Mouhamadou Diaby

Abderrahman Iggidr
email: abderrahmann.iggidr@inria.fr

Mamadou Sy

Abdou Sène

Global analysis of a schistosomiasis infection model with biological control

Keywords: Epidemic models, Nonlinear dynamical systems, Monotone systems, Global stability, Reproduction number, Schistosomiasis

In this paper, the global stability of a schistosomiasis infection model that involves human and intermediate snail hosts as well as an additional mammalian host and a competitor snail species is studied by constructing Lyapunov functions and using properties of K monotone systems. We derive the basic reproduction number R 0 for the deterministic model, and establish that the global dynamics are completely determined by the values of R 0 . We show that the disease can be eradicated when R 0 ≤ 1. In the case where R 0 > 1, we prove the existence, uniqueness and global asymptotic stability of an endemic steady state. This mathematical analysis of the model gives insight about the epidemiological consequences of the introduction of a competitor resistant snail species.

Introduction

Schistosomiasis also known as bilharzia is a parasite-induced disease. The prevalence of Schistosomiasis is high in tropical and sub-tropical countries, especially in poor rural regions without access to safe drinking water and adequate sanitation. The parasites, schistosomes, have to go through an intermediate host (snails in most cases) to complete their life cycle: from eggs, to miracidia, to cercaria, finally to adult worms. Schistosomes have two stages of reproduction: the first one consists in a sexual reproduction in the blood vessels of humans and the second stage is an asexual amplification in snails. Many strategies are used to prevent and control the schistosomiasis infection: chemotherapy, chemical molluscicides, improvement of sanitary conditions, snail control. For many endemic situations, mass drug administration with praziquantel is recommended by the WHO (World Health Organization) [START_REF] Who | Schistosomiasis[END_REF]. However, as reported by WHO, "a major limitation to schistosomiasis control has been access to praziquantel. Available data show that only 10% of people requiring treatment were reached in 2011". The price of a chemical antischistosomial is still too expensive for people in many developing countries. Biological control by introducing competitive snails of the intermediate snail hosts is a control strategy that may require relatively little funding. For instance, Schistosomiasis infection has declined in Puerto Rico due to the introduction of Thiara granifera, a snail which competes with the intermediate host snail of schistosomes Biomphalaria glabrata (see [START_REF] Pointier | Biological control of the snail hosts of schistosomiasis in areas of low transmission: the example of the caribbean area[END_REF], [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]). Mathematical modeling and analysis of schistosomiasis has drawn many attentions since the first paper by MacDonald in [START_REF] Macdonald | The dynamics of helminth infections, with special reference to schistosomes[END_REF]. Thereafter many others researchers built excellent models and developed a decent understanding of transmission mechanism of schistosomiasis (see [START_REF] Cohen | Mathematical models of schistosomiasis[END_REF][START_REF] Chen | Mathematical modelling and control of schistosomiasis in hubei province, china[END_REF][START_REF] Woolhouse | On the application of mathematical models of schistosome transmission dynamics. i. natural transmission[END_REF]). Recently, a schistosomiasis infection model described by E.J Allen and H.D Victory [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF] has been proposed. This model generalizes in some way, previous mathematical models such as those described by Anderson and May [START_REF] Anderson | Helminth infections of humans: mathematical models, population dynamics, and control[END_REF]; Kimbir [START_REF] Kimbir | A mathematical model of the transmission dynamics of schistosomiasis[END_REF].

Here, we consider the model proposed in [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. This model allows competition between the intermediate host snails and a resistant snail species in order to study the advantages of biological control. In [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF], computational simulations have been done to estimate some parameters and to study control of the infection by chemotherapy and biological control using snail competition. In this paper, we propose a complete mathematical analysis of the deterministic model of [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. A stability analysis is provided to study the epidemiological consequences of control strategies. Therefore, the specific objectives are to determine the threshold parameter that measure initial disease transmission and to analyze the steady states stability. This reproduction number can be used to assess the potential impact of biological control in the schistosomiasis disease eradication. We show that the DFE is globally asymptotically stable by constructing a Lyapunov function. The existence and uniqueness of a nontrivial equilibrium (endemic equilibrium) is shown by applying a fixed point Theorem due to Hethcote and Thieme [START_REF] Hethcote | Stability of the endemic equilibrium in epidemic models with subpopulations[END_REF] and its local asymptotic stability is investigated using a Krasnoselskii sublinearity trick. The global asymptotic stability of this unique endemic equilibrium is shown using properties of K-monotone systems [START_REF] Hirsch | Stability and convergence in strongly monotone dynamical systems[END_REF]. The paper is organized as follows. In Section 2 we present the model described by E.J Allen and H.D Victory [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. Its well-posedness is established and a reduced model is proposed. In Section 3, we determine R 0 the basic reproductive number of the model and we establish the global asymptotic stability of the disease-free equilibrium when the basic reproductive number is less or equal to one. Section 4 is devoted to the study of the system behavior when the basic reproductive number is larger than one. In this case, by using some properties of K-monotone systems (see [START_REF] Smith | Systems of Ordinary Differential Equations Which Generate an Order Preserving Flow. A Survey of Results[END_REF]), we prove the existence of a unique endemic equilibrium and we show its global asymptotic stability. Finally, in Section 5 , we present some discussions about the characteristics of the competitor resistant snail species that can be used to eliminate the disease.

Model framework

We consider the model presented in [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. In their work, four definite mammalian host sub-populations, three intermediate snail host sub-populations, and a population of resistant competitor snails are considered. We assume that the human births and deaths dynamics can be neglected compared to the infection dynamics. Further, it is assumed that infected snails and infected mammals do not recover from schistosomiasis as their life span are short in comparison to that for humans. The state variables of the model are:

• H s (t) the susceptible (uninfected) human population size.

• H i (t) the infected human population size.

• S s (t) the susceptible snail host population size.

• S e (t) the population size of the infected snails which are not yet shedding cercariae (latent population size).

• S i (t) the infected and shedding snail population size(shedding population size) .

• S rc (t) the competitor snail population size(resistant to infection).

• M s (t) the susceptible mammal population size.

• M i (t) the infected mammal population size.

In addition, the population of snails as well as mammals are assumed to be competitive. Birth and death rates for the various sub-populations will be denoted by b i and d i . The transmission parameters for the model are:

• t 15 = t HsS i = transmission rate from infected snails to uninfected humans,

• t 32 = t SsH i = transmission rate from infected humans to uninfected snails,

• t 38 = t SsM i = transmission rate from infected mammals to susceptible snail,

• t 75 = t MsS i = transmission rate from infected snails to susceptible mammals.

The time evolution of the different populations is governed by the following system of equations:

                                                                   dH s dt = -
dt = b 7 (M s + M i ) -t 75 S i M s -c 77 M s (M s + M i ) -d 7 M s , dM i dt = t 75 S i M s -d 8 M i -c 88 M i (M s + M i ).
(

dN Si dt = (b 3 -d 3 ) N Si -c 33 N 2 Si -c 36 S rc N Si .
It follows that

dN Si dt ≤ (b 3 -d 3 ) N Si -c 33 N 2 Si = b 3 -d 3 c 33 -N Si c 33 N Si .
The dynamics of the resistant snails population and the total mammals pop-ulation (N M = M s + M i ) are respectively governed by

dS rc dt = (b 6 -d 6 ) S rc -c 66 S 2 rc -c 64 S rc (S s + S e + S i ) ≤ b 6 -d 6 c 66 -S rc c 66 S rc , dN M dt = (b 7 -d 7 )N M -c 77 N 2 M = b 7 -d 7 c 77 -N M c 77 N M .
Thus the feasible region for the system (1) is

D = {(H s , H i , S s , S e , S i , S rc , M s , M i ) ∈ IR 8 + : N Si ≤ b 3 -d 3 c 33 , N M ≤ b 7 -d 7 c 77 , S rc ≤ b 6 -d 6 c 66 }.
Proposition 2.1. The compact set D is positively invariant and attracting under the flow described by [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF].

Proof. It is sufficient to consider the system on the faces of D and to show that for each face, the vector fields associated to the system points into the set D.

If N Si = 0 then dN Si dt = 0. If N Si = b 3 -d 3 c 33 then dN Si dt ≤ 0. If N M = 0 then dN M dt = 0. If N M = b 7 -d 7 c 77 then dN M dt ≤ 0. If S rc = 0 then dS rc dt = 0. If S rc = b 6 -d 6 c 66 then dS rc dt = -c 64 b 6 -d 6 c 66 (S s + S e + S i ) ≤ 0.
Furthermore, the model ( 1) is well-posed epidemiologically. Hence, it is sufficient to study the dynamics of the basic model in D.

Reduction of the system

We will reduce the stability analysis of (1), to the study of a smaller and simpler system. The following theorem(see [START_REF] Vidyasagar | Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability[END_REF] ) will permit us to reduce the stability analysis to a smaller system.

Theorem 2.1 ([22], Theorem 3.1). Consider the following

C 1 system            ẋ = f (x); x ∈ R n y ∈ R m , ẏ = g(x, y);
with an equilibrium point (x * , y * ) i.e., f (x * ) = 0 and g(x * , y * ) = 0.

(

) 2
If x * is globally asymptotically stable (GAS) in R n for the system ẋ = f (x), and if y * is GAS in IR m , for the system ẏ = g(x * , y), then (x * , y * ) is (locally) asymptotically stable for (2). Moreover, if all the trajectories of ( 2) are forward bounded, then (x * , y * ) is GAS for (2).

We define the proportions of the no resistant snails by:

s j = S j N Si for j =
s, e, i, and the proportions of mammals m j = M j N M for j = s, i.

Using ṡj = Ṡj N Si - ṄSi N Si s j = Ṡj N Si -(a 3 -c 33 N Si -c 36 S rc ) s j , ṁj = Ṁj N M - ṄM N M m j = Ṁj N M -(b 7 -d 7 -c 77 N M ) m j
, and the fact that

s s + s e + s i = 1, m s + m i = 1
, simple computations show that the new variables H i , s e , s i , m i , N Si , S rc , and N M satisfy:

                                                   dH i dt = t 15 (N H -H i ) s i N Si -r 12 H i , ds e dt = (t 32 H i + t 38 N M m i )(1 -s e -s i ) -(b 3 + r 54 ) s e , ds i dt = r 54 s e -b 3 s i , dm i dt = t 75 N Si s i (1 -m i ) -b 7 m i , dN Si dt = a 3 (b 3 -d 3 ) N Si -c 33 N 2 Si -c 36 S rc N Si = X 1 (N Si , S rc ), dS rc dt = a 6 (b 6 -d 6 ) S rc -c 64 S rc N Si -c 66 S rc S rc = X 2 (N Si , S rc ), dN M dt = (b 7 -d 7 )N M -c 77 N 2 M .
(

) 3
This system is triangular: the last three equations do not depend on the variables H i , s e , s i , m i .

The last equation has two equilibria: the trivial one which is unstable and a positive equilibrium

N * M = b 7 -d 7 c 77 which is GAS.
Let us consider the following subsystem:

         dN Si dt = a 3 (b 3 -d 3 ) N Si -c 33 N 2 Si -c 36 S rc N Si = X 1 (N Si , S rc ), dS rc dt = a 6 (b 6 -d 6 ) S rc -c 64 S rc N Si -c 66 S rc S rc = X 2 (N Si , S rc ), (4)
The equilibria of (4) are: (0, 0) which is unstable: two positive eigenvalues a 3 and a 6 .

E 1 = a 3 c 33 , 0 with eigenvalues -a 3 and a 6 - c 64 a 3 c 33 = c 33 a 6 -c 64 a 3 c 33 . E 2 = 0, a 6 c 66 with eigenvalues -a 6 and a 3 - c 36 a 6 c 66 = c 66 a 3 -c 36 a 6 c 66 . If c 33 a 6 -c 64 a 3 < 0 then E 1 is LAS. If c 66 a 3 -c 36 a 6 < 0 then E 2 is LAS.
The system (4) has a positive equilibrium E * :

       N * Si = c 36 a 6 -
We remark that

N * Si = c 66 a 3 -c 36 a 6 c 33 a 6 -c 64 a 3 S * rc (6)
The equilibrium E * = (N * Si , S * rc ) exists if and only if 1. 1st case: c 33 c 66 -c 36 c 64 > 0. In this case the existence of E * implies that E * is LAS and the other are unstable. 2. 2nd case: c 33 c 66 -c 36 c 64 < 0. In this case E * exists if c 33 a 6 -c 64 a 3 < 0 and c 66 a 3 -c 36 a 6 < 0. In this case E 1 and E 2 are LAS but E * is unstable.

We shall assume that E 1 and E 2 are unstable which implies that c 33 a 6 -c 64 a 3 > 0 and c 66 a 3 -c 36 a 6 > 0.

This implies c 33 c 66 -c 36 c 64 > 0.

In this case E * is LAS: eigenvalues with negative real part. Let V a lyapunov function defined as follows :

V = (N Si -N * Si log N Si ) + d (S rc -S * rc log S rc ).
Then, its derivative along the solutions of (4) satisfies :

V = (N Si -N * Si )(a 3 -c 33 N Si -c 36 S rc ) + d (S rc -S * rc )(a 6 -c 64 N Si -c 66 S rc ).
Using equilibria relations, we obtain:

V = (N Si -N * Si )(c 33 N * Si + c 36 S * rc -c 33 N Si -c 36 S rc ) + d (S rc -S * rc )(c 64 N * Si +c 66 S * rc -c 64 N Si -c 66 S rc ) = -c 33 (N Si -N * Si ) 2 -d c 66 (S rc -S * rc ) 2 -c 36 (N Si -N * Si )(S rc -S * rc ) -d c 64 (N Si -N * Si )(S rc -S * rc ) = -c 33 (N Si -N * Si ) 2 -d c 66 (S rc -S * rc ) 2 -(c 36 + d c 64 )(N Si -N * Si )(S rc -S * rc ).
We choose d = c 66 a 2 3 c 33 a 2

6

. With this and using [START_REF] Diaby | Analysis of a schistosomiasis infection model with spatial structure[END_REF] we can show

(c 36 + d c 64 ) 2 -4 d c 33 c 66 < 0, (9)
then V is definite negative and hence the equilibrium

(N * Si , S * rc , N * M ) is GAS. Then, under the condition (8), (N * Si , S * rc , N * M ) is GAS. Remark 2.1.
It is also possible to prove the GAS of (N * Si , S * rc ) by using Dulac criterion [START_REF] Hale | Ordinary differential equations[END_REF] with the function ρ(N Si , S rc ) = 1 N Si S rc defined on the

set U =]0, b 3 -d 3 c 33 [×]0, b 6 -d 6 c 66 [. We have ∂(ρX 1 ) ∂N Si + ∂(ρX 2 ) ∂S rc = - c 33 S rc + c 66 N Si < 0.
Therefore, under the assumption (8) and using Theorem 2.1, the stability properties of system [START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF] on the set D are the same as those of the following reduced system

                     dH i dt = t 15 (N H -H i ) N * Si s i -r 12 H i , ds e dt = (t 32 H i + t 38 N * M m i )(1 -s e -s i ) -(b 3 + r 54 ) s e , ds i dt = r 54 s e -b 3 s i , dm i dt = t 75 N * Si s i (1 -m i ) -b 7 m i , (10)
defined on the set

D 1 = {0 ≤ H i ≤ N H , 0 ≤ s e + s i ≤ 1, 0 ≤ m i ≤ 1}
3. The Disease-free equilibrium and the basic reproductive number

In this section, we will give an analytic expression for R 0 the basic reproductive number of the system (for more details concerning the definition and the computation of R 0 one can see [START_REF] Van Den Driessche | Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF][START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF][START_REF] Castillo-Chavez | On the computation of R 0 and its role on global stability[END_REF]), and completely answer the stability question for the disease-free equilibrium (DFE). As usual ρ(M ) is the spectral radius of the matrix M .

3.1. R 0 and the local stability of the DFE Proposition 3.1. The DFE for system [START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF] is the origin (H i , s e , s i , m i ) = (0, 0, 0, 0) = E 0 and the basic reproduction ratio is given by:

R 0 = r 54 N * Si (b 7 t 15 t 32 N H + r 12 t 38 t 75 N * M ) b 3 b 7 r 12 (b 3 + r 54 ) = T 1/2 0 . Moreover The DFE is LAS if T 0 < 1 and is unstable if T 0 > 1.
Proof. It is clear that the DFE is E 0 = (0, 0, 0, 0). Using, the now standard techniques [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF][START_REF] Van Den Driessche | Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF], it is easy to show that the basic reproduction ratio is

R 0 = ρ(             0 r 54 t 15 N H N * Si b 2 3 + r 54 b 3 t 15 N H N * Si b 3 0 t 32 r 12 0 0 t 38 N * M b 7 0 0 0 0 0 r 54 t 75 N * Si b 2 3 + r 54 b 3 t 75 N * Si b 3 0            
).

A simple computation gives:

R 0 = r 54 N * Si (b 7 t 15 t 32 N H + r 12 t 38 t 75 N * M ) b 3 b 7 r 12 (b 3 + r 54 ) = T 1/2 0 .
The Jacobian matrix of ( 3) at E 0 is

J 0 =     -r 12 0 t 15 N H N * Si 0 t 32 -(b 3 + r 54 ) 0 t 38 N * M 0 r 54 -b 3 0 0 0 t 75 N * Si -b 7     .
J 0 is a Metzler matrix and we can write

J 0 = F + V with F =     0 0 t 15 N H N * Si 0 t 32 0 0 t 38 N * M 0 0 0 0 0 0 t 75 N * Si 0     , V =     -r 12 0 0 0 0 -(b 3 + r 54 ) 0 0 0 r 54 -b 3 0 0 0 0 -b 7     .
We have F > 0 and V is Metzler stable, see [START_REF] Berman | Nonnegative matrices in the mathematical sciences[END_REF][START_REF] Jacquez | Qualitative theory of compartmental systems[END_REF][START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF][START_REF] Brauer | Mathematical models in population biology and epidemiology[END_REF]. Thanks to Varga's Theorem in [START_REF] Varga | Matrix iterative analysis[END_REF]:

s(J 0 ) ≤ 0 iff ρ(-F V -1 ) ≤ 1. Since ρ(-F V -1 ) = R 0 , we then deduce that E 0 is LAS if R 0 < 1 and is unstable if R 0 > 1.

Global stability of the DFE

We have the following global behavior for system (3) when R 0 ≤ 1.

Theorem 3.1. If R 0 ≤ 1 then the DFE is GAS.

Proof. Consider the candidate Lyapunov function:

V = t 32 r 12 H i + s e + b 3 + r 54 r 54 s i + t 38 N * M b 7 m i .
Its derivative along the solutions of (10) satisfies:

V = t 32 r 12 t 15 (N H -H i ) s i N * Si -t 32 H i +(t 32 H i + t 38 N * M m i )(1 -s e -s i ) -(b 3 + r 54 ) s e + b 3 + r 54 r 54 (r 54 s e -b 3 s i ) + t 38 N * M b 7 (t 75 N * Si s i (1 -m i ) -b 7 m i ) = -t 32 H i s i -t 38 N * M m i s i +s e b 3 + r 54 r 54 r 54 -(b 3 + r 54 ) -t 32 H i -t 38 N * M m i +s i t 32 r 12 t 15 N * Si (N H -H i ) - b 3 + r 54 r 54 b 3 + t 38 t 75 N * M b 7 N * Si (1 -m i ) = -(s e + s i ) (t 32 H i + t 38 N * M m i ) +s i t 32 r 12 t 15 N * Si (N H -H i ) - b 3 + r 54 r 54 b 3 + t 38 t 75 N * M b 7 N * Si (1 -m i ) . Hence V ≤ -(s e + s i ) (t 32 H i + t 38 N * M m i ) +s i t 32 r 12 t 15 N * Si N H - b 3 + r 54 r 54 b 3 + t 38 t 75 N * M b 7 N * Si = -(s e + s i ) (t 32 H i + t 38 N * M m i ) +s i b 3 + r 54 r 54 b 3 r 54 N * Si (b 7 t 15 t 32 N H + r 12 t 38 t 75 N * M ) b 3 (b 3 + r 54 ) b 7 r 12 -1 . Therefore V ≤ -(s e + s i ) (t 32 H i + t 38 N * M m i ) + b 3 + r 54 r 54 b 3 (R 2 0 -1) s i ≤ 0 if R 0 ≤ 1. If R 0 < 1, then V = 0 implies s i = s e = 0, or s i = m i = H i = 0 and if R 0 = 1,
then V = 0 implies s i = s e = 0, or m i = H i = 0. However, in each case the largest invariant subset of V = 0 is the singleton E 0 . Thus, by LaSalle Invariance Principle [START_REF] Lasalle | The stability of dynamical systems[END_REF], E 0 is GAS on D 1 when R 0 ≤ 1.

Endemic Equilibrium

Existence and uniqueness of the endemic equilibrium

We have shown that the disease-free equilibrium is globally asymptotically stable when R 0 ≤ 1. In the following, we will prove that system (10) has a unique endemic equilibrium when R 0 > 1. To this end we formulate the problem as a fixed point problem. Then we use a theorem due to Hethcote and Thieme ([19], Theorem 2.1) for the existence and uniqueness of a positive fixed point of a multi-variable function. To be self contained, we recall hereafter the result of Hethcote and Thieme:

An equilibrium point (H *

i , s * e , s * i , m * i ) for system [START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF] satisfies

s * i = r 54 b 3 s * e , t 15 (N H -H * i ) s * i N * Si -r 12 H * i = 0.
This implies

H * i = t 15 N * Si N H r 54 b 3 s * e r 12 + t 15 N * Si r 54 b 3 s * e .
Solving the equations in [START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF] at steady state gives

s * e = b 3 r 54 s * i , m * i = t 75 N * Si t 75 N * Si s * i + b 7 s * i , H * i = t 15 N H N * Si s * i t 15 N * Si s * i + r 12 . (11)
Thus

s * e = (t 32 H * i + t 38 N * M m * i ) (b 3 + r 54 ) + (t 32 H * i + t 38 N * M m * i )(1 + r 54 b 3 )
.

The fourth equation gives .

m * i = N * Si t 75
We write this as

U = F (U ),
where

U =   H i s e m i   , and
F =   F 1 F 2 F 3   =                     t 15 N * Si N H r 54 b 3 s e r 12 + t 15 N * Si r 54 b 3 s e (t 32 H i + t 38 N * M m i ) (b 3 + r 54 ) + (t 32 H i + t 38 N * M m i )(1 + r 54 b 3 ) N * Si t 75 r 54 b 3 s e b 7 + N * Si t 75 r 54 b 3 s e                     . It is easy to remark that F 1 ≤ N H , F 2 ≤ 1 and F 3 ≤ 1.
The equilibrium points satisfy the relation U = F (U ). We use this formulation to prove existence and uniqueness of an endemic equilibrium point.

Let Ω = {(H i , s e , m i ) : 0 ≤ H i ≤ N H , 0 ≤ s e ≤ 1, 0 ≤ m i ≤ 1}.
The function F is a continuous bounded function that maps Ω into itself and it is infinitely differentiable with Jacobian

F (U ) =   0 J 12 0 J 21 0 J 23 0 J 32 0  
where:

J 12 = b 3 r 12 r 54 t 15 N H N * Si (b 3 r 12 + r 54 t 15 N * Si s e ) 2 , J 21 = b 2 3 t 32 (b 3 + r 54 ) (b 3 + t 32 H i + t 38 N * M m i ) 2 , J 23 = b 2 3 t 38 N * M (b 3 + r 54 ) (b 3 + t 32 H i + t 38 N * M m i ) 2 , J 32 = b 3 b 7 r 54 t 75 N * Si (b 3 b 7 + r 54 t 75 N * Si s e ) 2 .
The off-diagonal elements of the matrix F (U ) are non-negative. Thus the function F (U ) is monotone non-decreasing. We have F (0) = 0 and ρ(F (0)) = R 0 > 1. Thanks to the graph theory, we claim that F (0) is irreducible because the associated graph of the matrix is strongly connected. Let us now prove that F is strictly sublinear in Ω, i.e., F (λ U ) > λ F (U ), for any U ∈ Ω with U > 0, and λ ∈ (0, 1). Some calculations give

λ F 1 (U ) F 1 (λ U ) = r 12 + t 15 N * Si r 54 b 3 r s e r 12 + t 15 N * Si r 54 b 3 s e < 1. λ F 2 (U ) F 2 (λ U ) = (b 3 + r 54 ) + (t 32 H i + t 38 N * M m i )(1 + r 54 b 3 ) (b 3 + r 54 ) + r (t 32 H i + t 38 N * M m i )(1 + r 54 b 3 ) < 1. λ F 3 (U ) F 3 (λ U ) = b 7 + N * Si t 75 r 54 b 3 λ s e b 7 + N * Si t 75 r 54 b 3 s e < 1.
So the function F (U ) is strictly sublinear. In this way we have proved the following theorem Theorem 4.2. If R 0 ≤ 1, the only equilibrium point of the system is the disease-free equilibrium E 0 . If R 0 > 1, there also exists a unique endemic equilibrium E * in int(Ω) whose coordinates satisfy [START_REF] Hirsch | Stability and convergence in strongly monotone dynamical systems[END_REF].

Local Stability of the Endemic Equilibrium

In this section, we shall prove the local stability of the endemic equilibrium when R 0 > 1. For this we shall follow the method given by Hethcote and Thieme, which is based on a Krasnoselskii technique [START_REF] Hethcote | Stability of the endemic equilibrium in epidemic models with subpopulations[END_REF]. A usual way to prove the local asymptotic stability of an equilibrium point x of the system of differential equations ẋ = f (x) is to prove that the linearized equation ż = Df (x) z has no solutions of the form

z(t) = Z exp(wt) (12)
with Z ∈ C n \ {0}, w ∈ C and Re(w) ≥ 0, i.e., w Z = Df (x) Z with Z ∈ C n \ {0}, w ∈ C implies Re(w) < 0. Substituting a solution of the form [START_REF] Jacquez | Qualitative theory of compartmental systems[END_REF] in the linearized equation at the endemic equilibrium, we obtain the following linear equations.

                           w Z 1 = -(t 15 s * i N * Si + r 12 ) Z 1 + t 15 (N H -H * i ) N * Si Z 3 , w Z 2 = (1 -s * e -s * i ) t 32 Z 1 -(t 32 H * i + t 38 N * M m * i ) Z 2 -(b 3 + r 54 ) Z 2 -(t 32 H * i + t 38 N * M m * i ) Z 3 +N * M t 38 (1 -s e -s i ) Z 4 , w Z 3 = r 54 Z 2 -b 3 Z 3 , w Z 4 = t 75 N * Si (1 -m * i ) Z 3 -(t 75 N * Si s * i + b 7 ) Z 4 . (13)
Solving for Z 3 from the third equation of ( 13), and substituting the result into the second equation (and simplifying), gives the equivalent system

                         1 + w + t 15 s * i N * Si r 12 Z 1 = t 15 (N H -H * i ) N * Si r 12 Z 3 , (1 + G 2 (w)) Z 2 = 1 -s * e -s * i b 3 + r 54 t 32 Z 1 + (t 38 N * M m * i ) Z 4 , 1 + w b 3 Z 3 = r 54 b 3 Z 2 , 1 + t 75 N * Si s * i b 7 Z 4 = t 75 N * Si (1 -m * i ) b 7 Z 3 ,
where

G 2 (w) = w b 3 + r 54 + (t 32 H i + t 38 N * M m i ) b 3 + r 54 1 + r 54 w + b 3 .
Denoting in the same way

G 1 (w) = w + t 15 N * Si s * i r 12 , G 3 (w) = w b 3 , G 4 (w) = w + t 75 N * Si s * i b 7 ,
we obtain the system

                 [1 + G 1 (w)] Z 1 = (H Z) 3 , [1 + G 2 (w)] Z 2 = (H Z) 1 + (H Z) 4 , [1 + G 3 (w)] Z 3 = (H Z) 2 , [1 + G 4 (w)] Z 4 = (H Z) 3 , (14)
with

H =           0 0 t 15 (N H -H * i ) N * Si r 12 0 t 32 (1 -s * e -s * i ) b 3 + r 54 0 0 t 38 N * M (1 -s * e -s * i ) b 3 + r 54 0 r 54 b 3 0 0 0 0 t 75 N * Si (1 -m * i ) b 7 0          
.

System [START_REF] Lasalle | The stability of dynamical systems[END_REF] can spells in a compact way as follows

(1 + G i (w)) Z i = (H Z) i ,
where Z = (Z 1 , Z 2 , Z 3 , Z 4 ) with the lexicographic order. Note that the notation (H Z) i (with i = 1, ..., 4)) denotes the ith coordinate of the vector H Z. It should further be noted that the matrix H has non-negative entries, and the equilibrium

E * = (H * i , s * e , s * i , m * i ) satisfies E * = H E * .
Furthermore, since the coordinates of E * are all positive, it follows then that if Z is a solution of [START_REF] Lasalle | The stability of dynamical systems[END_REF], then it is possible to find a minimal positive real number ρ, depending on Z, such that

| Z |≤ ρ E * , (15)
where

| Z |= (| Z 1 |, | Z 2 |, | Z 3 |, | Z 4 |) and | • | is a norm in C
. Now we want to show that Re(w) < 0. We assume that Re(w) ≥ 0. From this assumption one has two cases : w = 0 and w = 0. In the first case, the determinant ∆ of the homogeneous linear system (13) in the variable Z i (i = 1, ..., 4) corresponds to the determinant of the matrix

             -1 -G 1 (0) 0 t 15 N * Si (N H -H * i ) r 12 0 t 32 (1 -s * e -s * i ) b 3 + r 54 -1 -G 2 (0) 0 t 38 N * M (1 -s * e -s * i ) b 3 + r 54 0 r 54 b 3 -1 -G 3 (0) 0 0 0 t 75 N * Si (1 -m * i ) b 7 -1 -G 4 (0)              .
Tacking into account that G 3 (0) = 0, some elementary calculations give

∆ = 1 + G 4 (0) 1 + G 1 (0) + G 2 (0) + G 1 (0)G 2 (0) - 1 -s * e -s * i (b 3 + r 54 ) s * e t 32 H * i 1 + G 4 (0) + t 38 N * M m * i 1 + G 1 (0) . Denoting α = max{1 + G 1 (0), 1 + G 4 (0)
}, and using the equilibrium relation

(t 32 H * i + t 38 N * M m * i )(1 -s * e -s * i ) -(b 3 + r 54 ) s * e = 0, we have ∆ > 1 + G 4 (0) 1 + G 1 (0) 1 + G 2 (0) -α.
This shows that ∆ > 0 since G 1 (0), G 2 (0), and G 4 (0) are positive. Therefore, for w = 0, the only solution of the system ( 14) is the trivial one Z = 0 which implies that w = 0. Assume now that w = 0, and Re w ≥ 0. Let G(w) = min {|1 + G i (w)| , i = 1, ..., 4}. It is easy to prove that in this case |1 + G i (w)| > 1 for all i, and therefore G(w) > 1. Taking norms on both sides of ( 14), and using the fact that H is non-negative, we obtain the following inequality:

G(w) | Z |≤ H | Z | . (16)
Using ( 15) and ( 16), we get

G(w) | Z |≤ ρ H E * = ρ E * .
Which implies

| Z |≤ ρ G(w) E * < ρ E * .
But this contradicts the minimality of ρ. Therefore Re(w) < 0. We have then proved the following theorem.

Theorem 4.3. If R 0 > 1, then the positive endemic equilibrium E * of the system (10) is locally asymptotically stable.

Global Stability of the Endemic Equilibrium

In this section we will establish the global stability of the unique endemic equilibrium point when R 0 > 1. We shall use the properties of K-monotone systems for the analysis of our system (see [START_REF] Smith | Systems of Ordinary Differential Equations Which Generate an Order Preserving Flow. A Survey of Results[END_REF]). We begin by recalling the definition of K-monotone systems as well as a result concerning convergence properties of strongly monotone systems. Let

ẋ = f (x) ( 17
)
where f is a continuously differentiable function defined on a convex, open set U in IR n . Let q = (q 1 , ..., q n ), q i ∈ {0, 1}, 1 ≤ i ≤ n, and K = {x ∈ IR n : (-1) q i x i ≥ 0, 1 ≤ i ≤ n}. The solution φ t (x) of ( 17) preserves the partial ordering ≤ K (for t ≥ 0) and system ( 17) is said to be of type Kmonotone if whenever x, y ∈ U with x ≤ K y then φ t (x) ≤ K φ t (y) for all t ≥ 0 for which both φ t (x) and φ t (y) are defined. The following lemma gives necessary and sufficient conditions for [START_REF] Pointier | Biological control of the snail hosts of schistosomiasis in areas of low transmission: the example of the caribbean area[END_REF] to be a type K-monotone system in the case that f ∈ C 1 (U ) for an open convex set U ∈ IR n .

Lemma 4.1. (Smith [START_REF] Smith | Systems of Ordinary Differential Equations Which Generate an Order Preserving Flow. A Survey of Results[END_REF], Lemma 2.1)

If f ∈ C 1 (U )
where U is open and convex in IR n then φ t preserves the partial ordering ≤ K for t ≥ 0 if only if P q Df (x)P q has nonnegative off-diagonal elements for every x ∈ U , where P q = diag ((-1) q 1 , ..., (-1) qn ).

For strongly monotone systems, we recall the following useful result:

Theorem 4.4. (Hirsh [START_REF] Hirsch | Stability and convergence in strongly monotone dynamical systems[END_REF], theorem 10.3) Suppose X is an open subset of a strongly ordered topological vector space, and f generates a strongly monotone flow in X. Let W ⊂ X be an open set of points with compact orbit closures, and assume that there is a unique equilibrium p in x∈W ω(x). Then φ t (x) -→ p for all x ∈ W .

We shall use this global convergence result to prove the following result Theorem 4.5. If R 0 > 1, then the positive endemic equilibrium state E * of the system (10) is globally asymptotically stable in the interior of the set D 1 .

Proof. We rewrite the system (10) using the variables (H i , s s , s i , m i ) instead of (H i , s e , s i , m i ) used in the previous sections. We get the following system:

                       dH i dt = t 15 (N H -H i ) N * Si s i -r 12 H i , ds s dt = b 3 -(t 32 H i + t 38 N * M m i + b 3 ) s s , ds i dt = r 54 (1 -s s -s i ) -b 3 s i , dm i dt = t 75 N * Si s i (1 -m i ) -b 7 m i .                        = f (H i , s s , s i , m i ) = f (x). ( 18
) With Jacobian Df (x) =     -r 12 -t 15 N * Si s i 0 t 15 N * Si (N H -H i ) 0 -t 32 s s -b 3 -t 32 H i -t 38 N * M m i 0 -t 38 N * M s s 0 -r 54 -b 3 -r 54 0 0 0 t 75 N * Si (1 -m i ) -b 7 -t 75 N * Si s i     .
If we choose the matrix P as

P =     1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1     .
Then the matrix P Df (x) P is a Metzler matrix, i.e., it has nonnegative offdiagonal elements for all x ∈ D 1 . Thus, system (18) is K-monotone in D1 (the interior of D 1 ) with respect to the partial ordering defined by the orthant K = {H i ≥ 0, s s ≤ 0, s i ≥ 0, m i ≥ 0}. Moreover it is strongly monotone with respect to the orthant K because Df (x) is irreducible. Thanks Hirsch's theorem 4.4 and the fact that we have only one endemic equilibrium E * in D1 which is locally asymptotically stable when R 0 > 1 we can conclude that E * is globally asymptotically stable in D1 when R 0 > 1.

Biological control

The first way to eradicate the disease is to use a competitor resistant snail species that is able to eliminate the population of the intermediate host snails. This means that system (4) admits E 2 = 0, a 6 c 66 as a globally asymptotically stable equilibrium state. This equilibrium corresponds to the absence of the intermediate host snails: N Si = 0. According to the computations done in Paragraph 2.1, the parameters must satisfy c 66 a 3c 36 a 6 < 0 and the other equilibrium E 1 = a 3 c 33 , 0 must be unstable which implies that we must have c 33 a 6 -c 64 a 3 > 0. Remark that when the two above conditions are satisfied then the coexistence equilibrium E * does not exist.

To summarize, the population of the intermediate host snails can be eliminated if the parameters satisfy the following condition

a 6 a 3 = b 6 -d 6 b 3 -d 3 > max c 66 c 36 , c 64 c 33 ( 19
)
This situation is illustrated in Figure 1. The simulation has been done using parameter values from [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF].

However the above condition ( 19) is strong and may be hard to satisfy in practice. Therefore a second method to eradicate the disease without completely eliminating the intermediate host snails is to choose a competitor resistant snail species in such a way that R 0 ≤ 1. Recall that

The right-hand side expression of the above inequality (20) depends only on the disease characteristics as well as on the human , mammal and intermediate snail demographic parameters. It does not depend on the resistant competitor snail species.

The competitor characteristics have then to satisfy the inequality [START_REF] Van Den Driessche | Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF] in order to eradicate the disease. This means that the resistant competitor snail species has to be able to reduce the steady state size of the intermediate snail species under some quantity that depends on the disease characteristics (r 12 , r 54 , t 15 , t 32 , t 75 ), on the demographic parameters of the intermediate snail, and on the human and mammal populations.

Summary and conclusions

In this paper, we have presented a stability analysis of a deterministic model for the transmission dynamics of a schistosomiasis infection. Eight sub population sizes were considered: human host susceptible and infected, snail intermediate host susceptible, latent, and shedding, resistant competitor snail, mammal host susceptible and infected. The snails competition is used to control the transmission of the disease. Mathematical properties of the model are analyzed and used to reduce the dimension of the system under consideration. The reproductive number R 0 is then analytically and explicitly computed. We proved that the disease-free steady state E 0 is globally asymptotically stable if R 0 ≤ 1.

We have also established the existence and uniqueness of an endemic equilibrium E * in the case where R 0 > 1. Using some properties of monotone systems we have proved the global asymptotic stability of the endemic equilibrium when it exists i.e., when R 0 > 1.

In a more realistic situation, the speed of a river should affect the transmission dynamics of schistosomiasis. We are working on this perspective. A model with spatial structure involving the present ODE system coupled with a shallow water equations is being studied, see [START_REF] Diaby | Analysis of a schistosomiasis infection model with spatial structure[END_REF].

  ) It is assumed for simplicity that d 3 = d 4 = d 5 , d 7 = d 8 , c 33 = c 44 = c 55 , c 77 = c 88 and c 46 = c 56 = c 36 . The total human population N H = H s + H i is constant since dN H dt = 0. The total no resistant snails population is denoted N Si = S s + S e + S i . Its time-evolution is governed by

c 66 a 3 c

 3 36 c 64 -c 33 c 66 = c 66 a 3 -c 36 a 6 c 33 c 66 -c 36 c 64 , S * rc = c 64 a 3 -c 33 a 6 c 36 c 64 -c 33 c 66 = c 33 a 6 -c 64 a 3 c 33 c 66 -c 36 c 64 .

Theorem 4 . 1 .

 41 ([19], Theorem 2.1) Let F (x) be a continuous, monotone non-decreasing, strictly sublinear, bounded function which maps the nonnegative orthant IR n + = [0, ∞) into itself. Let F (0) = 0 and F (0) exists and be irreducible. Then F (x) does not have a non-trivial fixed point on the boundary of IR n + . Moreover, F (x) has a positive fixed point iff ρ(F (0)) > 1. If there is a positive fixed point, then it is unique.

r

  54 N * Si (b 7 t 15 t 32 N H + r 12 t 38 t 75 N * M ) b 3 b 7 r 12 (b 3 + r 54 ) .In the expression of R 0 , the only quantity that depends on the competitor resistant snail species is N * Si since (by relation (5)) N * Si = c 36 a 6 -c 66 a 3 c 36 c 64 -c 33 c 66 and the other terms do not depend on the competitor. We can remark that the introduction of a competitor reduces the value of R 0 since the value of the steady state size of the intermediate host snail in the absence of the competitor (which is equal to a 3 c 33 ) is larger than its value in the presence of the competitor. Now, R 0 ≤ 1 iff c 36 a 6 -c 66 a 3 c 36 c 64 -c 33 c 66 = N * Si ≤ b 3 b 7 r 12 (b 3 + r 54 ) r 54 (b 7 t 15 t 32 N H + r 12 t 38 t 75 N * M )

Figure 1 :

 1 Figure 1: Evolution of the latent S e (t) and infected S i (t) snails when the elimination condition (19) is satisfied.

  t 15 S i H s + r 12 H i , dH i dt = t 15 S i H s -r 12 H i , dS s dt = b 3 (S s + S e + S i ) -t 32 H i S s -d 3 S s -c 33 S s (S s + S e + S i ) -c 36 S s S rc -t 38 S s M i , dS e dt = t 32 H i S s + t 38 S s M i -d 4 S e -c 44 S e (S s + S e + S i ) -c 46 S e S rc -r 54 S e , dS i dt = r 54 S e -d 5 S i -c 55 S i (S s + S e + S i ) -c 56 S i S rc , dS rc dt = b 6 S rc -c 64 S rc (S s + S e + S i ) -c 66 S rc S rc -d 6 S rc , dM s


Lionel Velut

Christophe Vergez

Joel Gilbert

Fabrice Silva

Numerical Simulation of the Production of Pedal Notes in Brass Instruments

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


An Answer to the Growth of Mathematical Knowledge?

The Répertoire Bibliographique des Sciences Mathématiques

Laurent Rollet & Philippe Nabonnand * In March 1885, the Société Mathématique de France (SMF) conceived the project of a bibliographical catalogue of 19 th century mathematical sciences. Facing the constant increase of mathematical journals and the exponential growth of mathematical knowledge, the members of the SMF felt the lack of a bibliographical tool for advanced students and mathematicians. The Répertoire Bibliographique des Sciences Mathématiques was to be the answer of the SMF to the expansion of mathematics.

This long-term bibliographical enterprise -which was directed by Henri Poincarérapidly became an international endeavour: for almost 27 years, about 50 mathematicians from 16 different countries went through more than 300 mathematical journals. Between 1885 and 1912, more than 20,000 bibliographical references to mathematical works (articles, books, treatises, etc.) were identified, collected, and published, using a methodical and systematic classification.

This repertoire constitutes an important stage for the history of 19 th century science. The reconstitution of its history represents an interpretative key for the study of the internationalisation of science. Moreover, it may be an essential source for investigations concerning the organisation of disciplinary frontiers within mathematical sciences.

The aim of this article is to set out the first elements of an investigation devoted to the history and the operation of the Répertoire Bibliographique des Sciences Mathématiques.

First we will describe the scientific and intellectual context of its emergence. Secondly, we will relate the history of its creation. Thirdly, by using a few elements extracted from our computer database, we will try to give a general survey of the contribution of this bibliography to the history of mathematics.

I -19 th Century Science: A Growing Concern for Bibliography

During the second half of 19 th century, a large number of disruptions affected European science: an unprecedented increase of research and a growing specialisation in every domain;

the organisation of institutions in networks; an institutionalisation of research in most European countries via the creation of academies, learned societies and universities; an increase in the number of university students, notably in France after the 1880 and 1890 reforms; finally, the considerable acceleration of the internationalisation of science (for instance the organisation of the first international congresses).

Besides these disruptions, one should add an important phenomenon: the exponential growth of the number of scientific journals. At the end of 19 th century many scholars perceived the constant increase of printed matter as a serious problem. Two simple figures can give some idea of this situation: in 1890, international statistics of printed matters estimated that 100,000 items a year were published. In 1900, one estimated the annual production at

1868-1944

Bulletin des sciences mathématiques.

This journal was created by Darboux and Hoüel in 1870 as the Bulletin des sciences mathématiques et astronomique. In 1885, it became the Bulletin des sciences mathématiques.

1870-…

Revue semestrielle des publications mathématiques.

Published by the Amsterdam Mathematical Society (each volume recorded about 3000 bibliographical references).

1893-1934

Catalogue of Scientific Papers.

Published by the London Royal Society. It recorded about 40,000 publications (not exclusively devoted to mathematics).

1867-1925

International Catalogue of Scientific Literature (section A, Mathématiques).

This catalogue was the successor of the Catalogue of Scientific Papers. It collected about 25,000 bibliographical references.

1902-1921

Bibliographie mathématique de Valentin. This was compiled by Georg Valentin in Berlin. It probably contained more than 150,000 references but was never published.

1885-1910

4 Nevertheless, one can easily find such bibliographical endeavours before 19 th century. For instance, in the case of mathematics, at least two bibliographies are significant: J. E. Scheibel's Einleitung zur mathematischen Bücherkenntnis published in Breslau from 1769 to 1798; F. W. A. Murhard's Literatur der mathematischen Wissenschaften published in Leipzig from 1797 to 1805 (both recording about 10,000 mathematical works).

5 Concerning Otlet's bibliographical works, see [START_REF] Rasmussen | L'Internationale scientifique: 1890 -1914[END_REF]], [START_REF] Otlet | Transformations opérées dans l'appareil bibliographique des sciences[END_REF]], [Otlet1908], [START_REF] Otlet | Transformations opérées dans l'appareil bibliographique des sciences[END_REF]] and [Rayward (Ed.)

II -The first years: 1885-1894

The SMF formulated the idea of a bibliographical repertoire for mathematics in 1885. Very rapidly the society diffused a preliminary project within the international mathematical community. One can find a first mention of the project in a lettre circulaire of March 1885 entitled "Projet de répertoire bibliographique". This letter began with an acknowledgment of the incapacity of mathematicians regarding the increase of mathematical publications and went on with a presentation of the different steps of the project: (1) Compilation of bibliographical references on individual index cards, (2) ordering of the cards with the help of a suitable classification and (3) publication. According to Eneström this proposition was warmly welcomed; consequently, a month later, the SMF sent another circular that gave more precise instructions 9 .

Nevertheless, the project was defined and accepted four years later, in 1889. Moreover the first set of hundred printed index cards was not to be published before 1894. Such a time lag finds its main source in the complexity and the sophistication of the classification adopted.

6 For further information concerning the bibliographical repertoires devoted to various fields, see 7 [START_REF] Neuenschwander | Les Journaux mathématiques[END_REF]], p. 1836. During this congress -that was supported by the Ministry of Trade, Industry and Coloniesseveral decisions were taken:

1) It was decided that the bibliography would consist in an inventory of the titles of memoirs in pure and applied mathematics published from 1800 until 1889, as well as works concerning the history and the philosophy of mathematics from 1600 to 1889.

In concrete words, the purpose was to establish a systematic list of all articles and memoirs published in the most important mathematical journals.

2) It was decided that the titles of memoirs would be classified, not by the authors'

names, but according to the logical order of subjects.

3) The participants of the congress decided to exclude the publications on astronomy that had already been mentioned in the astronomical bibliography compiled by Houzeau and Lancaster in the 1880's.

4) The titles of the works written in languages other than French, Italian, German, Spanish or Latin were to be translated into French. Besides, the commission appealed to a generalized adoption of the classification index by mathematicians and mathematical journals. However, despite this strong international participation, one should not think that every country worked at the same level. It is indeed likely that several countries never sent any bibliographical reference to the Paris office: it is for instance the case for Great Britain and United States, partly because they were involved in other bibliographical projects. Moreover, the international dimension of the project was counterbalanced by the fact that the bibliography was created in Paris, directed from Paris, subsidized by French authorities and published in France. It is indeed very difficult to know whether it met with success or was distributed outside of France.

In the beginning, the SMF had planned to publish the bibliography as a book. Such a choice would have implied a very slow publication process since it would have been necessary to wait for the end of the perusal process before printing anything. Consequently Maurice d'Ocagne and Roland Bonaparte proposed to publish the mathematical bibliography as a collection of printed index cards. This option offered the possibility to start the publication as soon as a sufficient number of bibliographical items had been collected.

Despite this arrangement, the publication lasted for almost 20 years: the first set of cards was published in 1894 and the last one, the 20 th , in 1912).

The perusal of mathematical periodicals probably began in 1888 but it acquired its cruising speed after the 1889 congress. The following procedure was adopted for the preparation and the publication of the series: (1) All the memoirs and periodicals of a given country were placed under the responsibility of a member of the permanent commission living and working

Reproduction of a printed index card

Three main mathematical domains were distinguished in the classification of the index:

(1) Mathematical Analysis; (2) Geometry;

(3) Applied Mathematics. Within these domains, several classes were identified and designated by a capital letter (see table below). For instance, the class A contained works devoted to elementary algebra, theory of algebraic and transcendent equations, Galois groups, rational fractions and interpolation. Each class (A to X) was generally divided into subclasses, divisions, sections and, even, subsections. For instance, the code L 1 4c was used to designate the class of conics and surfaces of second degree (L), the subclass of conics ( 1), the division corresponding to tangents (4), the section dealing with tangents satisfying specific conditions (c) and the subsection corresponding to the case of right angles ().

Mathematical domain Classes

Mathematical Analysis A-J Geometry K-Q Applied Mathematics R-X

Repartition of the classes in each mathematical domain

The repertoire was published from 1894 until 1912. During these 18 years, 20 series of 100 index cards were published. Each series contained about 1000 bibliographical references and one can therefore estimate that these series represent a list of 20,000 mathematical works.

More precisely, one can distinguish three moments in the publication of the series; they correspond to the three attempts to cover the whole classification, as shown in the following The three periods of publication

III -Is the mathematical repertoire a new tool for historians of mathematics?

The intention of the SMF was to create a bibliographical tool for scholars in mathematics.

Some journals adopted the classification of the index (for instance the Bulletin de la Société Mathématique de France). Nevertheless, it is quite difficult to determine whether the mathematical community really used it. We are in possession of various documents concerning the creation and the publication of this bibliography, but there remain many points about which our information is still fragmentary. How many copies did the commission permanente print (and sell)? Was the bibliography largely distributed outside France? Why is it so difficult to find copies of the index cards in French or foreign libraries?

11 Moreover, one should add some other questions concerning the general context of its creation: it would be important to determine the relationships between this project and universalistic and internationalist trends at the turn of 19 th century. The mathematical repertoire constitutes a modest project compared to similar attempts in Germany, Belgium and Great Britain and one can consider that it had three major challengers:

( (2) The Universal Bibliographical Repertoire. The Belgians Paul Otlet and Henri

Lafontaine created this repertoire around 1895. It was conceived as a universal bibliography of human thought. This repertoire had some success for some years but French mathematicians had some serious reservations.

(3) The International Catalogue of Scientific Literature. The London Royal Society created this project in the 1890's. Its aim was to publish, at an international level, regular bibliographies on every field of scientific research. Although the French scientific community was involved in this project, professional mathematicians (especially in France) also expressed strong reservations 13 .

From a scientific point of view, the mathematical repertoire raises the problem of the internal structure of 19 th century mathematics. At first glance, this bibliography can constitute an interesting tool for the study of mathematical production over a large period; its 20,000 bibliographical references constitute a valuable source of investigation for historians of mathematics. However, this is probably not the main point. Indeed, with its specific and complex classification, the mathematical repertoire gives an outline of a specific conception of mathematics. It does not only give access to mathematics but also to the conception of mathematics that was constructed by the mathematical community at that time. Does it give an adequate description of 19 th century mathematics? In the light of recent works in history of science, can its classification be considered as conservative or should it be seen as modern?

Did the classification really take into account all the existing mathematical domains of 19 th century mathematics? Did the classification contain gaps and if so, what were they?

The creators of the project didn't only propose a thematic ordering of mathematical fields;

they also elaborated a personal definition of the mathematical corpus that the repertoire should take into account: they were for instance convinced that most of the works concerning pure and applied mathematics published since 1800 would be of interest for scholars. On the other hand, they excluded from their corpus classical works that didn't contain general results, notably those intended for students and for the preparation of examinations; in a similar way, they decided that the works in applied mathematics would be mentioned only in the case they Obviously, these 8 journals were published in Germany, Austria or France. Such a phenomenon can be explained by the strong mathematical domination of France and Germany 15 These figures only concern 2723 bibliographical references and it would be quite hazardous to consider that they provide an adequate account of the whole repertoire. Nevertheless, these items correspond to the first attempt to cover the whole classification (from A to X) and may be interesting, for two reasons: first, they furnish some indications about the complexity of this bibliography. Secondly, they give some idea of the richness of the classification and of the extent of the work that had been done in 1895. 16 This journal was published in Prague, a city that was part of the Austrian territory. 17 The repertoire also took mathematical monographs (books, treatises, etc.) into account. The first 3 series contain around 50 items of this type.

during 19 th century; as we will see, the presence of Austria finds probably its origin in the strong activity of Emil Weyr's crew at the beginning of the perusal work. As an illustration, when, for each class of the index, we only take into account the most important journals listed (more than 20 references for each periodical), the only remaining ones are those that were edited in France, Germany and Austria. It is then possible to give an overview of the repartition of these countries according to the three mathematical fields (graph below). and Spain correspond to the heading "Mémoires ou ouvrages séparés". These nations were taking part in the broadening of mathematical production and it is likely that they used the repertoire in order to gain some recognition at an international level. Moreover, one can assume that the mathematicians of these countries judged that their national periodicals weren't important enough and that monographs and books constituted the best illustration of their national production.

20 7200 references come from the Comptes rendus de l'Académie des Sciences. 21 The case of Sweden is also quite paradoxical. Eneström was the representative of the country within the Commission Permanente and he had written several articles in favour of the constitution of a mathematical bibliography during the 1880's. Nevertheless, he rapidly showed some serious reservations about the French project, partly because of its system of classification. His growing inertia towards the French repertoire is a plausible explanation for the absence of Sweden. . Concerning these 300 index files, it is important to notice that 113 of these 157 codes refer to bibliographical items that were just as well published before 1840 as after 1870. One can therefore conclude that the most part of the mathematical domains designated by these codes were active over a large period of 19 th century. Furthermore, 11 classification codes are exclusively devoted to mathematical works published after 1860; 8 of these codes concern mathematical domains that are emergent in the second half of 19 th century

24

. The presence of two other domainsone concerning the properties of general involutions 22 These figures come from Charles Laisant's report in 1900 [START_REF] Laisant | Sur l'état d'avancement des travaux du Répertoire bibliographique des sciences mathématiques[END_REF]].

23 Moreover, in a 1900 report, Charles Laisant confirmed that the first ten series that had been printed at that time didn't use more than 300 classification codes of the index [START_REF] Laisant | Sur l'état d'avancement des travaux du Répertoire bibliographique des sciences mathématiques[END_REF]]. To conclude, it might be interesting to notice that two major mathematical journals seem to be absent from the whole repertoire: the Mathematische Annalen and the Nouvelles annales de mathématiques. The case of the Mathematische Annalen is astonishing since Weyr announced the beginning of the perusal of this journal to Poincaré in 1892

28

. The case of the Nouvelles annales de mathématiques is quite different: it probably reveals the (intentional?) choice made by French mathematicians for a specific style of mathematical: i. e. 'official', academic and professional mathematics to the detriment of mathematics for education, a domain that was not very much recognized.

According to us, this mathematical bibliography can constitute a valuable source for historians of mathematics; in particular, it gives access to the conception of mathematics that was constructed by 19 th century mathematical community. Our analyses will be of course completed at the end of the data entry process. Moreover, a possible development of this research could consist in a comparison between this bibliography and some others repertoires, like the Belgian Répertoire Bibliographique Universel or the Jahrbuch über die Fortschritte der Mathematik. Finally, the understanding of the expectations, difficulties, illusions, etc.

raised by this first attempt to organize mathematical activity at an international level might be helpful for contemporary mathematicians, at a time when many international projects are devoted to the numerisation of past or present mathematical works.

IV -Bibliography

Besterman, Theodore 1949 A World Bibliography of Bibliographies and of Bibliographical Catalogues, Calendars, Abstracts, Digests, Indexes, and the Like, Genève, Societas Bibliographica. Third edition. Eneström, Gustaf 1890 "Sur les Bibliographies des sciences mathématiques", Bibliotheca mathematica 4, 2 nd series, pp. 37-42. 1895 "Répertoire Bibliographique des Sciences Mathématiques", Bibiotheca mathematica 9, 2 nd series, pp. 29.

Commission permanente du

8

  It is possible to find some echoes of these discussions in [Eneström 1890]. 9 [Eneström 1890]. The intention of the SMF was to differentiate its project from other bibliographical endeavours by proposing a logical classification of items based on a thematic index. This index probably started to circulate in an informal way in 1887 (the letters of Poincaré, Perott, Mathieu, Schlegel, Brioschi, Eneström or Mittag-Leffler contain numerous discussions about the logical organisation of mathematical domains in the index). A Projet de classification détaillée pour le Répertoire Bibliographique des Sciences Mathématiques was published by Gauthier-Villars on June 1888 (67 pages). Finally, the first edition of the index was to be published in 1893 10 . Taking occasion of the 1889 World Fair, the commission of the repertoire, which was chaired by Henri Poincaré, decided to organise the first Congrès International de Bibliographie des Sciences Mathématiques. The purpose of the organisers (Paul Appell, Gaston Darboux, Charles Rouché, Jules Tannery, Georges Humbert) was to gather in Paris all those involved in mathematical research and eager to contribute to the project at an international level.

Finally

  it was decided to establish a permanent commission that was based in Paris. Its composition was the following: Poincaré, Désiré André, Georges Humbert, Maurice d'Ocagne, Henry (France), E. Catalan (Belgium), D. Bierens de Haan (Holland), Glaisher (Great Britain), G. Gomes-Texeira (Portugal), Holst (Norway), Georg Valentin (Germany), E.Weyr (Austria), Guccia (Italy), G. Eneström (Sweden), J. P. Gram (Denmark), Liguine (Russia) and C. Stephanos (Greece).

  in the country. (2) This representative directed the perusal and the redaction of individual bibliographical index cards for each memoir and article that had been identified. For each memoir or article one index card containing its bibliographical references was made. (3) Each individual index card contained the common bibliographical information (title, name of the review, date, etc.) as well as a classification code corresponding to a specific domain of research (this code was furnished by the Index du Répertoire Bibliographique des Sciences Mathématiques). (5) Individual index cards were then regularly sent to Paris. They were centralized and prepared for publication. (6) The Paris secretariat directed the process of publication, which was very slow: he had to gather together individual cards according to their classification. Nevertheless the index cards concerning a specific division were not published before they contained a sufficient number of bibliographical references. For instance, if there weren't enough references for the code A1, no card was published and one had to wait for the sending of other bibliographical references concerning this classification code. Each printed card contained approximately ten references. Each set of 100 index cards contained about 1000 bibliographic items. The illustration below is a reproduction of a standard index card: the figure in the upper right corner indicates the number of the card and the framed code indicates the classification code according to the index.

1 )

 1 The mathematical bibliography compiled by Hermann Georg Valentin. Valentin was a chief librarian at the Berlin Library and he started the compilation of a mathematical bibliography in 1885, at the same time as the SMF. His aim was to collect all mathematical publications since the invention of typography. After several years of strained relations with the representatives of the French bibliography, Valentin finally became a member of the permanent commission 12 . His bibliography was probably never published, partly because of the First World War.

  implied a progress in pure mathematics. The introduction of such a separation between pure and applied mathematics is quite astonishing, particularly in the context of the development of engineer training in all European countries. What can historians of mathematics learn from such a bibliographical endeavour? In order to study precisely the structure of the catalogue, we created a computer database. Although we are quite far from the end of the data entry process 14 , we are in a position to give an outlook of the information that may be extracted from such a database.A general investigation covering 1700 index files (17 series) provides a piece of information concerning the repartition of bibliographical references within the 3 main mathematical domains: Mathematical Analysis (A to J) represents 50% of the references, Applied Mathematics (R to X) 33% and Geometry (K to Q) 17%. For Mathematical Analysis, the most important classes are those devoted to Algebra (A), Theory of Functions (D), Differential Equations (H) and Arithmetic (I). For Applied Mathematics, one can distinguish 3 significant classes: Mathematical Physics (T), General Mechanics (R) and Fluid Mechanics (S). Finally, the main classes in Geometry are those devoted to Elementary Trigonometry (K), Algebraic Curves and Surfaces (M) and Conics (L).Apart from these general figures, we are in a position to give some precise descriptive information concerning the first 300 index files (3 series) of the repertoire 15 . Very significantly, the figures of the repartition of items within the 3 mathematical fields is quite similar to the previous ones, which concerned 17 series: Mathematical Analysis represents 53%, Geometry 13% and Applied Mathematics 34%.The number of mathematicians listed in the first three series of the repertoire is 850.Although impressive, this figure requires further analysis: in fact, the most important part of these authors have between 1 and 5 bibliographical references and one can only find 120 mathematicians with more than 5 references. The prominent authors of the 3 series (20 items or more) are the following: A. Cauchy (101), N. H. Abel (55), L. Gegenbauer (49), C.-G.-J. Jacobi (42), M. Chasles (37), A. L. Crelle (31), Sophus Lie (29), F.-J. Studnicka (23), C. Duhamel (23), J. Liouville (22), P. Tannery (21), Th. Clausen (21), A. Cayley (21), E. Heine (20) and S. D. Poisson (20). The 2723 items listed come from 83 mathematical journals. This figure should be compared with the list of 227 journals of the Index du repertoire. Moreover, 62 of the 83 journals contain less than 20 bibliographical references. Only 8 periodicals contain more than 50 references and represent 71% of total number of items: Journal für die reine und angewandte Mathematik (712), Comptes Rendus des Séances de l'Académie des Sciences (500), Sitzungsberichte der Königliche Akademie der Wissenschaften in Wien (299), Journal de l'École Polytechnique (134), Bulletin de la Société Mathématique de France (117), Casopis pro pèstovàni mathematiky a fysiky 16 (80), Annales Scientifique de l'École Normale Supérieure (78), Journal de mathématiques pures et appliqués (58) 17 .

  memoirs and periodicals of a given country was placed under the responsibility of one or several mathematicians living and working in the country.Nevertheless, it is quite clear that, in the beginning of the project, very few countries put a lot of effort into the constitution of the bibliography. Apart from France, Germany and Austria 19 , only four countries sent more than 15 bibliographical references to the Paris office: Portugal (100 items, 3 journals perused), Russia (95 items, 4 journals), Italy (59 items, 1 journal) and Norway (52 items, 1 journal).In a 1900 report concerning the progress of the repertoire, Charles Laisant gave some information about the number of bibliographical references submitted by each country. At that time, only the half of the bibliography (ten series) had been printed and the most important suppliers were still France (10,682 items, 6 journals) 20 , Germany (3593 items, 7 journals) and Austria (1843 items, 8 journals). The graph below provides an overview of the participation of each country: Russia, Italy, Holland, Portugal, Belgium and Spain are the most significant suppliers. Although paradoxical, the absence of Great-Britain and United-States is not surprising since the two countries were engaged in a competing project, i. e. the International Catalogue of Scientific Literature.21 To a great extent, the references sent by Holland, Russia

  codes were used for the ordering of the first three series, although the Index du Répertoire Bibliographique des Sciences Mathématiques proposed around 2000 possible codes 23

24

  Some new results about determinants (2 codes), Dedekind's and Kronecker's arithmetical theory of algebraic functions (2 codes), properties of  (n) functions, Lie's theory of continuous groups, Gyldén's theory and nomography according to Chasles and the other one devoted to axonometric or isometric perspectivereveals the active role of Emil Weyr at the beginning of the perusal work since most of the bibliographical references come from the Sitzungsberichte der Königliche Akademie der Wissenschaften in Wien. The last domain (descriptive geometry on curves and surfaces) gives an illustration of the vitality of this old theory. This vivacity can be explained by the importance of descriptive geometry training in Engineer Schools and Faculties of Science. Who was in charge of the perusal process and how was it done? The most part of the German and Austrian contributions corresponds to the perusal of the Journal für die reine und angewandte Mathematik and of the Sitzungsberichte der Königliche Akademie der Wissenschaften in Wien. Emil Weyr essentially did this work 25 . He was certainly the most dynamic foreign contributor and his work ensured the international representation of the repertoire for the first three series. On the other hand, it seems that his work was not exhaustive at all. For instance, according to his collected works, Jacobi published 104 articles in the Journal für die reine und angewandte Mathematik; however, one can only find 36 references to Jacobis's works in the first three series of the repertoire. Judging from this example (and from several others), it appears that considerations concerning the mathematical domains, the length or the theoretical importance of publications were not automatically seen as relevant criteria. The essential part of the French contribution corresponds to the perusal of the Comptes rendus de l'Académie des sciences. This was made by Henri Brocard, who was the director of the Alger Meteorology Institute and one of the French representatives of the commission. His work only concerned publications in applied mathematics (classes R, S, T, U, V, X 26 ) but it was exhaustive (in his 1900 report, Laisant noticed that the Comptes rendus had been completely perused: "Il n'y a cependant peut-être que les Comptes rendus de l'Académie des Sciences de Paris qui soient tout à fait au courant, grâce au dévouement de M. le Commandant Brocard ; c'est lui qui a pris cette si lourde tâche, et il s'en est acquitté avec une exactitude et une régularité véritablement admirable") 27 .

  Répertoire bibliographique des sciences mathématiques 1888 Projet de classification détaillée pour le Répertoire bibliographique des sciences mathématiques. (Paris : Gauthier-Villars), 67 pages. 1893 Index du Répertoire bibliographique des sciences mathématiques. (Paris : Gauthier-Villars). 1894 Index du Répertoire bibliographique des sciences mathématiques -Errata, additions et modifications (octobre 1894) à joindre à l'édition (1893) de l'index. (Paris : Gauthier-Villars). 1898 Index du Répertoire bibliographique des sciences mathématiques. Seconde édition. (Paris : Gauthier-Villars). 1908 Index du Répertoire bibliographique des sciences mathématiques. Nouvelle édition. (Paris / Amsterdam : Gauthier-Villars / H. C. Delsman). 1916 Index du Répertoire bibliographique des sciences mathématiques. Troisième édition. (Paris / Amsterdam : Gauthier-Villars / H. C. Delsman).

1990]. The most important mathematical bibliographies (1850-1900)

			6
	All these bibliographical attempts constitute a sign for a general consensus regarding
	scientific research around 1900. Many mathematicians agreed that it was almost impossible to
	be certain that other scholars had not studied their ideas yet. Moreover, they were more and
	more upset by the time lag between the publication of an article and its registration in a
	mathematical bibliography (this discrepancy could reach 3 years in the Jahrbuch über die
	Fortschritte der Mathematik	7 or in the Catalogue of Scientific Papers). Consequently, many
	mathematical journals paid great attention to bibliographical issues in the 1880's and the
	1890's: what would be the best bibliographical system for mathematical sciences? What kind
	of classification should be adopted? Who should do the bibliographical work? Should
	mathematicians adopt an alphabetical classification or should they elaborate, at an
	international level, a systematic and consensual classification?	8

table :

 :

	Period	Number of series	Covered classes
		published
	1894-1895	3	A1 to X7
	1896-1905	12	A1 to V9
	1906-1912	5	B12 to X8

[Goldstein

2001] and[START_REF] Rasmussen | L'Internationale scientifique: 1890 -1914[END_REF]. See also[START_REF] Otlet | Transformations opérées dans l'appareil bibliographique des sciences[END_REF]].2 For instance the SMF was created in 1872 and the first issue of its journal, the Bulletin de la Société Mathématique de France, was published in 1873.3[START_REF] Neuenschwander | Les Journaux mathématiques[END_REF]], pp. 1534-1535; see also[START_REF] Gispert | Les journaux scientifiques en Europe[END_REF]].

Several edition of the index were printed. See [Commission permanente du Répertoire Bibliographique des Sciences Mathématiques 1893],[1898],[1908] and [1916].

At the present time, this database contain 3692 bibliographical items. This corresponds to the first 4 series of the repertoire, that is 400 index files over 2000. This database is accessible online thanks to Laurent Guillopé at the Cellule MathDoc: http://math-sahel.ujf-grenoble.fr/RBSM/.

Of course, these data are still fragmentary and must be confirmed by a more precise investigation of the whole set of 20 series.

920 bibliographical references concerned French mathematical journals, 783 were published in German journals and 539 in Austrian periodicals. Of course, these figures don't furnish any information about the authors' nationality!

This point would require further analysis since the situation is similar for other French journals.

[START_REF] Laisant | Sur l'état d'avancement des travaux du Répertoire bibliographique des sciences mathématiques[END_REF]], p. 248.

"M r le D r Krohn, Privatdozent dans notre université, s'occupera avec les Mathematische Annalen…"

He was the Austrian representative in the Commission permanente and the leader of a small crew. In a letter to Henri Poincaré (25/04/1892), Weyr wrote: "J'ai l'honneur de vous signaler qu'il nous sera possible de remplir votre demande au moins quant aux principaux recueils allemands. M. Le baron de Lichtenfels, professeur de mathématiques à l'École Polytechnique de Graz s'est chargé de dépouiller avec moi le Journal de Crelle. […] Cette collection de fiches déjà finie, concernant les Sitzungsbericht et les Denksschriften de Vienne et de Prague, les publications polonaises et quelques autres publications, vous parviendra en quelques jours par la poste".