Mohammed Louaked
email: louaked@math.unicaen.fr

Abdelkader Saïdi
email: saidi@math.unistra.fr

2D Control Problem and TVD-Particle Method for Water Treatment System

This work consists on the study of an optimal control problem relating to the water pollution. We analyze various questions: existence, uniqueness, control and the regularized formulation of the initial pointwise control problem. We propose also an implementation of an hybrid numerical scheme associated with an algorithm of descent.

Introduction

The improvement of the environmental conditions became, these last years, a major issue and in particular the safeguarding of the coastal regions. In this context, we need to predict some parameters, reflecting the quality of the aquatic media, and related to the propagation of the bacteria responsible for the infections transmitted by water or dependant on the consumption of some sea fruits.

To achieve this objective, we mainly address the evolution of waste water concentration based on some convection-reaction-diffusion equations with a pointwise source term modeling the action of the outlets where the convection term is deduced from the coupling with the shallow water system.

To improve the water quality, we introduce a control action at the points source by minimizing the deviation of fecal coliform distribution from standard value.

We firstly, analyze the existence and uniqueness of solution of the transport problem with a source term of Dirac type. For numerical reasons, we regularize the initial problem. We study then the regularized problem and show the convergence of the regular solutions towards that of the initial problem.

The simulation of the transport of biological systems are done by an hybrid numerical procedure combining particle method and finite difference technique.

The control problem is treated by a minimizing algorithm, where the gradient of the cost function is evaluated by adjoint techniques and a gradient type method as an iterative solution of the discrete control problem is chosen.

The structure of the paper is the following. We start by recalling the contaminant transport equation and the shallow water system. Next, we analyze the water quality problem and give some uniqueness, existence and regularity results concerning the state equation. After, we present the control problem and investigate the regularized version with an aim of numerical approximation. Finally, we provide numerical test cases.

Contaminant Transport Modeling

The model describing the evolution of water quality is given by an convection, reaction, diffusion equation where the concentration is solution of the following problem

∂C ∂t + u.∇C -ν∆C = -a 0 C + n j=1 f j (t)δ P j Q = (0, T ) × Ω, (1)
with initial and boundary conditions

C(0, x) = C 0 (x) in Ω, (2) ∂C ∂n (t, x) = 0, in Σ = Γ × (0, T ), (3)
in which C is the concentration of a pollutant, u is the external velocity, ν > 0 is the dispersion parameter, a 0 is the strictly positive coefficient for internal source corresponding to a reactive term, T is a specified positive number representing a time horizon and f j (t) is the mass flow rate at a discharge point P j at time t, δ P j (x) = δ(x-P j ) is a Dirac measure centred on fixed point P j .

The velocity field u = (u, v) is related to the water elevation by the shallow water equations (SWE)

∂U ∂t + ∂F (U ) ∂x + ∂G(U ) ∂y = S(U ). (4)
Where

U = [h, hu, hv] T , (5)
denotes the solution vector, and :

F (U ) = hu, hu 2 + 1 2 gh 2 , uvh T , (6)
G(U ) = hv, uvh, hv 2 + 1 2 gh 2 T , (7)
are the vector-valued flux functions. Where h, hu and hv, respectively, denote the height of water column and the discharge across the channel section, ν the diffusion coefficient and g is the gravitational acceleration. The source term S(U ) contains variable depth, bottom friction effects and also wind actions. An appropriate boundary and initial conditions are required to solve the SWE system.

The optimal control problem consists of finding a control f * that minimizes a cost functional

J(f ) = Φ(C(f )) + Ψ(|| f ||), (8)
where C(.) is the state satisfying constraints (1)-( 2)-( 3) and corresponding to control f (.) = (f j (.)) j=1...n . The functions Φ and Ψ are chosen according to the topology of the state space where C(.) is considered.

3 Analysis of the Water Quality Problem

Starting from the state equation described in the previous section and let us first assume the hypothesis :

H(3.1):
The velocity in the boundaries satisfies u(t, x) = 0 in (0, T ) × Γ.

H(3.2):

The function u belongs to L ∞ (Ω × (0, T )), C 0 ∈ L 2 (Ω) and f j ∈ L 2 (0, T ) for j = 1...n.

Let us multiply equation ( 1) by a test function Φ, integrate by part and combine conditions (3) and H(3.1), we obtain:

Ω C(t, x)Φ(t, x)dx T 0 + Q C(-Φ t -∇.( uΦ)-ν∆Φ+a 0 Φ)dtdx+ν Σ C ∂Φ ∂n dtdx = n j=1 T 0 f j (t)Φ(t, P j )dt.
(9) For a given Ψ ∈ L 2 (Q) we define the adjoint state equation by :

             -Φ t -∇( uΦ) -ν∆Φ + a 0 Φ = Ψ in Q, Φ(T, x) = 0 in Ω, ν ∂Φ ∂n (t, x) = 0 in (0, T ) × ∂Ω. ( 10
)
Remark 1 The weak formulation of the state equation ( 9) with the use of adjoint problem [START_REF] Louaked | Pointwise Control and Particle Analysis for Parabolic Equation[END_REF] is known as transposition method and can be formulated as [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]:

Find C ∈ L 2 (Q) such that: Q C(t, x)Ψ(t, x)dtdx = n j=1 T 0 f j (t)Φ(t, P j )dt + Ω C 0 (x)Φ(0, x)dx . (11)
Where Q = (0, T ) × Ω and Φ is a solution of the adjoint state equation [START_REF] Louaked | Pointwise Control and Particle Analysis for Parabolic Equation[END_REF].

For Ψ in L 2 (Q) the solution Φ of (10) is in L 2 (0, T ; H 2 (Ω)), since Ω ⊂ R 2 we have the inclusion H 2 (Ω) ⊂ C(Ω) space of continuous fonction on Ω such that Φ(t, P j ) is in L 2 (0, T ) and depends continuously on Ψ ∈ L 2 (Q) due to the adjoint problem [START_REF] Louaked | Pointwise Control and Particle Analysis for Parabolic Equation[END_REF]. The right hand side of the weak formulation [START_REF] Louaked | 2D Control problem for water treatment system[END_REF] :

l(Ψ) = n j=1 T 0 f j (t)Φ(t, P j )dt + Ω C 0 (x)Φ(0, x)dx is a linear continuous form on L 2 (Q).
As a consequence, we have the following result :

Theorem 3.1 Under the hypothesis H(3.1) and H(3.2) there is a unique solution C ∈ L 2 (Q) of problem [START_REF] Louaked | 2D Control problem for water treatment system[END_REF].

4 Analysis of the Control Problem

Performance function and gradient computation

To solve the optimal control problem, we have to find a mass flow rates f * ∈ G ad at a discharge points P j , j = 1 . . . n minimizing the concentration deviation of a pollutant distribution C from standard value C d :

J(f ) = 1 2 Q (C(t, x) -C d (t, x)) 2 dxdt + n j=1 T 0 µ j 2 (f j (t)) 2 dt. (12)
In the weighted L 2 norm penalization term, µ j is a small positive parameter. The vector control

f (t) = (f j (t)) j=1...n belongs to a set of admissible controls G ad = (L 2 (0, T )) n .
In what follows, we treat the question of existence of an optimal control f * and give the expression of the gradient of the cost function, from which one can set up a scheme to approximate the optimal control. Theorem 4.1 The above control problem (1) and ( 12) admits a unique solution f ∈ (L 2 (0, T )) n and the gradient of the cost function is identified as

∇J(f ) j = µ j f j (t) + P (t, P j ) j = 1, ..., n. (13)
Where the adjoint state function P satisfies the following form :

                 - ∂P ∂t -∇.( uP ) -ν∆P + a 0 P = C(t, x) -C d (t, x) in Q, P (T, x) = 0 in Ω, ∂P ∂n (t, x) = 0 in (0, T ) × Γ. ( 14
)
Proof.-For clearness reasons, we present our proof into two steps.

Step I: Existence of a minimum First of all, the operator f = (f j ) j=1...n → C(f ) is an affine function from (L 2 (0, T )) n to L 2 (Q).

One can set the functional to be optimized as :

J(f ) = 1 2 || C(f ) -C d || 2 L 2 (Q) + n i=1 µ j 2 || f j || 2 L 2 (0,T ) = 1 2 || (C(f ) -C(0)) + (C(0) -C d ) || 2 L 2 (Q) + n j=1 µ j 2 || f j || 2 L 2 (0,T ) = 1 2 b(f, f ) -2L(f )+ || C d -C(0) || 2 L 2 (Q) . ( 15
)
where b is a symmetric bilinear form over (L

2 (0, T )) n × (L 2 (0, T )) n defined by b(f, g) = (C(f ) -C(0), C(g) -C(0)) L 2 (Q) + n j=1 µ j 2 (f j , g j ) L 2 (0,T ) , ( 16
)
and L is a bounded operator on (L 2 (0, T )) n given by

L(f ) = (C d -C(0), C(f ) -C(0)) L 2 (Q) . (17)
The form b is (L 2 (0, T )) n -continuous and coercive by

b(f, f ) ≥ µ 2 || f || 2 (L 2 (0,T )) n with µ = min j=1...n µ j (18)
Hence, by Lax-Milgram theorem, one can conclude that J admits a unique optimal solution.

Step II: Gradient identification Since ∇J(f ) ∈ (L 2 (0, T )) n , we have :

δJ(f ) = n j=1 T 0 ∇J(f ) j δf j dt, ( 19
)
where δJ is the differential of J. It also follows from ( 1) and ( 12) that the differential satisfies

δJ(f ) = Q (C -C d ) δCdxdt + n i=1 µ j T 0 f j δf j dt, (20)
where δC solves the linearized convection-diffusion equation :

∂ δC ∂t + u∇ δC -ν∆ δC + a 0 δC = n j=1 δf j δ P j in Q, δC(0, x) = 0 in Ω, ∂ δC ∂n (t, x) = 0, in Σ = Γ × (0, T ). ( 21
)
By introducing the adjoint state P = P (t, x) and multiplying both sides of the differential equation in ( 21) by P and then integrating in both space and time we get :

Q ∂ δC ∂t P dxdt + Q u∇ δCP dxdt -ν Q ∆ δCP dxdt + Q a 0 δCP dxdt = n j=1 T 0 δf j (t)P (t, P j )dt. ( 22
)
If a suitable integration by parts is performed, in both space and time, by the help of the initial and boundary conditions of problem (21) and using hypothesis H(3.1), we find :

Ω δC(T,
                 - ∂P ∂t -∇.( uP ) -ν∆P + a 0 P = C -C d in Q, P (T, x) = 0 in Ω, ν ∂P ∂n (t, x) = 0 in Σ = Γ × (0, T ). ( 24
)
Then the gradient of the cost function is simply represented by :

∇J(f ) = (∇J(f ) j ) j=1...n with : ∇J(f ) j = P (t, P j ) + µ j f j (25)
The expression (13) of the gradient ∇J, using the adjoint problem [START_REF] Raviart | An Analysis of Particle Methods[END_REF], is directly exploited in the algorithm of descent to achieve the optimal control.

Regularized Control Problem

For numerical reasons, a regularized version of the control problem (1) is suitable

∂C ε ∂t + u∇C ε -ν∆C ε = -a 0 C ε + n j=1 f j (t)ρ j ε (x) in Q. ( 26
)
with initial and boundary conditions

C ε (0, x) = C 0 (x) in Ω, ( 27
)
∂C ε ∂n (t, x) = 0, in (0, T ) × Γ. ( 28
)
The Dirac measure δ P j (x) is replaced by a standard function. More precisely, we introduce for sufficiently small ε > 0 a set functions ρ j ε (x) ε as a mollifier which have the following property

∃ C 1 > 0 such that || δ P j (x) -ρ j ε || L 2 (Ω) ≤ C 1 ε (29)
The solution to the initial-boundary value problem is then defined as the limit ε ↓ 0 of the family {C ε } ε .

We may introduce a similar control analogous to the equation ( 12) associated to the regularized problem for the following cost function

J ε (f ) = 1 2 Q (C ε (t, x) -C d (t, x)) 2 dxdt + n j=1 µ j 2 T 0 | f j (t) | 2 dt. (30)
The optimal control f * ε is defined as a minimum of the cost function J ε (.).

Theorem 5.1 Under the conditions H(3.1)-H(3.2), one has the weak convergence

C ε ⇀ C in L 2 (Q). ( 31
)
Proof.-For a fixed point d = (d 1 , d 2 ) ∈ Ω and any point x = (x 1 , x 2 ) ∈ Ω the delta function is given by δ

(x) = δ(x 1 )δ(x 2 ) We first construct a mollifier {ρ ε (x)} ε = ρ 1 ε (x 1 )ρ 2 ε (x 2 )
with the property (29), we first start choosing a C ∞ function φ such that

Iε P (t)φ(t)dt = P (d) ∀P ∈ R 1 [X], (32)
where

I ε =]d -ε, d + ε[. Let put ρ ε (x) = ε -1 φ(T ε (x)) if x ∈ I ε 0 if not.
Here

T ε : x → x-d ε is the affine transformation from I ε onto ] -1, 1[. Taking ψ ∈ H 1 0 ]a, b[
and by the help of Bramble-Hilbert lemma, there exist a polynomial

P ∈ R 1 [X] such that : || ψ -P || L 2 (Iε) ≤ C 2 ε || ψ || H 1 (Iε) , || ψ -P || L ∞ (Iε) ≤ C 2 ε 1 2 || ψ || H 1 (Iε) .
Since

δ d (x) -ρ ε , ψ = δ d (x), ψ -ρ ε , P -ρ ε , ψ -P = ψ(d) -P (d) -ρ ε , ψ -P . (33)
We conclude that

| δ d (x) -ρ ε , ψ |≤ C 3 ε 1 2 || ψ || V , (34)
and also again

|| δ d (x) -ρ ε || V * ≤ C 3 ε 1 2 as ε -→ 0. ( 35
)
The difference

D ε = C -C ε is in L 2 (Q)
and is solution of the problem :

∂D ε ∂t + u∇D ε -ν∆D ε + a 0 D ε = n j=1 f j (t)(δ P j (x) -ρ j ε (x)) in Q, (36)
with initial and boundary conditions :

D ε (0, x) = 0 in Ω, ( 37
)
∂D ε ∂n (t, x) = 0, in Σ = Γ × (0, T ). ( 38
)
We multiply (36) by a test function Φ and integrating by part we have due to (37), (38) we obtain :

Ω D ε (T, x)Φ(T, x)dx + Q D ε (- ∂Φ ∂t -∇.(uΦ) -ν∆Φ + a 0 Φ)dtdx + ν Σ D ε (t, x) ∂Φ ∂n (t, x)dtdx = n j=1 Q f j (t)(δ P j (x) -ρ j ε (x))Φ(t, x)dtdx
(39) for a given Ψ ∈ L 2 (Q) we define the function Φ as a solution of the problem (10) and then we have :

Q D ε Ψdtdx = n j=1 Q f j (t)(δ P j (x) -ρ j ε (x))Φ(t, x)dtdx (40)
using the fact that Φ is solution of (10) we have :

Q (C -C ε )Ψdtdx ≤ C 0   n j=1 || f j (t)(δ P j (x) -ρ j ε (x)) || L 2 (Q) || Ψ || L 2 (Q)   -→ 0 when ε -→ 0 (41) f j (t)ρ j ε (x -P j ) -→ f j (t)δ P j (x) in L 2 (Q), (42)
we deduce the weak convergence in L 2 (Q).

Theorem 5.2 Under the assumptions H(3.1)-H(3.2), one has the strong convergence :

C ε -→ C in L 2 (Q). ( 43
)
Proof.-

We multiplay equation (36) by a test function P ε and integrate by part over Q we obtain using conditions (37), (38) :

Ω D ε (T, x)P ε (T, x)dx + Q D ε (- ∂P ε ∂t -∇.(uP ε ) -ν∆P ε + a 0 P ε )dtdx + ν Σ D ε (t, x) ∂P ε ∂n (t, x)dtdx = n j=1 Q f j (t)(δ P j (x) -ρ j ε (x))P ε (t, x)dtdx
(44) Let us know define P ε as the solution of the following adjoint problem :

                 - ∂P ε ∂t -∇.(uP ε ) -ν∆P ε + a 0 P ε = C -C ε in Q, P ε (T, x) = 0 in Ω, ν ∂P ε ∂n (t, x) = 0 in Σ = Γ × (0, T ). ( 45
)
We obtain from (45) and formal week formulation (44) :

Q D ε (C -C ε )dtdx = n j=1 Q f j (t)(δ P j (x) -ρ j ε (x))P ε (t, x)dtdx (46)
We have then the existence and uniqueness of a solution of (46) with the transposition method (ref) and for C -

C ε = D ε ∈ L 2 (Q) the solution of (45) is in L 2 (0, T ; H 2 (Ω)
) and depends continuously from D ε . As a consequence of that there exist a constante C 0 such that :

|| C -C ε || 2 L 2 (Q) = Q (D ε ) 2 dtdx ≤ C 0 (|| f j (t)(δ P j (x) -ρ j ε (x)) || L 2 (Q) || C -C ε || L 2 (Q) ( 47
)
and then we have :

|| C -C ε || L 2 (Q) ≤ C 0 ( n j=1 || f j (t)(δ P j (x) -ρ j ε (x)) || L 2 (Q) -→ 0 when ε -→ 0 (48)
The following result establishes the convergence of the regularized minima towards that of the initial problem.

Theorem 5.3 Let us set

j ε = J ε (f ε ) = Inf f ∈G ad J ε (f ) and j = J(f ⋆ ) = Inf f ∈G ad J(f ). ( 49
)
We assume that H(3.1)-H(3.2) hold. Then as ε -→ 0, we have

j ε -→ j in R. ( 50
)
f ⋆ ε -→ f ⋆ in G ad . ( 51
) Proof.-Since C ε -→ C in L 2 (Q), we have J ε (f ⋆ ) -→ J(f ⋆ ) ∀ f ⋆ fixed in G ad . (52)
In particular as ε ↓ 0

j ε ≤ J ε (f ⋆ ) -→ ε↓0 J(f ⋆ ) = j,
giving us lim sup j ε ≤ j, but

j ε = J ε (f ε ) ≥ n j=1 µ j 2 T 0 (f ε ) 2 j dt ≥ µ 2 || f ε || 2 (L 2 (0,T ) n with µ = min j=1...n µ j
we may extract a subsequence f jε , denoted again by f ε , such that

f ε ⇀ g in G ad , (53)
and by virtue of theorem 5.1 ( a faire .... )

lim inf J(f ε ) ≥ J(g).

But as lim

ε→0 (J(f ε ) -J ε (f ε )) = 0, we also have lim inf J(f ε ) ≤ lim inf J ε (f ε ), from which we obtain lim inf J(f ε ) ≥ J(g).
By the virtue of the inequalities lim sup j ε ≤ j, J(g) ≤ lim inf j ε ≤ j and by the uniqueness of the minimum, we get

g = f ⋆ . (54)
It follows j ≤ lim inf j ε ≤ j, and hence

j ε → j because j ≥ lim sup j ε , (55)
consequently j ε -→ j. (56)
It remains to prove the strong convergence of f ε towards f ⋆ . For such purpose we combine the facts

J ε (f ε ) = 1 2 Q (C ε (t, x) -C d (t, x)) 2 dxdt + n j=1 µ j 2 T 0 (f ε ) 2 j (t)dt = 1 2 Q (C ε (t, x)) 2 dxdt + n j=1 µ j 2 T 0 (f ε ) 2 j (t)dt - Q C ε (t, x)C d (t, x)dxdt + 1 2 Q (C d (t, x)) 2 dxdt -→ J(f ⋆ ) as ε → 0, (57)
and

J ε (0) -→ J(0), (58)
with the weak convergence of f ε towards f ⋆ (see equations (53-(54)) and the convergence (31), we obtain first

b(f ε , f ε ) -→ b(f ⋆ , f ⋆ ), (59)
and since the bilinear form b(g, g) defined by ( 16) is equivalent to || g || 2 (L 2 (0,T )) n , finally by virtue of the coercivity (18) we have the convergence f ε -→ f ⋆ in G ad .

Numerical Approximation

For the numerical resolution, a hybrid method seems to be well adapted. More precisely, a Lagrangian procedure is used to solve both the primal problem (concentration of pollutant) and its dual (adjoint state). On the other hand, a finite difference approximation (total variation diminishing (TVD)) for the shallow water system [START_REF] Louaked | TVD-Multiresolution Scheme for the Shallow Water equations[END_REF] is preferred. An algorithm of descent allows to estimate the optimal control where in each step, the resolution of the adjoint problem gives an access to the expression of the gradient.

In what follows, we give details concerning the solution for the 1D problem and for the 2D system, we use the well-known dimension by dimension technique.

The TVD Scheme

An explicit conservative difference scheme

U n+1 j = U n j -λ( F n i+ 1 2 -F n i-1 2 ), (60)
is serving as consistent approximation to the 1D shallow water system

U t + f (U ) x = 0. ( 61
)
Consider a uniform grid and let the grid spacing be denoted by ∆x such that x i = i∆x and ∆t be the time step. Let a l be the eigenvalues of A, matrix jacobian of F , and R is the matrix whose columns are right eigenvectors of A. The quantities a l , R, evaluated at U i+ 1 .

The difference of the characteristic variables in the local x-direction is defined by

α i+ 1 2 = R -1 i+ 1 2 (U i+1 -U i ). ( 62
)
The numerical flux Fi+ 1 2 with a diffusive term can be expressed as

F n i+ 1 2 = 1 2 [F n i + F n i+1 + R i+ 1 2 Φ i+ 1 2 ], (63)
where the elements of Φ i+ 1 2 are given by

Φ l i+ 1 2 = 1 2 σ(a l i+ 1 2 )(g l i + g l i+1 ) -Ψ(a l i+ 1 2 + γ l i+ 1 2 )α l i+ 1 2 , (64)
with

σ(z) = Ψ(z) - ∆t ∆x , z 2 (65)
γ l i+ 1 2 = 1 2 σ(a l i+ 1 2 )        (g l i+1 -g l i ) α l i+ 1 2 , α l i+ 1 2 = 0 0, α l i+ 1 2 = 0, (66)
the entropy correction function is of the form

Ψ(z) =    |z| |z| ≤ ε (z 2 + ε 2 ) 2ε |z| ≥ ε, ( 67
)
where ε is a positive small number.

A compressive limiter,

g l i = S.max 0, min(2 | α l i+ 1 2 |, S.α l i-1 2 ), min(| α l i+ 1 2 |, 2S.α l i-1 2 ) ; S = sgn(α l i+ 1 2 ), (68)
is chosen to deal with the linearly degenerate fields for better resolution of contact discontinuities and a less compressive limiter,

g l i = minmod(α l i-1 2 , α l i+ 1 2 ), (69)
to the genuinely nonlinear fields in order to avoid the steepening problem.

Particle Method

In this section, both regularized and adjoint state are solved with the use of the vortex method.

Vortex methods has a grid-free character with an appealing physical interpretation in which the particles carry a concentration of a pollutant and convect with the local flow velocity.

In order to define a numerical method to solve the problem

∂C ∂t + u.∇C -ν∆C = L(t, x) in (0, T ) × Ω, (70)
C(0, x) = C 0 (x) in Ω, (71)
∂C ∂n (t, x) = 0 in (0, T ) × Γ, (72)
we use a quadrature rule defined by a set of weighted points (x 0 p , α p ) p ∈ J 1 to provide a highorder approximation to the initial data, where x 0 p ∈ Ω and α 0 p > 0. We have, for any function Φ ∈ C 0 (Ω)

Ω Φ(x)dx ≈ p∈J 1 α 0 p Φ(x 0 p ). ( 73
)
The solution C is sought as a linear combination of Dirac distributions

c h (x, t) = p∈J 1 α p (t)c p (t)δ xp(t) (x), (74) 6.4

.1 Algorithm

Choose f (0) (t) given in (L 2 (0, T )) n 0. Compute an approximated solution C (0) (t, x) by solving (26) with f (t) = f (0) (t) Compute an approximated solution P (0) (t, x) by solving [START_REF] Raviart | An Analysis of Particle Methods[END_REF] with C = C (0) Compute the gradient of descent G (0) j (t) = µ j f (0) j (t) + P (0) (t, P j ). For k = 0, 1, 2, ... 1. Update the control by using the gradient f (k+1

) (t) = f (k) (t) + ρG (k) (t)
Compute an approximated solution C (k+1) (t, x) by solving (26) with f (t) = f (k+1) (t) Compute an approximated solution P (k+1) (t, x) by solving [START_REF] Raviart | An Analysis of Particle Methods[END_REF] with

C = C (k+1) Compute the gradient of descent G (k+1) j (t) = µ j f (k+1) j (t) + P (k+1) (t, P j ) 2. If ( | G (k+1) | | G (k) | ) > tol then put f (k) (t) = f (k+1) (t), put G (k) (t) = G (k+1) (t)
and goto 1. end

Numerical Tests

In this section, we present computation results using the hybrid method.

Peak in the concentration of pollutant

The hydrodynamic initialization is a dam break on flat bottom [START_REF] Audusse | Transport of Pollutant in Shallow Water. A Two Time Steps Kinetic Method[END_REF], due to the fact that this test includes several features of existing issues in the area of marine hydrodynamics and coastal engineering. The source term S, in the shallow water system, is equal to zero. There is a peak in the concentration of pollutant in the area [-0.7, -0.3]. Computation of the control problem is performed with the following values :

• Spatial domain : x ∈ [-1, 1].

• Physical time : T = 0.24.

• Discharge point : d = -0.5.

• Initial elevation : h(0, x) = 1 if x < 0 else h(0, x) = 0.5.

• Initial velocity : u(0, x) = 0 ∀x ∈ [-1, 1].

• Initial value for the concentration of pollutant : c(0, x) = 0.7 if x ∈ [-0.7, 0.7] else c(0, x) = 0.5.

• Standard value : c d (t, x) = 0.44 ∀x ∈ [-1, 1].

• Physical parameters : a 0 = 1 and ν = 0.01.

• Computational parameters : ∆x = 1/128, µ = 0.01 and ε = ∆x.

In term of setting, a tolerance of 10 -5 in the gradient algorithm is used as the stopping criterion or we stopped iterating after a fixed number of iterations. As can be seen in Fig 3, the pollutant distribution after convergence is reduced under the desired value.

Conclusion

This work study an environmental control model. The mathematical analysis of different related problems is firstly investigated. After, we treat the numerical part in which we propose an hybrid Particle-TVD method to solve the pollutant transport equations. Our results indicate that the particle strength exchange technique coupled appropriately to a TVD scheme is a robust and an efficient procedure for the direct control concentration pollutant problem.

2 are respectively denoted by a l i+ 1 2 and R i+ 1 2

 222

Figure 1 :

 1 Figure 1: Initial condition for pollutant, pollutant during iterative process, pollutant concentration after convergence at T = 0.24.

where the positions (x p (t)) p of the particles are defined according to the characteristic curves dx p dt (t) = u(x p (t), t), x p (0) = x 0 p ,

and the weights (c p (t)) p ) are determined by resolution of the ordinary differential system

The integral approximation of Laplace's operator is solved by the deterministic Particle Strength Exchange method (PSE)[4]

To define a more classical approximation of C, we introduce an even compactly supported function χ on R such that

and let χ ε (x) = ε -1 χ(x/ε). The more classical approximation c ε h of the function C than the measure c h is given by

The resolution of the convection-diffusion equation can be summarized by the resolution of a large system

(80)

Particle-Mesh interpolation

The assignment operators, particle strength to the mesh value and the interpolation of the particle values from the mesh, use the moment-conserving M

(81)

Optimal control method

In what follows, we present an algorithm of descent for the control estimation. The method is of gradient type and it is based on the use of an adjoint state given by the theorem 0.2.


Keywords: Orphaned and Vulnerable Children, Child morbidity, Rwanda, Graphical Chain Models, Demographic and Health Surveys

Introduction

This paper investigates the susceptibility of Orphaned and Vulnerable Children (OVC) towards fever and respiratory related illnesses and disentangles the complex associations that explain these health outcomes. Many children in Africa have become orphaned and vulnerable as a result of HIV/AIDS. By 2010, it is estimated that about 18 million children below age 18 would be classified as OVCs due to HIV/AIDS (UNICEF, 2005). In Rwanda, 220,000 children are estimated to have lost one or both parents because of HIV/AIDS (UNAIDS, 2008). The survival, growth and development of OVCs are greatly threatened as these children are subject to social neglect and discrimination (UNICEF, 2006).

The intra-household and environmental factors that affect the health status of OVCs are difficult to determine in Rwanda, which has not fully recovered from the impact of the 1994 genocide and periodic economic depression. The term "Orphaned and Vulnerable Children" has been used widely in the literature to describe children who have been orphaned or made vulnerable as a result of HIV/AIDS [START_REF] Schenk | Supporting orphans and vulnerable children affected by AIDS: using community-generated definitions to explore patterns of children's vulnerability in Zambia[END_REF][START_REF] Skinner | Towards a Definition of Orphaned and Vulnerable Children[END_REF][START_REF] Andrews | Epidemiology of health and vulnerability among children orphaned and made vulnerable by HIV/AIDS in sub-Saharan Africa[END_REF]Stover et al., 2006;[START_REF] Bhargava | Public policies and the orphans of AIDS in Africa[END_REF][START_REF] Matshalaga | Mass orphanhood in the era of HIV/AIDS[END_REF]. However, there is no universal consensus to its definition [START_REF] Skinner | Towards a Definition of Orphaned and Vulnerable Children[END_REF]. We follow the definition used by the Rwandan Demographic and Health Surveys (RDHS) based on UN recommendations (UNICEF, 2005). OVCs are defined as those who (i) have lost one or both parents, (ii) one or both parents have been chronically ill for at least three months within the 12 months preceding the survey, (iii) live in a household where at least one adult (18-59 We focus on recent morbidity experiences of children under five years, specifically diarrhoea, cough, fever and Acute Respiratory Infection (ARI) symptoms as recorded in the 2005 RDHS. The predictor variables at the individual and household levels were selected based on the child survival and growth framework which classifies different proximate determinants associated with malnutrition and infections (Hatt & Walters, 2006;[START_REF] Sreeramareddy | Care seeking behaviour for childhood illness-a questionnaire survey in western Nepal[END_REF]Erhardt et al., 2006;[START_REF] Filmer | Fever and its treatment among the more and less poor in sub-Saharan Africa[END_REF][START_REF] Boadi | Childhood diarrheal morbidity in the Accra Metropolitan Area, Ghana: socio-economic, environmental and behavioral risk determinants[END_REF]The Cebu Study Team, 1991;[START_REF] Mosley | An analytic framework for the study of child survival in developing countries[END_REF]. Socioeconomic and environmental factors mediate the synergic interactions between malnutrition and infections through a set of proximate determinants. This study addresses the direct and indirect pathways through which OVCs get exposed to morbidity risks taking into account the interactive effect of a range of household and individual (proximate) factors. been ill with a cough accompanied by short and rapid breathing. The response variables were dichotomous with children who experienced an illness coded "1" and "0" otherwise. The recall errors are assumed to be minimal since the events in question relate to a short period before the survey.

Data and Methods

Data

We used Graphical Chain Modelling (GCM) for the analysis. GCM is ideal for identifying direct and indirect determinants using regression models [START_REF] Cox | Multivariate Dependencies -Models, Analysis and Interpretation[END_REF]. This technique enables one to present the structure of associations in graphical form and enhances our understanding of the direct and indirect pathways of associations which are often difficult to capture in conventional regression modelling.

The reader is referred to Wermuth (2005) and [START_REF] Lauritzen | Graphical Models[END_REF] for a more detailed discussion on GCM. The explanatory variables were grouped into three blocks: (i) individual and household demographic and socio-economic factors; (ii) intermediate (proximate) factors and (iii) primary exposure variable (Figure 1). Prenatal and delivery care were selected as proxies to reflect access to healthcare (Whitehead et al., 1997). Source of drinking water and access to toilet facilities are considered proxies to ascertain the influence of health-related environment. Nutritional measures such as stunting and wasting considered important were not included in the analysis because anthropometric data for about 56% of children were recorded missing in the individual data file for various reasons, for example the child died or was away or living elsewhere at the time of survey. The statistical software used for the analysis was GraphFitI [START_REF] Blauth | GraphFitI -a computer program for graphical chain models[END_REF].

[---Figure 1

Results

Descriptive statistics

Table 1 shows the prevalence of diarrhoea, fever, cough and ARI symptoms among OVCs and non-OVCs. The distributions show that the prevalence of all four illnesses is higher among OVCs than non-OVCs. Chi-squared test revealed a significant difference in the prevalence of diarrhoea, fever and ARI symptoms between OVCs and non-OVCs at the 5% level but not for cough.

[---Table 1 about here---]

Associations among the outcome variables

Figure 2 show that there exist direct associations among diarrhoea, fever and ARI symptoms but not between ARI symptoms and cough. The associations between diarrhoea, fever and cough imply that children who experienced diarrhoea are also more likely to experience fever and ARI symptoms and vice versa. The estimates presented in Table 2 show a positive association between cough, fever and diarrhoea, implying that a child who had diarrhoea is more likely to have a cough and or experience feverish conditions. About 54% of children who had diarrhoea also had fever, while 60% of those who had diarrhoea also had a cough. There is also a positive association between ARI symptoms, fever and diarrhoea which suggests that children who experience ARI symptoms are more likely to suffer from diarrhoea and fever. About 71% of all children who experienced ARI symptoms also had a fever.

The associations are significant at the 1% level and remained significant even after adjusting for other relevant explanatory variables (Table 2). The green, pink, yellow and blue arrows show the direct associations of the pure explanatories, the intermediate factors and the primary exposure variable with diarrhoea, fever, ARI symptoms and cough, respectively. The figure shows that OVC status has a direct association with diarrhoea, fever and ARI symptoms but not with cough. The associations persist after accounting for demographic and socio-economic factors and are significant at the 5% level. OVCs have increased odds of experiencing diarrhoea (48%), fever (30%) and ARI symptoms (39%) compared to non-OVCs (Table 2).

The chain graph shows that, after accounting for OVC status, the intermediate factors directly associated with diarrhoea are breastfeeding, source of drinking water and prenatal care. Children who were not breastfed, those whose mothers received skilled prenatal care and those who live in households with access to safe drinking water are significantly less likely to have diarrhoea (Table 2). The pure explanatories directly associated with diarrhoea are age and sex of the child, the household structure and educational status of the mother/carer. Younger children have significantly higher odds of having diarrhoea compared to those aged 48-59 months (Table 2). The results also show that male children have increased odds (33%) of having diarrhoea compared to female children. Children who lived in households with fewer adults have increased odds of having diarrhoea compared to children who live in households with four or more adults. Also, if a child's mother/carer has no formal education, his/her odds of having diarrhoea increases by 37% compared to a child whose mother/carer has secondary or higher education.

The chain graph shows that the only intermediate factor that has a direct association with fever is prenatal care (Figure 2). As expected, a child whose mother 2). Children who lived with three or less adults, those residing in the lake and marshy Southern and Eastern regions are also more susceptible to fever compared to those who live with four or more adults and those who reside in the City of Kigali.

The intermediate factors directly associated with children showing ARI symptoms are breastfeeding and prenatal care received by the mother (Figure 3).

Currently breastfed children and those whose mothers received unskilled prenatal care have higher odds (45% and 22% respectively) of showing symptoms of ARI compared to non breastfed children and those whose mothers received skilled prenatal care (Table 2). After adjusting for OVC status and the intermediate factors, the pure explanatories directly associated with ARI symptoms are the age of the child, birth interval and household structure. Younger children, those with lower preceding birth interval and those living with fewer adults in the household have higher odds of showing ARI symptoms.

Cough unlike diarrhoea, fever and ARI symptoms does not have a direct association with OVC status, even after adjusting for other significant factors. The only intermediate factor directly associated with cough is mothers' prenatal care status (Figure 2). Children whose parents received skilled prenatal care have reduced odds of having a cough (Table 2). The pure explanatories directly associated with cough, after adjusting for the other morbidities and prenatal care, are the age of the child, the number of household members, the region and place of residence. Also, the odds of having a cough are higher among younger children, children in households with 7 or [---Table 2 andFigure 2 about here-

--]

Indirect determinants operating through OVC status

The red arrows in Figure 3 show the intermediate factors and pure explanatories which operate through OVC status to influence childhood morbidity (diarrhoea, fever and ARI symptoms). The chain graph shows that vaccination, type of toilet facility and prenatal care operate through OVC status to influence childhood morbidity. Note that prenatal care operates directly and also indirectly to influence childhood morbidity. However type of toilet facility and vaccination only operate indirectly through OVC status. This suggests that the poor health outcomes of OVCs are also attributable to their vaccination status, environmental hygiene and access to healthcare (Table 3).

The pure explanatories that operate through OVC status to influence health outcomes of children are the age and sex of the household head, educational status of the mother/carer, natural region and place of residence (Figure 3). OVCs are more likely to be found in households headed by elderly persons (40+ years) and females, where the mother/carer has inadequate education and are more likely to be in urban areas and the south and eastern regions of the country (Table 3). This is an indication that the household structure of OVCs, their mother's/carers educational status and place of residence also contribute to their poor health outcomes. The pure explanatories that operate through the intermediate factors and through OVC status to influence child morbidity are age of the child, age and sex of the household head, birth order, educational status of the mother/carer, number of household members, natural region and place of residence.

From Figure 3 (Box 1) it can be noted that vaccination status acts with prenatal care and type of toilet facility to influence childhood morbidity, through OVC status. The implications are that OVCs health outcomes are not only worsened because they are more likely to have not received any vaccination (Table 3) but also because of health utilisation and environmental related problems that confront them.

The pure explanatories which operate through vaccination and OVC status to influence child morbidity are age of the child, sex of the household head, the natural region and place of residence. Young and male children are less likely to have received any vaccination. Those in the South, West, North and Eastern regions and also from rural areas are also less likely to have received any vaccination compared to those in the City of Kigali and urban areas (Table 3). This shows that OVCs health outcomes are poor not only because they have poor access to healthcare but also lack facilities that prevent illnesses such as bednets.

The pure explanatories which operate through prenatal care and OVC status are age of the child, age and sex of the household head, birth order, number of Mothers who received unskilled prenatal care are more likely to belong to households where the head is elderly (40+ years) and female, the mother/carer has no formal education or has only primary education and resides in the Southern region of the country (Table 3).

Type of toilet facility is another indicator which other intermediate factors (vaccination, sleeping under a bednet and place of delivery) operate with to influence childhood morbidity through OVC status (Figure 3, Box 1). The estimates in Table 3 and edges in Figure 3 show that health, environmental and preventive measures are collectively important determinants of childhood morbidity which can operate through OVC status. The pure explanatories which operate through type of toilet facility and OVC status are the number of household members, educational status of the mother/carer, natural region and place of residence.

[---Table 3 and Figure 3 about here---]

Conclusions

The results show a wide range of significant direct and indirect associations interacting both within and between the various sets of factors predicting poor child health outcomes among OVCs in Rwanda. The advantage of using GCM is that the factors that have no direct association with the outcome variables which are usually ignored were proven to be important, operating indirectly through other variables. The results show that OVCs have relatively a higher burden of ill-health when compared to non-OVCs. OVC status influences childhood morbidity directly and is a conduit through which other factors influence childhood morbidity. The intra-block associations show that all morbidity indicators are associated except coughing which is conditionally independent of ARI symptoms. This study also demonstrates evidence When considering the direct associations it is worth noting that healthcare utilisation, environmental hygiene and childcare factors such as prenatal care, source of drinking water and breastfeeding are important predictors of childhood morbidity.

Nonetheless, vaccination status and type of toilet facility are not directly associated with childhood morbidity but they appear to have an indirect link to poor child health through OVC status. Vaccination status and type of toilet facility also act as a conduit through which a set of the socio-economic factors operate to influence poor child health through OVC status. This study presents evidence that poor healthcare utilisation, poor environmental conditions and lack of preventive measures contribute both directly and indirectly to poor health outcomes among OVCs. Socio-economic factors also aggravate the health outcomes of OVCs through a set of proximate factors and OVC status. Although previous studies had observed similar association between childhood morbidity and the proximate and socio-economic factors [START_REF] Kandala | Morbidity from diarrhoea, cough and fever among young children in Nigeria[END_REF]Kazembe et al., 2007;[START_REF] Kandala | The Spatial Epidemiology of Childhood Diseases in Malawi and Zambia[END_REF], this study contributed to a better understanding of the mediating effect of OVC status determining child health outcomes incorporating the interactive effect of proximate factors.

There are a few caveats in this study that are worth noting. Information on child morbidity was reported by mothers and not clinically validated. However, the reported morbidity experience is confined to the last two weeks preceding the survey.

We believe that mothers are capable of understanding childhood illnesses and they are likely to provide accurate responses in the survey. On the other hand, DHS data are the only source that provides nationally representative population-level data on childhood morbidity. There is also lack of data on direct and indirect costs associated with healthcare, quality of health services delivery, and perceptions and beliefs related From policy perspectives, an important finding of this study relates to the central role of OVC status in explaining poor child health outcomes. OVC status has a direct association with poor health outcomes; it also acts as a link through which many of the proximate and socio-economic factors operate to influence poor child health outcomes. The present study highlights the need to improve social and health care for OVCs, and the policy strategies that could be adopted to reduce childhood illnesses. More importantly, environmental hygiene, sanitation and preventive health measures should be given priority in programmes aimed at promoting better living conditions for OVCs.

F

Revision notes

We are very grateful to the referee for providing us with the comments and feedback on our manuscript. We have managed to address all the comments raised by the referee. The word count of the revised main text (without paper title and references) is 2988. The following notes explain how we reacted to each comment.

General comments

Comment 1: If this paper offers important new insights that might be useful in helping children, it does not report them in a way that is understandable, and it does not tell us how they might be used. "The aim of this paper is to disentangle the effect of underlying complex relationship between and within the proximate and socioeconomic factors that mediate poor health outcomes among children through their OVC status." This is OK, but you also need to say it in English, and then you need to say what it means. Why is it important?

Response 1: Thank you for this comment. We have made these clear in the revision. Please note that the revised text has been shortened following the instructions from the Editor. This study addresses the direct and indirect pathways through which OVCs get exposed to morbidity risks taking into account of the interactive effect of a range of household and individual (proximate) factors (explained on Page 2, last paragraph of the introduction). The relevance of the topic, the justifications and the rationale for using graphical chain techniques to model the mediating effect of OVC status and the complex association of factors in predicting child health outcomes have been addressed properly (also please see the conclusion section).

Comment 2: I confess that I have not fully understood the graphical chain analysis. The explanation is not all that clear. But more importantly, the paper provides me no reason for why I should spend more time and effort trying to understand it. OVC are more likely to be sick, and sick children are more likely to be sick. So what? How does your analysis help me address this? In the Introduction, you need to say why it's important to "disentangle the complex. …" in the way you propose.

Response 2:

The revision focuses more on the application and less on the methods. The methods section has been shortened, directing the reader to a few important references that provide a detailed explanation of graphical chain methods (Page 3, second paragraph). Also, please refer to Response 1. -2 -things we haven't known before? The paper needs a discussion section. Among issues that cry out for discussion.

Response 3: The analysis clearly show that OVC status influences childhood morbidity directly and also indirectly as a conduit through which other factors influence childhood morbidity. The advantage of using GCM is that the factors that appear to have no direct association with the outcome variables which are usually ignored were proven to be important in explaining the indirect associations. The present study highlights the need to improve social and health care for OVCs, and the policy strategies that could be adopted to reduce childhood illnesses. More importantly, environmental hygiene, sanitation and preventive health measures should be given priority in programmes aimed at promoting better living conditions for OVCs. These points are stressed in the conclusion section. The conclusion section provides a brief discussion of the key policy implications. A separate discussion section is not included due to word restrictions.

Specific comments

1. The differences in prevalence of morbidity between OVC and non-OVC are not huge.

Given the sample of 3741 children, the differences are statistically significant for all morbidity outcomes at p<0.05, except for cough. . Furthermore, over 20% more OVCs than non-OVCs have these three morbidities. Please refer to Table 1 and the multivariate results reported in Table 2.

2. A footnote says that there's no info on HIV status of children; this would seem to be a major point in a discussion, since many of the health outcomes could conceivably be explained by HIV status.

Yes, we agree with this comment. To avoid confusion, the footnote has been removed from the paper.

3. What does it mean that OVC status is associated with some illnesses but not with cough? Does this result make sense?

The analyses show that children suffering from cough alone are not statistically different between OVCs and non-OVCs. However, ARI symptoms are significant which include illness with a cough accompanied by short and rapid breathing. This might reflect the care given to OVCs in households with more adults (opportunity costs rather than economic costs).

Other specific comments:

Comment: Introduction -"These OVC live in deplorable conditions in households that are already poverty-stricken" -all of them? Response: We removed this sentence in the revision since the survey data are inadequate to comment on this observation. Also, there is no information on the tenure of OVCs in the households interviewed in the RDHS.

Comment:

The brief discussion of statistical methods should be in Methods, not the Introduction. Before we get to that, we should understand why this whole exercise is important.

Response: Thank you for this suggestion, we understand the point. The methods are discussed only in the 'data and methods' section.

Comment:

The two-week window for child health problems is very short. You don't include malaria as an outcome? Response: This is standard in all the Demographic and Health Surveys conducted across less developed countries. Most retrospective surveys rely on child morbidity data from the two weeks preceding the survey. This is the case even for specific follow-up studies, example the study by Goldman et al, 1998 on the use of calendars in measuring child morbidity (International Journal of Epidemiology (Vol. 27 Issue No. 3,. We believe that mothers are capable of understanding childhood illnesses and they are likely to provide accurate responses in the survey. On the other hand, DHS data are the only source that provides nationally representative population-level data on childhood morbidity. The RDHS did not ask malaria outcomes among children.

Comment: "The conceptual framework proposed by Mosley and Chen. …" This is deep in the Data and Methods section. It's needed for us to understand where we're going, should come much earlier.

Response: This is now discussed briefly in the introduction section (please see Page 2, last paragraph before the 'data and methods' section).

Comment: The exclusion of nutritional measures is too bad.

Response: Yes, we do agree with this point. Unfortunately, anthropometric data for about 56% of children were recorded missing in the individual data file for various reasons, for example the child died or was away or living elsewhere at the time of survey. Hence we could not include any nutritional measures.

Comment: In short, tell us why the study question is important, how your findings fit in with other studies (are they new? do they contradict? do they confirm?), and how they might be used to help kids. Response: Through graphical chain modelling, the study demonstrates new evidence that OVC status influences childhood morbidity directly and also a conduit through which other factors influence childhood morbidity. It concludes that the factors that have no direct association with the outcome variables which are usually ignored had indirect associations. These are elaborated in the revision. We gave careful attention to this comment while revising the manuscript.

***

  has been chronically ill for at least three months within the 12 months preceding the survey or (iv) live in a household where at least one adult (18-59 years old) has died in the past 12 months after being chronically ill for at least three months (Institut National de la Statistique du Rwanda & ORC Macro, 2005).

  on 3,747 children aged below 5 years extracted from the 2005 RDHS were considered for the present analysis. The survey collected detailed household data and other demographic and health information from women of reproductive ages. Data on child health were recorded for births that occurred during the five years preceding the survey (Institut National de la Statistique du Rwanda & ORC Macro, 2005). Children who were classified as OVCs constituted 9.7% of the sample.The main response variables of interest are diarrhoea, fever, cough and Acute Respiratory Infection (ARI) symptoms reported for the last two weeks preceding the survey. The RDHS asked if the child had any difficulties in breathing, specifically and unusual breaths. Children showing ARI symptoms were those who had

5

  Figure2shows the direct determinants of diarrhoea, fever, ARI symptoms and cough.

  care has reduced (20%) odds of having fever compared to a child whose mother received unskilled or no care. The pure explanatories that have direct association with fever are age of the child, household structure and natural region. The odds ratio shows that younger children are more (27%) susceptible to fever (Table

  those residing in urban areas and the Southern region and City of Kigali.

8

  Figure3show the pure explanatories that influence childhood morbidity through the

Figure 3 (

 3 Figure 3 (Box 1) shows that prenatal care operates collectively with sleeping

  educational status of the mother/carer and natural region.

  the proximate and socio-economic factors operate both directly and indirectly through OVC status to influence childhood morbidity.

  behaviour. Nonetheless, this study yielded policy relevant findings, particularly the role of direct and indirect factors which are methodologically difficult to establish otherwise.

Figure 3 .

 3 Figure 3. Chain graph showing the determinants that operate through the intermediate factors and OVC status

Table 1 .

 1 Prevalence (%) of morbidity among OVCs and non-OVCs,Rwanda, 2005

	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 2 .

 2 Estimated odds ratios and confidence intervals of the direct determinants of the morbidity indicators

	OR (95% CI)

Table 3

 3 Estimated odds ratios and confidence intervals of the intermediate factors and pure explanatories OR (95% CI)

			Skilled vs.	At least one	Improved vs.
		OVC vs.	unskilled prenatal	vaccination vs. no	unimproved
	Explanatory variables	non-OVC	care	vaccination	toilet facility
	Intermediate factors
	Vaccination			NA
	No vaccination	1.00			1.00
	At least one vaccination	0.71 (0.56, 0.90)			1.28 (1.14, 1.40)
	Sleeps under a bed net	NS		NS
	No		1.00		1.00
	Yes		1.34 (1.06, 1.69)		2.12 (1.79, 2.52)
	Type of prenatal care		NA
	Unskilled care	1.00		1.00
	Skilled care	0.58 (0.46, 0.73)		1.80 (0.67, 1.96)
	Place of delivery	NS
	Home		1.00	1.00	1.00
	Health facility		1.69 (1.34, 2.12)	1.08 (0.91, 1.29)	1.96 (1.65, 2.33)
	Toilet facility				NA
	Unimproved	1.00		1.00
	Improved	0.69 (0.52, 0.91)		1.77 (1.65, 1.92)
	Pure explanatories
	Age of child	NS			NS
	0 -23 months		6.80 (4.84, 9.55)	0.25 (0.20, 0.32)
	24 -47 months		2.19 (1.73, 2.78)	0.64 (0.53, 0.78)
	48 -59 months		1.00	1.00
	Age of head of household			NS	NS
	16-29 years	0.71 (0.52, 0.96)	1.69 (1.50, 1.95)
	30 -39 years	0.61 (0.47, 0.80)	1.67 (0.53, 1.84)
	40+ years	1.00	1.00
	Sex of household head
	Male	1.00	1.00	1.00	1.00
	Female	1.82 (1.43, 2.32)	0.46 (0.17, 0.82)	1.48 (1.25, 1.75)	0.92 (0.76, 1.11)
	Birth order	NS		NS
	First		0.26 (0.18, 0.38)		1.10 (0.84, 1.43)
	Second and third		0.54 (0.43, 0.70)		1.10 (0.91,1.33)
	Fourth or higher		1.00		1.00
	4+ adults
	Number of household members	NS		NS
	1 -3 members		7.59 (4.95, 11.61)		0.46 (0.33, 0.64)
	4 -6 members		2.03 (1.60, 2.58)		0.77 (0.64, 0.93)
	7+ members		1.00		1.00
	Educational status of mother/carer			NS
	No formal education	2.24 (1.31, 3.80)	0.68 (0.47, 0.97)		0.25 (0.19, 0.34)
	Primary	2.09 (1.27, 3.44)	0.72 (0.53, 0.99)		0.42 (0.33, 0.54)
	Secondary or higher	1.00	1.00		1.00
	Natural Region
	City of Kigali	1.00	1.00	1.00	1.00
	South	1.84 (1.16, 2.92)	0.70 (0.50, 0.96)	0.60 (0.45, 0.80)	1.03 (0.76, 1.39)
	West	1.55 (0.96, 2.51)	0.82 (0.58, 1.14)	0.65 (0.48, 0.88)	1.06 (0.78, 1.45)
	North	1.14 (0.67, 1.91)	0.93 (0.66, 1.33)	0.72 (0.53, 0.98)	0.48 (0.34, 0.67)
	East	1.77 (1.08, 2.91)	0.84 (0.59, 1.18)	0.56 (0.41, 0.77)	0.64 (0.46, 0.90)
	Place of residence		NS
	Urban	1.00		1.00	1.00
	Rural	0.52 (0.38, 0.70)		0.71 (0.57, 0.88)	0.78 (0.62, 0.97)
	NA -not applicable; NS -not significant at p<0.05

Socio-economic and demographic (pure explanatories)

  Age of child Sex of child Birth order Birth interval Age of household head Sex of household head Household structure Number of household members Educational status of respondent Family religion Natural region Place of residence

	Block 1: Pure explanatories
		BO	ED	RL	AH
	AG	SX	SH	HS	NR	HM	PR	BI
									Block 2: Intermediate variables
									PC	VC	TT	DC
									WT
									BR	BN
		F o r F o r
					Block 3: Status variable
				P P	OVC
				e e
					e r e r
						R FE R
		DI Block 4: Outcome variables		CO	e v i e e v i e	AR
								w w
		Block 4 DI -diarrhoea FE -fever AR -acute respiratory symptoms Block 3 OVC -orphaned and vulnerable children	O n l Block 1 BR -duration of breastfeeding Block 2 VC -child ever vaccinated BN -child sleeps under a bed net PC -prenatal care AG -age of the child SX -sex of the child BO -birth order BI -birth interval Block 1 BR -duration of breastfeeding Block 3 OVC -orphaned and vulnerable children Block 2 AG -age of the child O VC -child ever vaccinated BN -child sleeps under a bed net PC -prenatal care SX -sex of the child BO -birth order BI -birth interval n l DC -delivery care SZ -size of the child at birth
		CO -cough		y SZ -size of the child at birth AH -age of household head SH -sex of household head AH -age of household head WT -source of drinking water DC -delivery care TT -Toilet facility WT -source of drinking water TT -Toilet facility SH -sex of household head y HS -household structure
								HS -household structure HM -number of household members
								HM -number of household members ED -education of respondent
								ED -education of respondent RL -religion
								RL -religion NR -natural region
								NR -natural region PR -place of residence
									PR -place of residence
		Figure 2. Chain graph showing the direct determinants and the inter-block association for the
		morbidity indicators

  4. What does it mean that those were breastfed were MORE likely to be sick, and how does this fit with what is known about breastfeeding? What does it mean that households with more adults had less illness? Better child care and income (vs. cramped living conditions and less food)?

	5.
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	Breastfeeding status at the time of survey refers to current status for children under-
	five years. We could not include the duration of breastfeeding for censoring reasons
	as young children would have shorter duration when compared to their older
	-3 -

counterparts. Regarding the results, currently breastfed children are less likely to suffer from diarrhoea (which is an important result) but they are likely to be suffering from ARI symptoms. It is likely that mothers are relatively more likely to breastfeed children who suffer from illness than healthy children and hence reported as such.

. Resource needs to support orphans and vulnerable children in sub-Saharan Africa. Health Policy and Planning, 22, 21-27. The Cebu Study Team (1991). Underlying and proximate determinants of child health: The Cebu longitudinal health and nutritional study. American Journal of Epidemiology, 133, 185-200. UNAIDS. (2008). Report on the global AIDS epidemic. Joint United Nations Programme on HIV/AIDS, UNAIDS UNICEF. (2005). The state of the world's children: Childhood under threat. New York: UNICEF UNICEF. (2006). Africa's orphaned and vulnerable generations: children affected by aids. United Nations UNICEF. (2005). Guide to monitoring and evaluation of the national response for children orphaned and made vulnerable by HIV/AIDS. New York: UNICEF. Wermuth, N. (2005). Graphical chain models. In: Everitt B, Howell DC (Eds.), Encyclopaedia of Behavorial Statistics II. Wiley, Chichester. Whitehead, M., Evandrou, M., et al. (1997). As the health divide widens in Sweden and Britain, what's happening to access to care? British Medical Journal, 315, 1006-1009.


This book, written by a family physician, is intended to educate readers about the risks of sexually transmitted infections and the need for diagnosis and management through short stories featuring nineteen male and female characters. These short stories are easy to read but selectively explicit. They describe the nature of examinations and discussions between patients and their doctors in detail. When it comes to describing the sexual contact that led to the risk of infections, the writing errs on the side clinical descriptions rather than erotic. Indeed, this an apparent dilemma that the book creates for itself. It has a sexy title and cover page featuring red heeled shoes and discarded tie, yet does not describe the sexual contact that leads to the risk of infection in a way that might resonate with lay readers.

The book, however, certainly meets its aim of imparting information in each of the interspersed fact sheets about STIs, and these are factually correct and well simplified. However, the main feeling from reading the book is that patients who suspect they have STIs should go for tests, and they will meet non-judgemental, friendly clinicians. In terms of true prevention, it might have been helpful to read more detail of the stories of acquisition as they relate to the use of alcohol, drugs and other factors, and to draw in to patients how the indicator STI or fear of an STI might help them to modify risk taking behaviours fuelled by other adverse health behaviours.

In summary, this is a book which goes three quarters to meeting its objective, but misses an opportunity to really confront the determinants of sexual ill health, and perhaps through writing explore the reasons why people seek casual sexual contact and how, through knowledge and insight, they can reduce their risk of an adverse psychological and physical health outcome from these experiences.

Simon E. Barton, BSc MD FRCOG FRCP(Edin) FRCP

Clinical Director HIV/Genitourinary Medicine Services Chelsea & Westminster Hospital, London