Nathalie Verdière
email: nathalie.verdiere@univ-lehavre.fr

David Manceau

Shousheng Zhu

Lilianne Denis-Vidal

Inverse problem for a coupling model of reaction-diffusion and ordinary differential equations systems. Application to an epidemiological model

Keywords: Identifiability, PDEs and ODEs systems, Parameter estimation, Epidemiological models

This paper investigates an identifiability method for a class of systems of reaction diffusion equations in the L 2 framework. This class is composed of a master system of ordinary differential equations coupled with a slave system of diffusion equations. It can model two populations, the second one being diffusive contrary to the first one. The identifiability method is based on an elimination procedure providing relations called input-output polynomials and linking the unknown parameters, the inputs and the outputs of the model. These polynomials can also be used to estimate the parameters as shown in this article. To our best knowledge, such an identifiability method and a parameter estimation procedure have not yet been explored for such a system in the L 2 framework. This work is applied on an epidemiological model describing the propagation of the chikungunya in a local population.

Introduction

Many biological or physical processes are modeled using ordinary differential equations (ODE) or partial differential equations (PDE) to describe the (spatial) evolution over the time of quantities of interest. However, they depend on many parameters that need to be estimated from the measurements of the system to make the mathematical model useful. This kind of problem, called inverse problem, necessitates, in general, an identifiability study. An identifiability study ensures that the parameters of interest can be uniquely inferred from given measurements. Consequently, it is a necessary prerequisite before putting in place parameter identification procedures, otherwise, the latter can fail or give not realistic parameter values. This theoretical work is done in the 'best-case' that is from noise-free data.

Identifiability study for ODEs systems is well known and several methods exist in the literature for linear or nonlinear systems (see for example [START_REF] Fliess | Questioning some paradigms of signal processing via concret examples[END_REF][START_REF] Janzén | Three novel approaches to structural identifiability analysis in mixed-effects models[END_REF][START_REF] Ljung | On global identifiability for arbitrary model parametrizations[END_REF][START_REF] Miao | On identifiability of nonlinear ODE models and applications in viral dynamics[END_REF][START_REF] Pohjanpalo | System identifiability based on the power series expansion of the solution[END_REF][START_REF] Vajda | Similarity transformation approach to identifiability analysis of nonlinear compartmental models[END_REF][START_REF] Verdière | Identifiability and estimation of pharmacokinetic parameters for the ligands of the macrophage mannose receptor[END_REF][START_REF] Verdière | A systematic approach for doing an a priori identifiability study of dynamical nonlinear models[END_REF] and the references therein). In the case of PDEs systems (or equations), most of the work on inverse problems concerns a source problem (see [START_REF] Andrle | On an inverse source problem for the heat equation. Application to a pollution detection problem[END_REF][START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF][START_REF] Egger | Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation[END_REF][START_REF] Hamdi | Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations[END_REF][START_REF] Nakagiri | Review of Japanese work of the last ten years on identifiability in distributed parameter systems[END_REF][START_REF] Zhang | Parameter identifiability of Nonlinear systems with time-delay[END_REF]) or a Calderòn's problem (see the original paper [START_REF] Calderón | On an inverse boundary value problem[END_REF] and the review on the subject [START_REF] Uhlmann | Electrical impedance tomography and Calderón's problem[END_REF]). This kind of inverse problems consists in identifying a source term, or to determine a space variable parameter, from boundary measurements.

The inverse problem studied in this paper for PDEs systems is of different type to the one handled in the previous references. Indeed, the difference comes from the knowledge of one (or more) of its solutions (on all the domain or just a part of it). Even if this assumption seems to be less considered, some works can be found in this direction. An identifiability study has been proposed in [START_REF] Xiang | Solving the inverse problem of an SIS epidemic reactiondiffusion model by optimal control methods[END_REF] (see also reference [START_REF] Coronel | On the existence and uniqueness of an inverse problem in epidemiology, Applicable Analysis[END_REF] for a more recent reference) for a SIS (Susceptible -Infectious -Susceptible) epidemic reaction-diffusion model using an optimal control method. In [START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF], using input-output relationships, identifiability of a system of nonlinear integro-partial differential equations of transport type was proposed. An approach based on semigroup theory to identify the mortality rate in an age-structured population dynamics model have been proposed in [START_REF] Perasso | Identifiability problem for recovering the mortality rate in an age-structured population dynamics model[END_REF]. In the case of coupled ODEs and PDEs systems, a study of identifiability have been done in [START_REF] Lassoued | Stability results for the parameter identification inverse problem in cardiac electrophysiology[END_REF] by means of Carleman-type estimate, and in [START_REF] Tuncer | Structural and practical identifiability issues of immuno-epidemiological vector-host models with application to Rift Valley Fever[END_REF] using the differential algebra approach where the PDE model is an age-since-infection-structured population dynamics model.

In this paper, we propose an identifiability method, using the differential algebra approach, for some coupled nonlinear ODEs and reaction-diffusion systems composed of rational functions. This assumption is not so restrictive since by change of variables, lots of systems can be rewritten as rational ones. From the elimination theory developed in the case of ODEs systems, some input-output (IO) polynomials are obtained. The method that we propose is based on these specific relations permitting in the same way to propose a numerical procedure giving a first estimate of the parameters by direct measurements without any a priori knowledge. To our best knowledge, such an identifiability method has not yet been studied for a coupled nonlinear ODEs and reaction-diffusion systems in the L 2 framework.

For the application of the method, we propose to consider the model given in [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF] and describing the transmission of the chikungunya to the human population by the mosquito vector, the Aedes Albopictus. This disease is still current since this vector has developed capabilities to adapt to non tropical regions and a genetic mutation has reduced the delay between the contamination of the host and its propagation to humans. In 2014, several cases have been reported in the South of America, the Caraïbes and central America. In August 2017, it is the France and Italy that have been concerned. In [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF], the virus transmission to the human population has been modeled by a SI-SIR (Susceptible -Infectious and Susceptible -Infectious -Recovered) model taking into account the mosquito biological life cycle. An identifiability study of this model has been done in [START_REF] Zhu | Identifiability study in a model describing the propagation of the chikungunya to the human population[END_REF]. But this ODEs system does not consider an important factor favoring the re-emergence of diseases, that is the spatial mobility of humans. In order to consider this spatial mobility, a metapopulation model has been proposed in [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF]. This spatio-temporal model can be seen as a discrete model in space. In this paper, we model human mobility by a diffusion term in the same way as [START_REF] Murray | Mathematical Biology I: An Introduction[END_REF] or [START_REF] Abid | Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain[END_REF][START_REF] Lutambi | Mathematical modelling of mosquito dispersal in a heterogeneous environment[END_REF]. In [START_REF] Zhu | Identifiability analysis and parameter estimation of an epidemiologic model in a spatially continuous domain[END_REF], an identifiability study was proposed on this model in the continuous framework, that is with continuous initial datum. In this paper, we revisit this work with less assumptions on the regularity of the solutions. More precisely, we work in the L 2 framework.

The paper is structured as follows. In section 2, we present the coupling model which is considered in the sequel. Section 3 is devoted to general identifiability results based on the elimination theory and their applications on the coupling model. In section 4, the theoretical part is applied on a chikungunya epidemiological model and a parameter estimation method based on the input-output polynomials is proposed in section 5 to estimate two of the model parameters. Section 6 concludes the paper.

Coupling model

Notations

The notations are the following:

• Ω is a bounded domain of R n with C 2 -boundary.

• C l ([0, ∞); C(Ω)) denotes the space of piecewise continuous functions from [0, ∞) to C(Ω) and C 1 l ([0, ∞); C(Ω)) the space of piecewise differentiable functions from [0, ∞) to C(Ω).

• Given a local Lebesgue-integrable function F , T F defines its regular distribution.

• H 2 N (Ω) := {v ∈ H 2 (Ω) | ∂v ∂ν = 0 on ∂Ω},
where ν is the outward unit normal of ∂Ω.

• For any function f ∈ L 1 (Ω), we set f := 1 | Ω | Ω f (x)dx.

Coupling model

The model we investigate in this paper is a master-slave model of two species where the slave one diffuses, contrary to the master one. The master model is a system of classical ordinary differential equations defined on the space C(Ω) given by:

∂ t ζ = F (t, ζ), in (0, ∞), ζ(0) = ζ 0 . (1)
where ζ 0 ∈ C(Ω) and F : R × R p → R p is piecewise continuous with respect to the first component and uniformly Lipschitz continuous with respect to the second component. Then, from the Cauchy-Lipschitz theorem, there exists a unique piecewise differentiable solution ζ from any time interval of existence to

C(Ω), i.e. ζ ∈ C 1 l [0, ∞); C(Ω) .
The slave model is a system of reaction-diffusion equations defined on the Banach space X := C(Ω; R q 0 ) × L 2 (Ω; R q 1 ) and which depends on the solution ζ of the master model:

∂ t U + BU = G(t, ζ, U ), in (0, ∞), U (0) = U 0 , (2)
where U 0 ∈ X and the diagonal operator

B := diag(B i ) on X satisfies
• for all i ∈ {1, . . . , q 0 }, B i := I,

• for all i ∈ {q 0 + 1, . . . , q} (where q := q 0 + q 1 ), In other words, (2) is composed of q 0 ODEs and q 1 PDEs which are all together coupled. Moreover, G(t, ζ, •) is a non-linear operator from D(B η ) into X (see [START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF]), where 0 < η < 1, which satisfies the following Lipschitz type condition: there exists an increasing function φ such that, for any compact time interval I where ζ is differentiable and for all (t, U ),(s, V ) ∈ I × D(B η ), we have

B i := d i A + b i with d i , b i > 0 and
G(t, ζ, U ) -G(s, ζ, V ) X ≤ φ( U X + V X ) × B η (U -V ) X + ( B η U X + B η V X + 1)(|t -s| + U -V X ) , (3)
where • X denotes the norm of X.

Then, a simple adaptation of Theorem 4.4 of [START_REF] Yagi | Abstract Parabolic Evolution Equations and their Applications[END_REF] (p. 188, and p. 199 for the non autonomous case) implies that there exists T U 0 > 0 such that (2) admits a unique local solution

U ∈ C l ([0, T U 0 ]; D(B)) ∩ C 1 l ((0, T U 0 ]; X)
. Note that U is piecewise continuous and piecewise differentiable due to the dependence with respect to ζ.

Moreover, if there exists C U 0 > 0 such that the local solution U satisfies:

∀ t ∈ I, U (t) 2 ≤ C U 0 , (4)
for any compact time interval I where U is continuous with respect to the time variable, then U is a global solution:

U ∈ C l ([0, ∞); D(B)) ∩ C 1 l ((0, ∞); X).
Such systems depend on parameters not always directly accessible from measurements or by mean of a costly and experimental scheme. However, they must be estimated to make the model usable. Several numerical procedures, based on measurements, can be found in the literature. They carried out by optimizing some criterion function over the parameter space. However, an identifiability study should be done since it is a pre-condition for safely running a parameter estimation algorithm. Indeed, it will ensure that the algorithm will not converge towards an unrealistic solution.

More precisely, we introduce a parameter vector function θ := (θ a , θ b ) ∈ L ∞ (Ω; U p ), where U p is an open subset of R p , and some corresponding model outputs y(t, θ) := (y a (t, θ a ), y b (t, θ b )). Thus, we rewrite models (1) and (2) in the following form:

(Γ a ) ∂ t ζ(t, θ a ) = f (t, ζ, θ a ), t ∈ [0, ∞), y a (t, θ a ) = h a (ζ), t ∈ [0, ∞), (5)
completed with ζ(t = 0, θ a ) = ζ 0 , and

(Γ b ) ∂ t U (t, θ b ) + B(θ b )U (t, θ b ) = g(ζ, U, θ b ), t ∈ [0, ∞), y b (t, θ b ) = h b (ζ, U ), t ∈ [0, ∞), (6)
completed with U (t = 0, θ b ) = U 0 . The notations are the following:

• f := (f 1 , . . . , f p ) where f i , 1 ≤ i ≤ p, is a rational function with respect to ζ and θ a ,

• g := (g 1 , . . . , g q ) where g i , 1 ≤ i ≤ q, is a rational function with respect to ζ, U and θ b ,

• h a := (h a,1 , . . . , h a,ma ) (resp. h b := (h b,1 , . . . , h b,m b )) where h a,i , 1 ≤ i ≤ m a (resp. h b,i , 1 ≤ i ≤ m b )
, are rational functions with respect to ζ (resp. ζ and U ).

Afterwards, (Γ θ ) will design either system (Γ a ) or (Γ b ).

Identifiability

General Case

The identifiability definition considered in this paper is the following: Definition 3.1. We say that the model (Γ θ ) is identifiable with respect to θ if there exists a time t 1 > 0 and a subset Ω of Ω such that, for all θ, θ ∈ L ∞ (Ω; U p ), we have

∀ (t, x) ∈ [0, t 1 ] × Ω, y(t, θ)(x) = y(t, θ)(x) =⇒ θ = θ.
Definition 3.1 states that there exists a time interval and a space domain such that two distinct parameters will not generate identical model outputs. This definition can be easily extended to piecewise continuous outputs by considering the model's identifiability on each subinterval on which the outputs are defined.

We propose to extend an identifiability method based on the differential algebra theory applied to ODEs with constant parameters [START_REF] Boulier | DifferentialAlgebra. Package of MapleSoft MAPLE standard library since[END_REF][START_REF] Boulier | On Symbolic Approaches to Integro-Differential Equations[END_REF][START_REF] Denis-Vidal | Some effective approaches to check identifiability of uncontrolled nonlinear systems[END_REF][START_REF] Denis-Vidal | An algorithm to test identifiability of non-linear systems[END_REF]. This method is based on some differential polynomials obtained owing to an elimination procedure. The latter consists in eliminating unobservable variables for the benefit of the outputs and the parameters of the model.

Differential algebra theory provides an automatic elimination tool for performing specific relations depending only on the inputs, the outputs and the parameters of the system. These relations are considered as polynomials whose indeterminates are the inputs and the outputs of the model and are also called input-ouptut polynomials. In certain cases, the elimination procedure can be done by hand, however, dedicated packages are available, for example DifferentialAlgebra implemented in Maple [START_REF] Boulier | DifferentialAlgebra. Package of MapleSoft MAPLE standard library since[END_REF][START_REF] Boulier | On Symbolic Approaches to Integro-Differential Equations[END_REF]. This method necessitates most of the times derivation operations and, in the case of the use of a dedicated software, the elimination is done in a formal way, that is, without any consideration of the solution space. The relations can also not be defined with respect to the solution regularity of the model. Nevertheless, new well-defined relations can be obtained in considering the previous relations in the distribution sense or from the mean value, so that an identifiability result can be deduced on the studied system. Suppose that, from an elimination procedure considering only the derivation with respect to the time, the differential polynomials P i (y θ , θ, u) depending on θ, y θ and the input function u of the model have been obtained. The number of these polynomials is the number of model outputs according to [START_REF] Denis-Vidal | Some effective approaches to check identifiability of uncontrolled nonlinear systems[END_REF]. To simplify the notations, we consider afterwards that we have only one output, that is one differential relation P (y θ , θ, u) (which correspond to the case of the application of section 4). This differential polynomial can be expressed as (see [START_REF] Denis-Vidal | Some effective approaches to check identifiability of uncontrolled nonlinear systems[END_REF]):

P (y θ , θ, u) = m 0 (y θ , u) + n k=1
γ k (θ) m k (y θ , u) and P (y θ , θ, u) = 0, [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF] where (γ k (θ)) 1≤k≤n (γ α = γ β if α = β and α, β ∈ {1, . . . , n}) are rational in θ, (m k (y θ , u)) 0≤k≤n are differential polynomials with respect to y θ and u, and m 0 = 0. 1. In the present case, a differential polynomial with respect to y θ means a multivariate polynomial of the variables y θ , ∂ t y θ , ∂ 2 t y θ , . . .. 2. The denominators of γ k (θ) are supposed not to vanish. 3. For system (Γ b ), the function ζ can be considered as an input function.

4. By using Rosenfeld-Groebner algorithm, one can obtain a relation of the form [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF] such that the m k (y θ , u) for k = 1, . . . , n are linearly independent (see [START_REF] Glad | Nonlinear input output relations and identifiability[END_REF]).

The family {γ 1 (θ), . . . , γ n (θ)} is called the exhaustive summary of P and the differential polynomial P is called an input-output polynomial.

Example 3.1. We illustrate the notion of input-output polynomial on a simple problem. Consider an ODEs system given by

ẋ1 = u, ẋ2 = x 1 + k u,
where k ∈ R is a parameter to identify (θ = k) and u is an input function.

Suppose u is C 1 and assume that y := x 2 is observable. Then, y is C 2 and we have

ÿ = ẋ1 + k u = u + k u. (8)
Thus, we deduce that we have an input-output polynomial P such that P (y, k, u) = 0 which is given by

P (y, k, u) := ÿ -u -k u.
Moreover, the exhaustive summary is simply {k}.

Note that we assumed that u is C 1 . We will see below how we can handle a more general case of an input function with less regularity.

Suppose that the elimination process respected the regularity of the solutions of the ODEs or the PDEs system. Then the following proposition gives a necessary condition to obtain the identifiability of the model from the differential polynomial [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF]. Proposition 3.1. Assumed that the family (m k (y θ , u)) 1≤k≤n given by ( 7) is lin-

early independent. If, for all (θ, θ) ∈ L ∞ (Ω, U p ) 2 , we have ∀ k ∈ {1, . . . , n}, γ k (θ) = γ k ( θ) =⇒ θ = θ, then the model is identifiable with respect to θ. Proof 3.1. Let (θ, θ) ∈ L ∞ (Ω, U p ) 2 be such that y θ (t, θ) = y θ (t, θ). From (7), one gets: n k=1 (γ k (θ) -γ k ( θ))m k (y θ , u) = 0. ( 9
)
According to the linear independence of the family

(m k (y θ , u)) 1≤k≤n , relation (9) implies ∀ k ∈ {1, . . . , n}, γ k (θ) = γ k ( θ).
Thus, by assumption, θ = θ.

By using dedicated software programs, relation ( 7) is obtained in a formal way by differentiation with respect to the time and combination of the equations of the initial system. Its derivative degrees may be too high with respect to the solution regularity.

Assume that, in [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF], the m k (y θ , u), k = 0, . . . , n, are well defined almost everywhere and are locally Lebesgue integrable. Then, one can replace the not well defined m k (y θ , u) by their distributions T m k (y θ ,u) . Thus, we obtain a new relation of the form

T m 0 (y θ ,u) + n k=1 γ k (θ)T m k (y θ ,u) = 0. (10)
In this case, similarly to Proposition 3.1, we have the following result in the distribution sense. Proposition 3.2. Assumed that the family of distributions (T m k (y θ ,u) ) 1≤k≤n given by ( 10) is linearly independent. If, for all (θ, θ) ∈ L ∞ (Ω, U p ) 2 , we have

∀ k ∈ {1, . . . , n}, γ k (θ) = γ k ( θ) =⇒ θ = θ,
then the model considered in the distribution sense is identifiable with respect to θ.

Proof 3.2. The proof is similar to the proof of Proposition 3.1.

Remark 3.2. It is easy to show that if the (m k (y θ , u)) k=1,...,n are linearly independent then the (T m k (y θ ,u) ) 1≤k≤n are linearly independent too. In the application of section 4, the linear independence assumption of Proposition 3.2 is substituted by the linear independence of the (m k (y θ , u)) k=1,...,n .

Identifiability result for

(Γ a ) when θ a ∈ L ∞
The solution ζ of system (Γ a ) being piecewise differentiable in time, the output y a is, in general, piecewise differentiable (or just continuous) in time. Using this property, one can obtain a more informative input-output relation taking account the discontinuous points of y a (recall that only derivation with respect to the time variable is considered). These new relations can lead to a positive identifiability result.

Let s be the time derivative of highest order of y a appearing in [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF]. If the function y (s) a can be expressed formally in [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF] as a term of the form d(y a , y a , . . . , y s-1 a )y

(s) a
where d is a differential polynomial with respect to y a and does not cancel, then [START_REF] Darbas | Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency[END_REF] can be rewritten as follows:

y (s) a = m0 (y a ) + n k=1 γ k (θ a ) mk (y a ), (11)
where, for all k = 0, . . . , n, mk (y a ) is a rational function assumed to be locally Lebesgue-integrable in time. Note that we omit the dependence on the input function u to simplify the notation, we will keep this notation in the sequel.

Assume that y a is s times piecewise continuously differentiable in time and let (t ν ) ν be the corresponding finite increasing family of discontinuities points. Due to the discontinuities points, the relation [START_REF] Fliess | Questioning some paradigms of signal processing via concret examples[END_REF] should be considered in the distribution sense:

T y (s) a = T m0 (ya) + n k=1 γ k (θ a )T mk (ya) . (12)
Now, for any j = 0, . . . , s, denote the jump at t ν of y

(j)
a by

σ j ya,ν (θ a ) := y (j) a (t + ν ) -y (j) a (t - ν ).
Note that the jump depends upon θ a since y a itself depends on θ a . Then, jump formula yields

T (s) ya = T y (s) a + ν s-1 j=0 σ j ya,ν δ (s-1-j) tν
, where δ tν denotes the Dirac distribution at point t ν . Thus, equation ( 12) can be rewritten

T (s) ya = T m0 (ya) - ν s-1 j=0 σ j ya,ν (θ a )δ (s-1-j) tν + n k=1 γ k (θ a )T mk (ya) . (13)
Remark 3.3.

1. These relations are more informative than [START_REF] Fliess | Questioning some paradigms of signal processing via concret examples[END_REF] since new constraints on the discontinuity points linking the parameters are added to the study of identifiability. 2. This operation can permit to obtain relations well-defined with respect to the solution regularity.

The following proposition gives also a sufficient condition to obtain the identifiability of the parameters. Proposition 3.3. Assume that the family of distributions (T mk (ya) ) 1≤k≤n is linearly independent. If, for all (θ a , θa ) ∈ L ∞ (Ω, U p ) 2 , we have

     γ k (θ a ) = γ k ( θa ) ∀ν, s-1 j=0 σ j ya,ν (θ a ) = s-1 j=0 σ j ya,ν ( θa )
=⇒ θ a = θa , then the model is identifiable with respect to θ a .

Proof 3.3. The proof is similar to the proof of Proposition 3.1 and follows from the linear independence of the families (δ

(j)
tν ) j=0,...,s and ( Tm k (y θ ) ) k=1,...,n .

All these input-output polynomials ( 7), ( 10), ( 13) can be used to obtain a first estimate of the identifiable parameters. This will be illustrated at section 5.

Example 3.2. Consider again Example 3.1 but now assume that u is a piecewise constant function given by

u(t) := u 1 if t ∈ (0, t 1 ) u 2 if t ∈ (t 1 , T ),
where

u 1 , u 2 ∈ R, u 1 = u 2 .
Then, equation [START_REF] Denis-Vidal | Some effective approaches to check identifiability of uncontrolled nonlinear systems[END_REF] written in distribution sense yields

T y = T u + k(u 2 -u 1 )δ t 1 .
So, in this way, we obtained a new relation that could permit to identifiate k.

Identifiability result on (Γ b ) when θ b is a constant parameter vector

In this part, we focus on the identifiability of system (Γ b ). We are interested in the identifiability of the parameter θ b := (θ c , θ d ) where θ c ∈ R q 0 and θ d := (d i ) i∈I with I ⊂ {q 0 + 1, . . . , q}. Let U := (U i ) i=1,...,q be the solution of (Γ b ) and y b be given by [START_REF] Coronel | On the existence and uniqueness of an inverse problem in epidemiology, Applicable Analysis[END_REF].

y d := (U i ) i∈I .
Assume there exists ν ∈ N such that, for all t ∈ (t ν , t ν+1 ), 1. there exists a family of positive reals

(λ i ) i∈I such that i∈I λ i (g i (ζ(t), U (t), θ d ) -∂ t U i ) is independent of θ d , 2. the family (U i (t, •)) i∈I
∀ (t, x) ∈ (t ν , t ν+1 ) × Ω, y b (t, x, θ d ) = y b (t, x, θd ).
Then, using assumption (1), we have

i∈I λ i B i (θ d )U i = i∈I λ i (g i (ζ(t), U (t), θ d ) -∂ t U i ) = i∈I λ i (g i (ζ(t), U (t), θd ) -∂ t U i ) = i∈I λ i B i ( θd )U i , which leads to i∈I λ i (d i -di )AU i = 0.
Thus, V := i∈I λ i (d i -di )U i is an eigenvector of A associated with zero eigenvalue. Thus, V is constant. Since, for all t ∈ (t ν , t ν+1 ), (U i (t, •)) i∈I is linearly independent up to an additive constant in Ω, we deduce that for all i ∈ I λ i (d i -di ) = 0, which leads to the result since λ i = 0.

It remains to obtain the identifiability with respect to θ c . To this end, Propositions 3.1 and 3.2 can be used. However, in order to test the linear independence of the family (m k (y b )) k=1,...,n , the calculus of the wronskian can not be possible due to the lack of regularity of the solutions. In the following lemma, we propose to substitute this test by another one.

n k=1 (γ k (θ c )) -γ k ( θc )) m k (y b ) = 0. (14)
A linear system of solutions in the block of parameters γ k (θ c ) -γ k ( θc ) is deduced.

According to Assumption 3, we obtain that γ k (θ c ) = γ k ( θc ) and according to Assumption 2, (Γ b ) is identifiable with respect to θ c .

Remark 3.5.

1. If the relation P obtained from ( 7) by taken its mean value is not well-defined with respect to the time variable, it can be considered in the distribution sense. In that case, for Assumption 3, it is sufficient to prove the linear independence of the family ( m k (y b ) ) k=1,...,n . Indeed, if the family ( m k (y b ) ) k=1,...,n is linearly independent, then the family (T m k (y b ) ) k=1,...,n is also linearly independent.

2.

Another way to deal with the linear independence assumption is to suppose that there exist n distinct points t 1 , . . . , t n such that the determinant

m 1 (y b ) (t 1 ) . . . m n (y b ) (t 1 ) m 1 (y b ) (t 2 ) . . . m n (y b ) (t 2 ) . . . . . . m 1 (y b ) (t n ) . . . m n (y b ) (t n ) (15)
is not zero. Indeed, from relation ( 14), the following linear system in the block of parameters composed of n equations is obtained:

n k=1 (γ k (θ c )) -γ k ( θc )) m k (y b ) (t j ) = 0, for j = 1, . . . , n.
Thus, since the determinant ( 15) is not zero, following the same argumentation, the identifiability of (Γ b ) at θ c is deduced.

In the numerical applications, the n distinct points t 1 , . . . , t n can be chosen randomly to evaluate this determinant.

Application to an epidemiological model

The identifiability method presented at the previous section is applied on a coupling model describing the transmission and the propagation of the chikungunya disease by Aedes mosquitoes to the human population (See. [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF]).

Chikungunya transmission model

The model of [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF] couples two sub-systems: the mosquito dynamics model and the transmission virus model between mosquitoes and humans populations. The first one describes the biological mosquito life cycle so that three stages are concerned: E m the number of eggs, L m the number of larvae/pupae and A m the number of adult females. This model is the following

           E m (t) = bA m (t)(1 - E m (t) K E ) -(s + d)E m (t), L m (t) = sE m (t)(1 - L m (t) K L ) -(s L + d L )L m (t), A m (t) = s L L m (t) -d m A m (t), (16)
where To construct the second model, that is the transmission virus one, the human and mosquito populations had been first considered in [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF] and divided in subpopulations before doing a change of variables to obtain densities. In this paper, only the final version concerning densities is studied. In this model, three stages are considered: I m the density of infective mosquitoes; S H the density of susceptible humans; I H the density of infected humans. The transmission virus model is

K E > 0 (resp. K L = K E /2) is the carrying capacity of E m (resp. carrying capacity of L m ), s L > 0 (resp. s > 0) is transfer rate from L m to A m (resp. from E m to L m )
         I m (t) = -(s L L m (t) A m (t) + β m I H (t))I m (t) + β m I H (t), S H (t) = -(b H + β H I m (t))S H (t) + b H , I H (t) = β H I m (t)S H (t) -(γ + b H )I H (t), (17)
where β H > 0 (resp. β m > 0) is the infectious contact rate between susceptible humans and vectors (resp. susceptible mosquitoes and infected humans), γ > 0 is the human recovery rate, b H > 0 is the human birth rate. Under the same conditions as in [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF], a continuous model in space is proposed in this paper. Contrary to the displacement of mosquitoes limited to a few meters, human mobility is considered using diffusion terms (See e.g. [START_REF] Murray | Mathematical Biology I: An Introduction[END_REF]). Hence, Model [START_REF] Lutambi | Mathematical modelling of mosquito dispersal in a heterogeneous environment[END_REF] is rewritten as follows:

                                          ∂ t E m (t, x) = bA m (t, x)(1 - E m (t, x) K E (t, x) ) -(s + d)E m (t, x), ∂ t L m (t, x) = sE m (t, x)(1 - L m (t, x) K L (t, x) ) -(s L + d L )L m (t, x), ∂ t A m (t, x) = s L L m (t, x) -d m A m (t, x), (a)
         ∂ t I m (t, x) = -(s L L m (t, x) A m (t, x) + β m I H (t, x))I m (t, x) + β m I H (t, x), ∂ t S H (t, x) = -(b H + β H I m (t, x))S H (t, x) + b H + d 1 ∆S H (t, x), ∂ t I H (t, x) = β H I m (t, x)S H (t, x) -(γ + b H )I H (t, x) + d 2 ∆I H (t, x), (b) (18)
where d 1 , d 2 > 0 are diffusion coefficients, x is in a bounded domain Ω ⊂ R n with C 2 -boundary and t > 0. The model is completed with the following initial conditions (E m (0, x), L m (0, x), A m (0, x)) = (E 0 (x), L 0 (x), A 0 (x)), (I m (0, x), S H (0, x), I H (0, x)) = (I m 0 (x), S H 0 (x), I H 0 (x)).

Furthermore, we assume that there there is no population flux across the boundary ∂Ω of Ω, thus the model is completed by homogeneous Neumann boundary conditions:

∂S H ∂ν = ∂I H ∂ν = 0 in R + × ∂Ω,
where ν is the unit outward normal at ∂Ω. An identifiability study of this model was proposed in [START_REF] Zhu | Identifiability analysis and parameter estimation of an epidemiologic model in a spatially continuous domain[END_REF] but with constant carrying capacities and continuous initial conditions and consequently in the framework of continuous solutions. In this paper, we generalize the results of [START_REF] Zhu | Identifiability analysis and parameter estimation of an epidemiologic model in a spatially continuous domain[END_REF] to the L 2 framework. More precisely, we make the following assumptions:

1. (E 0 , L 0 , A 0 ) ∈ C(Ω) 3 , I m 0 ∈ C(Ω) and (S H 0 , I H 0 ) ∈ L 2 (Ω) 2 . 2.
The carrying capacities of mosquitoes stages K E and K L satisfy

K E , K L ∈ L ∞ ([0, ∞)×Ω) with K L (t, x) = K E (t, x)/2 = 0, a.e. (t, x) ∈ R + ×Ω.
3. Moreover, K E and K L are piecewise constant functions in time that can correspond to a brutal change in the environment.

Existence and uniqueness of the solutions 4.2.1. The ODE system

We set

Σ := (E, L, A) ∈ C(Ω) 3 0 ≤ E ≤ k, 0 ≤ L ≤ k 2 , 0 ≤ A ≤ s L k 2d m ,
where, to simplify notations, we set k := K E ∞ > 0.

A simple adaptation of Lemmas 4.2 and 4.3 of [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF], from the case K E constant to the case K E depending of x and piecewise constant function in time, leads to the following result:

Proposition 4.1. Let (E 0 , L 0 , A 0 ) ∈ Σ. We set R := s L d m b s + d s s L + d L .
Then, if R > 1, system ( 18)-(a) admits a unique solution (E m , L m , A m ) in the space of differentiable piecewise functions C 1 l ([0, ∞); int(Σ)) 3 where int(Σ) is the interior of Σ. 1. Note that the discontinuous points of E m , L m and A m are the same and are those of K E . 2. From [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF], the condition of survival of eggs, larvae and adults populations of mosquitoes is given by R > 1. Afterwards, we suppose that R > 1 so that E m , L m , A m > 0. In particular, we make the following assumption: (H) R > 1 and there exists c > 0 such that, for any t ≥ 0,

ζ 3 (t) := A m (t, •) > c.

The PDE system

Since (H) is satisfied, we have

r := L m A m ∈ C 1 l [0, ∞); C(Ω) .
By setting U := (I m , S H , I H ), system ( 18)-(b) is of the form (2) where the Banach space X is given by

X := (u 1 , u 2 , u 3 ) | u 1 ∈ C(Ω), u 2 , u 3 ∈ L 2 (Ω) ,
and the operator B is defined by

B := diag(I, -d 1 A + b H , -d 2 A + γ + b H ),
where A is the realization of the Laplacian in L 2 (Ω) with Neuman homogeneous boundary condition on ∂Ω, and I is the identity operator of C(Ω). Furthermore, the nonlinear operator G(t, •) is given by

G(t, U ) := (-(s L r(t) + β m u 3 -1)u 1 + β m u 3 , -β H u 1 u 3 + b H , β H u 1 u 2 ).
We prove in the appendix, section 7, that G satisfies (3) and thus ( 18)-(b) admits a unique local solution

U ∈ C l ([0, T U 0 ]; D(B)) ∩ C 1 l ((0, T U 0 ]; X) where U 0 := (I m 0 , S H 0 , I 0 H ). Moreover, setting X + := {(u 1 , u 2 , u 3 ) ∈ X | u 1 , u 2 , u 3 ≥ 0} , (19)
we have the following theorem which states that the solution is global.

Theorem 4.1. Under the assumption (H), for any (I m 0 , S H 0 , I H 0 ) ∈ X + , ( 18)-(b) possesses a unique non-negative global solution such that

I m ∈ C 1 l ([0, ∞); C(Ω)), S H , I H ∈ C l ([0, ∞); H 2 N (Ω)) ∩ C 1 l ((0, ∞); L 2 (Ω)).
The proof of the theorem is given in the appendix, section 7.

Identifiability result

The parameters whose values are not directly accessible from the field are s, s L for the system ( 18)-(a) and β H , β m , d 1 , d 2 for the system ( 18)-(b).

In the case of the ODEs model ( 16)-( 17), an identifiable study had been done in [START_REF] Zhu | Identifiability study in a model describing the propagation of the chikungunya to the human population[END_REF] and a procedure based on input-output polynomials had been proposed in [START_REF] Verdière | A distribution input-output polynomial approach for estimating parameters in nonlinear models. Application to a Chikungunya model[END_REF]. In [START_REF] Zhu | Identifiability analysis and parameter estimation of an epidemiologic model in a spatially continuous domain[END_REF], the identifiability model ( 18) was studied but in the case of very regular functions. Namely, for system ( 18)-(a), the solutions were C ∞ and for system ( 18)-(b) the solutions were continuous. We consider the same assumptions on the measured outputs, that is L m , S H , I H can be measured.

Indeed, some weekly informations on the number of mosquitoes larvae can be obtained owing to the use of water tanks in which the mosquitoes female lay their eggs. Then, institutes for Health Care can collect periodically data on the number of new infections. Thus, we assume that the number of susceptible humans S H , infected humans I H and larvae L m can be considered as measured variables in Ω during a finite time interval.

The main result of the paper is the following: • for all (t, x) ∈ (t ν , t ν+1 )×Ω, L m (t, x), S H (t, x) and I H (t, x) are the outputs of the model,

• for any t ∈ (t ν , t ν+1 ), S H (t, •) and I H (t, •) are linearly independent up to an additive constant in Ω.

We recall that the notation < ., . > is the mean on all the space Ω. Then, system (18) is identifiable with respect to s, s L , A 0 , d 1 and d 2 .

Let Ĩm := β H I m . Under the same assumptions, if there exists t ∈ (t ν , t ν+1 ) such that I H S 2 H ( t) = 0 and Ĩm I H S 2 H ( t) = 0, then system ( 18) is identifiable with respect to β m and β H . 1. According to equation ( 18)-(b), Ĩm can be expressed almost everywhere only in function of the measured variable S H (or I H ) and of the identifiable parameter d 1 (or d 2 ). Consequently, Ĩm is considered as a measured variable almost everywhere. 2. Note that a supplementary result is obtained on the identifiability of the initial condition A 0 ∈ C(Ω). Therefore, by an optimization procedure based on the identifiability study, the values of A 0 and the other unknown parameters can be obtained in a unique way from the knowledge of L m , S H and I H . 3. An example of the estimation of d 1 and d 2 is given at section 5.

Proof of Theorem 4.2

We proceed in two steps. In the first step, we show that ( 18)-(a) is identifiable with respect to θ a := (s, s L , A 0 ). In the second step, we show that ( 18

)-(b) is identifiable with respect to θ b := (β m , β H , d 1 , d 2 ).
First step. For x fixed in Ω, let us introduce the following notations:

∀ t ∈ [0, ∞), ζ 1 (t) := E m (t, x) K E (t, x) , ζ 2 (t) := L m (t, x) K E (t, x) , ζ 3 (t) := A m (t, x) K E (t, x) .
In all the sequel, we denote by (t ν ) ν the finite increasing family of discontinuity time points of K E (and so of ζ 1 , ζ 2 and ζ 3 ). First, to obtain simpler polynomials, the third equation of system ( 18)-(a) is integrated before using the package DifferentialAlgebra of Maple. Integration of the third equation of system ( 18)-(a) on any interval (t ν , t ν+1 ) yields

∀ t ∈ (t ν , t ν+1 ), ζ 3 (t) = ζ 3 (0)e -dmt + s L t ν+1 tν ζ 2 (s, x)e -dm(t-s) ds.
Then, we set

∀ t ∈ (t ν , t ν+1 ), ζ2,ν (t, x) := t ν+1 tν ζ 2 (s,
x)e -dm(t-s) ds and λ(t) := e -dmt .

To simplify the notations, we let ζ2,ν = ζ2 . Since ζ2 depends only on d m and the measured variable ζ 2 , it can be also considered as a measured variable. Thus, we have three observations y a,1 := ζ 2 , y a,2 := ζ2 and y a,3 := λ.

Then, system ( 18)-(a) can be rewritten:

               ∂ t ζ 1 = bζ 3 (1 -ζ 1 ) -(s + d)ζ 1 ∂ t ζ 2 = sζ 1 (1 -2ζ 2 ) -(s L + d L )ζ 2 ζ 3 = ζ 3 (0)λ + s L ζ2 ∂ t ζ2 = ζ 2 -d m ζ2 ∂ t λ = -d m λ y a,1 = ζ 2 , y a,2 = ζ2 , y a,3 = λ (20)
with the initial conditions:

(ζ 1 (0, x), ζ 2 (0, x), ζ 3 (0, x), ζ2 (0, x), λ(0)) = (ζ 1,0 (x), ζ 2,0 (x), ζ 3,0 (x), 0, 1).
The Rosenfeld-Groebner algorithm1 is then applied with [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF] and the elimination order 2[

ζ 3,0 , s, s L ] ≺ [y a,1 , y a,2 , y a,3 ] ≺ [ζ 1 , ζ 2 , ζ 3 , ζ2 , λ]
The software gives the characteristic presentation composed of the following polynomials:

Q 1 = -ζ 3,0 y a,3 -s L y a,2 + ζ 3 , Q 2 = ζ 1 (2 y a,1 -1) s + y a,1 s L + d L y a,1 + ∂ t y a,1 and
P = -2 ∂ t (y 2 a,1 ) -∂ 2 t y a,1 + 2 y a,1 ∂ 2 t y a,1 + b s ζ 3,0 y a,3 + (d d L + d s L + d L s +s s L ) (2 y 2 a,1 -y a,1 ) + (2 b d L ζ 3,0 + 4 b s ζ 3,0 + 2 b s L ζ 3,0 ) y 2 a,1 y a,3 +(-b d L ζ 3,0 -4 b s ζ 3,0 -b s L ζ 3,0 ) y a,1 y a,3 + b s s L y a,2 +(-b d L s L -4 b s s L -b s 2 L ) y a,1 y a,2 + (2 b d L s L + 4 b s s L + 2 b s 2 L ) y 2 a,1 y a,2 +b ζ 3,0 (-y a,3 ∂ t y a,1 + 2 y a,1 y a,3 ∂ t y a,1 ) + (2 d + 2 s) y a,1 ∂ t y a,1 -(d + d L + s + s L ) ∂ t y a,1 + b s L (2 y a,1 ∂ t y a,1 y a,2 -∂ t y a,1 y a,2 ).
From the two first polynomials Q 1 , Q 2 , ζ 1 and ζ 3 can be expressed according to y a,1 , y a,2 , y a,3 . The third one, P , links the outputs and the parameters s, s L and ζ 3,0 of the model, it is also an input-output polynomial. Since solutions of [START_REF] Miao | On identifiability of nonlinear ODE models and applications in viral dynamics[END_REF] are in C 1 l ([0, ∞); C) and P contains derivatives of second order in time, P is not well-defined with respect to the regularity of the solution. Let us show how to obtain a regular relation. In considering P in the distribution sense and in changing the term y a,1 ∂ 2 t y a,1 to ∂ t (y a,1 ∂ t y a,1 ) -(∂ t y a,1 ) 2 the following polynomial P in the distribution sense is obtained:

P = -2 T ∂t(y 2 a,1 ) -T y a,1 + 2T y a,1 ∂ty a,1 -2T (∂ty a,1 ) 2 + γ 1 (θ a ) T y a,3 + γ 2 (θ a ) T 2 y 2 a,1 -y a,1 +γ 2 (θ a ) T y a,1 + γ 3 (θ a ) T y 2 a,1 y a,3 + γ 4 (θ a ) T y a,1 y a,3 + γ 5 (θ a ) T y a,2 + γ 6 (θ a ) T y a,1 y a,2 +γ 7 (θ a ) T y 2 a,1 y a,2 + γ 8 (θ a ) T -y a,3 ∂ty a,1 +2 y a,1 y a,3 ∂ty a,1 +γ 9 (θ a ) T y a,1 ∂ty a,1 -γ 10 (θ a ) T y a,1 + γ 11 (θ a ) T 2 y a,1 ∂ty a,1 y a,2 -∂ty a,1 y a,2 , where γ(θ a ) := (γ k (θ a )) 1≤k≤11 := b s ζ 3,0 , d d L + d s L + d L s + s s L , 2 b d L ζ 3,0 + 4 b s ζ 3,0 + 2 b s L ζ 3,0 , -b d L ζ 3,0 -4 b s ζ 3,0 -b s L ζ 3,0 , b s s L , -b d L s L -4 b s s L -b s 2 L , 2 b d L s L + 4 b s s L + 2 b s 2 L , b ζ 3,0 , 2 d + 2 s -(d + d L + s + s L ), b s L .
Remark 4.3. From the observation L m and by using P , the unknown parameters s, s L and A 0 (x) can be estimated for all x ∈ Ω.

Let us prove that the model is identifiable with Proposition 3.2. Using Maple, we obtain that the functional determinant of the functions {y a,3 , 2 y 2 a,1 -y a,1 , y 2 a,1 y a,3 , y a,1 y a,3 , y a,2 , y a,1 y a,2 , y 2 a,1 y a,2 , -y a,3 ẏa,1 + 2y a,1 y a,3 ẏa,1 , y a,1 ẏa,1 , ẏa,1 , 2y a,1 ẏa,1 y a,2 -ẏa,1 y a,2 } is not identically equal to zero. Then, solving the algebraic system γ(θ a ) = γ( θa ) with the Groebner basis implemented in maple leads to θ a = θa . Consequently, applying Proposition 3.2, the model is identifiable at the parameters s, s L , and ζ 3,0 (x) for each x ∈ Ω. Second step. Since S H and I H are assumed to be known for all (t, x) ∈ (t ν , t ν+1 )× Ω, the equalities y b,1 = S H and y b,2 = I H are added to system (18)-(b) which is rewritten in the following form:

             ∂ t I m = -(s L r + β m I H )I m + β m I H , ∂ t S H = -(b H + β H I m )S H + b H + d 1 ∆S H , ∂ t I H = β H I m S H -(γ + b H )I H + d 2 ∆I H , y b,1 = S H , y b,2 = I H .
where r := ζ 2 ζ 3 is known according to the previous subsection. The parameters to identify being

β m , β H , d 1 , d 2 , we denote θ b := (β m , β H , d 1 , d 2 ).
The initial conditions are in that case (I m (0, •), S H (0, •), I H (0, •)) = (I m 0 , S H 0 , I H 0 ). Following section 3.3, the identifiability of the parameters is done in two steps: first the identifiability study of d 1 and d 2 by using Lemma 3.1, then the identifiability study of β m and β H by using Proposition 3.3.

Identifiability of d 1 and d 2 .
Summing the second equation and the third equation of (Γ b ) yields:

d 1 ∆y b,1 (t, •) + d 2 ∆y b,2 (t, •) = ∂ t y b,1 (t, •) + ∂ t y b,2 (t, •) + b H (y b,1 (t, •) -1) + (γ + b H )y b,2 (t, •). (21)
The right hand-side of [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF] does not depend on d 1 and d 2 , and the functions y b,1 and y b,2 are supposed to be linearly independent. According to Lemma 3.1, the model is identifiable with respect to d 1 and d 2 .

Identifiability of β m and β H .

From the package DifferentialAlgebra of Maple, we obtain the following input-output polynomial: m 1 can be rewritten:

-(-b H s L r -s L rγ)y b,2 y b,1 -(γ + b H )y b,2 ∂ t y b,1 -(-s L r -b H -γ)y b,1 ∂ t y b,2 -d 2 rs L y b,1 ∆y b,2 -(d 2 ∂ t ∆y b,2 -∂ tt y b,2 )y b,1 -∂ t y b,2 ∂ t y b,1 + d 2 ∂ t y b,1 ∆y b,2 = β m ((-γ -b H )y 2 b,2 y b,1 -y b,2 y b,1 ∂ t y b,2 + d 2 y b,2 y b,1 ∆y b,2 ) + β H β m y b,2 y 2 b,1 (22)
m 1 = y b,2 y b,1 ((-γ -b H )y b,2 -∂ t y b,2 + d 2 ∆y b,2 ) = -y b,2 y 2 b,1 Ĩm .
Thus, the Wronksian of ( m 1 , m 2 ) is equal to:

y b,2 y 2 b,1 (∂ t Ĩm )y 2 1 y 2 .
By assumption, there exists a time point t ∈ (t ν , t ν+1 ) such that this Wronskian does not vanish. Thus, from Lemma 3.2 and Remark 3.5, the model is identifiable with respect to β m and β H .

Remark 4.4. The proof of Proposition 4.2 provides two input-output polynomials [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF] and [START_REF] Murray | Mathematical Biology I: An Introduction[END_REF]. To illustrate a way to use them in a parameter estimation procedure, we propose, in the following section, a method to estimate d 1 and d 2 based on [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF]. The procedure does not require any knowledge of the parameter values. This first estimate of d 1 and d 2 can be used by a local algorithm to improve their estimation.

Example of the estimation of the two parameters d 1 and d 2

This part is devoted to a numerical simulation estimating the parameters d 1 and d 2 based on [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF]. We follow the following procedure:

• We solve [START_REF] Miao | On identifiability of nonlinear ODE models and applications in viral dynamics[END_REF]

Simulations

We use the parameters values from Table 1 of [START_REF] Moulay | Optimal Control of Chikungunya Disease: Larvae Reduction, Treatment and Prevention[END_REF] (see also [START_REF] Moulay | A metapopulation model for chikungunya including populations mobility on a large-scale network[END_REF]) and recalled in Table 1. Moreover, we set d 1 = 0.1 and d 2 = 0.01 so that susceptible populations diffuse through the domain faster than the infected populations. The domain considered is Ω := (0, 1) 2 and the final time is T = 20.

At the initial time, the variables ζ i are taken constant while the infected humans are supposed to be concentrated in the center of the domain, which can correspond for example to an hospital, and the susceptible humans are located at the extremities of the domain. More precisely, the initial conditions are given by:

           ζ 1 (t = 0, x, y) = 0.1, ζ 2 (t = 0, x, y) = 0.04, ζ 3 (t = 0, x, y) = 0.01, I m (t = 0, x, y) = 0.2, S H (t = 0, x, y) = 1 -z(x, y), I H (t = 0, x, y) = z(x, y), where z(x, y) :=    e - 1 4 1 4 -(x-1 2 ) 2 -(y-1 2 ) 2 , if (x -1 2 ) 2 + (y -1 2 ) 2 < 1 4 0 else.
Then, we compute approximations of S H and I H by solving the system [START_REF] Miao | On identifiability of nonlinear ODE models and applications in viral dynamics[END_REF], with the exact parameter values, by an explicit finite difference method.

The measured states S meas H and I meas H are constructed by using the function randn of Matlab, where the coefficient σ is chosen in order to have a maximal value of the relative error between S H and S meas H (and between I H and I meas H ) equal to 1%, 5%, 10% and 20%.

The numerical input-output method for estimating d 1 and d 2

The relation ( 21) is given in distribution sense by:

d 1 Ω S H (t, x)∆ψ(x) dx + d 2 Ω I H (t, x)∆ψ(x) dx = Ω ∂ t S H (t, x) + ∂ t I H (t, x) ψ(x) dx +(γ + b H ) Ω I H (t, x) ψ(x) dx -b H Ω 1 -S H (t, x) ψ(x) dx,
for all ψ ∈ C c (Ω). Integrating this relation on (0, t), we obtain

d 1 t 0 Ω S H (τ, s)∆ψ(s) ds dτ + d 2 t 0 Ω I H (τ, s)∆ψ(s) ds dτ = Ω S H (t, s) + I H (t, s) ψ(s) ds - Ω S H (0, s) + I H (0, s) ψ(s) ds +(γ + b H ) t 0 Ω I H (τ, s) ψ(s) ds dτ -b H t 0 Ω 1 -S H (τ, s) ψ(s) ds dτ, (23)
for all ψ ∈ C c (Ω)and all t > 0.

We consider the following test function: One can see that the method gives good approximations of d 1 and d 2 . Moreover, this method also works with a strong noise.

ψ(x, y) := e 1 1-x 2 -y 2 if (x, y) ∈ Ω, 0 else, and a discretization 0 = t 1 < t 2 < . . . < t M +1 = T of [0, T ]. For k = 1,

Conclusion

In this paper, we have investigated an identifiability method applied up to now to ordinary differential equations systems to partial differential equations systems. Furthermore, some generalization results on the L 2 framework had been obtained. This method based on algebra tools had proved to provide interesting relations after regularizing them in using the distribution theory. As shown on the two examples describing the chikungunya disease, these relations permit to provide identifiability results. Furthermore, a numerical method deduced from the differential relations had been proposed and permitted to give a first initial guess of the unknown parameters. Future works concern the exploration of the proposed method to parameters in L p and for other class of systems.

Appendix

In this appendix we give the proof of Theorem 4.1. To this end it suffices to show that (3) is satisfied so that a local solution U exists and then to show that U is non negative and satisfies [START_REF] Boulier | On Symbolic Approaches to Integro-Differential Equations[END_REF].

We restrict ourselves to the case r ∈ C 1 [0, +∞); C(Ω) . The case r piecewise differentiable in time is a simple adaptation by considering only compact time intervals on which r is C 1 . Thus, since we assume (H) satisfied, in all the sequel we have

r := L m A m ∈ C 1 [0, ∞); C(Ω) .
Set η ∈ (3/4, 1). The domain of the fractional power B η of B is given by (see theorem 16.7 of [START_REF] Yagi | Abstract Parabolic Evolution Equations and their Applications[END_REF]):

D(B η ) = {(u 1 , u 2 , u 3 ) T ; u 1 ∈ C, u 2 , u 3 ∈ H 2η N (Ω)} (24)
and there exists c > 0 such that for i = 1, 2

1 c f H 2η N (Ω) ≤ B η i f L 2 (Ω) ≤ c f H 2η N (Ω) , (25)
for all f ∈ D(B η i ). Moreover, since 2η > 1 and Ω is bounded, from Sobolev embedding theorem, we have

H 2η N (Ω) ⊂ C( Ω) ⊂ L ∞ (Ω) ⊂ L 2 (Ω), (26)
with continuous embedding.

First step. Existence and uniqueness of a local solution.

In all the sequel, C will denote a positive constant whose value can change from line to line. Set U

:= (u 1 , u 2 , u 3 ) ∈ D(B η ), V := (v 1 , v 2 , v 3 ) ∈ D(B η ), T > 0 and s, t ∈ [0, T ]. We have G(t, U ) -G(s, Ũ ) 2 ≤ -(s L r(t) + β m u 3 -1)u 1 + β m u 3 + (s L r(s) + β m v 3 -1)ũ 1 -β m v 3 L 2 + β H u 1 u 3 -v 1 v 3 L 2 + β H u 1 u 2 -v 1 v 2 L 2 . ( 27
)
From ( 25) and ( 26), we deduce

u 1 u 2 -v 1 v 2 L 2 ≤ u 1 u 2 -u 1 v 2 + u 1 v 2 -v 1 v 2 L 2 ≤ u 1 (u 2 -v 2 ) L 2 + (u 1 -v 1 )v 2 L 2 ≤ u 1 ∞ u 2 -v 2 L 2 + v 2 ∞ u 1 -v 1 L 2 ≤ C u 1 H 2η u 2 -v 2 L 2 + v 2 H 2η u 1 -v 1 L 2 ≤ C( B η U L 2 + B η V L 2 ) U -V L 2 . ( 28
)
In the same way, we also have

u 1 u 3 -v 1 v 3 L 2 ≤ C( B η U L 2 + B η V L 2 ) U -V L 2 . ( 29
)
Moreover, since r ∈ C 1 [0, ∞); C(Ω) , arguing as previously we obtain

-(s L r(t) + β m u 3 -1)u 1 + β m u 3 + (s L r(s) + β m v 3 -1)v 1 -β m v 3 L 2 ≤ (s L r ∞ + 1) u 1 -v 1 L 2 + s L r ∞ u 1 L 2 |t -s| + β m u 1 u 3 -v 1 v 3 L 2 + β m u 3 -v 3 L 2 ≤ C ( B η (U -V ) L 2 + ( B η U L 2 + B η V L 2 + 1) (|t -s| + U -V 2 ) . ( 30
)
From inequalities ( 27), ( 28), ( 29) and (30), we get (3). Set U 0 ∈ X + , where X + is given by [START_REF] Moulay | Optimal Control of Chikungunya Disease: Larvae Reduction, Treatment and Prevention[END_REF]. From the first step and Theorem 4.4 of [START_REF] Yagi | Abstract Parabolic Evolution Equations and their Applications[END_REF], we obtain that there exists T U 0 > 0 such that (18)-(b) admits a unique local solution: U := (u 1 , u 2 , u 3 ) ∈ C ([0, T U 0 ]; D(B)) ∩ C 1 ((0, T U 0 ]; X) .

Second step. Non negativity of the local solution.

This proof is quite standard and can be found in [START_REF] Yagi | Abstract Parabolic Evolution Equations and their Applications[END_REF] (e.g. page 197), we set it here for the reader convenience. Let χ(f ) be a cutoff function defined by χ(f ) = 0 for f ∈ (-∞, 0) and χ(f ) = f for f ∈ [0, +∞). We set G(t, Ũ ) := (-(s L r(t) + β m ũ3 -1) ũ1 + β m ũ3 , -β H χ(ũ 1 ) ũ3 + b H , β H χ(ũ 1 ) ũ2 ).

(31) Then, we consider the following problem

∂ t Ũ + B Ũ = G(t, Ũ ), in (0, ∞), Ũ (0) = U 0 . ( 32
)
Using similar arguments as in step 1, one can prove that there exists TU 0 such that (32) admits a unique local solution Ũ := (ũ 1 , ũ2 , ũ3 ) ∈ C [0, TU 0 ]; D(B) ∩ C 1 (0, TU 0 ]; X .

Consider H(s) := 1 2 s 2 for s ∈ (-∞, 0) and H(s) := 0 for s ∈ [0, +∞). This function satisfies H(s) ≥ 0, H (s) ≤ 0 and H (s)s ≥ 0, H (s)s ≤ 2H(s).

(

) 33
We set φ 1 (t) := Ω H(ũ 2 (t, x)) dx.

Since x → H(ũ 2 (t, x)) is continuous for all t ∈ (0, TU 0 ], if φ 1 ≡ 0 then ũ2 ≥ 0. Since φ 1 is continuously differentiable and from (32), we have: Owing to [START_REF] Verdière | A systematic approach for doing an a priori identifiability study of dynamical nonlinear models[END_REF], one gets that Ω H (ũ 2 )dx ≤ 0, Ω H (ũ 2 )ũ 2 dx ≥ 0 and consequently that φ 1 (t) ≤ 0. Thus, for any t ∈ [0, TU 0 ], 0 ≤ φ 1 (t) ≤ φ 1 (0). Since U 0 ∈ X + , ũ0 ≥ 0 and so φ 1 (0) = 0. We deduce that ũ2 ≥ 0 in [0, TU 0 ].

φ 1 (t) =
In the same way, we obtain that ũ1 ≥ 0 and ũ3 ≥ 0 in [0, TU 0 ]. Set T := min(T U 0 , TU 0 ). Since ũ1 ≥ 0 in [0, T ], we deduce from the definition [START_REF] Verdière | Identifiability and estimation of pharmacokinetic parameters for the ligands of the macrophage mannose receptor[END_REF] of G that G( Ũ (t)) = G(U (t)) in [0, T ]. By uniqueness of the solution of ( 18 To prove that the solution is global, it suffices to show that the inequality (4) is satisfied.

  A is the realization of the Laplacian in L 2 (Ω) with Neumann homogeneous boundary condition on ∂Ω.

Lemma 3 . 1 .

 31 Let y b be given by y b := (y c , y d ) where

  is linearly independent up to an additive constant in Ω, 3. the b i defined in B i = d i A + b i are supposed to be known. Then, (Γ b ) is identifiable with respect to θ d . Remark 3.4. In other words, Lemma 3.1 gives a sufficient condition for the identifiability of diffusion coefficients (d i ) i∈I when the associated solutions (U i ) i∈I are known. Proof 3.4. Let θ d , θd be such that

Lemma 3 . 2 .

 32 Let P := P . Assume that 1. (Γ b ) is identifiable with respect to θ d , 2. the polynomial P defined in (7) with θ := θ c satisfy, for all k = 1, . . . , n, γ k (θ c ) = γ k ( θc ) =⇒ θ c = θc , 3. the family of functions (< m k (y b ) >) k=1,...,n are linearly independent. Then (Γ b ) is identifiable with respect to θ c . Proof 3.5. The relation P (y b , θ c ) -P (y b , θc ) gives:

  , b > 0 is the intrinsic rate, d > 0, d L > 0 and d m > 0 are the mortality rate for E m , L m and A m respectively.

Theorem 4 . 2 .

 42 Assume the constants b, d, d L , d m , b H , γ are known and there exists ν ∈ N such that

  Since the function y b,1 = S H and y b,2 = I H are in C 1 l ((0, ∞), L 2 (Ω)), the previous polynomial is considered in the distribution sense.We apply Lemma 3.2. Since the model is identifiable with respect to d 1 and d 2 , Assumption 1 of Lemma 3.2 is satisfied. The exhaustive summary of the inputoutput relation[START_REF] Murray | Mathematical Biology I: An Introduction[END_REF] being (β m , β H β m ), Assumption 2 of Lemma 3.2 is easily verified. It remains to show Assumption 3. We set m 1 := (-γ -b H )y 2 b,2 y b,1 -y b,2 y b,1 ∂ t y b,2 + d 2 y b,2 y b,1 ∆y b,2 and m 2 := y b,2 y 2 b,1 .

ΩH (ũ 2 1 ΩH (ũ 2 )

 212 (t, x))∂ t ũ2 (t, x)dx = d ∆ũ 2 dx + Ω H (ũ 2 )(b H -(b H + β H χ(ũ 1 ))ũ 2 )dx -b H Ω H (ũ 2 )ũ 2 dx -β H Ω H (ũ 2 ) ũ2 χ(ũ 1 )dx. Since ∂ ν ũ2 = 0 on ∂Ω, we get Ω H (ũ 2 )∆ũ 2 dx = -Ω ∇(H (ũ 2 )).∇ũ 2 dx = -Ω | ∇ũ 2 | 2 dx ≤ 0.

  )-(b), this leads to Ũ = U in [0, T ]. Thus, u 1 , u 2 , u 3 ≥ 0 in [0, T ].Third step. Global non negative solution.

Table 1 :

 1 with known parameters and thus we obtain approximations of S H and I H . H (resp. I H ) with variance (σS H ) 2 (resp. (σI H ) 2 ), where σ ∈ R + . The constant σ is chosen in order to obtain the desired error between S H and S meas Parameters values

	• These approximations permit to simulate measured states S meas H	(resp. I meas H	)
	as normal distributions around S H	, and also between I H and I meas H	.
	• We apply the parameter estimation procedure with the measured states S meas H
	and I meas H	to recover the parameters d 1 and d 2 .
	Let us first present the simulations data.

Table 2 :

 2 Ad = B, where d := (d 1 , d 2 ) T .We solve this system by the standard least squares method based upon QR factorization to obtain the parameters approximations d a 1 and d a 2 . The results are summarized in Table2where λ is the relative error between (S H , I H ) and (S meas Calculation results and errors

	2, . . . , M , we set
	t k+1
	A(k, 1) :=	S H (τ, s)∆ψ(s) ds dτ,
	0	Ω
	t k+1
	A(k, 2) :=	I H (τ, s)∆ψ(s) ds dτ,
	0	Ω

  1. Let us prove first that u 1 is bounded . Suppose that there exists t such that u 1 (t, x) > 1 and considert 1 := inf{t | u 1 (t, x) = 1 and u 1 (t + ε, x) > 1, ∀ε > 0}, then ∀t ∈ [0, t 1 ], u 1 (t, x) ≤ 1, u 1 (t 1 , x) = 1. Since ∂ t u 1 (t 1 , x) = -s L ζ 2 (t 1 , x) ζ 3 (t 1 , x) + β m u 3 (t 1 , x) u 1 (t 1 , x) + β m u 3 (t 1 , x)) = -s L ζ 2 (t 1 , x) ζ 3 (t 1 , x) ≤ 0. If ∂ t u 1 (t 1 , x) = 0, then ζ 2 (t 1 , x) = 0 so ∂ t ζ 2 (t 1 , x) = sζ 1 (t 1 , x) ≥ 0. Recall that ζ 3 (t 1 , x) = 0. • ∂ t ζ 2 (t 1 , x) > 0, by continuity, there exists δ 1 such that ζ 2 (t, x) is strictly increasing on (t 1 -δ 1 , t 1 + δ 1 ), then ζ 2 (t, x) < ζ 2 (t 1 , x) = 0, t ∈ (t 1 -δ 1 , t 1 ), which contradicts the positivity of ζ 2 .• ∂ t ζ 2 (t 1 , x) = 0, then ζ 1 (t 1 , x) = 0 and ∂ t ζ 1 (t 1 , x) = bζ 3 (t 1 , x) > 0,so by continuity, there exists δ 2 such that ζ 1 (t, x) is strictly increasing on (t 1 -δ 2 , t 1 + δ 2 ), then ζ 1 (t, x) < ζ 1 (t 1 , x) = 0, t ∈ (t 1 -δ 2 , t 1 ), which contradicts the positivity of ζ 1 .So we have ∂ t u 1 (t 1 , x) < 0, by continuity, there exists δ 3 such that u 1 (t, x) is strictly decreasing on (t 1 -δ 3 , t 1 + δ 3 ), thenu 1 (t 1 -δ 3 , x) > u 1 (t, x) > u 1 (t 1 , x), t ∈ (t 1 -δ 3 , t 1 ). Since u 1 (t, x) > u 1 (t 1 , x) = 1, t ∈ (t 1 -δ 3 , t 1 ), which contradicts the definition of t 1 . So we obtain ∀t ∈ [0, T ], u 1 (t) ∞ ≤ 1.(34) 2. Let t ∈ [0, T ]. Multiplying ∂ t u 2 = -(b H + β H u 1 )u 2 + b H + d 1 ∆u 2 by u 2 and integrating by parts over Ω, we get

	1 2	d dt	u 2 (t) 2 L 2 = -b H u 2 (t) 2 L 2 -β H	Ω	u 1 (t)u 2 2 (t) dx +	Ω	b H u 2 (t) dx
			-d 1

Ω |∇u 2 (t)| 2 dx.

Preprint submitted to Applied Mathematics and ComputationDecember 9, 2019

This algorithm is implemented in in the package DifferentialAlgebra of Maple.

Note that the equations ξ3,0 = 0, ṡ = 0 and ṡL = 0 had been added to (Γ a ) before the elimination procedure.

Since

Solving this differential inequality leads to

3. In the same way, from [START_REF] Xiang | Solving the inverse problem of an SIS epidemic reactiondiffusion model by optimal control methods[END_REF], we obtain that u 3 satisfies

From Cauchy-Schwarz inequality and Young's inequality, for any ε > 0, we get

Taking ε := (γ + b H )/β H , from [START_REF] Yagi | Abstract Parabolic Evolution Equations and their Applications[END_REF], we obtain

Solving this differential inequality leads to

From ( 34), [START_REF] Yagi | Abstract Parabolic Evolution Equations and their Applications[END_REF] and [START_REF] Zhang | Parameter identifiability of Nonlinear systems with time-delay[END_REF] we obtain that U satisfies (4), which leads to the result.


MD, PhD David Saadoun

MD Patrice Cacoub

OBSERVATION Tocilizumab-Induced Drug Reaction With Eosinophilia and Systemic Symptoms Syndrome in Adult-Onset Still Disease: A Case Report

Background: Adult-onset Still disease is a systemic inflammatory condition characterized by spiking fevers; polyarthritis; and an evanescent, salmon-pink, maculopapular rash. Affected persons have high ferritin levels, a high leukocyte count with neutrophil predominance, and low glycosylated ferritin levels. Tocilizumab is a humanized monoclonal antibody against the interleukin-6 receptor that is used to treat refractory adult-onset Still disease. This drug can cause adverse cutaneous events, such as mild maculopapular rash, urticaria, and pruritus [START_REF] Smolen | Effect of interleukin-6 receptor inhibition with tocilizumab in patients with rheumatoid arthritis (OPTION study): a double-blind, placebo-controlled, randomised trial[END_REF].

Objective: To describe what we believe is the first reported case of severe adverse cutaneous effects from tocilizumab.

Case Report: A 55-year-old woman was hospitalized for a 2-month history of spiking fevers, sore throat, arthralgia, evanescent maculopapular rash, leucocytosis (leukocyte count, 23 × 10 9 cells/L with 89% neutrophils), elevated levels of C-reactive protein (485.7 nmol/L) and ferritin (38 199 pmol/L), and decreased levels of glycosylated ferritin (7%). We excluded infectious diseases and lymphoma and diagnosed adult-onset Still disease.

Therapy with oral corticosteroids and methotrexate was not effective and was replaced with tocilizumab monotherapy, 8 mg/kg. Ten days after the first administration, the patient developed an unusual diffuse, persistent, pruritic maculopapular rash (Figure). Laboratory studies documented a decrease in the levels of C-reactive protein (123.8 nmol/L) and ferritin (7981 pmol/L) and the leukocyte count (14 × 10 9 cells/L), which suggested improvement. Skin biopsy results showed lymphocyte and eosinophil infiltration in perivascular areas, which supported a diagnosis of a drug-induced reaction. Topical corticosteroids helped alleviate the rash, but laboratory tests 10 days later revealed hypereosinophilia (eosinophil count, 2.3 × 10 9 cells/L) and liver enzyme levels up to 6 times the upper limit of normal. Polymerase chain reaction testing found a human herpesvirus-6 viral load of 105 copies/mL. We excluded parasitic and blood diseases and diagnosed tocilizumab-induced drug reaction with eosinophilia and systemic symptoms (DRESS) syndrome. The new manifestations that we attributed to this syndrome resolved 5 weeks after we discontinued tocilizumab therapy.

Discussion: The dress syndrome is a potentially lifethreatening combination of severe skin eruption, fever, hypereosinophilia, and internal organ involvement [START_REF] Cacoub | The DRESS syndrome: a literature review[END_REF]. It typically develops 2 to 6 weeks after initiation of the culprit drug. Manifestations also usually include a morbilliform rash with facial edema, lymphadenopathy, elevated liver enzyme levels, and leucocytosis [START_REF] Cacoub | The DRESS syndrome: a literature review[END_REF]. An immune response to herpesvirus reactivation (human herpesvirus types 6 and 7 and Epstein-Barr virus) is associated with a more prolonged and difficult course. There are few standards for diagnosis of DRESS syndrome, which relies on a compatible history and typical clinical manifestations and laboratory results in the absence of other possible conditions.

Diagnosis of this syndrome was difficult because adultonset Still disease could have caused the patient's elevated liver enzyme levels, leukocytosis, fever, and morbilliform rash. We believe that our diagnosis is justified because the patient did not have abnormal results on liver function tests before initiation of tocilizumab therapy; her new morbilliform rash differed from the previous rash caused by adult-onset Still disease; hypereosinophilia is not a typical feature of adultonset Still disease; and her usual manifestations of adult-onset Still disease (arthralgia, standard rash, and hyperferritinemia) had remitted when the new symptoms developed, presumably because tocilizumab therapy was successful. Tocilizumab has been proposed as the cause of other adverse cutaneous reactions, including 1 unconfirmed case of Stevens-Johnson syndrome (3), a few cases of psoriasiform eruptions (4), and 1 case of drug-related generalized exanthematous pustulosis [START_REF] Izquierdo | Acute generalized exanthematous pustulosis due to tocilizumab in a rheumatoid arthritis patient[END_REF].

Elina Zuelgaray, MD

  Assistance Publique-Ho ˆpitaux de Paris, Groupe Hospitalier Pitie ´-Salpe ˆtrie `re Paris, France Fanny Domont, MD Assistance Publique-Ho ˆpitaux de Paris, Groupe Hospitalier Pitie ´-Salpe ˆtrie `re, and Sorbonne Universite ´s Paris, France Nathan Peiffer-Smadja, MD Assistance Publique-Ho ˆpitaux de Paris, Groupe Hospitalier Pitie ´-Salpe ˆtrie `re Paris, France This article was published at Annals.org on 23 May 2017.

Figure .

 . Figure. Rash that developed 10 d after the first administration of tocilizumab.

Annals of Internal Medicine Annals.org Downloaded From: http://annals.org/pdfaccess.ashx?url=/data/journals/aim/0/ by a University of California San Diego User on 05/23/2017


Nathan Peiffer-Smadja

Thomas Hanslik

Pierre Mordant

François-Xavier Lescure

Cultural differences between surgical and medical teams: is it time for comanagement?

Dear Editor,

We have read with much interest the article of Charani and colleagues entitled "The Differences in Antibiotic Decision-making Between Acute Surgical and Acute Medical Teams: An Ethnographic Study of Culture and Team Dynamics". [START_REF] Charani | The Differences in Antibiotic Decision-making Between Acute Surgical and Acute Medical Teams: An Ethnographic Study of Culture and Team Dynamics[END_REF] They demonstrate important differences in the functioning of medical and surgical teams, including a more individualistic approach, a more acute fear of blame and a shorter span of available time for the ward round in surgery.

In their article, they propose such solutions as adapting to surgeons' working hours or engaging in communication platforms used by surgeons. The associated editorial underlines the need for stewardship interventions tailored to the context of surgery. 2 Such solutions are essential, however, most the causes put forward to account for these differences seem to us structural. Indeed, the fact that the surgeons have to share their time between the ward, clinics and the operating room, which forces them to do the round very quickly and makes it difficult to decide collectively, is not easily modifiable. Likewise, the "interventionist" side of surgery and consequently, "the impetus "to "do something"" is part and parcel of surgery.

We propose developing comanagement between surgeons and medical physicians as another solution. In this model, a physician trained in internal medicine takes care of chronic medical comorbidities and acute complications of the surgical patient, including infections and antibiotic prescribing. [START_REF] Whinney | Surgical comanagement: A natural evolution of hospitalist practice[END_REF] Comanagement is different from standard consultation where the surgeon retains sole authority and responsibility for identifying patients who would need a medical evaluation and can ignore the internist's recommendations. [START_REF] Klein | The preoperative consultation. Response to internists' recommendations[END_REF] Comanagement is the result of a formal and negotiated agreement, ideally written down between the surgeon and the internist (sometimes called "hospitalist") to share the responsibility, authority and accountability for the care of surgical patients. The internist takes care of patients without a specific referral by the surgeon and is able to write out medical orders, communicate with the patient and other specialists including pharmacists.

As we are of the opinion that Charani and colleagues' findings are not limited to antibiotic prescribing and concern many medical decisions in surgical wards, comanagement seems a broader solution.

The practice of comanagement is gaining popularity, mainly in the USA. [START_REF] Sharma | Comanagement of hospitalized surgical patients by medicine physicians in the United States[END_REF] The only randomized controlled trial to date found that comanagement reduced the number of complications before hospital discharge. [START_REF] Huddleston | Medical and surgical comanagement after elective hip and knee arthroplasty: a randomized, controlled trial[END_REF] Other studies suggest that comanagement improves patient outcomes, including decreasing mortality, [START_REF] Tadros | The effect of a hospitalist comanagement service on vascular surgery inpatients[END_REF][START_REF] Hinami | Potential role of comanagement in "rescue" of surgical patients[END_REF] and reduced length of hospital stay. [START_REF] Rohatgi | Surgical Comanagement by Hospitalists Improves Patient Outcomes: A Propensity Score Analysis[END_REF] Moreover, surgeons and nurses sound more satisfied with comanagement than medical consultation and it could lead to reduced costs. [START_REF] Huddleston | Medical and surgical comanagement after elective hip and knee arthroplasty: a randomized, controlled trial[END_REF][START_REF] Auerbach | Allison University user on 02 January 2019 patients between neurosurgeons and hospitalists[END_REF] To our knowledge, no study so far has addressed the impact of comanagement of surgical patients on antibiotic prescribing.

In France and the UK, comanagement is not widely used but we think that everybody would benefit from it, starting with surgeons who could spend more time in the operating room while of course participating in medical decisions. Antimicrobial stewardship teams would keep a crucial role advising internists working in surgical wards.

To meet this call, we think that a discussion at national levels should be started on comanagement of surgical patients between internists, surgeons and antimicrobial prescribing specialists. Such a proposal would need the support of public health specialists and political authorities to train more physicians in the medical management of surgical patients.

All authors confirm that they have no potential conflicts of interest to disclose

Downloaded from https://academic.oup.com/cid/advance-article-abstract/doi/10.1093/cid/ciy1133/5267865 by Mount Allison University user on 02 January 2019


Thành H Phan

Nicolas Nam

Rougerie

Nam Ành

Nicolas And

IMPROVED STABILITY FOR 2D ATTRACTIVE BOSE GASES

Improved stability for 2D attractive Bose gases

Introduction

Consider a 2D non-relativistic gas of bosonic particles trapped in the external potential Mathematically this means looking at the action of the many-body Schrödinger operator

V : R 2 → R + , V (x) -→
H N := N j=1 (-i∇ j + A(x j )) 2 + V (x j ) + 1 N -1 1 i<j N N 2β w(N β (x i -x j )) (1.2)
on the symmetric space

H N = L 2 sym (R 2N ) = N sym L 2 (R 2 ).
For nice data V, w, A, this operator is bounded from below with the core domain H N ∩ C ∞ c (R 2N ), and thus can be extended to be a self-adjoint operator by Friedrichs' method. The non-trivial question is then that of stability of the second kind [START_REF] Lieb | The Stability of Matter in Quantum Mechanics[END_REF], i.e. whether H N -CN for a constant C independent of N . Note that when the negative part w -:= min{w, 0} is nonzero, the interaction energy with the rescaled potential in (1.1) may be very negative, and we will see that the stability question cannot be answered using merely the uncertainty principle.

Date: September, 2019.

1

Out of H N one constructs a corresponding non-linear Schrödinger (NLS) functional

E nls [u] = lim N →∞ 1 N u ⊗N |H N |u ⊗N = u|h|u L 2 + b 2 R 2 |u| 4 (1.3) where b = R 2 w and h = (-i∇ + A(x)) 2 + V (1.4)
is the one-body Hamiltonian. Clearly, E nls is bounded from below under the constraint

u L 2 (R 2 ) = 1 if and only if b -a * (1.5)
where a * is the optimal constant for the Gagiardo-Nirenberg inequality

R 2 |u| 2 R 2 |∇u| 2 a * 2 R 2 |u| 4 , ∀u ∈ H 1 (R 2 ). (1.6)
See [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Guo | Symmetry breaking and collapse in Bose-Einstein condensates with attractive interactions[END_REF][START_REF] Maeda | On the symmetry of the ground states of nonlinear Schrödinger equations with potential[END_REF][START_REF] Kwong | Uniqueness of positive solutions of ∆u -u + u p = 0 in R n[END_REF][START_REF] Frank | Ground states of semi-linear PDEs[END_REF] for references. Given the variational construction (1.3), Condition (1.5) is necessary for H N to be stable of the second kind. Sufficient conditions are in fact more stringent. Simple considerations show that we must demand more than (1.5). Here we work with the condition

R 2 |w -| < a * (1.7)
and refer to [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF] for a refinement (Hartree stability).

In this note we are interested in the range of diluteness parameter β > 0 for which the stability of the second kind can be shown to hold. Indeed, β measures how fast the interactions converge to point-like ones. Implicit in the above is that the reference lengthscale of the system is fixed, set by that of the one-body Hamiltonian h (think of particles in a fixed box if you wish). Hence N -β measures the range of the interaction potential in units of the reference length scale. From the point of view of interactions, N β-1/2 is the average number of particles a tagged one interacts with at a time, for N -1/2 is the mean inter-particle distance.

It is well-known that large quantum interacting systems are harder to deal with for large values of β, in particular, in our 2D case, for β > 1/2 which is the threshold to have a dilute system (few but strong inter-particle collisions). In this note we make the remark that a combination of the tools in [START_REF]A note on 2D focusing many-boson systems[END_REF][START_REF]Non linear schrödinger limit of bosonic ground states, again[END_REF] allows to prove stability of the second kind under the condition β < 1. This extends the range of validity of methods [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF][START_REF]A note on 2D focusing many-boson systems[END_REF][START_REF] Chen | The rigorous derivation of the 2D cubic focusing NLS from quantum manybody evolution[END_REF][START_REF] Jeblick | Derivation of the time dependent two dimensional focusing NLS equation[END_REF] dealing with the large-N limit of H N .

Main result

For simplicity we assume that V, A, w are smooth. The actual condition on the interaction potential we need is

w ∈ L 1 (R 2 ) ∩ L 2 (R 2 ).
(2.1) We also assume that there exist positive constants s > 0 and c > 0 such that

V (x) c -1 |x| s -c (
H N = L 2 sym (R 2N ). Remarks.
1. The best result preceding the above is that from [START_REF]A note on 2D focusing many-boson systems[END_REF] which covers β < s + 1 s + 2 with s the exponent in (2.2). This had the merit of allowing a dilute gas, β > 1/2, whereas the previous results [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF] were limited to β < 1/2. It is not our aim to pretend that increasing β > 0 is an undertaking that should go on forever, but there are reasons that make us feel the above is noteworthy. First, the annoying dependence on the trapping potential gets dispensed with (provided the growth is still polynomial). Second, the proof is somewhat cleaner. Third, a natural barrier seems to have been reached: the condition β < 1 is that needed to obtain a second-moment estimate following the techniques of [START_REF] Erdös | Derivation of the nonlinear Schrödinger equation from a many body Coulomb system[END_REF][START_REF] Nam | Ground states of large Bose systems: The Gross-Pitaevskii limit revisited[END_REF][START_REF]A note on 2D focusing many-boson systems[END_REF].

2.

It is not obvious to us whether there should exist a physically natural upper bound on β in the attractive case. For 3D repulsive gases this would be given by the Gross-Pitaevskii [START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF][START_REF]Scaling limits of bosonic ground states[END_REF] limit β = 1. For 2D repulsive gases, any β > 0 should be allowed, for the GP limit [START_REF] Lieb | A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas[END_REF][START_REF] Jeblick | Derivation of the time dependent gross-pitaevskii equation in two dimensions[END_REF] corresponds to an interaction scaled exponentially with N .

3. The proof proceeds by combining the second moment estimate of [START_REF]A note on 2D focusing many-boson systems[END_REF] with the informationtheoretic quantum de Finetti theorem [START_REF] Brandão | Quantum de Finetti Theorems under Local Measurements with Applications[END_REF][START_REF] Li | Quantum de Finetti Theorems under fully-one-way adaptative measurements[END_REF] of Brandão-Harrow. The interest of the latter for large bosonic systems [START_REF]Non linear schrödinger limit of bosonic ground states, again[END_REF] is that it (almost) gives a quantitative de Finetti theorem when the one-body state-space is infinite dimensional (more precisely the dimension should be still finite but only its logarithm enters relevant estimates), thus bypassing the main technical limitation of the tools used in [START_REF]A note on 2D focusing many-boson systems[END_REF].

4.

For the above stability result we may assume A = 0, thanks to the diamagnetic inequality [16,Theorem 7.21], as well as ignore the bosonic symmetry (i.e. the lower bound holds true on the full space L 2 (R 2N )). However, the presence of the magnetic field and the Bose-Einstein statistics are meaningful for the Corollary 2.2 below.

The stability result has two corollaries regarding the large N limit of the system at hand. First for ground states: Corollary 2.2 (NLS limit for ground states). Under the above assumptions, the ground state energy per particle

E(N ) N = 1 N inf σ H N (H N ) = 1 N inf{ Ψ, H N Ψ : Ψ H N = 1}
converges when N → ∞ to the ground state energy of the NLS functional in (1.3), i.e.

E nls = inf{E nls (u) : u L 2 (R 2 ) = 1}.
Moreover, the reduced density matrices {γ

(k)
Ψ N } N of ground states Ψ N of H N converge to convex combinations of projections on NLS minimizers, namely there exists a Borel probability measure µ supported on the minimizers of E nls such that, along a subsequence N → ∞,

lim N →∞ Tr γ (k) Ψ N -|u ⊗k u ⊗k |dµ(u) = 0, ∀k ∈ N.
(2.5)

Next, for dynamics:

Corollary 2.3 (NLS limit for dynamics).

Under the above assumptions and with A ≡ 0, the many-body Schrödinger dynamics

i∂ t Ψ N (t) = H N Ψ N (t)
starting from a well-prepared initial datum, e.g. Ψ N (0) = u(0) ⊗N with u(0) smooth, converges when N → ∞ to the NLS dynamics

i∂ t u(t) = hu(t) + b|u(t)| 2 u(t)
in the sense of reduced density matrices

lim N →∞ Tr γ (k) Ψ N (t) -|u(t) ⊗k u(t) ⊗k | = 0, ∀k ∈ N, ∀t ∈ R. ( 2
u ⊗N = |u ⊗k u ⊗k |.
Thus the convergences (2.5), (2.6) tells us that the corresponding many-body wave functions are close to product states in a weak sense (it is well known that they are not close in L 2 -norm).

2.

Although the statements cover also repulsive gases (w 0), the novelty lies mainly in the attractive case. If w 0 the proof is much simpler indeed and one can reach much higher values of β; see [START_REF] Lieb | A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas[END_REF] for ground states and [START_REF] Jeblick | Derivation of the time dependent gross-pitaevskii equation in two dimensions[END_REF] for dynamics.

3.

We do not even sketch the proof of Corollary 2.2. The reader should have no difficulty in figuring out that, given Theorem 2.1, the proof of [14, Theorem 1] applies mutatis mutandis for β < 1, at least under the additional condition |A(x)| 2

V (x). The latter technical condition can be replaced by (2.3) following arguments from [22, Section 4.2].

4.

For the dynamical statement, the method of [START_REF] Jeblick | Derivation of the time dependent two dimensional focusing NLS equation[END_REF] allows to derive NLS dynamics provided (2.4) holds. Hence, Theorem 2.1 extends the range of validity of their main result to all β < 1. Such a range was obtained previously in [START_REF] Nam | Norm approximation for many-body quantum dynamics: focusing case in low dimensions[END_REF] (without using (2.4)), and a more restricted one in [START_REF] Chen | The rigorous derivation of the 2D cubic focusing NLS from quantum manybody evolution[END_REF] (in the special case s = 2). We also refer to these references for precise descriptions of the well-prepared initial data that can be covered.

Remark 2.4 (Classically stable case in 3D). Stability of the second kind is also an issue for 3D Bose gases with potentials

N 3β-1 w(N β x)
having an attractive part, in the dilute regime β > 1/3. Because of the respective scalings of the mean-field interaction and kinetic energy one must then assume [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF][START_REF] Triay | Derivation of the dipolar Gross-Pitaevskii energy[END_REF] that the unscaled potential w is classically stable

1 i<j N w(x i -x j ) -CN
for all x 1 , . . . , x N ∈ R d . This holds for exampe if w 0.

Our method also covers this case, but we refrain from stating details, for the range of β we are allowed to reach is 1/3 < β < 9/26. Triay, in his study of the dipolar Bose gas [START_REF] Triay | Derivation of the dipolar Gross-Pitaevskii energy[END_REF], has already obtained 1/3 < β < 1/3 + s/(45 + 42s) where s is the exponent in (2.2). Only for rather small values of s does our estimate improve on his. The 3D case for a restricted class of potentials with an attractive part has also been considered [START_REF] Lee | Ground state energy of dilute Bose gases in small negative potential case[END_REF][START_REF] Yin | The ground state energy of the dilute Bose gas in potentials with positive scattering length[END_REF] for homogeneous gases in the thermodynamic limit. ⋄

Proof

This being for a large part an improvement on [START_REF]A note on 2D focusing many-boson systems[END_REF][START_REF]Non linear schrödinger limit of bosonic ground states, again[END_REF], we shall be brief. See [START_REF] Benedikter | Effective Evolution Equations from Quantum Dynamics[END_REF][START_REF] Golse | On the Dynamics of Large Particle Systems in the Mean Field Limit[END_REF][START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF][START_REF]Scaling limits of bosonic ground states[END_REF][START_REF] Rougerie | De Finetti theorems, mean-field limits and Bose-Einstein condensation[END_REF][START_REF] De | limites de champ moyen et condensation de Bose-Einstein, Les cours Peccot[END_REF][START_REF] Schlein | Derivation of effective evolution equations from microscopic quantum dynamics[END_REF] for general background on large N limits of bosonic quantum systems.

Recall that we write the proof for A ≡ 0, which implies the general statement using the diamagnetic inequality [16, Theorem 7.21].

Old arguments. Consider the spectral projectors

P = 1 h Λ , Q = 1 -P (3.1)
with Λ > 0 a (large enough) one-body energy cut-off. Let

H 2 := h 1 + h 2 + N 2β w(N β (x 1 -x 2 ))
be the two-body Hamiltonian associated with (1.2). Let Ψ N be a ground state of H N and let γ

(k)

N be its k-body density matrix. Then

1 N E(N ) = 1 N Ψ N |H N |Ψ N = 1 2 Tr H 2 γ (2) N . (3.2)
We start with a localization lemma, essentially a restatement of [START_REF]A note on 2D focusing many-boson systems[END_REF]Equation (46)]:

Lemma 3.1 (Localization).
With the above notation, for any δ > 1/2 there exists a C δ > 0 such that

Tr H 2 -P ⊗2 H 2 P ⊗2 γ (2) Ψ N -C δ Λ (δ-1)/2 Tr h γ (1) Ψ N (1-δ)/2 Tr h ⊗ h γ (2) Ψ N δ .
To put the above to good use we need a priori bounds on the first and second moments of the one-body Hamiltonian. The following is [14, Lemma 5], and this is where we use our main assumption β < 1 and also the requirement w ∈ L 2 (R 2 ).

Lemma 3.2 (Moments).

Let 0 < β < 1. For all ε ∈ (0, 1) we have

Tr h γ (1) Ψ N C 1 + |e N,ε | ε and Tr h ⊗ h γ (2) Ψ N C 1 + |e N,ε | ε 2 (3.3)
where

e N,ε := N -1 inf Ψ∈H N , Ψ =1 Ψ H N -ε N j=1 h j Ψ . (3.4)
3.2. New argument. We rely on a version of the quantum de Finetti theorem from [START_REF] Brandão | Quantum de Finetti Theorems under Local Measurements with Applications[END_REF][START_REF] Li | Quantum de Finetti Theorems under fully-one-way adaptative measurements[END_REF]:

Lemma 3.3 (de Finetti).
Let H be a complex separable Hilbert space, and H N = H ⊗symN the corresponding bosonic space. Let γ

N be the 2-body reduced density matrix of a N -body state vector Ψ N ∈ H N (or a general mixed state).

Let P be a finite dimensional orthogonal projector. There exists a Borel measure µ

(2) N with total mass 1 on the set of one-body mixed states

S P := {γ positive trace-class operator on P H, Tr γ = 1} (3.5)
such that

Tr A ⊗ B P ⊗2 γ (2)
N P ⊗2 -γ ⊗2 dµ (2)
N (γ) C log(dim(P )) N ||A|| ||B|| (3.6)
for all A, B self-adjoint operators on P H. The norm in the right-hand side is the operator norm.

Proof. The proof of [START_REF] Brandão | Quantum de Finetti Theorems under Local Measurements with Applications[END_REF] gives the statement with A, B replaced by quantum measurements.

In [START_REF]Non linear schrödinger limit of bosonic ground states, again[END_REF]Proposition 3.2] it is explained how the statement with A, B positive operators follows. The full result is obtained by decomposing self-adjoint operators in the manner

A = 1 A<0 A + 1 A 0 A
as used already in [START_REF] Girardot | Average field approximation for almost bosonic anyons in a magnetic field[END_REF].

To apply the above we shall, as in [START_REF]Non linear schrödinger limit of bosonic ground states, again[END_REF], decompose the interaction operator using the Fourier transform in the manner

N 2β w(N β (x -y)) = R 2 N 2β w(N β •)(k)e ik•x e -ik•y dk = R 2 w(N -β k) (cos (k • x) cos (k • y) + sin (k • x) sin (k • y)) dk (3.7)
and apply Lemma 3.3 for each k. We interject a simple control of the involved multiplication operators: Lemma 3.4 (Multiplication by plane waves). Let k ∈ R 2 , k = 0 and e k be the multiplication operator on L 2 (R 2 ) by either cos(k • x) or sin(k • x). Let P be the spectral projector in (3.1). As operators

± P e k P min 1, C Λ 1/2 |k| . (3.8)
Proof. The first upper bound 1 is obvious as |e k | 1. We write the proof of the second bound for e k the multiplication by cos(k • x) (the case of sin(k • x) is similar). For any smooth compactly supported function f , integrating by parts,

f |e k |f = - R 2 sin(k • x) |k| k |k| • ∇ |f | 2 , whence | f |e k |f | 2 |k| R 2 |f ||∇f | 2 |k| R 2 |f | 2 1/2 R 2 |∇f | 2 1/2 .
Applying the above with f = P g, g ∈ L 2 (R d ) proves the claim (recall we work without a magnetic field).

We can finally give the Proof of Theorem 2.1. According to Lemma 3.1 and Lemma 3.2 we have

Tr H 2 γ (2) N Tr P ⊗2 H 2 P ⊗2 γ (2) N -C δ Λ (δ-1)/2 Tr hγ (1) N (1-δ)/2 Tr h ⊗ hγ (2) N δ Tr P ⊗2 H 2 P ⊗2 γ (2) N -C δ Λ (δ-1)/2 1 + |e N,ε | ε (1+3δ)/2
(3.9)

for any 1/2 < δ 1 and 0 < ε < 1. On the other hand we may use Lemma 3.3: denoting

γ N (2) := γ ⊗2 dµ (2)
N (γ)

we have

Tr P ⊗2 H 2 P ⊗2 γ (2) N Tr P ⊗2 H 2 P ⊗2 γ N (2) -C log(Λ) N   Λ + e k ∈{cos(k•x),sin(k•x)} R 2 ||P e k P || 2 | w(N -β k)|dk   . (3.10)
Here we have decomposed the interaction term as in (3.7), used the triangle inequality with (3.6) and recalled that dim(P ) depends at worst polynomially on Λ as recalled e.g. in [START_REF] Lewin | The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases[END_REF]Lemma 3.3] (this is the place where Condition (2.2) is used). The error term also contains the operator norm 2Λ of P ⊗2 (h 1 + h 2 )P ⊗2 . The second term on the right side of (3.10) can be estimated using Lemma 3.4 and Condition (2.1) (the latter ensures that w ∈ L 2 ∩ L ∞ ):

R 2 ||P e k P || 2 | w(N -β k)|dk R 2 min{1, CΛ|k| -2 }| w(N -β k)|dk |k| 1 w L ∞ dk + C 1<|k| N β Λ|k| -2 w L ∞ dk + C |k|>N β Λ|k| -2 | w(N -β k)|dk C + CΛ log N + CΛ.
As regards the first term on the right-hand side of (3.10) we write

Tr P ⊗2 H 2 P ⊗2 γ N (2) = E H [γ]dµ (2)
N (γ)

where

E H [γ] = Tr (hγ) + 1 2 R 2 ×R 2 ρ γ (x)N 2β w(N β ((x -y))ρ γ (y)dxdy.
Here the density ρ γ (x) = γ(x, x) (defined properly by the spectral decomposition) satisfies

R 2 ρ γ = Tr γ = 1.
Using the diamagnetic inequality [16,Theorem 7.21] and the convexity of the kinetic energy [16,Theorem 7.8] we have the Hoffmann-Ostenhof-type inequality

Tr (hγ) R 2 |∇ √ ρ γ | 2 .
Then inserting the Cauchy-Schwarz inequality in the interaction term,

N 2β w(N β ((x -y))ρ γ (x)ρ γ (y)dxdy -N 2β |w -(N β ((x -y))| ρ γ (x) 2 + ρ γ (y) 2 2 ,
and combining with the Gagiardo-Nirenberg inequality (1.6) and Assumption (1.7), we get

E H [γ] R 2 |∇ √ ρ γ | 2 - 1 2 |w -| R 2 ρ 2 γ 0. (3.11)
Thus the first term on the right side of (3.10) is nonnegative. Therefore, (3.9) reduces to

e N = Tr H 2 γ (2) N -C log(Λ) N Λ log N -C δ Λ (δ-1)/2 1 + |e N,ε | ε (1+3δ)/2
.

Moreover by a simple trial state argument we know that e N C. In summary, we have

|e N | C + C log(Λ) N Λ log N + C δ Λ (δ-1)/2 1 + |e N,ε | ε (1+3δ)/2
(3.12) for all 1/2 < δ 1 and 0 < ε < 1. Now we bootstrap the above arguments. Let ε 0 ∈ (0, 1) be a fixed constant such that Clearly we can pick 0 < a < 1/2 and b > 0 small enough so that this implies

|e N,ε | C ε 1 + N α-c
for a fixed constant c > 0 (independent of N, ε, α). Thus we have shown that if (3.13) holds with α > 0, it also holds with α replaced by max(α -c, 0). After finitely many steps of this procedure, we deduce that actually |e N,ε | is bounded independently of N , which implies the theorem.

  an external magnetic field B of vector potential A : R 2 → R 2 such that curl A = B. The particles interact via the pair potential of the form 1 N -1 N 2β w(N β x), (1.1) with a fixed (N -independent) parameter β > 0 and a fixed function w : R 2 → R, w(x) = w(-x), w(x) -→ |x|→∞ 0.

. 6 ). 1 .

 61 RemarksRecall that the k-particles reduced density matrix of any Ψ ∈ H N is defined asγ (k) Ψ := Tr k+1→N |Ψ Ψ|.Here Tr k+1→N means the partial trace over N -k factors of H N = N sym L 2 (R 2 ). In particular, for the product state we have γ (k)

R 2 w

 2 -> -a * (1 -ε 0 ). Assume we know that there is a α > 0 such that|e N,ε | C ε,α N α for all 0 < ε < ε 0 . (3.13)We can start the bootstrap from α = 2β, using the simple one-body inequality1 -∆ x -N 2β w(N β x) -CN 2βThen, we can apply the above arguments to the ε-perturbed Hamiltonian in(3.4) to bound e N,ε . Note that the lower bound (3.11) remains valid with w replaced by (1 -ε) -1 w, provided that 1 < ε < ε 0 . Combining with (3.13) we obtain|e N,ε | C ε + C ε log(Λ) N Λ log N + C δ,ε Λ (δ-1)/2 N α(1+3δ)/2 , ∀0 < ε < ε 0 .If we choose Λ = N α+a and δ = 1/2 + b ∈ (1/2, 1) in the above we obtain |e N,ε | C ε,a,b 1 + N α+a-1/2 log N + N α+2bα+ab/2-a/4 .

Here we do not need the improved bound O(N

2β-1 ) in [14,Lemma 2] which requires w ∈ L 1 .

Acknowledgements. We thank Mathieu Lewin and Fernando G.S.L. Brandão for helpful discussions. We received fundings from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (Grant agreement CORFRONMAT No 758620), and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy (EXC-2111-390814868).