * (•) ), 1) then yields (through application of Gronwall's lemma) exponential decrease of Ω ψ S * (•) (x)[e(t, x) + θi(t, x)] dx when t → +∞. One sees easily that the subsystem in (e, i) of ( 23) is a monotone system. The eigenfunction ψ S * (•) being positive, the functional (e(•), i(•))

→ Ω ψ S * (•) (x)[e(•) + θi(t•)]
dx is thus a Lyapunov functional for this subsystem, and the origin of the latter is asymptotically stable. Due to the block-triangular structure mentioned earlier, the complete system ( 23) is (simply) stable.

Assume now that r(K S * (•) ) > 1. Denoting ϕ S * (•) the principal eigenvector of K S * (•) , define

e * (x) := r(K S * (•) ) + 1 2 ϕ S * (•) (x) α(x) , i * (x) := ϕ S * (•) (x) γ(x) .
Let us evaluate the value of the right-hand sides corresponding to ∂e ∂t and ∂i ∂t at the point (e * , i * ). Then, for any x ∈ Ω,

S * (x) Ω β(x, y)i * (y) dy -α(x)e * (x) = S * (x) Ω β(x, y) ϕ S * (•) (y) γ(y) dy - r(K S * (•) ) + 1 2 ϕ S * (•) (x) = K S * (•) [ϕ S * (•) ](x) - r(K S * (•) ) + 1 2 ϕ S * (•) (x) = r(K S * (•) ) -1 2 ϕ S * (•) (x) > r(K S * (•) ) -1 r(K S * (•) ) + 1 αe * (x), while similarly α(x)e * (x) -γ(x)i * (x) = r(K S * (•) ) + 1 2 ϕ S * (•) (x) -ϕ S * (•) (x) = r(K S * (•) ) -1 2 ϕ S * (•) (x) > r(K S * (•) ) -1 2 γi * (x).
The subsystem describing the evolution of the components (e, i) being a monotone system, this condition is sufficient to ensure that any trajectory of this subsystem departing from a point (e 0 , i 0 ) ≥ λ(e * , i * ), λ > 0, is increasing. This shows the instability of the origin of ( 23), and achieves the proof of Lemma 3.5.

General contact matrices with diffusion

We now come to system [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF] where diffusion is included, and prove Theorem 2.2. We adapt for this the methodology developed in Section 3.

Conserved quantities and long time limit

Let us introduce the solution of the nonlinear elliptic equation

-∆Φ(t, x) + γΦ(t, x) = S(t, x) + E(t, x) + I(t, x), on Ω, ∂ n Φ(t, x) = 0 on (0, ∞) × ∂Ω, ( 24
)
where n is the outward normal unit at the boundary of Ω.

Lemma 4.1 Under assumption (3), the solution of (9) verifies that the quantity Ω (S(t, x) + E(t, x) + I(t, x) + R(t, x)) dx is constant with respect to time, as well as the quantities ln S(t, x) -Ω β(x, y)Φ(t, y) dy for a.e. x ∈ Ω.

Proof. The first conservation relation is obtained straightforwardly by adding and integrating on Ω the equations in [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF]. For the second, we compute, as usual,

∂ t ln S(t, x) = - Ω β(x, y)I(t, y) dy. (25)
Secondly

-∆∂ t Φ(t, x) + γ(x)∂ t Φ = ∂ ∂t (S + E + I)(t, x) = -γ(x)I(t, x) + ∆I(t, x).
Therefore, by the uniqueness of solution of [START_REF] Artem | On the spread of epidemics in a closed heterogeneous population[END_REF], which is ensured by the fact that γ > 0 (see ( 3)), we obtain

∂ t Φ(t, x) = -I(t, x).
Inserting this expression of I in equation ( 25) then yields

∂ ∂t -ln S(t, x) + Ω β(x, y)Φ(t, y)dy = 0. ( 26
)
This achieves the proof of Lemma 4.1.

As before, we may deduce existence of limits from the previous result, as stated now.

Lemma 4.2 Assume that (3) and (4) hold. Then the solution of (9) verifies :

There exist S ∞ , R ∞ ∈ L ∞ + (Ω), S ∞ ≤ S 0 , such that, lim t→+∞ S(t, x) = S ∞ (x), lim t→+∞ R(t, x) = R ∞ (x), and lim t→+∞ E(t, x) = lim t→+∞ I(t, x) = 0,
uniformly, for a.e. x ∈ Ω. Moreover, for a.e. x ∈ Ω, Φ(t, x) converges to Φ ∞ (x) as t goes to +∞ where

-∆Φ ∞ (x) + γΦ ∞ (x) = S ∞ (x), ∂ n Φ ∞ (x) = 0 on (0, ∞) × ∂Ω. (27)
Proof. As above, for a.e. x ∈ Ω, since t → S(t, x) is decreasing, it has a limit S ∞ (x), and because t → S(t, x) + E(t, x) is decreasing, t → E(t, x) has a limit which vanishes as can be seen adding from (9a)-(9b). By the dominated convergence theorem, one deduces also that these convergences hold in L 1 (Ω). Then it follows from equations (9a) and (9b) by application of Arzela-Ascoli's theorem that the convergences of S(t, •) towards S ∞ and of E(t, •) towards 0 is uniform a.e. over Ω. Inserting this information in equation (9c), we conclude that, I(t, •) converges towards 0 uniformly a.e. over Ω as t → +∞. Finally, we pass to the limit in equation ( 24) to get the limit Φ ∞ of Φ.

The next-generation operator

Following the approach of Section 3, we define the next-generation operator. Definition 4 For any t ≥ 0 and S(t, •) ∈ L 1 (Ω) ∩ L ∞ + (Ω) solution of (9), we call next-generation operator K S(t,•) associated to (9), the operator defined over L 2 (Ω) by

K ∆ S(t,•) [φ](x) := S(t, x) Ω β(x, y) γ(y) φ(y) dy + ∆ φ(x) γ(x) , (28)
with Neumann boundary conditions. Its adjoint is defined as

K ∆ * S(t,•) [ψ](y) := Ω S(t, x) β(x, y) γ(y) ψ(x) dx + 1 γ(y) ∆ψ(y), (29)
also with Neumann boundary conditions.

The existence of a principal eigenvalue for the next-generation operator is established in the next result, whose proof is postponed to A.1.

Lemma 4.3 Let S ∈ L 1 (Ω) ∩ L ∞ + (Ω). The operator K ∆ S(•) admits a principal eigenvalue, that is, there exists r(K ∆ S(•) ) > 0 and functions ϕ S(•) ∈ L 2 + (Ω) and ψ S(•) ∈ L 2 + (Ω) such that K ∆ S(•) [ϕ S(•) ] = r(K ∆ S(•) )ϕ S(•) , in Ω, ∂ n ϕ S(•) = 0, on ∂Ω, and
K ∆ * S(•) [ψ S(•) ] = r(K ∆ S(•) )ψ S(•) , in Ω, ∂ n ψ S(•) = 0, on ∂Ω.
The basic reproduction number and the herd immunity domain may then be defined as in Definition 3. Notice that an analogue of Lemma 3.5, which links herd immunity and stability, may also be written when diffusion is present.

From Lemma 4.2, there exists S ∞ ∈ L 1 (Ω) ∩ L ∞ + (Ω), which is the limit of S(t, •). We use the shorthand notation

K ∆ ∞ := K ∆ S∞(•) and K ∆ * ∞ := K ∆ * S∞(•) ; its principal eigenvalue is denoted r ∆ ∞ := r(K ∆ S∞(•)
) and the associated eigenfunctions are respectively ϕ ∞ := ϕ S∞(•) and ψ ∞ := ψ S∞(•) . The properties described in Lemma 3.3 still hold in this framework: Lemma 4.4 Assume that (3), (4) hold, and that r(K ∆ S 0 (•) ) > 1. Then, t → r(K ∆ S(t,•) ) is decreasing, continuous; lim t→+∞ r(K ∆ S(t,•) ) < 1 and there exists a unique T 0 > 0 such that r(K ∆ S(T 0 ,•) ) = 1. Proof. The first point follows exactly the proof of Lemma 3.3. For the existence of T 0 , as above, we multiply (9b) and (9c) by ψ ∞ , and integrating, one finds

d dt Ω ψ ∞ (x) E(t, x) + I(t, x) dx = Ω×Ω ψ ∞ (x)S(t, x)β(x, y)I(t, y) dydx - Ω γ(x)ψ ∞ (x)I(t, x) dx + Ω ψ ∞ (x)∆I(t, x) dx = Ω×Ω K ∆ * S(t,•) [ψ ∞ ](y) γ(y)I(t, y) dy - Ω ψ ∞ (x)γ(x)I(t, x) dx = r ∆ ∞ -1 Ω γ(x)ψ ∞ (x)I(t, x) dx.
As in the proof of Lemma 3.3, this proves that there is T 0 > 0 such that herd immunity is reached, that is r(K ∆ S(T 0 ,•) ) = 1.

Long time behavior and final size equation -Proof of Theorem 2.2

We have now the material to prove Theorem 2.2. We follow and adapt to the case at hand the proof in Section 3.

Step 1. Existence of limits. The long time convergence has been studied in Lemma 3.3. Then, integrating (26) between t and +∞, we get, for a.e. x ∈ Ω, t ≥ 0.

ln S(t, x) -

Ω β(x, y)Φ(t, y) dy = ln S ∞ (x) - Ω β(x, y)Φ ∞ (y) dy. (30)
It may be rewritten straightforwardly

S ∞ (x) = S(t, x) exp Ω β(x, y)(Φ ∞ (y) -Φ(t, y)) dy , for a.e. x ∈ Ω, t ≥ 0. (31)
Injecting this identity into the equation for Φ ∞ , and taking t = 0, we deduce that the long time limit is characterized by (11b).

Step 2. Exponential rate of decay. When (7) holds, exponential convergence also follows, as in Section 3. We just indicate the main steps. Firstly, from (30), and because Φ(t, x) converges to Φ ∞ (x) almost everywhere, we conclude that for all ε and for t ≥ T ε , we have S(t, x) ≤ (1 + ε)S ∞ (x). Secondly, adapting formula [START_REF] Kermack | Contributions to the mathematical theory of epidemics -I[END_REF] to the present setting and using the definition of the adjoint operator K ∆ * S(t,•) given in (29), we compute, for θ < 1 but close to 1 d dt Ω ψ ∞ (x) E(t, x) + θI(t, x) dx

≤ (1 + ε) Ω×Ω ψ ∞ (x)S ∞ (x)β(x, y)I(t, y)dydx -(1 -θ) Ω ψ ∞ (x)α(x)E(t, x)dx -θ Ω ψ ∞ (x)γ(x)I(t, x)dx + θ Ω ψ ∞ (x)∆I(t, x)dx ≤ θ Ω×Ω K ∆ * S(t,•) [ψ ∞ ](y) γ(y)I(t, y)dy -(1 -θ) Ω ψ ∞ (x)α(x)E(t, x)dx -θ Ω ψ ∞ (x)γ(x)I(t, x)dx + (1 + ε -θ) Ω×Ω ψ ∞ (x)S ∞ (x)β(x, y)I(t, y)dydx ≤ -θ(1 -r(K ∆ S(t,•) )) Ω ψ ∞ (x)γ(x)I(t, x)dx -(1 -θ) Ω ψ ∞ (x)α(x)E(t, x)dx + (1 + ε -θ) Ω×Ω ψ ∞ (x)S ∞ (x)β(x, y)I(t, y)dydx.
Taking t > T 0 in such a way that r(K ∆ S(t,•) ) < 1, choosing ε > 0 small enough and 0 < θ < 1 close enough to 1 to have 1 + ε -θ < 0, and using the fact (assumed in (3)) that inf x∈Ω α(x) and inf x∈Ω γ(x) are positive together with the positivity of the principal eigenvector ψ ∞ , allows to use again Gronwall lemma, and to conclude to the exponential rate of convergence of Ω ψ ∞ (x)E(t, x) dx and Ω ψ ∞ (x)I(t, x) dx. In the same spirit as in the proof of Lemma 3.3, we have r(K ∆ * ∞ ) > 0 and from the relation

K ∆ * ∞ [ψ ∞ ] = r(K ∆ * ∞ )ψ ∞ , we deduce r(K ∆ * ∞ )ψ ∞ - 1 γ ∆ψ ∞ ≥ β ess inf Ω S ∞ ess sup Ω γ Ω ψ ∞ (y) dy > 0.
By comparison principle, we get ess inf Ω ψ ∞ > 0. Hence, we have the exponential convergence of Ω E(t, x) dx and Ω I(t, x) dx. From (9b), we deduce the exponential convergence of E(t, x) towards 0 as t → +∞ for a.e. x ∈ Ω. And from (9c), we have the exponential convergence of I(t, x). Then, we may define, for a.e. x ∈ Ω,

R ∞ (x) = R 0 (x) + +∞ 0 I(s, x) ds.
From (9d), we have

R ∞ (x) -R(t, x) = +∞ t I(s, x) ds.
Then, the exponential convergence of I implies the exponential convergence of R(t, x) to R ∞ (x) for a.e. x ∈ Ω as t → +∞. Exponential convergence of S is then obtained as in Section 3. It concludes the proof of the convergences in (5) at exponential rate.

Step 3. Uniqueness for t large enough. From Lemma 4.4 we may choose T > 0 large enough such that r(K ∆ S(T,•) ) < 1. From (31) and ( 27), we deduce

-∆Φ ∞ (x) + γ(x)Φ ∞ (x) = S(T, x) exp Ω β(x, y)(Φ ∞ (y) -Φ(T, y)) dy ,
complemented with Neumann boundary conditions. Recall that Φ(t, •) is defined for any t by [START_REF] Artem | On the spread of epidemics in a closed heterogeneous population[END_REF]. We notice that since t → S(t, x) is decreasing, we have S ∞ ≤ S(T ) and by the maximum principle applied to [START_REF] Artem | On the spread of epidemics in a closed heterogeneous population[END_REF], we deduce also that Φ ∞ ≤ Φ(T ).

Let us consider the problem

   -∆ u(x) γ(x) + u(x) = S(T, x) exp Ω β(x, y) u(y) γ(y) -Φ(T, y) dy , ∂ n u(x) γ(x) = 0 on (0, ∞) × ∂Ω. (32)
We show the uniqueness of a solution of this problem in the set {0 ≤ u ≤ γΦ(T, •) a.e.}. Let us assume that there are two solutions u and v. We have as in Section 3.3, using the fact that the exponential is 1-Lipschitz in the set of nonpositive real numbers,

-∆ (u -v)(x) γ(x) + (u -v)(x) ≤ S(T, x) Ω β(x, y) γ(y) |u(y) -v(y)| dy.
From Kato's inequality [START_REF] Kato | Schrödinger operators with singular potentials[END_REF], we deduce

-∆ |u -v|(x) γ(x) + |u -v|(x) ≤ S(T, x) Ω β(x, y) γ(y) |u(y) -v(y)| dy. (33)
We recall that, from Lemma 4.3, there exists ψ S(T,•) such that

K ∆ * S(T,•) [ψ S(T,•) ](y) = Ω S(T, x) β(x, y) γ(y) ψ S(T,•) (x) dx + 1 γ(y) ∆ψ S(T,•) (y) = r(K ∆ S(T,•) )ψ S(T,•) (y).
We use the norm

u T := Ω ψ S(T,•) (x)|u(x)| dx.
Multiplying (33) by ψ S(T,•) and integrating, we get

Ω ψ S(T,•) (x)|u -v|(x) dx ≤ Ω×Ω S(T, x)ψ S(T,•) (x) β(x, y) γ(y) |u(y) -v(y)| dydx + Ω ∆ψ S(T,•) (x) |u -v|(x) γ(x) dx ≤ r(K ∆ S(T,•) ) u -v T .
Since r(K ∆ S(T,•) ) < 1, we conclude to the uniqueness of the solution of (32) in the set {0 ≤ u ≤ γΦ(T, •)}, and thus to the uniqueness of Φ ∞ , obtained by dividing by γ, in the set {0 ≤ Φ ≤ Φ(T, •)}.

Step 4. Uniqueness below Φ 0 . It remains to extend the uniqueness result to all functions smaller than Φ 0 . Analogously to the argument conducted in Section 3, we make use of the following result. Lemma 4.5 Among the functions Φ ≤ Φ 0 , there is a solution Φ of (11b) which is larger than all subsolutions of (11b), and Φ ∞ = Φ. Lemma 4.5 implies that any other solution Φ of (11b) satisfies Φ ≤ Φ ∞ , then obviously Φ ≤ Φ(T, •), but by the uniqueness result below Φ(T, •) obtained in Step 3, we deduce that Φ = Φ ∞ . To achieve the proof of Theorem 2.2, it now only remains to prove the previous lemma.

Proof. [Proof of Lemma 4.5] The proof follows closely the proof of Lemma 3.4 in Section 3.3. For the sake of completeness, we repeat the argument in this framework. Firstly, we build a sequence Φ k (x) departing from Φ 0 and defined by induction by, for k ≥ 0, Φ k+1 (x) is a solution to

-∆Φ k+1 + γΦ k+1 = S 0 exp Ω β(x, y)(Φ k (y) -Φ 0 (y)) dy , ∂ n Φ k+1 = 0 on ∂Ω.
Clearly, from [START_REF] Di Lauro | The impact of network properties and mixing on control measures and disease-induced herd immunity in epidemic models: a mean-field model perspective[END_REF] and the maximum principle, we deduce that Φ 1 ≤ Φ 0 on Ω. Then iterating we deduce that Φ k+1 ≤ Φ k for any k ≥ 0. Hence (Φ k ) k is a non-increasing sequence which converges to a solution of (11b) denoted by Φ.

Secondly, if Φ is a subsolution of (11b) which verifies Φ ≤ Φ 0 . We have

-∆Φ + γΦ ≤ S 0 exp Ω β(x, y)(Φ(y) -Φ 0 (y)) dy ≤ S 0 = -∆Φ 1 + γΦ 1 .
By the maximum principle, we deduce that Φ ≤ Φ 1 . Iterating, we deduce that Φ ≤ Φ k for any k ≥ 0. Thus Φ ≤ Φ. In particular, we have Φ ∞ ≤ Φ. Thirdly, we prove that Φ ≤ Φ ∞ . We have Φ ≤ Φ 0 . Let us show that Φ ≤ Φ(t) for all t ≥ 0. By contradiction, if it is not true, there exists a first time t 0 for which Φ(t 0 ) touches Φ, i.e. Φ ≤ Φ(t 0 ) and for some x 0 ∈ Ω, Φ(x 0 ) = Φ(t 0 , x 0 ). Then, ∆Φ(x 0 ) ≤ ∆Φ(t 0 , x 0 ) and we have

S(t 0 , x 0 ) + E(t 0 , x 0 ) + I(t 0 , x 0 ) = -∆Φ(t 0 , x 0 ) + γΦ(t 0 , x 0 ) ≤ -∆Φ(x 0 ) + γΦ(x 0 ) = S 0 exp Ω β(x 0 , y)(Φ(y) -Φ 0 (y)) dy .
Moreover, from (26), we deduce

S(t 0 , x 0 ) = S 0 exp Ω β(x 0 , y) Φ(t 0 , y) -Φ 0 (y) dy ≥ S 0 exp Ω β(x 0 , y) Φ(y) -Φ 0 (y) dy .
It is a contradiction which concludes the proof of Lemma 4.5 and thus of uniqueness of Φ ∞ .

Examples: finite rank contact matrices

The results of Section 2 apply to finite rank contact matrices which have been studied widely with several applications as distributed susceptibility, [START_REF] Horst | Mathematics in population biology[END_REF][START_REF] Horst | Distributed susceptibility: a challenge to persistence theory in infectious disease models[END_REF]. We show in Section 5.1 that for rank-1 contact matrices, the general formulas previously found may be simplified. An alternative approach is presented in Section 5.2 in the case where γ is constant, and generalized to finite rank matrices in Section 5.3.

SEIR system with rank-1 matrices

The case of rank-1 contact matrices refers to matrices β(x, y) which we may be split into β(x, y) = β(x)p(y).

Then system (1) reads Ṡ(t, x) = -β(x)S(t, x) Ω p(y)I(t, y) dy, S(0, x) = S 0 (x), (34a)

Ė(t, x) = β(x)S(t, x) Ω p(y)I(t, y) dy -α(x)E(t, x), E(0, x) = E 0 (x), (34b)
İ(t, x) = α(x)E(t, x) -γ(x)I(t, x), I(0, x) = I 0 (x), (34c) Ṙ(t, x) = γ(x)I(t, x), R(0, x) = R 0 (x), (34d)
where α and γ satisfy assumptions (3), β and p are bounded and positive, and the initial data are given non-negative functions satisfying the conditions in (4). The result in Theorem 2.1 applies. We can identify explicitly the next-generation operator K S(t,•) in ( 15) as

K S(t,•) [φ](x) = β(x)S(t, x) Ω p(y) γ(y) φ(y) dy, K ⋆ S(t,•) [ψ](y) = p(y) γ(y) Ω β(x)S(t, x)ψ(x) dx.
It is worth noticing that, for this operator, the principal eigenelements are (up to normalisation)

   r(K S(t,•) ) = Ω β(y)p(y) γ(y) S(t, y) dy, ϕ S(t,•) (x) = β(x) S(t, x), ψ S(t,•) (y) = p(y) γ(y) .
Indeed, both ϕ S(t,•) and ψ S(t,•) are positive and satisfy the eigenvalue property

K S(t,•) [βS(t, •)](x) = r(K S(t,•) ) β(x)S(t, x), K ⋆ S(t,•) p(•) γ(•) (y) = r(K S(t,•) ) p(y) γ(y) .
Hence, in this case, the basic reproduction number introduced in Definition 3 just reads:

R 0 := Ω S 0 (x) β(x)p(x) γ(x) dx, (35)
and the herd immunity domain is reached for distributions S such that

Ω S(x) β(x)p(x) γ(x) dx < 1.
Lastly, the computation of the final size reduces to

S ∞ (x) = S 0 (x) exp β(x) Ω p(y) γ(y) (S ∞ (y) -S 0 (y) -E 0 (y) -I 0 (y)) dy ,
with S ∞ ≤ S 0 .

An alternative approach for SIR system with rank-1 matrices

In the simpler case of the SIR system with finite rank contact matrices, another method can be used to study the final size. We begin with rank-1 matrices, for which the formulas are simpler, and assume that γ(x) is constant. Analogously to (34), the system reads Ṡ(t, x) = -β(x)S(t, x) Ω p(y)I(t, y) dy, S(0, x) = S 0 (x), (36a) İ(t, x) = β(x)S(t, x) Ω p(y)I(t, y) dy -γI(t, x), I(0, x) = I 0 (x), (36b) Ṙ(t, x) = γI(t, x), R(0, x) = R 0 (x). (36c)

In that case, one can reduce the system to a single ODE by introducing the quantity m(t) := which, for any I 0 > 0 has a single root m H > 0 which is attractive. Indeed, F is a smooth and convex function on R + such that F (0) < Ω p(x)[I 0 + S 0 ](x) dx and lim m→+∞ F (m) = +∞. For I 0 small, m H may be close or far from S 0 depending on the stability of I ≡ 0 which is determined by R 0 = 1 -F ′ (0) γ .

SIR system with rank-N matrices

The reduction to a single equation for rank-1 matrices can be extended to rank-N contact matrices, i.e., which may be written β(x, y) = N i=1 β i (x)p i (y).

Following the above construction, we introduce the matrix M defined by M i,j := Ω β i (x)p j (x) γ(x) S 0 (x) dx.

This matrix has positive coefficients and thus we can define its principal eigenvalue λ and the associated eigenvector X = (X i ) i∈{1,...,N } , with X i > 0 for all i = 1, ..., N . which proves the result for φ S 0 (•) , the same calculation can be performed for ψ S 0 (•) .

Notice that the characterization of R 0 given in the statement extends (35).

For rank-N matrices, it is also noticeable that one can also reduce the SIR system to N differential equations as we did in the rank-1 case. We assume that γ is independent of x and define for j = 1, ..., N , m j (t) = We have Ṡ(t, x) = -S(t, x) Q(t, x), S(t, x) = S 0 (x)e -Q(t,x) .

From the equation for infected, we compute

d dt ṁi (t) = Ω p i (x)S(t, x) Q(t, x)dx -γ ṁi (t)
= Ω p i (x)S 0 (x)e -Q(t,x) Q(t, x)dx -γ ṁi (t).

Integrating once in t, we obtain the equations, for i = 1, ..., N , ṁi (t) = -Ω p i (x)S 0 (x)e -Q(t,x) dx -γm i (t) + Ω p i (x)[S 0 (x) + I 0 (x)]dx, m i (0) = 0. (39)

Notice that (39) extends (38). Conceptually, this is a system of N differential nonlinear equations, which can be computed at least numerically.

Conclusion

We considered an epidemic in a heterogeneous population modelled by a SEIR system with a continuous structure variable and a general contact matrix. We investigated question of the final size of the epidemic both with and without diffusion. In this general framework, our main contribution is to prove uniqueness results for the equation characterizing the final size of the epidemic. The proof combines a contraction property close to the steady state with a monotonicity argument which localized roughly the final state.

Although we presented our results for a SEIR model, they can be extended to other compartmental models, like the SIR model and to more general equations than the simple diffusion.

The main limitation comes from the specific form of the SEIR model excluding birth and loss of immunity. In those models, up to our knowledge, proving merely convergence to a steady state is an open problem.

F

  y) exp(u(y) -Φ(T, y)) dy, in such a way that the fixed point problem[START_REF] Ma | Generality of the final size formula for an epidemic of a newly invading infectious disease[END_REF] reads Φ ∞ = F(Φ ∞ ). Let us denote

  of fact, the equation for S(t, x) can be solved as ln S(t, x) = ln S 0 (x) -β(x)m(t).Inserting expression (37) in the equation (36b) for I, and integrating against the weight p(•), we findd dt ṁ(t) = d dt Ω p(x)I(t, x) dx = ṁ(t) Ω β(x)p(x)S(t, x) dx -γ ṁ(t) = ṁ(t) Ω β(x)p(x)S 0 (x)e -β(x)m(t) dx -γ ṁ(t).Integrating once, one finds, since ṁ(t = 0) = Ω p(y)I 0 (y) dy,ṁ(t) = Ω p(x)[I 0 + S 0 ](x) dx -Ω p(x)S 0 (x)e -β(x)m(t) dx -γm(t), m(0) = 0. (38)Herd immunity is reached when ṁ = 0, that is when Ω p(x)[I 0 + S 0 ](x) dx = Ω p(x)S 0 (x)e -β(x)m H dx + γm H =: F (m H )

Lemma 5 . 1

 51 The basic reproduction number, defined in Definition 3 is given by R 0 = r S 0 (•) = λ and (up to normalisation) the corresponding eigenfunctions areφ S 0 (•) (x) := N i=1 X i β i (x)S 0 (x),ψ S 0 (•) (y) := For the given expression of φ S 0 (•) , we computeK S 0 (•) [φ S 0 (•) ](x) = N i,j=1 S 0 (x)β i (x) y p i (y) γ(y) β j (y)X j dy = N i,j=1 S 0 (x)β i (x)M i,j X j = λ N i=1S 0 (x)β i (x)X i = λφ S 0 (•)

t 0 p

 0 j (y)I(t, y)dy, Q(t, x) = N k=1 β k (x)m k (t).

Acknowledgment

The authors acknowledge the unknown referee for his/her comments and suggestions which allow to improve this paper.

B.P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623).

‡ B.P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623).

Appendix

A.1 Proof of Lemma 4.3 In this technical part, we show that the operator defined on L 2 (Ω) by

admits a principal eigenvalue, that is, that there exist λ > 0 and a function ϕ > 0 such that

It is more convenient to work with φ = ϕ/γ. Let the operator R be defined as Rφ(x) := S(t, x) γ(x) Ω β(x, y)φ(y) dy.

We define for all M > 0 the operator:

This operator is clearly compact and strongly positive, from L 2 (Ω) to L 2 (Ω). The Krein-Rutman theorem thus guarantees the existence of a principal eigenvalue Λ M , associated with a positive eigenfunction φ M that we normalize by

Moreover, the function M → Λ M is clearly continuous by Kato's regularity theory. On the one hand, multiplying by φ M and integrating by parts, one finds

On the other hand, one has

for some constant C > 0 due to (3) and S(t, x) ≤ S 0 (x). Hence, using the Cauchy-Schwarz inequality, one gets (increasing C if necessary) C Λ M ≥ M and thus Λ M → 0 as M → +∞. Moreover, integrating equation (A.1) and using again our hypotheses (3), one gets

for some positive constant C > 0. Hence, CM ≥ 1/Λ M and thus Λ M → +∞ as M → 0. We conclude that there exists M > 0 such that Λ M = 1 by the intermediate value theorem, and thus (A.1) exactly yields, by letting ϕ = γφ M :

with ϕ > 0 in Ω. Letting λ := M ends the proof.


Corinne Marois
email: c.marois@ploufragan.afssa.fr

Roland Cariolet

Hervé Morvan

Marylène Kobisch

Transmission of pathogenic respiratory bacteria to Specific Pathogen Free pigs at slaughter

Keywords: SPF pigs, slaughterhouse, scalding tank water, pathogenic respiratory bacteria

The purpose of this study was to evaluate the transmission of pathogenic respiratory bacteria to thirteen 5 month-old Specific Pathogen Free (SPF) pigs, during the slaughtering process in a commercial slaughterhouse. Before transportation, the SPF pigs and the lorry were checked to confirm the absence of pathogenic respiratory bacteria.

Nine SPF pigs (group 1) were in contact in a conventional slaughterhouse with finishing pigs, during 4 hours before slaughtering. Four SPF pigs (group 2) were slaughtered immediately at arrival in the slaughterhouse.

Five bacterial pathogens (Mycoplasma hyopneumoniae, Actinobacillus pleuropneumoniae, Pasteurella multocida, Haemophilus parasuis and Streptococcus suis) were detected by PCR, after slaughtering, from nasal cavities, tonsils and trachea in the two groups of pigs.

Lung samples were PCR negative. Three and four bacterial species were isolated from the pigs of group 2 and group 1, respectively. Cultures were negative from the lungs.

All the bacterial species present in the SPF pigs were detected by PCR. P. multocida was isolated, from three samples of scalding water before the onset of slaughtering.

Our results suggest that the SPF pigs became contaminated mainly by the slaughterhouse environment and the scalding water. Histological examinations revealed that during scalding, contaminated water could reach the trachea and the lungs of pigs. Checks conducted at slaughter for respiratory disorders have to be carried on, but nasal cavities and tonsils are not appropriate for bacteriological investigations. Moreover, bacteriological results obtained from the lungs of slaughtered pigs have to be used with carefulness.

Introduction

Mycoplasma hyopneumoniae, a widespread respiratory pathogen in pigs, is the primary agent of swine enzootic pneumonia. In combination with various other infectious agents, such as bacteria and viruses, this chronic disease is implicated in a syndrome known as the porcine respiratory disease complex (PRDC) causing substantial losses to the pig industry (Thacker et al., 1999;Thacker et al., 2001;[START_REF] Opriessnig | Experimental reproduction of postweaning multisystemic wasting syndrome in pigs by dual infection with Mycoplasma hyopneumoniae and porcine circovirus type 2[END_REF]Thanawongnuwech et al., 2004). Among the pathogenic respiratory bacteria, Actinobacillus pleuropneumoniae, Pasteurella multocida, Haemophilus parasuis and Streptococcus suis are commonly associated with M. hyopneumoniae and can induce pathologic alterations in the thoracic organs of the pig. Lesions such as pneumonia, pleuropneumonia, pleuritis, abscesses and pericarditis can be observed in growing pigs during slaughter examinations that are an important diagnostic tool for surveillance of herd health at regular intervals. Microbiological investigations have been performed on slaughter. Detection of pathogenic bacteria involved in the lung lesions is the first step to control and to prevent respiratory disorders in pig herds [START_REF] Pijoan | Severe pleuritis associated with certain strains of Pasteurella multocida in swine[END_REF][START_REF] Andreasen | No simple association between time elapsed from seroconversion until slaughter and the extent of lung lesions in Danish swine[END_REF]. Pathogenic respiratory bacteria are generally detected by culture or PCR from the most effective samples carried out for each bacterial species, after macroscopic examinations and quantification of the extend of lung lesions [START_REF] Calsamiglia | Application of a nested polymerase chain reaction assay to detect Mycoplasma hyopneumoniae from nasal swabs[END_REF]Schaller et al., 2001;[START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infections[END_REF][START_REF] Marois | Multiplex PCR assay for detection of Streptococcus suis species and serotypes 2 and 1/2 in tonsils of live and dead pigs[END_REF].

It is widely accepted that respiratory pathogens are transmitted by direct contact from pig to pig or by coughing and sneezing of infected pigs in close proximity. Pathogen circulation can also occur from infected to naïve pigs, in short distances by airborne transmission [START_REF] Otake | Evaluation of aerosol transmission of porcine reproductive and respiratory syndrome virus under controlled field conditions[END_REF]Thacker, 2006;[START_REF] Gottschalk | Actinobacillus pleuropneumoniae[END_REF].

At the end of the fattening period, pigs are collected from different farms and transported to the slaughterhouse, sometimes in the same vehicle. They wait in lairage at the slaughterhouse before slaughtering. The lairage duration is carried out during at least three hours, the time required before slaughtering for animal welfare and meat quality reasons.

To our knowledge, unlike bacterial species isolated from the digestive tract, no information is available about the risk factors influencing the spread of respiratory pathogens at slaughter.

The main objectives of this work were to study (1) the impact of contact between pigs before slaughtering on transmission of pathogenic bacteria from the respiratory tract of conventional finishing pigs to Specific Pathogen Free (SPF) pigs, (2) the samples allowing the recovery of the pathogenic respiratory bacteria from the SPF pigs and (3) the role of the slaughterhouse environment and of the scalding tank water.

Materials and methods

Animals

Thirteen 5 month-old hysterectomy derived SPF pigs were obtained from the porcine experimental unit of AFSSA Ploufragan (Agreement B-22-745-1). Very strict biosecurity measures were implemented in order to avoid undesirable contamination of the pigs: air filtration system and airlocks for each unit, unit-specific clothes and compulsory showering before and after visiting the pigs [START_REF] Cariolet | Rappel des différentes méthodes d obtention de porcelets assainis : conditions de maintient du statut sanitaire et valorisation de ces animaux[END_REF].

The selected SPF pigs were ear tagged and carried to the slaughterhouse in a cleaned and disinfected conventional lorry. Forty-eight hours before departure, bacteriological analyses of the respiratory tract were performed on the animals. Drag-swab samples were also taken from different places in the lorry.

A first group of nine SPF pigs (average weight: 114.6 kg ( 9.9 kg)) was brought, very early in the morning, just before the onset of slaughter, to a cleaned commercial slaughterhouse, where they were in contact during 4 h with finishing pigs coming from several conventional commercial herds. One week later, a second group of four SPF pigs (average weight: 111 kg ( 16.5 kg)) was slaughtered 4 h after the onset of slaughter. Pigs of group 2 were slaughtered immediately at arrival, without any contact with conventional finishing pigs.

Samples

SPF pigs

Before departure for the slaughterhouse, the SPF pigs were swabbed. CytoBrushs (VWR International, Fontenay-sous-Bois, France) were used for nasal cavities and palatine tonsils. Sterile catheters (Laboratoire Euromedis, Neuilly sous Clermont, France) were used for trachea swabbing [START_REF] Marois | Isolation of Mycoplasma hyopneumoniae from different sampling sites in experimentally infected and contact SPF piglets[END_REF]. Drag-swabs (Sodibox, La Forêt Fouesnant, France), humidified with 5 mL of Buffered Pepton Water Broth (BP), were also taken from different places in the lorry.

At the end of the slaughterline, the thoracic organs of the SPF pigs were checked, especially for respiratory disorders. Nasal cavities and tonsils were swabbed. Lungs with trachea and whole palatine tonsils were recovered and carried to our laboratory. Trachea was swabbed with a CytoBrush. Whole tonsils and pieces of the right middle lobe of lungs were cauterised on the surface with a hot spatula and swabbed after several parallel incisions. Each swab was placed in 2 mL of BP (Initial Suspension: IS). The drag-swabs were placed in 10 mL of BP and the supernatant were centrifuged at 5.000 x g for 45 min.

The pellet was resuspended into 2 mL of BP (IS).

Scalding tank water

Just before slaughtering of the SPF pigs, 50 mL water samples from the scalding tank were collected, with sterile bottles, thrice at a depth of 0.20 m, before the onset of slaughter (T0), as soon as the SPF pigs arrived (T+4 h) and 4 h later (T+8 h), at 5 different places (every 10 m) in the tank (50 m long and 1 m wide). In this slaughterhouse, the scalding tank was washed and cleaned every day. The temperature of the scalding water was measured during each sampling.

Post-mortem examinations of finishing pigs from conventional herds

In order to have some information about the score of lung lesions during the day of slaughter, post-mortem examinations were carried out on 320 finishing pigs slaughtered one hour before the SPF pigs. The lung lesions were scored according to the method previously described by [START_REF] Madec | Fréquence, intensité et localisation des lésions pulmonaires chez le porc charcutier[END_REF]. According to this method, the maximum total score possible for each lung was 28.

Bacteriological investigations

Samples were prepared for PCR assays as described by [START_REF] Kellog | Detection of human immunodeficiency virus[END_REF].

Briefly, 1 mL of each IS was centrifuged (12.000 g, 4°C, 20 min) and the pellets were resuspended in 800 µL of lysis solution. Samples were incubated for 1 h at 60°C, 10 min at 95°C and kept at -20°C. When an inhibition of the PCR reaction was observed, DNA was reextracted with phenol/chloroform/isoamyl(ic) alcohol (25/24/1) [START_REF] Marois | Multiplex PCR assay for detection of Streptococcus suis species and serotypes 2 and 1/2 in tonsils of live and dead pigs[END_REF]. PCR assays were carried out as previously described for M. hyopneumoniae [START_REF] Calsamiglia | Application of a nested polymerase chain reaction assay to detect Mycoplasma hyopneumoniae from nasal swabs[END_REF], A. pleuropneumoniae (Schaller et al., 2001), Heamophilus parasuis [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infections[END_REF] and Streptococcus suis [START_REF] Marois | Multiplex PCR assay for detection of Streptococcus suis species and serotypes 2 and 1/2 in tonsils of live and dead pigs[END_REF]. A new PCR test was developed for P. multocida detection. PCR mixture contained PCR buffer (67 mM Tris-HCl, 16 mM (NH 4 ) 2 SO 4 , 0.01% Tween 20, 2.5 mM MgCl 2 [pH 8.8]), 500 µM of each deoxyribonucleoside triphosphate (Eurobio, Les Ulis, France), 600 nM of Pm16S/3-f (AAGGGATGTTGTTAAATAGATAGC) and Pm16S/6-r (GCTTCGGGCACCAAGCATAT) primers, 1.5 units of Taq DNA polymerase (Eurobio), and 5 µl of the DNA template. Amplification was performed in GeneAmp PCR system 9700 (Applied Biosystem). The reaction procedure consisted of initial denaturation at 94°C for 5 min, 40 cycles of denaturation at 94°C for 30 s, primer annealing at 60°C for 30 s, and extension at 72°C for 60 s and final elongation at 72°C for 10 min. The amplified product (411 pb) was revealed in a 2% agarose gel in TBE buffer (90 mM Tris, 90 mM borate, 2.5 mM EDTA [pH 8]) for 1 h at a constant voltage of 125 V. The specificity of this PCR assay was tested with a collection of 75 strains representing 33 toxigenic and non toxigenic P. multocida subsp. multocida as well as seven P. multocida subsp. septica or gallicida and 35 other bacterial species isolated from the respiratory tract of pigs, among them A. pleuropneumoniae, H. parasuis, M. hyopneumoniae and S. suis. The detection threshold was 3 P. multocida CFU/assay. Firstly, all the samples were analysed by PCR, then the positive samples were cultured. In order to isolate A. pleuropneumoniae, P. multocida, H. parasuis and S. suis, 10 µL of each IS were placed onto PPLO agar (Difco, Cergy Pontoise, France), supplemented with 10 µg/ml of nicotinamide dinucleotide ( -NAD), 1 mg/ml of glucose and 5% of decomplemented horse serum agar [START_REF] Jobert | Experimental aerosol transmission of Actinobacillus pleuropneumoniae to pigs[END_REF]. Media were incubated overnight at 37°C in 5% CO 2 and all bacterial colonies were identified by PCR. M. hyopneumoniae was cultivated on modified Friis medium [START_REF] Marois | Isolation of Mycoplasma hyopneumoniae from different sampling sites in experimentally infected and contact SPF piglets[END_REF].

Microscopic examinations

Microscopic lung examinations were conducted on each SPF pig. Lung samples (a piece of the right middle lobe) were fixed in 10% buffered formalin: paraffin-embedded sections were cut at 5 µm, stained by a trichrome coloration (hematoxylin, eosin and safran) and examined by light microscopy.

Results

Post-mortem examinations of the lungs at slaughter

No macroscopic lung lesions were observed in the SPF pigs.

During the four-hour wait of SPF pigs of group 1, 3.828 finishing pigs were slaughtered and post-mortem examinations of lungs were carried out on 160 randomly selected pigs originating from 7 herds, slaughtered one hour before the SPF pigs.

Pneumonia was observed in 70 % of lungs and the mean of pneumonic scores was 3.30 (± 4.20). Recovering lesions (corresponding to early infections) were also noticed in 34 % of lungs. Pleuritis and abscesses were observed in 5.6 % and 3.2 % of the pigs, respectively.

Post-mortem examinations were also carried out on 160 randomly selected lungs of 3.627 conventional finishing pigs from 8 farms, slaughtered one hour before the arrival of the second group of SPF pigs (4 SPF pigs without any contact with conventional finishing pigs).

Pneumonia was observed in 80 % of lungs and the mean score was 5.10 (± 5.30). Thirteen per cent of them had also recovering lesions. Pleuritis and abscesses were observed in 11.3 % and 1.3 % of the pigs, respectively.

Detection of pathogenic respiratory bacteria

Animals and lorry

Using culture and PCR tests, pathogenic respiratory bacteria were neither detected from the SPF pigs nor from the lorry before departure for the slaughterhouse.

Results of the bacteriological investigations performed on the SPF pigs at the slaughterhouse, are presented in Table 1. Major pathogenic respiratory bacteria (M. hyopneumoniae, A. pleuropneumoniae, P. multocida, H. parasuis, and S. suis) were detected in the SPF pigs by PCR, whatever the group of SPF pigs (group 1: with contact or group 2: without any contact, with finishing pigs at slaughter). One to five bacterial species were detected in each SPF pig. Except for M. hyopneumoniae, the percentages of positive samples were higher in pigs of group 2. The total number of positive results, obtained with tracheal swabs, was higher than with the other samples. Nevertheless, M. hyopneumoniae, A. pleuropneumoniae, H. parasuis and S. suis, were also detected from nasal cavities and palatine tonsils, particularly from the pigs of group 1. PCR assays were negative from the right middle lobe of the lungs.

The culture of the microorganisms was fastidious, nevertheless, three and four bacterial species were isolated in group 2 and in group 1, respectively. H. parasuis and S. suis were the most frequent bacteria in the two groups. Except for M. hyopneumoniae, never isolated from the respiratory tract of the SPF pigs, the most appropriate sites for bacteria cultivation were nasal cavities, tonsils and trachea. Respiratory bacteria were not isolated from the lungs.

Scalding tank water

During the collection of samples A (corresponding to group 1) and samples B (corresponding to group 2), the average temperatures inside the scalding tank water were 61.1°C ( 0.4°C) and 60.7°C ( 0.3°C), respectively. The pigs stayed in the water for about 7 min.

Bacteriological results are shown in Table 2. All bacterial species, followed during this study, were detected in the scalding tank water by PCR tests. Before the onset of slaughter, M. hyopneumoniae, A. pleuropneumoniae and S. suis were detected in five samples A, while samples B were PCR negative at the same moment. The following samples (T+4 h and T+8 h) were PCR positive for the five bacterial species, except for H. parasuis at T+8 h. P. multocida was isolated from samples B, in four cases (T+4 h and T+8 h).

Microscopic examinations

Microscopic lung examinations of the SPF pigs at the end of the slaughter process, revealed severe alterations of the respiratory tract of 4/9 pigs of group 1 (Fig. 1). In these pigs, an exfoliation of the bronchial epithelial cells and a dilation of the capillary vessels were observed (Fig. 1 B). Moreover, the examinations showed cellular debris, scales and bacterial accumulations in the bronchial lumen (Fig. 1 C andD). In 9/13 pigs, the structure of the respiratory tract was normal (Fig. 1 A).

4-Discussion

In the conditions described in this study, major pathogenic respiratory bacteria (M. hyopneumoniae, A. pleuropneumoniae, P. multocida, H. parasuis and S. suis) were detected in the respiratory tract of SPF pigs slaughtered in a conventional slaughterhouse. It is widely accepted that transportation of pigs, from farms to slaughterhouses, is a crucial risk factor of cross-contaminations between pigs. In our study, transportation could not have any influence on the transmission of respiratory pathogens to the SPF pigs.

The bacteriological results suggest that the SPF pigs became contaminated (1) by the slaughterhouse environment or (2) by contact with infected finishing pigs during the lairage period (pigs of group 1 only) or (3) by the scalding water.

A rapid contamination occured in the two groups of SPF pigs in the slaughterhouse. PCR, more sensitive than culture, was able to detect five bacterial species. Trachea was the most appropriate site to detect these pathogens. Moreover, four and three species were isolated in groups 1 and 2, respectively. One cause of contamination of SPF pigs was probably the microbial flora present in the slaughterhouse environment. Previous studies indicated that M. hyopneumoniae and A. pleuropneumoniae could be transmitted by aerosols over short distances [START_REF] Jobert | Experimental aerosol transmission of Actinobacillus pleuropneumoniae to pigs[END_REF][START_REF] Otake | Evaluation of aerosol transmission of porcine reproductive and respiratory syndrome virus under controlled field conditions[END_REF][START_REF] Gottschalk | Actinobacillus pleuropneumoniae[END_REF]Thacker, 2006). The potential role of air for dissemination of pathogens is accepted, for instance, M. hyopneumoniae was detected in air samples from pig herds (Stärk et al., 1998). Effective control of respiratory pathogens depends on an optimal environment (air quality, temperature, etc) but factors, such as high relative humidity, mixed infections and stress may encourage the development and the spread of respiratory pathogens in pigs and in their environment [START_REF] Gottschalk | Actinobacillus pleuropneumoniae[END_REF][START_REF] Pijoan | Pneumonic Pasteurellosis[END_REF].

The main route of spread of respiratory pathogens is by direct contact from pig to pig (Thacker, 2006). The time spent in lairage at the slaughterhouse (four hours in our study) might be a risk factor for naïve pigs, as described for Salmonella (Beloeil et al., 2004). In experimental conditions, M. hyopneumoniae was quickly transmitted to sentinel pigs and was detected in nasal, tracheal and bronchial swabs [START_REF] Meyns | Quantification of the spread of Mycoplasma hyopneumoniae in nursery pigs using transmission experiments[END_REF][START_REF] Fano | Evaluation of the aerosol transmission of a mixed infection of Mycoplasma hyopneumoniae and porcine reproductive and respiratory syndrome virus[END_REF][START_REF] Marois | Isolation of Mycoplasma hyopneumoniae from different sampling sites in experimentally infected and contact SPF piglets[END_REF]. Similar results were described with S. suis and A. pleuropneumoniae. [START_REF] Jobert | Experimental aerosol transmission of Actinobacillus pleuropneumoniae to pigs[END_REF]Berthelot-Hérault et al., 2001[START_REF] Marois | Multiplex PCR assay for detection of Streptococcus suis species and serotypes 2 and 1/2 in tonsils of live and dead pigs[END_REF] and with P. multocida [START_REF] De Jong | Progressive and non progressive atrophic rhinitis[END_REF]. Nevertheless, our results showed that, except for M. hyopneumoniae, percentages of PCR positive samples were higher in pigs of group 2. Percentages of positive cultures were similar in the two groups, but A. pleuropneumoniae was not isolated from pigs of group 2. These results suggest that a contact between the SPF pigs and the finishing pigs, during 4 h in the slaughterhouse, had little or no influence on the transmission of respiratory pathogens.

The scalding tank appeared to be a critical point of the slaughtering process. Before the onset of slaughter, DNA from three major respiratory pathogens was detected in water of the scalding tank (samples A). After the passage of finishing pigs in the scalding water (more than 3.600 pigs in each case), and before the slaughtering of SPF pigs, the number of positive samples of water increased and five bacterial species were recovered by PCR. Four hours later, H. parasuis was not detected from the water samples. Post-mortem examinations of 320 lungs of finishing pigs, before the slaughtering of SPF pigs, showed lung lesions in a large proportion of them. No bacteriological analysis was carried out from these lungs. Nevertheless, it is well known that the bacterial pathogens involved in respiratory diseases of pigs are the species detected from pigs and scalding water in the present study.

Thus, the scalding tank water was contaminated by infected finishing pigs during the day of slaughter. Interestingly, viable P. multocida were detected from four samples of water. In the scalding tank water, P. multocida was probably surrounded by organic material and was protected against a high temperature. According to [START_REF] De Jong | Progressive and non progressive atrophic rhinitis[END_REF], P. multocida can survive 10 min at 60°C. During the scalding, some water contaminated by respiratory pathogens, can reach the trachea and the lungs of recently killed pigs. Histological results were in agreement with this hypothesis. Contaminated water from the scalding tank water invaded bronchi of some SPF pigs and bacterial accumulations were observed in the bronchial lumen.

To conclude, this experimental study showed that bacterial population present in slaughterhouses might contribute to transmit pathogens between pigs. Nevertheless, the scalding tank water was the critical point of the slaughtering process and an important source of contamination for pigs. Checks routinely conducted at slaughter, for respiratory disorders in finishing pigs, are very useful for veterinarians and have to be carried on, but nasal cavities, tonsils and trachea are not appropriate for bacteriological investigations. Despite we could not detect pathogenic respiratory bacteria from the lungs of the SPF pigs, bacteriological investigations at slaughter have to be used with carefulness. Schaller, A., Djordjevic, S.P., Eamens, G.J., Forbes, W.A., Kuhn, R., Kuhnert, P., Gottschalk, M., Nicolet, J., Frey, J., 2001

  . Identification and detection of Actinobacillus pleuropneumoniae by PCR based on the gene apxIVA. Vet.Microbiol. 79, 47-62. Stark, K.D., Nicolet, J., Frey, J., 1998. Detection of Mycoplasma hyopneumoniae by air sampling with a nested PCR assay.Appl. Environ. Microbiol. 64, 543-548. Thacker, E.L., Halbur, P.G., Ross, R.F., Thanawongnuwech, R., Thacker, B.J., 1999. Mycoplasma hyopneumoniae potentiation of porcine reproductive and respiratory syndrome virus-induced pneumonia. J. Clin. Microbiol. 37, 620-627.Thacker, E.L., Thacker, B.J.,Janke, B.H., 2001. Interaction between Mycoplasma hyopneumoniae and swine influenza virus. J. Clin. Microbiol. 39, 2525-2530. Thacker, E.L., 2006. Mycoplasmal Diseases. In: Straw, B.E., Zimmerman, J., D Allaire, S., Taylor, D.J. (Eds), Diseases of swine, 9th ed. Iowa State University Press, Ames, IA, pp. 701-717. Thanawongnuwech, R., Thacker, B., Halbur, P., Thacker, E.L., 2004. Increased production of proinflammatory cytokines following infection with porcine reproductive and respiratory syndrome virus and Mycoplasma hyopneumoniae. Clin. Diagn. Lab. Immunol. 11, 901-908. Table 1: Detection of pathogenic respiratory bacteria from 13 SPF pigs, at the end of the slaughter process. NS: nasal swabs; TOS: tonsillar swabs; WT: whole tonsils; TS: tracheal swabs; LS: lung swabs *4 h before slaughtering Mhp: Mycoplasma hyopneumoniae, App: Actinobacillus pleuropneumoniae, Pm: Pasteurella multocida, Hps: Haemophilus parasuis, S. suis : Streptococcus suis. Table 2: Detection of pathogenic respiratory bacteria from the scalding tank water (water samples taken thrice (T0, T+4, T+8) at 5 different places in the tank: A (9 SPF pigs), B (4 SPF pigs). Mhp: Mycoplasma hyopneumoniae, App: Actinobacillus pleuropneumoniae, Pm: Pasteurella multocida, Hps: Haemophilus parasuis, S. suis : Streptococcus suis.

Figure 1 :

 1 Figure 1: Microscopic examinations in the lungs of three SPF pigs (group 1) checked at the

Mhp: Mycoplasma hyopneumoniae, App: Actinobacillus pleuropneumoniae, Pm: Pasteurella

multocida, Hps: Haemophilus parasuis, S. suis : Streptococcus suis.

Acknowledgement

The authors thank Pr. M. Gottschalk, Dr A. Bouchardon and Dr. F. Madec for helpful discussions and Pierre Ecobichon, André Kéranflec h and Jean-Claude Rault and the slaughter staff for skilled technical assistance.

Total (n=20) 3