Miquel Oliu-Barton

Guillaume Vigeral

A uniform Tauberian theorem in optimal control

In an optimal control framework, we consider the value V T (x) of the problem starting from state x with finite horizon T , as well as the value W λ (x) of the λ-discounted problem starting from x. We prove that uniform convergence (on the set of states) of the values V T (•) as T tends to infinity is equivalent to uniform convergence of the values W λ (•) as λ tends to 0, and that the limits are identical. An example is also provided to show that the result does not hold for pointwise convergence. This work is an extension, using similar techniques, of a related result by Lehrer and Sorin in a discrete-time framework.

Erratum

We thank very warmly Dmitry Khlopin for the following observation : in section 2.1, in addition to the property of concatenation of trajectories X • s Y ∈ Γ(x), we also need to assume that the following property holds:

If X ∈ Γ(x), then for any t > 0, the trajectory s → X(t + s) is in Γ(X(t)). That is, any trajectory obtained by cutting the beginning of a feasible trajectory is also feasible.

This property was used implicitly in both parts of the proof of Theorem 6:

. Thus one needs this additional assumption for Theorem 6 to hold in full generality.

Introduction

Finite horizon problem of optimal control have been studied intensively since the pioneer work of Stekhov, Pontryagin, Boltyanskii [START_REF] Pontryiagin | The mathematical theory of optimal processes[END_REF], Hestenes [START_REF] Hestenes | A General Problem in the Calculus of Variations with Applications to the Paths of Least Time[END_REF], Bellman [START_REF] Bellman | On the theory of dynamic programming[END_REF] and Isaacs [START_REF] Isaacs | Games of Pursuit[END_REF][START_REF] Isaacs | Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization[END_REF] during the cold war -see for instance [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF][START_REF] Kirk | Optimal Control Theory: an Introduction[END_REF][START_REF] Lee | Foundations of Optimal Control Theory[END_REF] for major references, or [START_REF] Evans | An Introduction to Mathematical Optimal Control Theory[END_REF] for a short, clear introduction. A classical model considers the following controlled dynamic over R+ y (s) = f (y(s), u(s)) y(0) = y 0 (1.1) where y is a function from R+ to R n , y 0 is a point in R n , u is the control function which belongs to U, the set of Lebesgue-measurable functions from R+ to a metric space U and the function f : R n × U → R n satisfies the usual conditions, that is: Lipschitz with respect to the state variable, continuous with respect to the control variable and bounded by a linear function of the state variable, for any control u.

Together with the dynamic, an objective function g is given, interpreted as the cost function which is to be minimized and assumed to be Borel-measurable from R n × U to [0, 1]. For each finite horizon t ∈]0, +∞[, the average value of the optimal control problem with horizon t is defined as It is quite natural to define, whenever the trajectories considered are infinite, for any discount factor λ > 0, the λ-discounted value of the optimal control problem, as

V t (y 0 ) = inf
W λ (y 0 ) = inf u∈U λ +∞ 0
e -λs g(y(s, u, y 0 ), u(s))ds (1.3) In this framework the problem was initially to know whether, for a given finite horizon T and a given starting point y 0 , a minimizing control u existed, solution of the optimal control problem (T, y 0 ). Systems with large, but fixed horizons were considered and, in particular, the class of "ergodic" systems (that is, those in which any starting point in the state space Ω is controllable to any point in Ω) has been thoroughly studied [START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations[END_REF][START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation I[END_REF][START_REF] Arisawa | On ergodic stochastic control[END_REF][START_REF] Artstein | The value function of singularly perturbed control systems[END_REF][START_REF] Barles | Some homogenization results for non-coercive Hamilton-Jacobi equations[END_REF][START_REF] Cardaliaguet | Ergodicity of Hamilton-Jacobi equations with a non coercive non convex Hamiltonian in R 2 /Z 2[END_REF][START_REF] Lions | Homogenization of Hamilton-Jacobi Equations[END_REF]. These systems are asymptotically independent of the starting point as the horizon goes to infinite. When the horizon is infinite, the literature on optimal control has mainly focussed on properties of given trajectories as the time tends to infinity. This approach corresponds to the uniform approach in a game theoretical framework and is often opposed to the asymptotic approach (described below), which we have considered in what follows, and which has received considerably less attention.

In a game-theoretical, discrete time framework, the same kind of problem was considered since [START_REF] Shapley | Stochastic games[END_REF], but with several differences in the approach: 1) the starting point may be chosen at random, i.e, a probability µ may be given on Ω, which randomly determines the point from which the controller will start the play; 2) the controllability-ergodicity condition is generally not assumed; 3) because of the inherent recursive structure of process played in discrete time, the problem is generally considered for all initial states and time horizons.

For these reasons, what is called the "asymptotic approach" -the behavior of V t (•) as the horizon t tends to infinity, or of W λ (•) as the discount factor λ tends to zero -has been more studied in this discrete-time setup. Moreover, when it is considered in Optimal Control, in most cases [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation II[END_REF][START_REF] Bettiol | On ergodic problem for Hamilton-Jacobi-Isaacs equations[END_REF] an ergodic assumption is made which not only ensures the convergence of V t (y 0 ) to some V , but also forces the limit function V to be independent of the starting point y 0 . The general asymptotic case, in which no ergodicity condition is assumed, has been to our knowledge studied for the first time recently. In [START_REF] Cardaliaguet | Ergodicity of Hamilton-Jacobi equations with a non coercive non convex Hamiltonian in R 2 /Z 2[END_REF][START_REF] Quincampoix | On the existence of a limit value in some non expansive optimal control problems[END_REF] the authors prove in different frameworks the convergence of V t (•) and W λ (•) to some non-constant function V (y 0 ). Some important, closely related questions are the following : does the convergence of V t (•) imply the convergence of W λ (•) ? Or vice versa ? If they both converge, does the limit coincide ? A partial answer to these questions goes back to the beginning of the 20 th century, when Hardy and Littlewood proved (see [START_REF] Hardy | Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive[END_REF]) that for any sequence of bounded real numbers, the convergence of the Cesaro means is equivalent to the convergence of their Abel means, and that the limits are then the same : Theorem 1 (Hardy-Littlewood 1914) For any bounded sequence of reals {a n } n≥1 , define

V n = 1 n n i=1 a i and W λ = λ +∞ i=1 (1 -λ) i-1 a i . Then, lim inf n→+∞ V n ≤ lim inf λ→0 W λ ≤ lim sup λ→0 W λ ≤ lim sup n→+∞ V n .
Moreover, if the central inequality is an equality, then all inequalities are equalities.

Noticing that {a n } can be viewed as a sequence of costs for some deterministic (uncontrolled) dynamic in discrete-time, this results gives the equivalence between the convergence of V t and the convergence of W λ , to the same limit. In 1971, setting V t =1 t t 0 g(s)ds and W λ = λ +∞ 0 e -λs g(s)ds, for a given Lebesgue-measurable, bounded, real function g, Feller proved that the same result holds for continuous-time uncontrolled dynamics (particular case of Theorem 2, p.445 in [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]).

Theorem 2 [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF])

lim inf n→+∞ V n ≤ lim inf λ→0 W λ ≤ lim sup λ→0 W λ ≤ lim sup n→+∞ V n .
Moreover, if the central inequality is an equality, then all inequalities are equalities.

In 1992, Lehrer and Sorin [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF] considered a discrete-time controlled dynamic, defined by a correspondence Γ : Ω ⇒ Ω, with nonempty values, and by g, a bounded real cost function defined on Ω. A feasible play at z ∈ Ω is an infinite sequence y = {y n } n≥1 such that y 1 = z and y n+1 ∈ Γ(y n ). The value functions are defined by

V n (z) = inf 1 n n i=1 g(y i ) and respectively W λ (y 0 ) = inf λ +∞ i=1 (1 -λ) i-1 g(y i )
, where the infima are taken over the feasible plays at z.

Theorem 3 (Lehrer-Sorin 1992)

lim n→+∞ V n (z) = V (z) uniformly on Ω ⇐⇒ lim λ→0 W λ (z) = V (z) uniformly on Ω.
This result establishes the equivalence between uniform convergence of W λ (y 0 ) when λ tends to 0 and uniform convergence of V n (y 0 ) as n tends to infinity, in the general case where the limit may depend on the starting point y 0 . The uniform condition is necessary: in the same article, the authors provide an example where only pointwise convergence holds and the limits differs.

In 1998, Arisawa (see [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation II[END_REF]) considered a continuous-time controlled dynamic and proved the equivalence between the uniform convergence of W λ and the uniform convergence of V t in the specific case of limits independent of the starting point.

Theorem 4 (Arisawa 1998) Let d ∈ R, then lim t→+∞ V t (z) = d, uniformly on Ω ⇐⇒ lim λ→0+ W λ (z) = d, uniformly on Ω.
This does not settle the general case, in which the limit function may depend on the starting point 1 . For a continuous-time controlled dynamic in which V t (y 0 ) converges to some function V (y 0 ), dependent on the state variable y 0 , as t goes to infinity, we prove the following Theorem 5 V t (y 0 ) converges to V (y 0 ) uniformly on Ω, if and only if W λ (y 0 ) converges to V (y 0 ) uniformly on Ω.

In fact, we will prove this result in a more general framework, as described in section 2. Some basic lemmas which occur to be important tools will also be proven on that section. Section 3 will be devoted to the proof of our main result. Section 4 will conclude by pointing out, via an example, the fact that uniform convergence is a necessary requirement for the Theorem 5 to hold. A very simple dynamic is described, in which the pointwise limits of V t (•) and W λ (•) exist but differ. It should be noted that our proofs (as well as the counterexample in Section 4) are adaptations in this continuous-time framework of ideas employed in a discrete-time setting in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF]. In the appendix we also point out that an alternative proof of our theorem is obtained using the main theorem in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF] as well as a discrete/continuous equivalence argument.

For completeness, let us mention briefly this other approach, mentioned above as the uniform approach, and which has also been deeply studied, see for exemple [START_REF] Carlson | Optimal Control on Infinite Time Horizon[END_REF][START_REF] Colonius | Infinite Time Optimal Control and Periodicity[END_REF][START_REF] Grune | On the Relation between Discounted and Average Optimal Value Functions[END_REF]. In these models, the optimal average cost value, i.e the V t , is not taken over a finite period of time [0, t], which is then studied for t growing to infinite, as in [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation II[END_REF][START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF][START_REF] Hardy | Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive[END_REF][START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF][START_REF] Quincampoix | On the existence of a limit value in some non expansive optimal control problems[END_REF] or in our framework. On the contrary, only infinite trajectories are considered, among which the value V t is defined as inf u∈U sup τ ≥t 1 τ τ 0 g(y(s, u, y 0 ), u(s))ds, or some other closely related variation. The asymptotic behavior, as t tends to infinity, of the function V t has also been studied. In [START_REF] Grune | On the Relation between Discounted and Average Optimal Value Functions[END_REF], both λ-discounted and average evaluations of an infinite trajectory are considered and their limits are compared. However, we stress out that the asymptotic behavior of those quantities is in general2 not related to the asymptotic behavior of V t and W λ .

Finally, let us point out that in the framework of zero-sum differential games, that is when the dynamic is controlled by two players with opposite goals, a Tauberian theorem is given in the ergodic case by Theorem 2.1 in [START_REF] Alvarez | Ergodic Problems in Differential Games[END_REF]. However, the general, non ergodic case is still an open problem in both the discrete and the continuous settings.

Model

General framework

We consider a deterministic dynamic programming problem in continuous time, defined by a measurable set of states Ω, a subset T of Borel-measurable functions from R + to Ω, and a bounded Borel-measurable real-valued function g defined on Ω. Without loss of generality we assume g : Ω → [0, 1]. For a given state x, define Γ(x) := {X ∈ T , X(0) = x} the set of all feasible trajectories starting from x. We assume Γ(x) to be non empty, for all x ∈ Ω. Furthermore, the correspondence Γ is closed under concatenation: given a trajectory X ∈ Γ(x) with X(s) = y, and a trajectory Y ∈ Γ(y), the concatenation of X and Y at time s is

X • s Y := X(t) if t ≤ s Y (t -s) if t ≥ s (2.1)
and we assume that

X • s Y ∈ Γ(x).
We are interested in the asymptotic behavior of the average and the discounted values. It is useful to denote the average payoff of a play (or trajectory) X ∈ Γ(x) by:

γ t (X) := 1 t t 0 g(X(s))ds (2.2) ν λ (X) := λ +∞ 0 e -λs g(X(s))ds (2.3)
This is defined for t, λ ∈ (0, ∞). Naturally, we define the values as:

V t (x) = inf X∈Γ(x) γ t (X) (2.4) W λ (x) = inf X∈Γ(x) ν λ (X) (2.5)
Our main contribution is the following :

Theorem 6 (A) W λ -→ λ→0 V , uniformly on Ω ⇐⇒ (B) V t -→ t→∞ V , uniformly on Ω.
Notice that our model is a natural adaptation to the continuous-time framework of deterministic dynamic programming problems played in discrete time ; as it was pointed out during the introduction, this theorem is an extension to the continuous-time framework of the main result of [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], and our proof use similar technics.

This result can be applied to the model presented in section 1: denote Ω = R d × U and for any (y 0 , u 0 ) ∈ Ω, define Γ(y 0 , u 0 ) = {(y(•), u(•)) |u ∈ U, u(0) = u 0 and y is the solution of (1.1).} Then Ω, Γ and g satisfy the assumptions of this section. Defining V t and W λ as in (2.4) and (2.5) respectively, since the solution of (1.1) does not depend on u(0) we get that

V t (y 0 , u 0 ) = V t (y 0 ) W λ (y 0 , u 0 ) = W λ (y 0 ).
Theorem 6 applied to V and W thus implies that V t converges uniformly to a function V in Ω if and only if W λ converges uniformly to V in Ω.

Preliminary results

We follow the ideas of [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], and start by proving two simple lemmas yet important tools, that will be used in the proof. The first establishes that the value increases along the trajectories. Then, we prove a convexity result linking the finite horizon average payoffs and the discounted evaluations on any given trajectory.

Lemma 1 Monotonicity (compare with Proposition 1 in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF]) For all X ∈ T , for all s ≥ 0, we have

lim inf t→∞ V t (X(0)) ≤ lim inf t→∞ V t (X(s)) (2.6) lim inf λ→0 W λ (X(0)) ≤ lim inf λ→0 W λ (X(s)) (2.7)
Proof. Set y := X(s) and x := X(0). For ε > 0, take T ∈ R + such that s s+T < ε. Let t > T and take a ε-optimal trajectory for V t , i.e. Y ∈ Γ(y) such that γ t (Y ) ≤ V t (y) + ε. Define the concatenation of X and Y at time s as in (2.1), where X • s Y is in ∈ Γ(x) by our hypothesis. Hence

V t+s (x) ≤ γ t+s (X • s Y ) = s t + s γ s (X) + t t + s γ t (Y ) ≤ ε + γ t (Y ) ≤ 2ε + V t (y).
Since this is true for any t ≥ T the result follows.

Similarly, for the discounted case let λ 0 > 0 be such that λ 0 s 0 e -λ 0 r dr = 1 -e λ 0 s < ε. Let λ ∈ (0, λ 0 ) and take Y ∈ Γ(y) a ε-optimal trajectory for W λ (y). Then:

W λ (x) ≤ ν λ (X • s Y ) = λ s 0 e -λr g(X(r))dr + λ ∞ s e -λr g(Y (r -s))dr ≤ ε + e -λs ν λ (Y ) ≤ 2ε + W λ (y).
Again, this is true for any λ ∈ (0, λ 0 ), and the result follows.

Lemma 2 Convexity (compare with equation ( 1) in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF]) For any play X ∈ T , for any λ > 0:

ν λ (X) = ∞ 0 γ s (X)µ λ (s)ds, (2.8)
where µ λ (s)ds := λ 2 se -λs ds is a probability density on [0, +∞].

Proof. It is enough to notice that the following relation holds, by integration by parts :

ν λ (X) = λ ∞ 0 e -λs g(X(s))ds = λ 2 ∞ 0 se -λs 1 s s 0 g(X(r))dr ds,
and that ∞ 0 λ 2 se -λs ds = 1. The probability measure µ λ will play an important role in the rest of the paper. Denoting M (α, β; λ) := β α µ λ (s)ds = e -λα (1 + λα) -e -λβ (1 + λβ), we prove here two estimates that will be helpful in the next section.

Lemma 3

The two following results hold (compare with Lemma 3 in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF]):

(i) ∀t > 0, ∃ 0 such that ∀ ≤ 0 , M ((1 -)t, t; 1/t) ≥ /2e. (ii) ∀δ > 0, ∃ 0 such that ∀ ≤ 0 , ∀t > 0, M ( t, (1 -)t; 1/t √ ) ≥ 1 -δ.
Proof. Notice that in these particular cases, M does not depend on t:

(i) M (t(1 -), t; 1/t) = (2 -)e -1+ -2e -1 = 1 e ( + o( )) ≥ 2e , for small enough. (ii) M (t , t(1 -); 1/ √ t) = (1 + √ )e - √ -(1 -1/ √ + √ )e -1/ √ + √
. This expression tends to 1 as → 0, hence the result.

3 Proof of Theorem 6

3.1 From V t to W λ Assume (B) : V t (•) converges to some V (•) as t goes to infinity, uniformly on Ω. Our proof follows Proposition 4 and Lemmas 8 and 9 in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF].

Proposition 4 For all > 0, there exists λ 0 > 0 such that W λ (x) ≥ V (x) -ε for every x ∈ Ω and for all λ ∈ (0, λ 0 ].

Proof. Let T be such that V t -V ∞ ≤ /2 for every t ≥ T . Choose λ 0 > 0 such that λ 2 ∞ T se -λs ds = 1 -(1 + λT )e -λT ≥ 1 -/4
for every λ ∈]0, λ 0 [. Fix λ ∈]0, λ 0 [ and take a play Y ∈ Γ(x) which is /4-optimal play for W λ (x). Since γ s (X) ≥ 0, the convexity formula (2.8) from Lemma 2 gives:

W λ (x) + /4 ≥ ν λ (Y ) ≥ 0 + λ 2 ∞ T se -λs γ s (Y ) ≥ λ 2 ∞ T se -λs V s (x) ≥ (1 -/4)(V (x) -/2) = V (x) -/4V (x) -/2 + 2 /8 ≥ V (x) -3/4 Lemma 5 ∀ > 0, ∃M such that for all t ≥ M, ∀x ∈ Ω, there is a play X ∈ Γ(x) such that γ s (X) ≤ V (x) + for all s ∈ [ t, (1 -)t].
Proof. By (B) there exists M such that V r -V ≤ 2 3 for all r ≥ M . Given t ≥ M and x ∈ Ω, let X ∈ Γ(x) be a play (from x) such that γ t (X)

≤ V t (x) + 2 3 . For any s ≤ (1 -)t, t -s ≥ t ≥ M so Proposition 1 (Monotonicity) imply that V t-s (X(s)) ≥ V (X(s)) - 2 3 ≥ V (x) - 2 3 . (3.1)
Since V (x) + 2 /3 ≥ V t (x), we also have:

t V (x) + 2 2 3 ≥ t V t (x) + 2 3 ≥ tγ t (X) = s 0 g(X(r))dr + t s g(X(r))dr ≥ sγ s (X) + (t -s)V t-s (X(s)) ≥ sγ s (X) + (t -s) V (x) - 2 3 by (3.1).
Isolating γ s (X) we get:

γ s (X) ≤ V (x) + 2 t s ≤ V (x) + , for s/ ≥ t,
and we have proved the result for all s ∈ [ t, (1 -)t].

Proposition 6 ∀δ > 0, ∃λ 0 such that ∀x ∈ Ω, for all λ ∈ (0, λ 0 ], we have W λ (x) ≤ V (x) + δ.

Proof. By Lemma 3 (ii), one can choose small enough such that M ( t, (1 -)t;

1 t √ ) ≥ 1 -δ
2 , for any t. In particular, we can take ≤ δ 2 . Using Lemma 5 with δ/2, we get that for t ≥ t 0 (and thus for λ t := 1 t √ ≤ 1 t 0 √ ) and for any x ∈ Ω, there exists a play X ∈ Γ(x) such that

ν λt (X) ≤ δ/2 + λ t 2 (1-)t t se λts γ s (X)ds ≤ δ/2 + 1 • (V (x) + δ/2).
Propositions 4 and 6 establish the first part of Theorem 6: (B) ⇒ (A).

From W λ to V t

Now assume (A) : W λ (•) converges to some W (•) as λ goes to 0, uniformly on Ω. Our proof follows Proposition 2 and Lemmas 6 and 7 in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF]. Start by a technical Lemma: Lemma 7 Let > 0. For all x ∈ Ω and t > 0, and for any trajectory Y ∈ Γ(x) which is /2-optimal for the problem with horizon t, there is a time L ∈ [0, t(1 -/2)] such that, for all T ∈]0, t -L]:

1

T L+T L g(Y (s))ds ≤ V t (x) + . Proof. Fix Y ∈ Γ(x) some /2-optimal play for V t (x). The function s → γ s (Y ) is continuous on ]0, t] and satisfies γ t (Y ) ≤ V t (x) + /2. The bound on g implies that γ r (Y ) ≤ V t (x) + for all r ∈ [t(1 -/2), t]. Consider now the set {s ∈]0, t] | γ s (Y ) > V t (x) + }.
If this set is empty, then take L = 0 and observe that 1

r r 0 g(Y (s))ds ≤ V t (x) + , ∀r ∈]0, t].
Otherwise, let L be the superior bound of this set. Notice that L < t(1 -/2) and that by

continuity γ L (Y ) = V t (x) + . Now, for any T ∈ [0, t -L], V t (x) + ≥ γ L+T (Y ) = L L + T γ L (Y ) + T L + T 1 T L+T L g(Y (s))ds = L L + T (V t (x) + ) + T L + T 1 T L+T L g(Y (s))ds
and the result follows.

Proposition 8 ∀ > 0, ∃T such that for all t ≥ T we have V t (x) ≥ W (x) -, for all x ∈ Ω.

Proof. Let λ be such that W λ -W ≤ /8, and T such that λ 2 ∞ T /4 se -λs ds < /8. Proceed by contradiction and suppose that > 0 is such that for every T , there exists t 0 ≥ T and a state

x 0 ∈ Ω such that V t 0 (x 0 ) < W (x 0 ) -.
Using Lemma 7 with /2, we get a play Y ∈ Γ(x 0 ) and a time L ∈ [0, t 0 (1 -/4)] such that, ∀s ∈ [0, t 0 -L] (and, in particular, ∀s ∈ [0, t 0 /4]):

1 s L+s L g (Y (r)) dr ≤ V t 0 (x 0 ) + /2 < W (x 0 ) -/2.
Thus,

W (Y (L)) -/8 ≤ W λ (Y (L)) ≤ λ ∞ 0 e -λs g(Y (L + s))ds ≤ λ 2 t 0 /4 0 se -λs 1 s L+s L g (Y (r)) dr ds + /8 ≤ W (x 0 ) -/2 + /8 = W (x 0 ) -3 /8.
This gives us W (Y (L)) ≤ W (x 0 ) -/4, contradicting Proposition 1 (Monotonicity).

Proposition 9 ∀ > 0, ∃T such that for all t ≥ T we have V t (x) ≤ W (x) + , for all x ∈ Ω.

Proof. Otherwise, ∃ > 0 such that ∀T, ∃t ≥ T and x ∈ Ω with V t (x) > W (x) + . For any

X ∈ Γ(x) consider the (continuous in s) payoff function γ s (X) = 1 s s 0 g(X(r))dr. Of course, γ t (X) ≥ V t (x) > W (x) + . Furthermore, because of the bound on g, γ r (X) ≥ W (x) + /2, ∀r ∈ [t(1 -/2), t].
By Lemma 3, we can take small enough, so that for all t, M (t(1 -/2), t; 1/t) ≥ /4e holds. We set δ := /4e.

By Proposition 8, there is a K such that V t ≥ W (x) -δ /8, for all t ≥ K.

For K fixed, we consider M (0, K; 1/t) = 1 -e -K/t (1 + K/t) as a function of t. Clearly, it tends to 0 as t tends to infinity, so let t be such that this quantity is smaller than δ /16. Also, let t be big enough so that W 1/t -W < δ /5, which is a consequence of assumption (A).

On the following, we set λ := 1/t and consider the λ-payoff of some play X ∈ Γ(x). We'll split the integral over [0, +∞] in three parts :

K = [0, K], R = [t(1 -/2), t], and (K ∪ R) c .
The three parts are clearly disjoint since t >> K. We have seen that µ λ (s)ds = λ 2 se -λs ds is a probability measure on [0, +∞], for any λ > 0. Then by the Convexity formula (2.8), we can write:

ν λ(X ) = K γ s (X)µ λ(ds) + R γ s (X)µ λ(ds) + (K∪R) c γ s (X)µ λ(ds) . Recall that γ s (X) |K ≥ 0 γ s (X) |(K∪R) c ≥ W (x) -δ /8 γ s (X) |R ≥ W (x) + /2
It is straightforward that

ν λ(X ) ≥ 0 + δ • (W (x) + /2) + (1 -δ -δ /16) • (W (x) -δ /8) = W (x) + δ 1 2 - 1 16 - 1 8 - δ 8 + δ 64 ≥ W (x) + δ /4.
This is true for any play, so its infimum also satisfies W λ(x) ≥ W (x) + δ /4, which is a contradiction, for we assumed that W λ < W (x) + δ /5. Propositions 8 and 9 establish the second half of Theorem 6 : (A) ⇒ (B).

A counter example for pointwise convergence

In this section we give an example of an optimal control problem in which both V t (•) and W λ (•) converge pointwise on the state space, but to two different limits. As implied by Theorem 6, the convergence is not uniform on the state space.

Lehrer and Sorin were the first to construct such an example [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], in the discrete-time framework. We consider here one of its adaptation in continuous time, which was studied as Example 5 in [START_REF] Quincampoix | On the existence of a limit value in some non expansive optimal control problems[END_REF] 3 , where the notations are the same that in Section 1:

-The state space is Ω = R 2 + .

-The payoff function is given by g(x, y) = 0 if x ∈ [1, 2], 1 otherwise.

-The set of control is U = [0, 1].

-The dynamic is given by f (x, y, u) = (y, u) (thus Ω is forward invariant.)

An interpretation is that the couple (x(t), y(t)) represents the position and the speed of some mobile moving along an axis, and whose acceleration u(t) is controlled. Observe that since U = [0, 1], the speed y(t) increases during a play. We claim that for any (x 0 , y 0 ) ∈ R 2 + , V t (x 0 , y 0 ) (resp W λ (x 0 , y 0 )) converges to V (x 0 , y 0 ) as t goes to infinity (resp. converges to W (x 0 , y 0 ) as λ tends to 0, where:

V (x 0 , y 0 ) =      1 if y 0 > 0 or x 0 > 2 0 if y 0 = 0 and 1 ≤ x 0 ≤ 2 1-x 0 2-x 0 if y 0 = 0 and x 0 < 1 W (x 0 , y 0 ) =        1 if y 0 > 0 or x 0 > 2 0 if y 0 = 0 and 1 ≤ x 0 ≤ 2 1 -(1-x 0 ) 1-x 0 (2-x 0 ) 2-x 0
if y 0 = 0 and x 0 < 1

Here we only prove that V (0, 0) = 1 2 and W (0, 0) = 3 4 ; the proof for y 0 = 0 and 0 < x 0 < 1 is similar and the other cases are easy.

First of all we prove that for any t or λ and any admissible trajectory (that is, any function X(t) = (x(t), y(t)) compatible with a control u(t)), starting from (0, 0), γ t (X) ≥ 1 2 and ν λ (X) ≥ . This is clear if x(t) is identically 0, so consider this is not the case. Since the speed y(t) is increasing, we can define t 1 and t 2 as the time at which x(t 1 ) = 1 and x(t 2 ) = 2 respectively, and moreover we have t 2 ≤ 2t 1 . Then,

γ t (X) = 1 t min(t,t 1 ) 0 ds + t min(t,t 2 ) ds = 1 + min 1, t 1 t -min 1, t 2 t ≥ 1 + min 1, t 2 2t -min 1, t 2 t ≥ 1 2 and ν λ (X) = t 1 0 λe -λs ds + +∞ t 2 λe -λs ds = 1 -e -λt 1 + e -λt 2 ≥ 1 -e -λt 1 + e -2λt 1 ≥ min a>0 {1 -a + a 2 } ≥ 3 4 .
On the other hand, one can prove [START_REF] Quincampoix | On the existence of a limit value in some non expansive optimal control problems[END_REF] that lim sup V t (0, 0) ≤ 1 2 : in the problem with horizon t, consider the control "u(s) = 1 until s = 2 t and then 0". Similarly one proves that lim sup W λ (0, 0) ≤ 3 4 : in the λ-discounted problem, consider the control "u(s) = 1 until s = λ ln 2

and then 0". So the functions V t and W λ converge pointwise on Ω, but their limits V and W are different, since we have just shown V (0, 0) = W (0, 0). One can verify that neither convergence is uniform on Ω by considering V t (1, ε) and W λ (1, ε) for small positive ε.

Remark 1 One may object that this example is not very regular since the payoff g is not continuous and the state space is not compact. However a related, smoother example can easily be constructed:

1. The set of controls is still [0, 1].

2. The continuous cost g(x) is equal to 1 outside the segment [0.9,2.1], to 0 on [START_REF] Alvarez | Ergodic Problems in Differential Games[END_REF][START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations[END_REF], and linear on the two remainings intervals.

The compact state space is

Ω = {(x, y)|0 ≤ y ≤ √ 2x ≤ 2 √ 2}.
4. The dynamic is the same that in the original example for x ∈ [0, 3], and f (x, y, u) = ((4 -x)y, (4 -x)u) for 3 ≤ x ≤ 4. The inequality y(t)y (t) ≤ x (t) is thus satisfied on any trajectory, which implies that Ω is forward invariant under this dynamic.

With these changes the values V t (•) and W λ (•) still both converge pointwise on Ω to some V (•) and W (•) respectively, and V (0, 0) = W (0, 0).

Possible extensions

-We considered the finite horizon problem and the discounted one, but it should be possible to establish similar Tauberian theorems for other, more complex, evaluations of the payoff. This was settled in the discrete time case in [START_REF] Monderer | Asymptotic Properties in Dynamic Programming[END_REF].

-It would be very fruitful to establish necessary or sufficient conditions for uniform convergence to hold. In this direction we mention [START_REF] Quincampoix | On the existence of a limit value in some non expansive optimal control problems[END_REF] in which sufficient conditions for the stronger notion of Uniform Value (meaning that there are controls that are nearly optimal no matter the horizon, provided it is large enough) are given in a general setting.

-In the discrete case an example is constructed in [START_REF] Monderer | Asymptotic Properties in Dynamic Programming[END_REF] in which there is no uniform value despite uniform convergence of the families V t and W λ . It would be of interest to construct such an example in continuous time, in particular in the framework of section 1.

-It would be very interesting to study Tauberian theorems for dynamic systems that are controlled by two conflicting controllers. In the framework of differential games this has been done recently (Theorem 2.1 in [START_REF] Alvarez | Ergodic Problems in Differential Games[END_REF]): an extension of Theorem 4 has been accomplished for two player games in which the limit of V T or W λ is assumed to be independent of the starting point. The similar result in the discrete time framework is a consequence of Theorems 1.1 and 3.5 in [START_REF] Kohlberg | Asymptotic behavior of nonexpansive mappings in normed linear spaces[END_REF]. Existence of Tauberian theorems in the general setup of twopersons zero-sum games with no ergodicity condition remains open in both the discrete and the continuous settings.

  (s, u, y 0 ), u(s))ds(1.2)

Hence, by equation ( 2 . 5 )( 1 -Lemma 10 ( 1 -Proof. Since λ +∞ 0 ( 1 -( 1 e

 251101011 and the bound on the cost function, for any λ ∈]0, 1],|W λ (ω) -w λ (ω, x)| ≤ λ +∞ 0 λ) t -e -λt dtwhich tends uniformly (with respect to x and ω) to 0 as λ goes to 0 by virtue of the following lemma. The functionλ -→ λ +∞ 0 λ) t -e -λt dtconverges to 0 as λ tends to 0 λ) t = λ +∞ 0 e -λt dt = 1, for any λ > 0, the lemma is equivalent to the convergence to 0 ofE(λ) := λ +∞ 0 -λ) t -e -λt + dtwhere [x] + denotes the positive part of x. Now, from the relation 1 -λ ≤ e -λ , true for any λ, one can easily deduce that, for any λ > 0, t ≥ 0, the relation (1 -λ) t e λt ≤ e λ holds. Hence,E(λ) = λ +∞ 0 e -λt (1 -λ) t e λt -1 -λt (e λ -1)dt = e λ -1which converges to 0 as λ tends to 0.

Lemma 6 and Theorem 8 in[START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation II[END_REF] deal with this general setting, but we believe them to be incorrect since they are stated for pointwise convergence and, consequently, are contradicted by the example in Section 4.

The reader may verify that this is indeed not the case in the example of Section 4.

We thank Marc Quincampoix for pointing out this example to us, which is simpler that our original one.

Acknowledgments

This article was done as part of the PhD of the first author. Both authors wish to express their thanks to Sylvain Sorin for his numerous comments and his great help. We also thank Helena Frankowska and Marc Quincampoix for helpful remarks on earlier drafts.

Appendix

We give here another proof 4 of Theorem 6 by using the analoguous result in discrete time [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF] as well as an argument of equivalence between discrete and continuous dynamic.

Consider a deterministic dynamic programming problem in continuous time as defined in section 2.1, with a state space Ω, a payoff g and a dynamic Γ. Recall that, for any ω ∈ Ω, Γ(ω) is the non empty set of feasible trajectories, starting from ω. We construct an associated deterministic dynamic programming problem in discrete time as follows.

Let Ω = Ω × [0, 1] be the new state space and let g be the new cost function, given by g(ω, x) = x. We define a multivalued-function with nonempty values Γ : Ω ⇒ Ω by

Following [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], we define, for any initial state ω = (ω, x)

where the infima are taken over the set of sequences { ω i } i∈N such that ω 0 = ω and ω i+1 ∈ Γ( ω i ) for every i ≥ 0. Theorem 6 is then the consequence of the following three facts. Firstly, the main theorem of Lehrer and Sorin in [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], which states that uniform convergence (on Ω) of v n to some v is equivalent to uniform convergence of w λ to the same v.

Secondly, the concatenation hypothesis (2.1) on Γ implies that for any (ω,

where V t (ω) = inf X∈Γ(ω) 1 t n 0 g(X(s))ds, as defined in equation (2.4). Consequently, because of the bound on g, for any t ∈ R+ we have

where t stands for the integral part of t.

Finally, again because of hypothesis (2.1), for any λ ∈]0, 1],

(1 -λ) t g(X(t))dt. 4 We thank Frédéric Bonnans for the idea of this proof


Jean-Claude Berthélemy
email: jean-claude.berthelemy@univ-paris1.fr

Jean Gaudart
email: jean.gaudart@univ-amu.fr

Josselin Thuilliez
email: josselin.thuilliez@univ-paris1.fr

Prevention or Treatment? The Case of Malaria. *

Keywords: Economic epidemiology, Malaria, Treatment, Prevention, Price JEL: O12, I15, I25

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

Introduction

In spite of the recent decreases in malaria mortality and morbidity [START_REF] Feachem | Shrinking the malaria map: progress and prospects[END_REF][START_REF] Jamison | Global health 2035: a world converging within a generation[END_REF], this disease persists in large regions of the world. This can be partly explained by the high heterogeneity in disease environment. Furthermore, the relationship between economic development and malaria elimination follows a bidirectional relationship, which has been extensively discussed in the economic literature. Long-lasting insecticidal nets (LLINs) have been shown to be efficient preventive tools leading to ambitious campaigns of LLINs dissemination. However, in spite of their mechanical and chemical efficiency, LLINs are partially used by the populations and the efficiency decreases through time. This hinders malaria elimination. A malaria trap defined as the result of malaria reinforcing poverty, while poverty reduces the ability to control malaria, has been found to be one possible explanation for such paradox [START_REF] Berthélemy | Malaria and protective behaviours: is there a malaria trap?[END_REF].

A review of studies on price-elasticity of health products [START_REF] Bates | The Price is Wrong[END_REF], including malaria specific measures, shows that charging small fees in an attempt to balance access and "sustainability" may not be a good solution, as relative to free distribution, charging even very small user fees is highly disincentivizing. Behavioral obstacles to elimination of malaria also appear to have been underestimated. LLINs reduced malaria transmission and child morbidity in short-term trials. However, it has been shown [START_REF] Rhee | Use of insecticide-treated nets (ITNs) following a malaria education intervention in Piron, Mali: a control trial with systematic allocation of households[END_REF]Toé et al., 2009) that, even in experimental contexts, the use of bednets remained low.

The second strategy, which has been involved in the malaria burden decrease, is the access improvement to rapid diagnostic tests (RDT) and treatment. Actually, artemisinin-based chemotherapy (ACT) are very efficacious, in spite some cases of resistances in Asia (Tulloch et al., 2013). Many countries have pushed up elimination policies based on free access to ACT. Different studies have studied the impact of the RDT use or knowledge on behaviors [START_REF] Cohen | Price subsidies, diagnostic tests, and targeting of malaria treatment: evidence from a randomized controlled trial[END_REF][START_REF] Adhvaryu | Learning, Misallocation, and Technology Adoption: Evidence from New Malaria Therapy in Tanzania[END_REF] showing that many patients are taking ACT even with negative RDT. All studies recommend to improve both LLIN use and ACT access. But, none have studied the impact of treatment-based policies on LLIN use.

In this paper, we present a simple theoretical model of preventive and treatment behavior related to malaria prevalence. The framework follows the economic epidemiology literature [START_REF] Philipson | Economic epidemiology and infectious diseases[END_REF][START_REF] Gersovitz | Infectious diseases, public policy, and the marriage of economics and epidemiology[END_REF], 2004, 2005). The novelty is to 1 include a trade-off between prevention and treatment which depends on the relative cost of treatment with respect to prevention. Such behavioral response was also observed following the introduction of ARV treatments in the US in 1996 [START_REF] Geoffard | Comportements sexuels risqués et incitations[END_REF]. In this context, the benefit that is associated with the scaling-up of ARV treatments in terms of a decreased number of new infections could be at least partly offset by a relapse in the preventive behaviors among the general population. In extreme scenarios, the number of new infections could increase after a larger diffusion of ARV treatments, similar to what has been modeled for the hypothetical release of a vaccine [START_REF] Blower | Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco[END_REF][START_REF] Bogard | The impact of a partially effective HIV vaccine on a population of intravenous drug users in Bangkok, Thailand: a dynamic model[END_REF].

The rest of this paper is organized as follows. Section 2 briefly describes the epidemiological framework. Protection and treatment are added in 3 and 4. Section 5 outlines the main implications.

Standard epidemiological model of malaria transmission

A classical approach was used to model transmission of malaria between human populations and mosquito (Anopheles) populations [START_REF] Smith | Statics and dynamics of malaria infection in Anopheles mosquitoes[END_REF], based on the McDonald and Ross malaria transmission model. The following classical assumptions have te be beared for both human and mosquito populations: constant population sizes through time, uniform contacts, no superinfection or immunity. The time variation of malaria prevalences among human (eq.1) and mosquito (eq.2) populations are defined as follow:

Ẋ = mabZ(1 -X) -rX (1) Ż = acX(e -gn -Z) -gZ (2)
where m is the mosquito density (ratio of mosquitoes per human), a is the number of bits per unit of time and per mosquito, b is the proportion of infected bites that produce infection among humans, Z is the proportion of infectious mosquitoes, r is the clearance rate of malaria in humans, c is the proportion of bites on infectious humans that produce transmission to mosquitoes, g is the death rate of mosquitoes, and n is the length of sporogonic cycle.

Assuming that the life-time period of humans is long enough, malaria prevalence reaches a steady state equilibrium defined by a concave function Q(X, m) (see [START_REF] Berthélemy | Malaria and protective behaviours: is there a malaria trap?[END_REF]), for which the slope at origine is the basic reprodution rate, R 0 , and:

   Q(0, m) = 0, Q(1, m) < 1 (3)
The function Q(X, m) converges towards the trivial disease free stable steady state, if R 0 ≤ 1. This case is not considered in what follows, as it does not coincide with the persistence of malaria in large regions of the developing world. Conversely, Q(X, m) converges towards a stable steady state characterized by a strictly positive prevalence of malaria.

Economic epidemiological model with protection

The aim of LLIN-based policies is that protection tools such as LLIN could reduce malaria transmission, leading to a disease free stable steady state. But these policy only address the problem of LLIN distribution but rarely the LLIN use. Protection behavior has been added

to the previous epidemiological transmission model, based on economic mechanisms and fully described in [START_REF] Berthélemy | Malaria and protective behaviours: is there a malaria trap?[END_REF]. Two protective behaviors are defined: LLIN use (h = 1) or no use (h = 0) leading to malaria exposure. It was assumed that the only mean to prevent from infection is to use a LLIN, LLIN use was supposed to provide complete protection. These assumptions can be relaxed without affecting the main findings of the model [START_REF] Berthélemy | Malaria and protective behaviours: is there a malaria trap?[END_REF]. At any time, depending on the use of LLIN before, the health status of the individual, can be susceptible, σ(h) = S , or infected, σ(h) = I. The probability of being infected at any time, conditionally to the absence of protection before, can then be written as:

π I = P (σ(h) = I|h = 0) = Q(X, m) (4)
and then:

X = (1 -H)π I ( 5
)
where H is the proportion of LLIN use among (1 -X) uninfected.

The mosquito density, m, is modified by LLIN use: the exposed human population decreased, being only the proportion 1-H of non-protected population; the absolute number of mosquitoes decreases with H as LLINs kill mosquitoes (knock down effect). Hence m, which was a parameter in the pure epidemiological model, can be written as a function of H as follows :

m(H) = m(0) 1 -H (1 -γ(H)) (6)
where γ(H) is the proportion of mosquitoes killed by LLINs, an increasing function of H, and m(0) is the value of mosquito density in case of no protection. It follows that, at the steady state, Q(X, m(H), and then the probability of being infected pi I depend on H.

At the microeconomic level, the choice of protection is determined by maximizing the expected utility of each individual. The decision h of protection affects individuals' utility through two path: (i) an expected positive impact on the health status in case of protection and (ii) a private cost, called κ. Protection decision is described through the following maximization program:

max h E[u(σ(h))] -κW (ω)h (7)
where u(S) or u(I) are the utility levels attached to the health status (susceptible or infected thus depending on h, the use of a protection), with 0 < u(I) < u(S); ω is the individual income; W (ω) is the marginal utility of the income, supposed as usual to decrease with income. The expected utility (the expected positive impact of using LLIN on the health status) can be estimated using the following probabilities of being susceptible or infected, conditionally to the use of protection:

         P (σ(h) = S|h = 1) = 1, P (σ(h) = S|h = 0) = 1 -π I , P (σ(h) = I|h = 0) = π I . (8)
As in standard economic epidemiological models, the individual will use protective tools when W (ω) is lower than the expected utility loss associated with the risk of infection that occurs in the absence of protection:

E[u(σ(1)) -u(σ(0))] ≥ κW (ω)h (9)
Thus protection occurs if and only if:

π I ≥ κW (ω) u(S) -u(I) (10)
Indeed, a person will use LLIN if the utility of being non-infected is greater than the utility of paying for a protective tool, according to the income and the probability of being infected without using any protection.

The key point in this approach is that the threshold probability of infection depends on the marginal income utility loss associated with using the LLIN, κW (ω). The threshold function, linking π I to ω, termed C(ω), is monotonic and C (ω) < 0. In addition, the function C() is increasing with κ. Consequently:

   h = 1 if ω ≥ C -1 (π I ), h = 0 else (11)
and the income threshold conditioning protection, C -1 (π I ), decreases with κ. Knowing individual protection behaviors, the percentage of protected persons can be computed as follows:

H = +∞ C -1 (π I ) f (ω)dω ( 12
)
where f is the probability density function of ω, describing the income distribution of the population. The long term properties of the model are described in [START_REF] Berthélemy | Malaria and protective behaviours: is there a malaria trap?[END_REF]. In the next section, we include treatment in addition to prevention.

Economic epidemiological model with protection and treatment

Until now, we have focused on the dynamics without the presence of a treatment choice.

Assuming that an infected individual can buy medical treatment at a unit cost, χ, which reduces his/her illness, then the treatment does not play exactly the same role as protection.

There are two scenarios.

• Firstly, malaria chemoprophylaxis is a preventive measure and can be assimilated to the use of ITNs/LLINs in the model rather than to curative measures.

• Secondly, curative measures (or medical treatment in this case) will not play the same role as protection by ITNs in the model.

Treatment does not have an immediate effect on the infectious status of the individual but the decision tree changes as, in subsequent period, the infected individual will decide whether to buy medical treatment or not. The decision can be considered sequential, given that the decision to buy medical treatment is only taken once infection has been observed.

However, the possibility of buying medical treatment will affect the protection decision before infection. A decision variable η, is therefore introduced, which is equal to 1 if the individual buys medical treatment and 0 if he/she does not.

In the subsequent period the treatment decision is represented as:

η = 1 if and only if u(I|η = 1) -u(I|η = 0) ≥ λW (ω) (13)
where u(I|η = 0) is the value of the health status when infected and non-treated, u(I|η = 1) is the value of the health status when infected and treated, with u(I|η = 0) ≤ u(I|η = 1).

λ is the unit cost of the treatment. An individual is susceptible after the treatment but the value of health is different from u(S) as the individual has been sick, so that u(S) > u(I|η = 1).

There are now two thresholds on π I , depending on the value of η , denoted C η (ω), with η = 0 or η = 1.

The condition given in equation ( 13) can be written as:

η = 1 if and only if ω ≥ W -1 u(I|η = 1) -u(I|η = 0) λ (14)
For simplicity reason, the right-hand side expression is denoted ω 1 . It depends only on parameters of the model (values attached to different health status) and on the price of the treatment. The higher the price of treatment, the higher ω 1 , and the smaller the number of infected individuals who will decide to buy the treatment.

Once this second period decision rule has been established, an will choose in the first period to use the protection device according to a rule comparable to that of the previous section,

h = 1 if and only if ω ≥ C -1 η (π I ) for η = 0 or 1 ( 15
)
where

C 0 (ω) = λW (ω) u(S)-u(I|η=0) and C 1 (ω) = λW (ω)
u(S)-u(I|η=1) . The complete solution then depends on how ω 1 , C -1 0 (π I ) and C -1 1 (π I ) compare. From u(I|η = 0) ≤ u(I|η = 1), it can be inferred that C -1 0 (π I ) ≤ C -1 1 (π I ), ∀π I . Then, there are only three cases to consider:

• Case 1: ω 1 ≤ C -1 0 (π I ) ≤ C -1 1 (π I ) • Case 2: C -1 0 (π I ) ≤ ω 1 ≤ C -1 1 (π I ) • Case 3: C -1 0 (π I ) ≤ C -1 1 (π I ) ≤ ω 1
For case 1, h = 1 implies η = 1 and is obtained if and only if C -1 1 (π I ) ≤ ω, implying:

H = +∞ C -1 1 (π I ) f (ω)dω, denoted H 1 (π I ). ( 16
)
For case 2, h = 1 either when C -1 1 (π I ) ≤ ω (and η = 1) or when C -1 0 (π I ) ≤ ω ≤ ω 1 (and η = 0), then:

H = H 1 (π I ) + ω 1 C -1 0 (π I ) f (ω)dω (17)
For case 3, h = 1 when C -1 0 (π I ) ≤ ω (and η = 0), then:

H = +∞ C -1 0 (π I ) f (ω)dω, denoted H 0 (π I ). (18)
Given the definitions of H 0 (π I ) and H 1 (π I ) it is clear that H 1 (π I ) ≤H 0 (π I ). Overall, H can be represented as a function of π I as illustrated in Figure 1. This function of π I is continuous everywhere but not differentiable at C 0 (ω 1 ) or C 0 (ω 1 ). For π I ∈ [0, C 0 (ω 1 )[, its partial derivative is equal to:

δH δπ I = f (C -1 1 (π I )) C 1 (π I ) (19)
while for π I ∈ [C 0 (ω 1 ), C 1 (ω 1 )] , its partial derivative is equal to:

δH δπ I = f (C -1 1 (π I )) C 1 (π I ) - f (C -1 0 (π I )) C 0 (π I ) (20)
and for π I ∈ [C 1 (ω 1 ), +∞[, its partial derivative is equal to:

δH δπ I = f (C -1 0 (π I )) C 0 (π I ) (21)
Figure 1 illustrates how the model changes in relation with the introduction of a possible treatment. Specifically, it introduces changes in the protection behavior depending on the price of treatment λ.

When the price of treatment, λ, increases, ω 1 increases and C -1 0 (π I ) and C -1 1 (π I ) decrease. If the price is very high or very low, the kinks of Figure 1 are moved leftward or upward.

For a price equal to 0 or infinite, they disappear.

Conclusions

The main conclusions of the model are the following. First, a low price of treatment (case 1 ) implies that individuals will buy treatment in case of infection, and H = H 1 (π I ). As a consequence there is less protection than in absence of a treatment, but protective behaviors are locally invariant with λ.

Second, An intermediate price of treatment (case 2 ) implies that some, but not all, infected individuals buy treatment. As a consequence there is less protection than in absence of a treatment, and protective behaviors depends positively on λ.

Third, a high price of treatment (case 3 ) implies that individuals do not buy treatment when they are infected, and H = H 0 (π I ). As a consequence the protective behavior is locally the same as in absence of treatment.

Calibrating the model with known parameters from the literature constitutes avenues for future research. This could enable us to show how the results generate different predictions regarding the equilibrium levels of infection and the usage of treatment and prevention from epidemiological models which do not take this tradeoff explicitly into account.

We are very grateful to the Fondation pour and University Paris 1 for financial support. This work also benefited from the support of the French National Research Agency, as part of the program "Investissements d'avenir", ANR-10-LABX-14-01. The usual caveat applies.