Sophie Mercier
email: sophie.mercier@univ-mlv.fr

Numerical bounds for semi-markovian quantities and application to reliability

Keywords: Continuous and discrete time homogeneous semi-Markov processes, Markov renewal equations, Numerical algorithms AMS: 60K15, 90B25

We propose new easily computable bounds for di¤erent quantities which are solutions of Markov renewal equations linked to some continuoustime semi-Markov process (SMP). The idea is to construct two new discretetime SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge.

To illustrate the results, numerical bounds are provided for two quantities from the reliability …eld: mean sojourn times and probability transitions.

Introduction

Semi-Markov processes (SMP) are used in reliability to model the time-evolution of a system with a …nite (or countable) state space divided into up-and down-1 states. Among quantities of interest to measure the performance of the system are di¤erent time-dependent ones, such as: the time-dependent availability, namely the probability that the system is in an up-state at some time t; the mean number of failures of the system on some time interval [0; t]; the mean cumulated up-and down-times on [0; t] (see [START_REF] Csenki | Transient Analysis of Semi-Markov Reliability Models -A Tutorial Review with Emphasis on Discrete-Parameter Approaches[END_REF]] e.g. for other quantities). If asymptotic quantities are often easily analytically computable for semi-markovian systems, it is usually not the case for time-dependent ones. Indeed, lots of them are solutions of Markov renewal equations, namely Volterra integral equations of the second kind, and only their Laplace transforms are usually analytically reachable. Due to the instability of the numerical inversion of the Laplace transform [START_REF] Cocozza-Thivent | Processus stochastiques et …abilité des systèmes[END_REF]] or [START_REF] Csenki | Transient Analysis of Semi-Markov Reliability Models -A Tutorial Review with Emphasis on Discrete-Parameter Approaches[END_REF]], this is not always a very good method for their numerical computations. Other methods have then been developed such as the phase method, where the idea is to approach an SMP by a Markov process for which computations are much simpler. The main problem here is with the identi…cation of the phases and their number. Other methods consist in the numerical resolution of the Volterra integral equations, see [START_REF] Csenki | Transient Analysis of Semi-Markov Reliability Models -A Tutorial Review with Emphasis on Discrete-Parameter Approaches[END_REF]] with lots of reference therein or [Fritz, Posgai & Bertsche (2000)].

An algorithm for the computation of the marginal distribution of an SMP has also been developed recently, using …nite volume methods [Cocozza-Thivent & Eymard (2004)].

Other methods consist in approaching continuous time SMP by discrete time SMP for which computations are much simpler (see [START_REF] Csenki | Transient Analysis of Semi-Markov Reliability Models -A Tutorial Review with Emphasis on Discrete-Parameter Approaches[END_REF]] with reference therein). In such methods, the main problem is that the accuracy of the results is generally not known.

In the present paper, we propose some method of the last category, namely approaching continuous time SMP by discrete time SMP. The precision of the approximation is here known, and may be made as small as wanted (at least theoretically): for a given continuous-time (homogeneous) SMP, the principle is to construct two new SMP with the same Markov chain as the initial SMP, but with inter-arrival times which bound the initial ones. This allows to bound solutions of Markov renewal equations linked to the initial SMP by solutions of Markov renewal equations linked to the two new SMP. The convergence of the bounds is established with minimal assumptions on the initial semi-Markov kernel (no assumption of density with respect to Lebesgue measure; no need for the inter-arrival times to be almost surely strictly positive; countable but not necessarily …nite state space). The two new SMP are discrete time processes and jump only at points kh (k 2 N), which allows exact calculation for the resolution of the associated Markov renewal equations as in other methods of the same kind. This provides bounds for di¤erent time-dependent quantities linked to some general SMP.

A similar method has been used in [START_REF] Elkins | On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds[END_REF]] to bound the Markov renewal function (a special case of ours) in case of a …nite state space, though not described in terms of discrete SMP as here. However, this previous paper is mainly concerned with the numerical computation of the bounds and its implementation, and does not take in consideration the mathematical convergence.

An approximating discrete time SMP has also been used in [START_REF] Blasi | Numerical treatment of homogeneous and nonhomogeneous semi-Markov reliability models[END_REF]], [START_REF] Janssen | Numerical solution of non-homogeneous semi-Markov processes in transient case[END_REF]] and [Corradi, Janssen & Manca (2004)], where the almost sure convergence of the approximating discrete time SMP towards the initial continuous-time SMP is proved in the sense of the Skorokhod topology.

Contrary to the present paper, the non-homogeneous case is also envisionned in such papers but the initial semi-Markov kernel is assumed to admit density with respect to Lebesgue measure, the state space is …nite and the inter-arrival times are assumed to be almost surely positive. Under such assumptions, their approximating discrete time SMP roughly meet with one of those considered here. However, the construction of a second one allows us to get here some bounds for the goal quantities whereas the accuracy of the numerical results is not provided in the quoted papers. Also, the link is not made there between the a.s. convergence in the sense of the Skorokhod topology and the convergence of the appproximation for the goal quantity, the transition probabilities in their case (a special case of ours).

Finally, the same bounding method as in the present paper has already been used in [START_REF] Mercier | Encadrement de variables aléatoires réelles et application à la …abilité[END_REF]] and [START_REF] Mercier | Discrete random bounds for general random variables and applications to reliability[END_REF]] to bounds a few other performance measures from the reliability …eld, such as 1) cumulative density functions of sums of i.i.d. nonnegative random variables, 2) renewal functions and 3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables.

Such performance measures may all be interpreted in terms of SMP so that they can be seen as special cases of the present paper.

The paper is organized as follows: the notations and assumptions are given in Section 2, as well as some recalls on SMP. The two discrete SMP are constructed in Section 3 and bounds for solutions of Markov renewal equations associated to the initial SMP are derived. The convergence of the bounds when the step size h goes to 0 is also proved in this section. Solutions for Markov renewal equations associated to the two new discrete SMP are given in Section 4. Applications and numerical examples are provided in Section 5 and numerical bounds are computed for two quantities of interest for semi-markovian systems: mean sojourn times and transition semi-group. Conclusions are derived in Section 6.

Notations, assumptions and recalls

Let E be some …nite or countable space and let (Y n ; T n ) n2N be a (homogeneous) Markov renewal process with T 0 = 0 and Y n 2 E for all n 2 N. Also, let (q (i; j; dt)) i;j2E be the associated semi-Markov kernel, namely some family of non-negative measures such that P j2E q (i; j; dt) is a probability measure (all i 2 E) and such that:

P (Y n+1 = j \ T n+1 T n tjY 0 = i 0 ; T 1 = t 1 ; :::; Y n 1 = i n 1 ; T n = t n ; Y n = i) = P (Y n+1 = j \ T n+1 T n tjY n = i) = P (Y 1 = j \ T 1 tjY 0 = i) = Z [0;t]
q (i; j; du) = q (i; j; [0; t]) ; for all n 2 N, all t 0, all t 1 ; ::; t n 2 R + , all i; j 2 E, all i 0 ; :::; i n 1 2 E such that the conditional probability exists.

The transition matrix for the Markov chain (Y n ) n2N then is

P i;j = P (Y 1 = jjY 0 = i) = P i (Y 1 = j) = q (i; j; R + )
for all i; j 2 E, where P i stands for the conditional probability distribution

P ( jY 0 = i).
We assume that (T n ) n2N is such that sup n2N T n = +1 a.e. (regularity assumption). We recall from [START_REF] Cinlar | Introduction to stochastic processes[END_REF]] that su¢ cient conditions for that are: P i (T 0 = ::: = T n = ::: = 0) = 0 for all i 2 E and …nite number of transient states for the Markov chain (Y n ) n2N , or existence of b; c > 0 such that P i (T 1 > b) > c for all i 2 E (other conditions in [START_REF] Cinlar | Introduction to stochastic processes[END_REF]]).

Symbol (X t ) t 0 stands for the continuous-time (homogeneous) SMP associated with (Y n ; T n ) n2N :

X t = Y n if T n t < T n+1 ;
for all n 2 N, all t 0. The transition semi-group for (X t ) t 0 is (P t ) t 0 with:

P t (i; j) = P i (X t = j) ;
for all i; j 2 E, all t 0. Also, the Markov renewal function (i; j; [0; t]) associated to (X t ) t 0 is

(i; j; [0; t]) = E i 0 @ X n 0 1 fTn tg 1 fYn=jg 1 A
for all i; j 2 E, all t 0, where 1 fg stands for the indicator function.

(i; j; [0; t])
is the mean number of visits to j on [0; t] for (X t ) t 0 starting from i. Note that due to assumption sup n2N T n = +1 a.e., we have P i (T 1 = T 2 = ::: = T n = ::: = 0) = 0 for all i 2 E, so that (i; j; [0; t]) < +1 for all i; j 2 E, all t 0, see [START_REF] Cinlar | Introduction to stochastic processes[END_REF] For all f 2 B + , we set:

(dq f ) (i; t) = X j2E Z [0;t] f (j; t u) q (i; j; du) = X j2E E i f (j; t T 1 ) 1 fT1 tg 1 fY1=jg = E i f (Y 1 ; t T 1 ) 1 fT1 tg and (d f ) (i; t) = X j2E Z [0;t] f (j; t s) (i; j; ds) = X j2E X n 0 E i f (j; t T n ) 1 fTn tg 1 fYn=jg = X n 0 E i f (Y n ; t T n ) 1 fTn tg (1)
with dq f 2 B + and d f 2 B + .

We recall from [START_REF] Cinlar | Introduction to stochastic processes[END_REF]] that, due to the assumption sup n2N T n = +1

a.e., for all g 2 B + , the equation

f = g + dq f
has one single solution f g 2 B + which is:

f g = d g (2)
with f g < +1.

3 Bounding f g 3.1 Construction of X h t t 0 and of X h+ t t 0

Let b:::c be the ‡oor function, namely the function from R to Z such that, for all x 2 R, bxc is the single integer such that:

bxc x < bxc + 1
For any h > 0 and any random variable (r.v.) U with general distribution, we may set:

U h = h U h and U h+ = h U h + h
(same notation in all the paper) with

U h U < U h+ and lim h!0 + U h = lim h!0 + U h+ = U namely: U h (!) U (!) < U h+ (!) and lim h!0 + U h (!) = lim h!0 + U h+ (!) = U (!) for all ! in the domain of U .
Now, let q h (i; j; dt) i;j2E and q h+ (i; j; dt) i;j2E be the semi-Markov kernels associated to h > 0 and (q (i; j; dt)) i;j2E ; and such that:

q h (i; j; [0; t]) = P Y 1 = j \ T h 1 tjY 0 = i (3) q h+ (i; j; [0; t]) = P Y 1 = j \ T h+ 1 tjY 0 = i (4) with T h 1 = h T1 h and T h+ 1 = h T1 h + h = T h 1 + h.
We can then construct two new SMP X h t t 0 and X h+ t t 0 respectively associated to q h (i; j; dt) i;j2E and q h+ (i; j; dt) i;j2E with the same initial distribution as (X t ) t 0 . Both of those SMP have semi-Markov kernels supported by hN. In other words, they are discrete-time SMP. The associated Markov chains are "copies" of the initial Markov chain (Y n ) n2N associated to (X t ) t 0 in the sense that they have the same initial distribution and the same transition matrix, due to q h (i; j; R + ) = q h+ (i; j; R + ) = q (i; j; R + ) = P i;j for all i; j 2 E. The visited states for X h t t 0 and X h+ t t 0 actually are the same for X h t t 0 and X h+ t t 0 as for the initial SMP (X t ) t 0 . Those "copies" of (Y n ) n2N are still denoted by (Y n ) n2N in all the following.

The inter-arrival times for X h t t 0 and X h+ t t 0 respectively are

(T n+1 T n ) h n2N and (T n+1 T n ) h+ n2N
where

(T n+1 T n ) h = h T n+1 T n h T n+1 T n (T n+1 T n ) h+ = (T n+1 T n ) h +h:
Then, the SMP X h t t 0 stays shorter in each state than (X t ) t 0 whereas the SMP X h+ t t 0 stays longer. Note that the approximation is made on the inter-arrival times T n+1 T n and not on the arrival-times T n . Indeed, in order to construct an approximate SMP, the approximate n th inter-arrival time ]

T n+1 f T n given Y 0 = i 0 ; T0 = t 0 ; :::; Y n 1 = i n 1 ; Tn 1 = t n 1 ; Y n = i; Tn = t n and Y n+1 = j should be conditionaly distributed as f T 1 given Y 0 = i and Y 1 = j. This is true if ] T n+1 f T n
depends on T n and on T n+1 only through their increment T n+1 T n (all n 2 N). It is not true any more if the approximation is taken on T n and T n+1 themselves. (For instance, taking f

T n = h Tn h would not lead to a SMP because h j Tn+1 h k h Tn h 6 = h j Tn+1 Tn h k ).
The respective arrival times for X h t t 0 and X h+ t t 0 actually are:

T (h) n = n 1 X i=0 (T i+1 T i ) h T (h+) n = n 1 X i=0 (T i+1 T i ) h+ = T (h) n + nh (5) (note the nh in T (h+) n
).

In the case where E is …nite and q (i; j; dt) admits some density with respect to Lebesgue measure, X h+ t t 0 meets with the approximating discrete time SMP studied in [START_REF] Blasi | Numerical treatment of homogeneous and nonhomogeneous semi-Markov reliability models[END_REF]] and [Corradi, Janssen & Manca (2004)] (with approximate arrival times substituted by (5)).

For all other notations than Y n , T , we add some subscripts h and h+ when referring to the SMP X h t t 0 and X h+ t t 0 , respectively. For instance, for g 2 B + , symbol f h g stands for the single solution of f = g + dq h f (under assumption which ensures existence and singleness, see the following).

We …nally express the semi-Markov kernels q h (i; j; dt) i;j2E and q h+ (i; j; dt) i;j2E with respect to the initial one (q (i; j; dt)) i;j2E .

Lemma 1 For all i; j 2 E, all h > 0, the measures q h (i; j; dt) and q h+ (i; j; dt) have supports included in hN with

q h (i; j; kh) = q (i; j; [kh; (k + 1) h[) q h+ (i; j; kh) = 1 fk 1g q (i; j; [(k 1) h; kh[) = 1 fk 1g q h (i; j; (k 1) h) for all k 2 N.
Proof. It is clear that the supports are included in hN. Besides, for all k 2 N, we have:

h T 1 h kh , (T 1 < h (k + 1))
We get:

q h (i; j; kh) = q h (i; j; [0; kh]) 1 fk 1g q h (i; j; [0; (k 1) h]) = q (i; j; [0; (k + 1) h[) 1 fk 1g q (i; j; [0; kh[) = q (i; j; [kh; (k + 1) h[)
and similar results for q h+ .

A technical lemma

We now give a technical condition which ensures us with sup n2N T

(h) n = +1 a.e.
and in particular with existence and uniqueness of solutions to Markov renewal equations associated to q h (i; j; dt) i;j2E (clear for q h+ (i; j; dt) i;j2E due to

sup n2N T (h+) n sup n2N nh = +1, see (5)). Lemma 2 If C = sup f(i;j):Pi;j 6 =0g E i (exp ( T 1 ) jY 1 = j) < 1, then sup n2N T (h) n = +1
a.e. for all 0 < h < ln C. In case E is …nite, the condition somewhat simpler min f(i;j):Pi;j 6 =0g E i (T 1 jY 1 = j) > 0 is equivalent to the condition C < 1.

Proof. For all n 2 N and t 0, we have:

P T (h) n t P (T n t + nh) = P e (t+nh)
e Tn

e t+nh E e Tn due to Markov inequality. Besides:

E e Tn = E exp n 1 X k=0 (T k+1 T k ) !! = E E n 1 Y k=0 exp ( (T k+1 T k )) jY 0 ; Y 1 ; :::; Y n !! = E n 1 Y k=0 E (exp ( (T k+1 T k )) jY 0 ; Y 1 ; :::; Y n ) ! due to independence of all T k+1 T k (with 0 k n 1) given Y 0 ; Y 1 ; :::; Y n .
By assumption:

E (exp ( (T k+1 T k )) jY 0 ; Y 1 ; :::; Y n ) C < 1
Whence:

P T (h) n t e t+nh C n = e t e n(h+ln C) (6)
and by monotony:

lim n!+1 # P T (h) n t = P sup n2N T (h) n t = 0
for all 0 < h < ln C and t 0. We derive by monotony again that sup n2N T (h) n = +1 a.e. for all 0 < h < ln C. Finally, it is easy to see that, for all i; j 2 E such that P i;j 6 = 0, the condition E i (exp ( T 1 ) jY 1 = j) < 1 is equivalent to

P i (exp ( T 1 ) < 1jY 1 = j) = P i (T 1 > 0jY 1 = j) > 0, which is also equivalent to E i (T 1 jY 1 = j) > 0.
Whence the result in case E is …nite.

Bounds for f g and convergence of the bounds

We now turn to the main result of this paper which provides us with bounds for f g and the convergence of the bounds when h goes to 0.

Theorem 3 Let (X t ) t 0 be a semi-Markov process such that C = sup f(i;j):P (i;j)6 =0g E i (exp ( T 1 ) jY 1 = j) < 1. For all g 2 B + :

1. if t 7 ! g (i; t) is non-decreasing for all i 2 E, then, for all 0 < h < ln C:

f h+ g = d h+ g f g = d g f h g = d h g < +1 (7)
2. if g is of the shape g = g 1 g 2 with g 1 ; g 2 2 B + and t 7 ! g j (i; t) nondecreasing for j = 1; 2 (all i 2 E), then, for all 0 < h < ln C:

f h+ g1 f h g2 f g = f g1 f g2 f h g1 f h+ g2 < +1 (8)
3. if g is uniformly continuous on E [0; t] where t 0 (namely setting " g ( ) = sup j2E sup jx yj

x;y2[0;t] jg (j; x) g (j; y)j, we have lim !0 + " ( ) = 0), then:

lim h!0 + f h g (i; t) = lim h!0 + f h+ g (i; t) = f g (i; t)
Proof. Uniqueness, existence and …niteness of f h+ g , f g and f h g is clear due to Lemma 2 and assumption on (X t ) t 0 , for all 0 < h < ln C. Besides, inequality (7) may easily be derived from the de…nition of d g (see (1)), from the non-increasingness of u 7 ! g (j; t u) 1 fu tg and from the fact that

T (h) n T n < T (h+) n
. This gives the …rst point. The second point is a direct consequence from the …rst point using f g = f g1 f g2 . As for the third point, for 0 < h < ln C and n 0 2 N …xed, we may write:

f h g (i; t) f g (i; t) = X n 0 E i 1 n T (h) n t o g Y n ; t T (h) n 1 fTn tg g (Y n ; t T n ) S 1 (n 0 ; h) + S 2 (n 0 ; h) with S 1 (n 0 ; h) = n0 X n=0 E i 1 n T (h) n t o g Y n ; t T (h) n 1 fTn tg g (Y n ; t T n ) S 2 (n 0 ; h) = X n>n0 E i 1 n T (h) n t o g Y n ; t T (h) n 1 fTn tg g (Y n ; t T n ) Writing 1 n T (h) n t o g Y n ; t T (h) n 1 fTn tg g (Y n ; t T n ) = 1 n T (h) n t o 1 fTn tg g Y n ; t T (h) n + 1 fTn tg g Y n ; t T (h) n g (Y n ; t T n )
in S 1 (n 0 ; h), we get

S 1 (n 0 ; h) n0 X n=0 E i 1 n T (h) n t o 1 fTn tg g Y n ; t T (h) n + n0 X n=0 E i 1 fTn tg g Y n ; t T (h) n g (Y n ; t T n )
Noting that

1 n T (h) n t o 1 fTn tg = 1 n T (h) n t o 1 fTn tg 1 ft<Tn<t+nhg and T n T (h)
n nh n 0 h for all 0 n n 0 ;

and using the function " g ( ) de…ned in the theorem and similar argument as for (6), we get:

S 1 (n 0 ; h) kgk [0;t] n0 X n=0 P i (t < T n < t + nh) + " g (n 0 h) n0 X n=0 P i (T n t) kgk [0;t] n0 X n=0 P i (t < T n < t + nh) + (n 0 + 1) " g (n 0 h) with lim h!0 + P i (t < T n < t + nh) = P i (t < T n t) = 0
for all 0 n n 0 . We derive:

lim h!0 + (S 1 (n 0 ; h)) = 0
for all …xed n 0 . Moreover:

S 2 (n 0 ; h) kgk [0;t] X n>n0 h P i T (h) n t + P i (T n t) i kgk [0;t] X n>n0 C n e t+nh + C n e t kgk [0;t] e h(n0+1) 1 Ce h + 1 1 C e t C n0+1
We derive:

lim sup h!0 + f h g (i; t) f g (i; t) kgk [0;t] 2 1 C e t C n0+1
for all …xed n 0 . Taking the limit when n 0 ! +1, we get lim sup h!0 + f h g (i; t) f g (i; t) = 0 and the result for f h g . The result for f h+ g may be proved similarly, which completes the proof.

In case where the kernel (q (i; j; dt)) i;j2E is triangular, one can provide alternate bounds in some special cases:

Proposition 4 Let (X t ) t 0 be a semi-Markov process such that C = sup f(i;j):P (i;j)6 =0g E i (exp ( T 1 ) jY 1 = j) < 1. Then let v : E ! R + and let

w : E R + ! R + be such that w (i; t) = v (i) P i (T 1 > t)
. Assume E to be equipped with some order relation and the kernel (q (i; j; dt)) i;j2E to be upper triangular. Then if v is non decreasing, we have, for all 0 < h < ln (C), t 0, i 2 E :

f h+ w h+ (i; t) f w (i; t) = E i (v (X t )) f h w h (i; t) (9) with w h (i; t) = v (i) P i T h 1 > t = v (i) P i T 1 t h+ w h+ (i; t) = v (i) P i T h+ 1 > t = v (i) P i T 1 t h
Inequalities are reversed in case (q (i; j; dt)) i;j2E is lower triangular or v is non increasing.

Proof. Setting F Yn (t) = P Yn (T 1 > t) = E 1 fTn+1 Tn>tg jY n , we have:

f w (i; t) = X n2N E i 1 fTn tg v (Y n ) F Yn (t T n ) = X n2N E i 1 fTn tg v (Y n ) 1 fTn+1 Tn>t Tng = X n2N E i v (Y n ) 1 fTn t<Tn+1g (10) = E i (v (X t ))
Beside, starting again from (10):

f w (i; t) = X n2N E i v (Y n ) 1 ft<Tn+1g 1 ft<Tng = X n2N E i (v (Y n ) v (Y n+1 )) 1 ft<Tn+1g
If (q (i; j; dt)) i;j2E is upper triangular, we know that: Y n Y n+1 and consequently, assuming v to be non decreasing, v (Y n ) v (Y n+1 ) 0 for all n 2 N.

Due to

T (h) n T n < T (h+) n
for all n 2 N, we derive:

1 n t<T (h) n+1 o 1 ft<Tn+1g 1 n t<T (h+) n+1
o and (9).

Remark 5 Note that t 7 ! w (i; t) is here non increasing whatever the monotony of v is, so that (7) cannot be applied to g = w. In the case where (q (i; j; dt)) i;j2E

is triangular and v is monotone, (9) provides simpler bounds than (8).

Remark 6

In case E = f1; :::; m + 1g with only possible transitions 1 ! 2 ! ::: ! m ! m + 1, taking v (i) = 1 fm ig and U i a random variable with distribution q (i; i + 1; dt) for 1 i m (U 1 , ..., U m independent), (9) then provides bounds for E 1 (v (X t )) = P 1 (X t m) = P (U 1 + ::: + U m t) which are similar to those obtained in [START_REF] Mercier | Discrete random bounds for general random variables and applications to reliability[END_REF]].

Numerical computations

Under assumptions of Theorem 3, it is known that for all g 2 B + and 0 < h < ln C, the equation f = g + dq h f or equivalently

f h g (i; N h) = g (i; N h)+ X j2E N X k=0 q h (i; j; (N k) h) f h g (j; kh) for all N 0, all i 2 E (11)
has got one single solution f h g = d h g (the same for dq h+ ). We here provide algorithms very easy to emplement for the recursive computation of f h g and f h+ g in case E is …nite (recursion on N ).

Proposition 7 Let us assume E to be …nite and min f(i;j):Pi;j 6 =0g E i (T 1 jY 1 = j) > 0. Let g 2 B + and 0 < h < ln C where C = max fi;j2E:P (i;j)6 =0g E i (exp ( T 1 ) jY 1 = j) <

1. For all n 2 N, we set: q h ( ; ; N ) = q h (i; j; N ) i;j2E , f h g ( ; N h) = f h g (i; N h) i2E columnwise, the same for g ( ; N h) and f h+ g ( ; N h). Let I be the identity matrix with size cardinal(E). The matrix I q h ( ; ; 0) then is non singular and:

f h g ( ; 0) = I q h ( ; ; 0) 1 g ( ; 0) f h+ g ( ; 0) = g ( ; 0)
For all N 2 N, we have:

f h g ( ; N h) = I q h ( ; ; 0) 1 " g ( ; N h) + N 1 X k=0 q h ( ; ; (N k) h) f h g ( ; kh) # (12) f h+ g ( ; N h) = g ( ; N h) + N 1 X k=0 q h ( ; ; (N k 1) h) f h+ g ( ; kh) (13)
Proof. Let be a eigen value of q h ( ; ; 0). We know that:

j j max i2E 0 @ X j2E q h (i; j; 0) 1 A = max i2E 0 @ X j2E P i (T 1 < hjY 1 = j) P (i; j) 1 A = max i2E 0 @ X j2E P i e T1 > e h jY 1 = j P (i; j) 1 A e h C < 1
for all 0 < h < ln C. We derive that 1 is not an eigenvalue of q h ( ; ; 0) so that I q h ( ; ; 0) is non singular. The other results may easily be derived from (11) and from q h+ ( ; ; N ) = 1 fN 1g q h ( ; ; N 1) for all N 2 N.

Remark 8 The algorithms provided here are the most natural and might surely be improved (see [START_REF] Elkins | On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds[END_REF]]). As for the computation of f h+ g ( ; N h), a similar algorithm as the present one may also be found in [Barbu, Boussemart, Limnios (2004)] e.g.. As for the computation of f h g ( ; N h), the matrix inversion ( I q h ( ; ; 0) 1 ) that appears is actually not computed: an LU factorization is provided for I q h ( ; ; 0) once for all at the beginning of the numerical computations and two successive triangular systems are then solved each time quantities like I q h ( ; ; 0) 1 y are computed (with y a column vector). This saves much computation time.

Applications

We …nally provide numerical bounds for two di¤erent performance measures from the reliability …eld: one in the case where t 7 ! g (i; t) is non-decreasing (the cumulated mean sojourn time), one in the case where g = g 1 g 2 with both g j non-decreasing with respect of t (the transition probabilities).

In all this section, E is …nite, (X t ) t 0 is such that min f(i;j):Pi;j 6 =0g E i (T 1 jY 1 = j) > 0 and h is such that 0 < h < ln (C) = ln max fi;j2E:P (i;j)6 =0g E i (exp ( T 1 ) jY 1 = j) (see Lemma 2).

Cumulated mean sojourn time

For i 2 E, A E and t 0, we are interested in the cumulated mean sojourn time in A on [0; t] starting from i de…ned by:

C A (i; t) = E i Z [0;t] 1 fXu2Ag du ! = X j2A Z [0;t] P u (i; j) du Setting g A (i; t) = 1 fi2Ag Z [0;t] P i (T 1 > u) du
for all i 2 E, all t 0 and using standard methods, one easily gets:

C A = g A + dq C A
and hence:

C A = d g A
where t 7 ! g A (i; t) is non-decreasing for all i 2 E and where u 7 ! g (i; u) is uniformly continuous on [0; t]. We derive from Theorem 3:

f h+ g A C A = f g A f h
g A and the convergence of both bounds when h ! 0 + . The bounds are then computed with the results of Proposition 7.

Example 9 A semi-Markov reparable system is considered, which may be up (i = 1; 2; 3) or down (i = 4; 5) at time t. (The system may go on degrading when down). The system is initially in state 1 and we are interested in the cumulated mean down time on [0; t] starting from 1, namely C A (1; t) with A = f4; 5g. We take:

q ( ; ; dt) = 0 B B B B B B B B B @ 0 (1 ) w 1 (t) dt + 0 0 0 0 0 0 (1 ) w 2 (t) dt + 0 0 0 0 0 0 (1 ) w 3 (t) dt + 0 0 2 3 (1 ) w 4 (t) 1 6 (1 ) w 4 (t) 0 0 1 6 (1 ) w 4 (t) + 0 2 3 w 5 (t) 1 6 w 5 (t) 0 1 6 w 5 (t) 0 1 C C C C C C C C C A
where 0 stands for the Dirac mass at 0 and w i (t) stands for the Weibull p.d.f.of W ( i ; i ):

w i (t) = i i t i 1 e it i 1 R+ (t)
Note that there may be some instantaneous degradation with probability so that q ( ; ; dt) does not admit density with respect of Lebesgue measure.

We here have:

g A (i; t) = 1 fi=4g (1 ) Z [0;t] e 4t 4 dt + 1 fi=5g Z [0;t]
e 5t 5 dt for all i 2 E, t 0 and we take:

1 = 10 6 ; 1 = 1:6; 2 = 10 5 ; 2 = 2:2; 3 = 10 4 ; 3 = 3:3; 4 = 10 3 ; 4 = 4:5; 5 = 10 3 ; 5 = 3:5; = 10 1 which respective means m i and coe¢ cients of variations cv i : m 1 ' 5:042 10 3 ; cv 1 ' 0:64; m 2 ' 1:659 10 2 ; cv 2 ' 0:48; m 3 ' 14:62; cv 3 ' 0:33; m 4 ' 4:236; cv 4 ' 0:25; m 5 ' 6:48; cv 5 ' 0:3165

The numerical results are displayed in Figure 1 for h = 6 as well as the asymptotic direction (see [START_REF] Cinlar | Introduction to stochastic processes[END_REF]] e.g.):

C A (i; t) t!+1 X j2A E j (T 1 ) (1; j; [0; t]) t!+1 t P 5 j=4 E j (T 1 ) (j) P 5 k=1 (k) E k (T 1 ) ' 1:49 10 3 t
where is the stationary distribution for the embedded Markov chain (Y n ) n2N and E k (T 1 ) = ( 1) m k for 1 k 4, E 4 (T 1 ) = m 5 . In order to better study the precision of the results, one also provides the results for di¤ erent values of h and t in Table 1 We can see in such a table that, as expected, the smaller the discretization step h, the tighter the bounds are. Also, for h …xed, the error …rst decreases when t increases (for t . 900) and then increases with t. For small t, one may think that the lack of precision is due to some round-o¤ error. For bigger t, the loss of precision when t increases is clearly due to the method.

Transition semi-group

We are now interested in computing the transiton semi-group P t (i; j) and we set: f j (i; t) = P t (i; j) for i; j 2 E and t 0. We recall that, setting g j (i; t) = 1 fi=jg P i (T 1 > t) for all i; j 2 E, all t 0, we have: [START_REF] Cocozza-Thivent | Processus stochastiques et …abilité des systèmes[END_REF]] e.g. or [START_REF] Cinlar | Introduction to stochastic processes[END_REF]].

f j = g j + dq f j namely f j = f gj = d g j , see
Noting that g j = I j u j with I j (i; t) = 1 fi=jg and u j (i; t) = 1 fi=jg P i (T 1 t), we get P t (i; j) = f gj (i; t) = f Ij (i; t) f uj (i; t) with f Ij (i; t) = (i; j; [0; t]). As I j and u j are both non-decreasing with respect of t for all j 2 E, we derive from (8) :

h+ (i; j; [0; t]) f h uj (i; t) P t (i; j) = f gj (i; t) h (i; j; [0; t]) f h+ uj (i; t) (14)

and the convergence of both bounds in case t 7 ! P i (T 1 t) is continuous.

The computation of the numerical bounds provided by ( 14) is here much longer than in the previous case where t 7 ! g j (i; t) were non-decreasing. This has lead us to adapt the program in order to take into account only non trivial terms in (12 13). Indeed, in reliability theory, mean down times are often much shorter than mean up times so that non trivial terms (at a given precision) among the q h (i; j; kh)'s may highly depend on i and j. Keeping only the non trivial terms in P N 1 k=0 q h (i; j; (N k) h) f h g (j; kh) then save much computation time.

Example 10 We here consider an example extracted from [Cocozza-Thivent & Eymard (to appear)] denoted by [CTE] in the following: a system is considered with two components in cold stand-by redundancy (see [CTE] for details) and its evolution is described by a SMP with state space E = f1; 2; 3; 4g and semi-Markov kernel:

q ( ; ; dt) = 0 B B B B B B @ 0 w 1 (t) W 0 1 (t) 0 w 0 1 (t) W 1 (t) l 2 (t) W 0 2 (t) 0 w 0 2 (t) L 2 (t) 0 0 0 0 0 0 0 0 0 1 C C C C C C A dt
where W i (t) and L i (t) stand for the respective survival functions of W ( i ; i )

and of log-normal distribution LN ( i ; i ) with p.d.f.

l i; i (t) = 1 t p 2 e (ln(t) i ) 2 2 2 1 R + (t) ;
the same for W 0 i (t) and L 0 i (t) with i and i substituted by 0 i and 0 i . Following [CTE], we take: 1 = 1=2216 1:5 ' 9:59 10 6 ; 1 = 1:5; 0 1 = 1=11284 2 ' 7:85 10 9 ; 0 1 = 2; 2 = 4:5; 2 = 0:5; 0 2 = 1=2000 Here again, the smaller the discretization step h , the tighter the bounds are.

Also, for h …xed, the error …rst increases with t and then decreases.

As for the comparison between the method from [CTE] and the present one, one can look at the asymptotic value: we know that P t (1; 3) is increasing with respect of t (see Figure 2) and that lim t!+1 P t (1; 3) ' 0:386 365 (easy computation using standard method). In Table 2, we can see that the value from

[CTE] for P t (1; 3) and t = 1:99 10 5 is slightly too big (0:395 72) whereas our bounds (0:385 96 and 0:386 06) are coherent with the asymptotic value. A similar remark is valid for P t (1; 4) with P t (1; 4) slightly too small in [CTE] for t = 1:99 10 5 whereas our bounds are coherent.

We can then conclude that, in this example, our bounds seem slightly more accurate than the results from [CTE] (which may however be improved using some smaller discretization step).

In case where the semi-Markov kernel is triangular, we …nally provide a last numerical example to compare the bounds provided by Theorem 3 and by Proposition 4.

Example 11 A semi-Markov unreparable system is considered with state space E = f1; :::; 5g, U = f1; 2; 3g, D = f4; 5g. We want to evaluate the unreliability of the system starting from 1, namely F (t) = P 1 (T t) where T is the time to failure of the system. The semi-Markov kernel is:

q ( ; ; dt) = 0 B B B B B B B B B @ 0 (1 ) w 1 (t) dt + 0 0 0 0 0 0 (1 ) w 2 (t) dt + 0 0 0 0 0 0 (1 ) w 3 (t) dt + 0 0 0 0 0 0 (1 ) w 4 (t) dt + 0 0 0 0 0 0 1 C C C C C C C C C A
with w 1 , w 2 , w 3 , w 4 and the same as in Example 9.

Due to the triangular shape of the kernel, we here have: F (t) = P 1 (X t 4) = f w (t) with w (i; t) = v (i) P i (T 1 t) and v (i) = 1 fi 4g , all i 2 E (see Proposition 4 and Remark 6). As v is non-decreasing and q ( ; ; dt) is upper triangular, we may apply (9). The bounds for the unreliability are plotted in Figure 5 for h = 5 as provided by (14) (bounds 1) and by (9) (bounds 2). We can see that, as expected, the bounds provided by (9) are tighter than those provided by (14). Beside, the computations are quicker for (9) than for (14). Then, it is better to use (9) than (14) when possible.

Conclusion

We have proposed, in this paper, easily computable bounds for di¤erent timedependent quantities for semi-markovian system. The idea used here, namely approximating continuous-time SMP by discrete-time SMP is not new. However, contrary to most of previous works (apart from [START_REF] Elkins | On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds[END_REF]], to our knowledge), the present construction of the approximating processes has allowed us to control the error between the goal quantities and their approximations. The theoretical results have been tested on a few numerical examples, showing the accuracy of the method.

We have focused here in solutions of Markov renewal equations. Note however that the bounding method might be generalized and that for any function non-decreasing e.g. with respect to all T n and such that the expectations exist, we would have: n + nh. More generally, one could also consider h = (h i ) i2E and the semi-Markov kernels q h (i; :; dt) and q h+ (i; :; dt) (both supported by h i N) with: q h (i; j; kh i ) = P i (Y 1 = j; kh i T 1 < (k + 1) h i ) q h+ (i; j; kh i ) = 1 fk 1g q h (i; j; (k 1) h i )

E (Y n ) n2N ; T (h) n n2N E (Y n ) n2N ; (T n ) n2N E (Y n ) n2N ; T (h+)
Here again, the bounding would remain true. However, more general than in the present paper and/or non constant step size h would lead to more complicated computations of the approximate quantities, which only require here a few lines of implementation.

Figure 1 .

 1 Figure 1. Example 9, Cumulated Mean Down Time on [0; t], h = 6:

Figure 5 .

 5 Figure 5. Example 11, F (t) by the two methods, h = 20

Figure 1 .

 1 Figure 1. Example 9, Cumulated Mean Down Time on [0; t], h = 6:

Figure 2 .Figure 3 .

 23 Figure 2. Example 10, P t (1; j) for j = 1::4:

Table 1 .

 1 . Example 9: Cumulated Mean Down Time with respect of h and t

	t	h Lower bound Upper bound	t	h Lower bound Upper bound
		20	0.4929	0.6487		20	2.2879	2.4144
		12	0.5431	0.6344		12	2.3215	2.3977
	300	6	0.5803	0.6251	2400 6	2.3480	2.3863
		3	0.5949	0.6171		3	2.3585	2.3776
		1	0.6048	0.6121		1	2.3658	2.3722
		20	0.8088	0.8856		20	3.3479	3.5063
		12	0.8325	0.8773		12	3.3899	3.4853
	600	6	0.8496	0.8716	3300 6	3.4229	3.4708
		3	0.8560	0.8670		3	3.4359	3.4598
		1	0.8605	0.8641		1	3.4450	3.4530
		20	0.9928	1.0585		20	4.5284	4.7165
		12	1.0111	1.0503		12	4.5781	4.6913
	900	6	1.0251	1.0447	4200 6	4.6171	4.6739
		3		1.0404		3	4.6323	4.6607
		1	1.0344	1.0376		1	4.6431	4.6526
		20	1.1894	1.2651		20	5.7825	5.9991
		12	1.2095	1.2552		12	5.8395	5.9699
	1200 6	1.2255	1.2484	5100 6	5.8842	5.9495
		3	1.2318	1.2433		3	5.9015	5.9342
		1	1.2363	1.2401		1	5.9139	5.9248
		20	1.4195	1.5087		20	7.0787	7.3236
		12	1.4430	1.4969		12	7.1429	7.2903
	1500 6	1.4618	1.4889	6000 6	7.1931	7.2670
		3	1.4693	1.4828		3	7.2124	7.2494
		1	1.4745	1.4790		1	7.2260	7.2387

Table 2 .

 2 = 2 and h = 40 for j = 3 and 4. We can see in such a …gure that our results seem to more or less coïncide with those from[CTE].In order to better compare the results, we now give in Table2the results from [CET] and our bounds for P t (1; 3), for di¤ erent values of h and t. Example 10, P t (1; 3) with respect of h and comparison with the results from[CTE]

					1:2 ' 1:09 10 4 ; 0 2 = 1:2
	with
		m 1 ' 2000; cv 1 ' 0:68; m 0 1 ' 10 4 ; cv 0 1 ' 0:52;
		m 2 ' 102; cv 2 ' 0:53; m 0 2 ' 1881; cv 0 2 ' 0:84
	Bounds for P t (1; j) (j = 1; ::; 4) as well as numerical results from [CTE] and
	asymptotic values are plotted in Figure 2 with 0 t 2 10 5 , h = 4 for j = 1,
	h = 2 0 0.2 0.4 0.6 1 3 for j 0 ← P t P t (1,1) (1,2) ↓ 0.8	0.5 ← P t	(1,4)	1	1.5 Upper Bound ↑ P t (1,3) Lower Bound Cocozza-Eymard Asympt. Probab.	2
					x 10	5
				t

Figure 2. Example 10, P t (1; j) for j = 1::4:

Acknowledgement 12 The author thanks C. Cocozza-Thivent and R. Eymard for allowing her to use the data of a …gure from [Cocozza-Thivent & Eymard (to appear)].


Denis Boulanger
email: denis.boulanger@ird.sn.

Yemou Dieng

Badara Cissé

Franck Remoue

Frédéric Capuano

Jean-Louis Dieme

Tofène Ndiaye

Cheikh Sokhna

Jean-François Trape

Brian Greenwood

Badara Cisse

Frederic Capuano

Tofene Ndiaye

Francois Simondon

Antischistosomal efficacy of artesunate combination therapies administered as curative treatments for malaria attacks

Keywords: artesunate, Schistosoma haematobium, malaria, chemotherapy, Senegal

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Provision of an effective anti-malarial for the treatment of uncomplicated malaria is the corner stone of antimalarial control programmes in Africa [START_REF] Who | WHO Expert Committee on Malaria[END_REF]. Because of progressive loss of efficacy of previously widely used antimalarials such as chloroquine and sulphadoxine/pyrimethamine, most African national malaria control programmes and the World Health Organisation now recommend artemisinin-based combination therapies (ACTs) as the first line of treatment for uncomplicated malaria in Africa [START_REF] Olliaro | Controlling malaria: challenges and solutions[END_REF]. Numerous clinical trials have demonstrated their safety and efficacy. Surprisingly, both artesunate and artemether have also proved to have activity against several species of schistosomes [START_REF] Xiao | Development of antischistosomal drugs in China, with particular consideration to praziquantel and the artemisinins[END_REF]. There is, however, debate about their use on a large scale to control the second most prevalent parasitic disease worldwide. Because of the overlapping distributions of schistosomiasis and malaria, the risk that widespread use of artemisinins for the treatment of schistosomiasis could favour the emergence of resistant strains of Plasmodium must be seriously considered [START_REF] Menard | Association of failures of seven-day courses of artesunate in a non-immune population in Bangui, Central African Republic with decreased sensitivity of Plasmodium falciparum[END_REF][START_REF] Utzinger | Current progress in the development and use of artemether for chemoprophylaxis of major human schistosome parasites[END_REF]. Nevertheless, there is a possibility that use of artemisinins to treat malaria could have, as a secondary effect, a beneficial effect on the burden of schistosomiasis [START_REF] Utzinger | Combination chemotherapy of schistosomiasis in laboratory studies and clinical trials[END_REF] but there is no information so far on how effective artemisinins might be against schistosome infections when used in this way. Like in many other African countries, chloroquinoresistance has reached in Senegal such a level [START_REF] Trape | Impact of chloroquine resistance on malaria mortality[END_REF] that the National Program had to consider a change of malaria case management. Therefore, we have taken advantage of a clinical trial of two different forms of ACTs conducted in a rural area to evaluate if artesunate, when used for the treatment of uncomplicated malaria, also has an impact on the schistosome burden of young Senegalese children.

Materials and Methods

Study area and population

The study took place between November 2003 and January 2004, at the end of the rainy season, in four villages located within the community of Niakhar site, situated 145 kms east from Dakar, where regular demographic surveillance has been maintained since 1963 [START_REF] Delaunay | Decline of infant and child mortality rates in rural Senegal over a 37-year period (1963-1999)[END_REF]. These villages are situated in an area where water collections, either temporary (marshy lands) or semi-permanent (ponds used for agricultural purposes), are potential sites for the breeding of Anopheles mosquito larvae and of Bulinus snails, resulting in a high level of both Plasmodium falciparum malaria and of urinary schistosomosis in the community [START_REF] Briand | Coinfection with Plasmodium falciparum and Schistosoma haematobium: protective effect of schistosomiasis on malaria in senegalese children?[END_REF]. 518 children, aged between 15 and 74 months old at the start of the study, were eligible.

Detection of malaria

Malaria cases were detected through both passive surveillance at health centres and weekly domiciliary visits. Malaria was defined as an illness associated with an auricular temperature of 37.5° C or greater and/or a history of fever or vomiting during the previous 24 hours, accompanied by the detection of asexual malaria parasites on blood film examination. Blood smears were performed 3, 7, 14 and 28 days post-treatment to detect any persistence of parasites after treatment.

Chemotherapy

Children with malaria were randomised to receive treatment with artesunate (AS, Sanofi, Paris, France) given either with sulfadoxine/pyrimethamine (SP) or amodiaquine (AQ), both provided by the Senegalese National Pharmaceuticals Company. SP (25 mg/kg sulfadoxine + 1.25 mg/kg pyrimethamine) was administered once on the first day of treatment. Amodiaquine was used at a dose of 10 mg/kg for the first 2 days, then 5 mg/kg on the third day. Artesunate (4 mg/kg) was given once daily for three consecutive days. All treatments were given under direct observation.

Detection of schistosome infection

As soon as a malaria case was microscopically confirmed, a field worker visited the child, started chemotherapy and left a container with the child's mother together with instruction to collect the urine from a whole micturition the following morning (Day 0). In the laboratory, the volume of the sample was recorded (range : 3 -325 ml), the urine was filtered through a nylon disc filter (Nytrel®) with a 30  mesh and the numbers of eggs trapped on the filter counted under a dissecting microscope. The same procedure was repeated on Days 1, 2, 28, 29 and 30 after the start of antimalarial treatment, before any re-infection larva could have become an adult fertile schistosome.

Data processing

Data were independently recorded by two data entry clerks and then checked for concordance. Aberrant values were identified and corrected.

Levels of egg excretion were expressed as the number of eggs per 10 ml of urine. Malaria cases were considered as eligible for inclusion in the study if the results of at least two filtrations before and after treatment were available. At least one of the pretreatment samples had to be positive before a case was included in the efficacy analysis. Cure rate was expressed as the percentage of children who became completely negative after treatment.

Geometric means minus 1 of the available daily counts plus 1 were used to calculate the mean levels of individual egg excretions before (mean of Days 0-1-2) and after (mean of Days 28-29-30) treatment. Group mean egg excretion was calculated as the geometric mean minus 1 of the individual mean counts plus 1.

Statistical analysis

Yates' corrected chi-2 was used to compare cure rates. Exact 95% confidence intervals (CI) were calculated for proportions. Mann-Whitney U test and Wilcoxon signed-rank test were run to assess the significance of the differences between means of unpaired and paired values, respectively.

Ethical clearance

Informed consent was obtained from the children's mothers and the global protocol received approval from the Senegalese National Ethics Committee. All children who were still excreting schistosome eggs on either Day 28-29-30 were treated with praziquantel (BILTRICID®) at a 40 mg/kg dosage.

Results

108 cases of malaria meeting the study definition were detected among 518 children during the period of the survey. Twenty-seven of these children also met the study criterion for a diagnosis of urinary schistosomiasis, 12 and 15 in the AQ3+AS3 and SP1+AS3 groups respectively. Fifteen were male and their age varied from 26 to 74 months. All of them were successfully cured of Plasmodium infection, as shown by negative 3, 7, 14 and 28 day blood smears. The mean level of egg excretion before treatment was higher in the SP1+AS3 group than in the AQ3 +AS3 group (Table ) but the difference between the groups was not significant (p = 0.097). Twentyfive children were totally negative for schistosome eggs after treatment, an overall cure rate of 92.6 %. Cure rates did not differ significantly between the two groups (p = 0.84). All children in the AQ3+AS3 were cured (p < 0.005). In the SP1+AS3 group, the mean number of eggs was reduced by more than 90% (pre-versus post-treatment means comparison, p < 0.001).

Discussion

The main goal of this study was to determine if artesunate, when administered for the treatment of young children against malaria, also had an effect on Schistosoma haematobium egg output. So far, three randomised clinical trials have investigated the capacity of artesunate to cure established S. haematobium infections but with divergent results. The first one, done in Gabon using a dosage of 4 mg/kg/day of artesunate, once a day over three days [START_REF] Borrmann | Artesunate and praziquantel for the treatment of Schistosoma haematobium infections: a double-blind, randomized, placebo-controlled study[END_REF], concluded that it was not effective at all. In contrast, 8 tablets of 50 mg over 5 days delivered in Senegal [START_REF] De Clercq | Efficacy of artesunate and praziquantel in Schistosoma haematobium infected schoolchildren[END_REF] or two doses of 6mg/kg administered two weeks apart in Nigeria [START_REF] Inyang-Etoh | Efficacy of artesunate in the treatment of urinary schistosomiasis, in an endemic community in Nigeria[END_REF] provided good evidence of efficacy with mean reductions of egg excretion ranging from 55 to 89%, 5 to 6 weeks after treatment. However, these three studies involved schoolchildren between 5 and 18 years who were harbouring heavy infections. Our study is the first undertaken in children less than 6 years old who had malaria at the time that they were treated. Both artesunatecontaining combination therapies led to marked reductions in egg counts, above 90%. The association of artesunate and amodiaquine completely cured all children who received this combination which ranks number two behind artemether/lumefantrine in WHO recommendations as first line of treatment of uncomplicated malaria [START_REF] Majori | Combined antimalarial therapy using artemisinin[END_REF]. The combination of SP with artesunate was slightly less effective but numbers were small. It is unlikely that the partners of artesunate contributed to the activity seen against schistosome infections and one can reasonably assume that artesunate was the active drug.

This pilot study has a number of weaknesses. Firstly, there was no control group and some of the negative counts obtained at the end of the trial could have resulted from natural fluctuations in egg excretion. However, these could not explain the very high levels of 28 day cure obtained and repeated filtrations of a whole micturition are a way to prevent false negatives.

Secondly, the study was conducted in an area of relatively low schistosome endemicity with an overall mean egg count of 0.5 eggs/10 ml of urine. It cannot be speculated that such satisfactory results would be obtained in children with much higher levels of infection but, in Sahelian countries, malaria strikes mainly during the first five years of age at a time when heavy schistosome loads are unfrequent. Finally, the number of children in the study was small, making comparisons between groups able to detect only major differences. Despite these reservations, our findings suggest that adoption of artemisinin-based combination therapies as first lines of treatment for malaria in many African countries, including Senegal, may have an added benefit by reducing the burden of schistosomiais in the many areas where both infections are found. Moreover, such bitherapies are also strongly supported as a basis for intermittent preventive treatment, especially in children less than 5 years [START_REF] Cisse | Seasonal intermittent treatment with artesunate and sulfadoxinepyrimethamine for prevention of malaria in Senegalese children: a randomised, placebo-controlled, double-blind trial[END_REF]. It would be of great value to assess if, within the framework of a global strategy, both curative and preventive, against malaria, artesunate could also reduce schistosomiasis impact. However, our study was only a pilot trial involving children under the age of five years participating in a malaria study. There

Acknowledgments

We would like to specially thank the population of the Niakhar site for its willingness to participate to the study and the IRD Service Unit 009 for maintenance of the Niakhar database. Skillful participation of Ernest Faye, Ibou Cissé and Agnès Bakhoum is greatly acknowledged. This work was funded by the French Ministry of Research and Technology (PAL+ contract #2829), by the University Cheikh Anta Diop of Dakar, by the Bill and Melinda Gates Foundation and by the Institut de Recherche pour le Développement. Sanofi donated artesunate tablets.

is a need to confirm our findings in other areas where ACTs are used as first line treatment for malaria. In addition, there is a need for studies in older children, in whom schistosome burdens are likely to be heavier than was the case in this study, and for trials in areas where S. mansoni is the dominant species.

Conflicts of interest statement

The authors declare to have no known conflict of interest. The sponsors and the artesunate provider (Sanofi) did not interact in the management of the study nor in the data analysis.