Jean-François Delmas
email: jean-francois.delmas@enpc.fr

Dylan Dronnier
email: dylan.dronnier@enpc.fr

TARGETED VACCINATION STRATEGIES FOR AN INFINITE-DIMENSIONAL SIS MODEL

Keywords: . 2010 Mathematics Subject Classification. 92D30, 58E17, 47B34, 34D20 SIS Model, infinite dimensional ODE, kernel operator, vaccination strategy, effective reproduction number, multi-objective optimization, Pareto frontier

We formalize and study the problem of optimal allocation strategies for a (perfect) vaccine in the infinite-dimensional SIS model. The question may be viewed as a bi-objective minimization problem, where one tries to minimize simultaneously the cost of the vaccination, and a loss that may be either the effective reproduction number, or the overall proportion of infected individuals in the endemic state. We prove the existence of Pareto optimal strategies for both loss functions.

We also show that vaccinating according to the profile of the endemic state is a critical allocation, in the sense that, if the initial reproduction number is larger than 1, then this vaccination strategy yields an effective reproduction number equal to 1.

I n t ro d u c t i o n

1.1. Motivation. Increasing the prevalence of immunity from contagious disease in a population limits the circulation of the infection among the individuals who lack immunity. This so-called "herd effect" plays a fundamental role in epidemiology as it has had a major impact in the eradication of smallpox and rinderpest or the near eradication of poliomyelitis; see [START_REF] Fine | Herd Immunity: A Rough Guide[END_REF]. Targeted vaccination strategies, based on the heterogeneity of the infection spreading in the population, are designed to increase the level of immunity of the population with a limited quantity of vaccine. These strategies rely on identifying groups of individuals that should be vaccinated in priority in order to slow down or eradicate the disease.

In this article, we establish a theoretical framework to study targeted vaccination strategies for the deterministic infinite-dimensional SIS model introduced in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], that encompasses as particular cases the SIS model on graphs or on stochastic block models. In companion papers, we provide a series of general and specific examples that complete and illustrate the present work: see Section 1.5 for more detail. 1.2. Herd immunity and targeted vaccination strategies. Let us start by recalling a few classical results in mathematical epidemiology; we refer to Keeling and Rohani's monograph [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF] for an extensive introduction to this field, including details on the various classical models (SIS, SIR, etc.)

In an homogeneous population, the basic reproduction number of an infection, denoted by R 0 , is defined as the number of secondary cases one individual generates on average over the course of its infectious period, in an otherwise uninfected (susceptible) population. This number plays a fundamental role in epidemiology as it provides a scale to measure how difficult an infectious disease is to control. Intuitively, the disease should die out if R 0 < 1 and invade the population if R 0 > 1. For many classical mathematical models of epidemiology, such as SIS or S(E)IR, this intuition can be made rigorous: the quantity R 0 may be computed from the parameters of the model, and the threshold phenomenon occurs.

Assuming R 0 > 1 in an homogeneous population, suppose now that only a proportion η uni of the population can catch the disease, the rest being immunized. An infected individual will now only generate η uni R 0 new cases, since a proportion (1η uni ) of previously successful infections will be prevented. Therefore, the new effective reproduction number is equal to R e (η uni ) = η uni R 0 . This fact led to the recognition by Smith in 1970 [START_REF] Smith | Prospects for the control of infectious disease[END_REF] and Dietz in 1975 [START_REF] Dietz | Transmission and control of arbovirus diseases[END_REF] of a simple threshold theorem: the incidence of an infection declines if the proportion of nonimmune individuals is reduced below η uni crit = 1/R 0 . This effect is called herd immunity, and the corresponding percentage 1η uni crit of people that have to be vaccinated is called herd immunity threshold ; see for instance [START_REF] Smith | Concepts of herd protection and immunity[END_REF][START_REF] Somerville | Public health and epidemiology at a glance[END_REF].

It is of course unrealistic to depict human populations as homogeneous, and many generalizations of the homogeneous model have been studied; see [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF]Chapter 3] for examples and further references. For most of these generalizations, it is still possible to define a meaningful reproduction number R 0 , as the number of secondary cases generated by a typical infectious individual when all other individuals are uninfected; see [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF]. After a vaccination campaign, let the vaccination strategy η denote the (non necessarily homogeneous) proportion of the non-vaccinated population, and let the effective reproduction number R e (η) denote the corresponding reproduction number of the non-vaccinated population. The vaccination strategy η is critical if R e (η) = 1. The possible choices of η naturally raises a question that may be expressed as the following informal optimization problem: [START_REF] Anselone | Collectively compact operator approximation theory and applications to integral equations[END_REF] Minimize: the quantity of vaccine to administrate subject to: herd immunity is reached, that is, R e ≤ 1.

If the quantity of available vaccine is limited, then one is also interested in:

(2) Minimize: the effective reproduction number R e subject to: a given quantity of available vaccine.

Interestingly enough, the strategy η uni crit , which consists in delivering the vaccine uniformly to the population, without taking inhomogeneity into account, leaves a proportion η uni crit = 1/R 0 of the population unprotected, and is therefore critical since R e (η uni crit ) = 1. In particular it is admissible for the optimization problem [START_REF] Anselone | Collectively compact operator approximation theory and applications to integral equations[END_REF].

However, herd immunity may be achieved even if the proportion of unprotected people is greater than 1/R 0 , by targeting certain group(s) within the population; see Figure 3.3 in [START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF]. For example, the discussion of vaccination control of gonorrhea in [START_REF] Hethcote | Gonorrhea Transmission Dynamics and Control[END_REF]Section 4.5] suggests that it may be better to prioritize the vaccination of people that have already caught the disease: this lead us to consider a vaccination strategy guided by the equilibrium state. This strategy denoted by η equi will be defined formally below. Let us mention here an observation in the same vein made by Britton, Ball and Trapman in [START_REF] Britton | A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2[END_REF]. Recall that in the S(E)IR model, immunity can be obtained through infection. Using parameters from real-world data, these authors noticed that the disease-induced herd immunity level can, for some models, be substantially lower than the classical herd immunity threshold 1 -1/R 0 . This can be reformulated in term of targeted vaccination strategies: prioritizing the individuals that are more likely to get infected in a S(E)IR epidemic may be more efficient than distributing uniformly the vaccine in the population.

The main goal of this paper is two-fold: formalize the optimization problems (1) and (2) for a particular infinite dimensional SIS model, recasting them more generally as a bi-objective optimization problem; and give existence and properties of solutions to this bi-objective problem. We will also consider a closely related problem, where one wishes to minimize the size of the epidemic rather than the reproduction number. We will in passing provide insight on the efficiency of classical vaccination strategies such as η uni crit or η equi .

1.3. Literature on targeted vaccination strategies. Targeted vaccination problems have mainly been studied using two different mathematical frameworks.

1.3.1. On meta-populations models. Problems (1) and ( 2) have been examined in depth for deterministic meta-population models, that is, models in which an heterogeneous population is stratified into a finite number of homogeneous sub-populations (by age group, gender, . . . ). Such models are specified by choosing the sizes of the subpopulations and quantifying the degree of interactions between them, in terms of various mixing parameters. In this setting, R 0 can often be identified as the spectral radius of a next-generation matrix whose coefficients depend on the subpopulation sizes, and the mixing parameters. It turns out that the next generation matrices take similar forms for many dynamics (SIS, SIR, SEIR,...); see the discussion in [25, Section 10]. Vaccination strategies are defined as the levels at which each sub-population is immunized. After vaccination, the next-generation matrix is changed and its new spectral radius corresponds to the effective reproduction number R e . Problem [START_REF] Anselone | Collectively compact operator approximation theory and applications to integral equations[END_REF] has been studied in this setting by Hill and Longini [START_REF] Hill | The critical vaccination fraction for heterogeneous epidemic models[END_REF]. These authors study the geometric properties of the so-called threshold hypersurface, that is the vaccination allocations for which R e = 1. They also compute the vaccination belonging to this surface with minimal cost for an Influenza A model. Making structural assumptions on the mixing parameters, Poghotayan, Feng, Glasser and Hill derive in [START_REF] Poghotanyan | Constrained minimization problems for the reproduction number in meta-population models[END_REF] an analytical formula for the solutions of Problem [START_REF] Anselone | Spectral properties of integral operators with nonnegative kernels[END_REF], for populations divided in two groups. Many papers also contain numerical studies of the optimization problems ( 1) and ( 2) on real-world data using gradient techniques or similar methods; see for example [START_REF] Duijzer | The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect[END_REF][START_REF] Feng | Evaluating targeted interventions via meta-population models with multi-level mixing[END_REF][START_REF] Feng | An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing[END_REF][START_REF] Goldstein | Distribution of vaccine/antivirals and the 'least spread line' in a stratified population[END_REF][START_REF] Zhao | Identifying optimal vaccination strategies via economic and epidemiological modeling[END_REF].

Finally, the effective reproduction number is not the only reasonable way of quantifying a population's vulnerability to an infection. For an SIR infection for example, the proportion of individuals that eventually catch (and recover from) the disease, often referred to as the attack rate, is broadly used. We refer to [START_REF] Duijzer | The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect[END_REF][START_REF] Duijzer | Dose-optimal vaccine allocation over multiple populations[END_REF] for further discussion on this topic. 1.3.2. On networks. Whereas the previously cited works typically consider a small number of subpopulations, often with a "dense" structure of interaction (every subpopulation may directly infect all the others), other research communities have looked into a similar problem for graphs. Indeed, given a (large), possibly random graph, with epidemic dynamics on it, and supposing that we are able to suppress vertices by vaccinating, one may ask for the best way to choose the vertices to remove.

The importance of the spectral radius of the network has been rapidly identified as its value determines if the epidemic dies out quickly or survives for a long time [START_REF] Ganesh | The effect of network topology on the spread of epidemics[END_REF][START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF]. Since Van Mieghem et al. proved in [START_REF] Van Mieghem | Decreasing the spectral radius of a graph by link removals[END_REF] that the problem of minimizing spectral radius of a graph by removing a given number of vertices is NP-complete (and therefore unfeasible in practice), many computational heuristics have been put forward to give approximate solutions; see for example [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF] and references therein.

1.4. Main results. The differential equations governing the epidemic dynamics in metapopulation SIS models were developed by Lajmanovich and Yorke in their pioneer paper [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF]. In [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], we introduced a natural generalization of their equation, which can also be viewed as the limit equation of the stochastic SIS dynamic on network, in an infinite-dimensional space Ω, where x ∈ Ω represents a feature and the probability measure µ(dx) represents the fraction of the population with feature x.

1.4.1. Regularity of the effective reproduction function R e . We consider the effective reproduction function in a general operator framework which we call the kernel model. This model is characterized by a probability space (Ω, F , µ) and a measurable non-negative kernel k : Ω×Ω → R + .

Let T k be the corresponding integral operator defined by:

T k (h)(x) = Ω k(x, y)h(y) µ(dy).
In the setting of [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] (see in particular Equation (11) therein), T k is the so-called next generation operator, where the kernel k is defined in terms of a transmission rate kernel k(x, y) and a recovery rate function γ by the product k(x, y) = k(x, y)/γ(y); and the reproduction number R 0 is then the spectral radius ρ(T k ) of T k .

Following [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]Section 5], we represent a vaccination strategy by a function η : Ω → [0, 1], where η(x) represents the fraction of non-vaccinated individuals with feature x; the effective reproduction number associated to η is then given by

(3) R e (η) = ρ(T kη ),
where ρ stands for the spectral radius and kη stands for the kernel (kη)(x, y) = k(x, y)η(y). If R 0 ≥ 1, then a vaccination strategy η is called critical if it achieves precisely the herd immunity threshold, that is R e (η) = 1.

In particular, the "strategy" that consists in vaccinating no one corresponds to η ≡ 1, and of course R e (1) = R 0 . As the spectral radius is positively homogeneous, we also get, when R 0 ≥ 1, that the uniform strategy that corresponds to the constant function:

η uni crit ≡ 1 R 0 is critical, as R e (η uni crit ) = 1
. This is consistent with results obtained in the homogeneous model given in Section 1.2.

Let ∆ be the set of strategies, that is the set of [0, 1]-valued functions defined on Ω. The usual technique to obtain the existence of solutions to optimization problems like (1) or ( 2) is to prove that the function R e is continuous with respect to a topology for which the set of strategies ∆ is compact. It is natural to try and prove this continuity by writing R e as the composition of the spectral radius ρ and the map η → T kη . The spectral radius is indeed continuous at compact operators (and T kη is in fact compact under a technical integrability assumption on the kernel k formalized on page 10 as Assumption 1), if we endow the set of bounded operators with the operator norm topology; see [START_REF] Burlando | Continuity of spectrum and spectral radius in Banach algebras[END_REF][START_REF] Newburgh | The variation of spectra[END_REF]. However, this would require choosing the uniform topology on ∆, which then is not compact.

We instead endow ∆ with the weak topology, see Section 3.1, for which compactness holds; see Lemma 3.1. This forces us to equip the space of bounded operators with the strong topology, for which the spectral radius is in general not continuous; see [29, p. 431]. However, the family of operators (T kη , η ∈ ∆) is collectively compact which enables us to recover continuity, using a serie of results obtained by Anselone [START_REF] Anselone | Collectively compact operator approximation theory and applications to integral equations[END_REF]. This leads to the following result, proved in Theorem 4.2 below. We recall that Assumption 1, formulated on page 10, provides an integrability condition on the kernel k.

Theorem 1.1 (Continuity of the spectral radius). Under Assumption 1 on the kernel k, the function R e : ∆ → R + is continuous with respect to the weak topology on ∆.

In fact, we also prove the continuity of the spectrum with respect to the Hausdorff distance on the set of compact subsets of C. We shall write R e [k] to stress the dependence of the function R e in the kernel k. In Proposition 4.3, we prove the stability of R e , by giving natural sufficient conditions on a sequence of kernels (k n , n ∈ N) converging to k which imply that R e [k n ] converges uniformly towards R e [k]. This result has both theoretical and practical interest: the next-generation operator is unknown in practice, and has to be estimated from data. Thanks to this result, the value of R e computed from the estimated operator should converge to the true value. 1.4.2. On the maximal endemic equilibrium in the SIS model. We consider the SIS model from [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]. This model is characterized by a probability space (Ω, F , µ), the transmission kernel k : Ω × Ω → R + and the recovery rate γ : Ω → R * + . We suppose in the following that the technical Assumption 2, formulated on page 11, holds, so that the SIS dynamical evolution is well defined. This evolution is encoded as u = (u t , t ∈ R + ), where u t ∈ ∆ for all t and u t (x) represents the probability of an individual with feature x ∈ Ω to be infected at time t ≥ 0, and follows the equation:

(4) ∂ t u t = F (u t ) for t ∈ R + , where F (g) = (1 -g)T k (g) -γg for g ∈ ∆,
with an initial condition u 0 ∈ ∆ and with T k the integral operator corresponding to the kernel k acting on the set of bounded measurable functions, see [START_REF] Esser | On the spectrum of a complete multipartite graph[END_REF]. It is proved in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] that such a solution u exists and is unique under Assumption 2. An equilibrium of ( 4) is a function g ∈ ∆ such that F (g) = 0. According to [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], there exists a maximal equilibrium g, i.e., an equilibrium such that all other equilibria h ∈ ∆ are dominated by g: h ≤ g. Furthermore, we have R 0 ≤ 1 if and only if g = 0. In the connected case (for example if k > 0), then 0 and g are the only equilibria; besides g is the long-time distribution of infected individuals in the population: lim t→+∞ u t = g as soon as the initial condition is non-zero; see [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]Theorem 4.14].

As hinted in [START_REF] Hethcote | Gonorrhea Transmission Dynamics and Control[END_REF]Section 4.5] for vaccination control of gonorrhea, it is interesting to consider vaccinating people with feature x with probability g(x); this corresponds to the strategy based on the maximal equilibrium:

η equi = 1 -g.
The following result entails that this strategy is critical and thus achieves the herd immunity threshold. Recall that Assumption 2, formulated page 11, provides technical conditions on the parameters k and γ of the SIS model. The effective reproduction number of the SIS model is the function R e defined in (3) with the kernel k = k/γ.

Theorem 1.2 (The maximal equilibrium yields a critical vaccination). Suppose Assumption 2 holds. If R 0 ≥ 1, then the vaccination strategy η equi is critical, that is, R e (η equi ) = 1.

This result will be proved below as a part of Proposition 8.2. Let us finally describe informally another consequence of this Proposition. We were able to prove in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]Theorem 4.14] that, in the connected case, if R 0 > 1, the disease-free equilibrium u = 0 is unstable. Proposition 8.2 gives spectral information on the formal linearization of the dynamics (4) near any equilibrium h; in particular if h = g then h is linearly unstable. 1.4.3. Regularity of the total proportion of infected population function I. According to [7, Section 5.3.], the SIS equation with vaccination strategy η is given by ( 4), where F is replaced by F η defined by:

F η (g) = (1 -g)T kη (g) -γg.
and u t now describes the proportion of infected among the non-vaccinated population. We denote by g η the corresponding maximal equilibrium (thus considering η ≡ 1 gives g = g 1 ), so that F η (g η ) = 0. Since the probability for an individual x to be infected in the stationary regime is g η (x) η(x), the fraction of infected individuals at equilibrium, I(η), is thus given by:

(5)

I(η) = Ω g η η dµ = Ω g η (x) η(x) µ(dx).
As mentioned above, for a SIR model, distributing vaccine so as to minimize the attack rate is at least as natural as trying to minimize the reproduction number, and this problem has been studied for example in [START_REF] Duijzer | The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect[END_REF][START_REF] Duijzer | Dose-optimal vaccine allocation over multiple populations[END_REF]. In the SIS model the quantity I appears as a natural analogue of the attack rate, and is therefore a natural optimization objective.

We obtain results on I that are very similar to the ones on R e . Recall that Assumption 2 on page 11 ensures that the infinite-dimensional SIS model, given by equation ( 4), is well defined. The next theorem corresponds to Theorem 4.6.

Theorem 1.3 (Continuity of the equilibrium infection size). Under Assumption 2, the function I : ∆ → R + is continuous with respect to the weak topology on ∆.

In Proposition 4.7, we prove the stability of I, by giving natural sufficient condition on a sequence of kernels and functions ((k n , γ n ), n ∈ N) converging to (k, γ) which imply that I[k n , γ n ] converges uniformly towards I[k, γ]. We also prove that the loss functions L = R e and L = I are both non-decreasing (η ≤ η implies L(η) ≤ L(η )), and sub-homogeneous (L(λη) ≤ λL(η) for all λ ∈ [0, 1]); see Propositions 4.1 and 4.5.

1.4.4.

Optimizing the protection of the population. Consider a cost function C : ∆ → [0, 1] which measures the cost for the society of a vaccination strategy (production and diffusion). Since the vaccination strategy η represents the non-vaccinated population, the cost function C should be decreasing (roughly speaking η < η implies C(η) > C(η ); see Definition 5.1). We shall also assume that C is continuous with respect to the weak topology on ∆, and that doing nothing costs nothing, that is, C(1) = 0. A simple and natural choice is the uniform cost C uni given by the overall proportion of vaccinated individuals:

C uni (η) = Ω (1 -η) dµ = 1 - Ω η dµ.

See Remark 5.2 for comments on other examples of cost functions.

Our problem may now be seen as a bi-objective minimization problem: we wish to minimize both the loss L(η) and the cost C(η), subject to η ∈ ∆, with the loss function L being either R e or I. Following classical terminology for multi-objective optimisation problems [START_REF] Miettinen | Nonlinear multiobjective optimization[END_REF], we call a strategy η Pareto optimal if no other strategy is strictly better:

C(η) < C(η ) =⇒ L(η) > L(η ) and L(η) < L(η ) =⇒ C(η) > C(η ).
The set of Pareto optimal strategies will be denoted by P L , and we define the Pareto frontier as the set of Pareto optimal outcomes:

F L = {(C(η ), L(η )) : η ∈ P L }.
Notice that, with this definition, the Pareto frontier is empty when there is no Pareto optimal strategy.

For any strategy η, the cost and loss of η vary between the following bounds: Let L be the optimal loss function and C ,L the optimal cost function defined by:

0 = C(1) ≤ C(η) ≤ C(0) = c max = cost
L (c) = inf { L(η) : η ∈ ∆, C(η) ≤ c } for c ∈ [0, c max ], C ,L ( ) = inf { C(η) : η ∈ ∆, L(η) ≤ } for ∈ [0, max ].
We simply write C for C ,L when no confusion on the loss function can arise. Proposition 5.5 (in a more general framework in particular for the cost function) and Lemma 5.

F L = {(c, L (c)) : c ∈ [0, C ,L (0)]} = {(C ,L ( ), ) : ∈ [0, max ]}.
We also establish that P L is compact in ∆ for the weak topology in Corollary 5.7; that the set of outcomes or feasible region F = {(C(η), L(η)), η ∈ ∆} has no holes in Proposition 6.1; and that the Pareto frontier is convex if C and L are convex in Proposition 6.6. We study in Proposition 6.2 the stability of the Pareto frontier and the set of Pareto optima when the parameters vary.

In a sense the Pareto optimal strategies are intuitively the "best" strategies. Similarly, we also study the "worst" strategies, which we call anti-Pareto optimal strategies, and describe the corresponding anti-Pareto frontier. Understanding the "worst strategies" also helps to avoid pitfalls when one has to consider sub-optimal strategies: for example, we prove in [START_REF] Delmas | Effective reproduction number: convexity, invariance and cordons sanitaires[END_REF] that disconnecting strategies are not the "worst" strategies, and we provide in [START_REF] Delmas | Optimal vaccination: various (counter) intuitive examples[END_REF]Section 4] an elementary example where the same strategies can be "best" or "worst" according to model parameters values. Surprisingly, proving properties of the anti-Pareto frontier sometimes necessitates stronger assumptions than in the Pareto case: for example, the connectedness of the anti-Pareto frontier is only proved under a quasi-irreducibility assumption on the kernel, see Lemmas 5.11 and 5.12.

Remark 1.5 (Eradication strategies do not depend on the loss). In [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], we proved that, for all η ∈ ∆, the equilibrium infection size I(η) is non zero if and only if R e (η) > 1. Consider the uniform cost C = C uni . First, this implies that P I is a subset of {η ∈ ∆ : R e (η) ≥ 1}. Secondly, a vaccination strategy η ∈ ∆ is Pareto optimal for the objectives (R e , C) and satisfies R e (η) = 1 if and only if η is Pareto optimal for the objectives (I, C) and satisfies I(η) = 0: [START_REF] Conway | A course in functional analysis[END_REF] η ∈ P Re and R e (η) = 1 ⇐⇒ η ∈ P I and I(η) = 0.

Remark 1.6 (Minimal cost of eradication). Assume R 0 > 1 and the uniform cost C = C uni . The equivalence [START_REF] Conway | A course in functional analysis[END_REF] implies directly that:

C ,Re (1) = C ,I (0)
.

Thus, this latter quantity can be seen as the minimal cost (or minimum percentage of people that have to be vaccinated) required to eradicate the infection. Recall the critical vaccination strategies η uni crit ≡ 1/R 0 and η equi = 1-g (as R e (η uni crit ) = R e (η equi ) = 1). Since C(η uni crit ) = 1-1/R 0 and C(η equi ) = Ω g dµ = I(1), we obtain the following upper bounds of the minimal cost required to eradicate the infection:

C ,Re (1) = C ,I (0) ≤ min 1 - 1 R 0 , Ω g dµ .
1.4.5. Equivalence of models. Our last results address a natural question stemming from our choice of a very general framework to modelize the infection. Since our models are infinite dimensional and depend on the choices of the probability space (Ω, F , µ), the kernel k (for the kernel model) and the kernel k and recovery rate γ (for the SIS model), the are different equivalent ways to model the same situation. We study in Section 7 a way to ensure that, even if the parameters are different, we end up with the same Pareto frontiers. This situation is similar to random variables having the same law in probability theory, or to equivalent graphons in graphon theory. In particular it allows us to treat the same meta-population model in either a discrete or a continuous setting, see Figure 4 for an illustration and Example 1.7.

1.4.6. An illustrative example: the multipartite graphon. Let us illustrate some of our results on an example, which will be discussed in details in a forthcoming companion paper [START_REF] Delmas | Optimal vaccination: various (counter) intuitive examples[END_REF].

Example 1.7 (Multipartite graphon). Graphs that can be colored with colors, so that no two endpoints of an edge have the same color are known as -partite graphs. In a biological setting, this corresponds to a population of groups, such that individuals in a group can not contaminate individuals of the same group. Let us generalize and assume there is an infinity of groups, = ∞ of respective size (2 -n , n ∈ N * ) and that the next generation kernel k is equal to the constant κ > 0 between individuals of different groups and equal to 0 between individuals of the same group (so there is no intra-group contamination). Using the equivalence of models from Section 7, we can represent this model by using a continuous state space Ω = [0, 1], endowed with µ the Lebesgue measure on Ω, the group n being represented by the interval

I n = [1 -2 -n+1 , 1 -2 -n ) for n ∈ N * . The kernel k is then given by k = κ(1 -n∈N * 1 In×In ); it is represented in Figure 1( a ).
Consider the loss L = R e and the cost C = C uni giving the overall proportion of vaccinated individuals. Based on the results of [START_REF] Esser | On the spectrum of a complete multipartite graph[END_REF][START_REF] Stevanović | On spectral radius and energy of complete multipartite graphs[END_REF], we prove in [START_REF] Delmas | Optimal vaccination: various (counter) intuitive examples[END_REF] that the vaccination strategies

1 [0,1-c] , with cost C(1 [0,1-c] ) = c ∈ [0, 1/2], are Pareto optimal.
Remembering that the natural definition of the degree in a continuous graph is given by deg(x) = Ω k(x, y) µ(dy), we note that the vaccination strategy 1 [0,1-c] corresponds to vaccinating individuals with feature x ∈ (1c, 1], that is, the individuals with the highest degree. In Figure 1( b ), the corresponding Pareto frontier (i.e., the outcome of the "best" vaccination strategies) is drawn as the solid red line; the blue-colored zone corresponds to the feasible region that is, all the possible values of (C(η), R e (η)), where η ranges over ∆; the dotted line corresponds to the outcome of the uniform vaccination strategy η ≡ c, that is (C(η), R e (η)) = (c, (1c)R 0 ) where c ranges over [0, 1]; and the red dashed curve corresponds to the anti-Pareto frontier (i.e., the outcome of the "worst" vaccination strategies), which for this model correspond to the uniform vaccination of the nodes with the updated lower degree; see [START_REF] Delmas | Optimal vaccination: various (counter) intuitive examples[END_REF]. Notice that the path (1 [0,1-c] , c ∈ [0, 1/2]) is an increasing continuous (for the topology of the simple convergence and thus the L 1 (µ) topology) path of Pareto optima which gives a complete parametrization of the Pareto frontier. The latter has been computed numerically using the power iteration method. In particular, we obtained the following value: R 0 0.697κ. 1.5. On the companion papers. We detail some developments in forthcoming papers where only the uniform cost C = C uni is considered. In [START_REF] Delmas | Effective reproduction number: convexity, invariance and cordons sanitaires[END_REF], motivated by the conjecture formulated by Hill and Longini in finite dimension [START_REF] Hill | The critical vaccination fraction for heterogeneous epidemic models[END_REF]Conjecture 8.1], we investigate the convexity and concavity of the effective reproduction function R e . We also prove that a disconnecting strategy is better than the worst, i.e., is not anti-Pareto optimal.

In [START_REF] Delmas | Vaccinating higly connected people is (sometimes) optimal[END_REF], under monotonicity properties of the kernel, satisfied for example by the configuration model, it is proven that vaccinating the individuals with the highest (resp. lowest) number of contacts is Pareto (resp. anti-Pareto) optimal. In this case the greedy algorithm, which performs infinitesimal locally optimal steps, is optimal as it browses continuously the set of Pareto (resp. anti-Pareto) optimal strategies, providing an increasing parametrization of the Pareto (resp. anti-Pareto) frontier. In this setting, we provide some examples of SIS models where the set of Pareto optimal strategies coincide for the losses R e and I: [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] P I = P Re ∩ {η ∈ ∆ : R e (η) ≥ 1}.

In [START_REF] Delmas | Optimal vaccination: various (counter) intuitive examples[END_REF], which includes a detailed study of the multipartite kernel of Example 1.7, we study the optimal vaccination when the individuals have the same number of contacts. This provides examples where the uniform vaccination is Pareto optimal, or anti-Pareto optimal, or not optimal for either problem. We also provide an example where the set P Re has a countable number of connected components (and is thus not connected). This implies in particular that the greedy algorithm is not optimal in this case.

In [START_REF] Delmas | Optimal vaccination for a 2 sub-populations SIS model[END_REF], we give a comprehensive treatment of the two groups model, Ω = {1, 2}, for L = R e , and some partial results for L = I. Despite its apparent simplicity, the derivation of formulae for the Pareto optimal strategies is non trivial, see also [START_REF] Poghotanyan | Constrained minimization problems for the reproduction number in meta-population models[END_REF]. In addition, this model is rich enough to give examples of various interesting behaviours:

• On the critical strategies η uni crit and η equi . Depending on the parameters, the strategies η uni crit and/or η equi may or may not be Pareto optimal, and the cost C(η uni crit ) may be larger than, smaller than or equal to C(η equi ).

• Vaccinating people with highest contacts. The intuitive idea of vaccinating the individuals with the highest number of contacts may or may not provide the optimal strategies, depending on the parameters. • Dependence on the choice of the loss function. For examples where R 0 > 1, the optimal strategies for the losses I and R e may coincide, so that (7) holds, or not at all, so that P I ∩ P Re ∩ {η ∈ ∆ : 1 < R e (η) < R 0 } = ∅, depending on the parameters.

1.6. Structure of the paper. Section 2 is dedicated to the presentation of the vaccination model and the various assumptions on the parameters. We also define properly the so-called loss functions R e and I. After recalling a few topological facts in Section 3, we study the regularity properties of R e and I in Section 4. We present the multi-objective optimization problem in Section 5 under general condition on the loss function L and cost function C and prove the results on the Pareto frontier. This is completed in Section 6 with miscellaneous properties of the Pareto frontier. In Section 7, we discuss the equivalent representation of models with different parameters. Proofs of a few technical results are gathered in Section 8.

S e t t i n g a n d n o tat i o n

2.1. Spaces, operators, spectra. All metric spaces (S, d) are endowed with their Borel σ-field denoted by B(S). The set K of compact subsets of C endowed with the Hausdorff distance d H is a metric space, and the function rad from K to R + defined by rad(K) = max{|λ| , λ ∈ K} is Lipschitz continuous from (K , d H ) to R endowed with its usual Euclidean distance.

Let (Ω, F , µ) be a probability space. We denote by L ∞ , the Banach spaces of bounded real-valued measurable functions defined on Ω equipped with the sup-norm, L ∞ + the subset of L ∞ of non-negative function, and ∆ = {f ∈ L ∞ : 0 ≤ f ≤ 1} the subset of non-negative functions bounded by 1. For f and g real-valued functions defined on Ω, we may write f, g or Ω f g dµ for Ω f (x)g(x) µ(dx) whenever the latter is meaningful. For p ∈ [1, +∞], we denote by L p = L p (µ) = L p (Ω, µ) the space of real-valued measurable functions g defined Ω such that g p = |g| p dµ 1/p (with the convention that g ∞ is the µ-essential supremum of |g|) is finite, where functions which agree µ-almost surely are identified. We denote by L p + the subset of L p of non-negative functions.

Let (E, • ) be a Banach space. We denote by • E the operator norm on L(E) the Banach algebra of bounded operators. The spectrum Spec(T ) of T ∈ L(E) is the set of λ ∈ C such that T -λId does not have a bounded inverse operator, where Id is the identity operator on E. Recall that Spec(T ) is a compact subset of C, and that the spectral radius of T is given by: ( 8)

ρ(T ) = rad(Spec(T )) = lim n→∞ T n 1/n E .
The element λ ∈ Spec(T ) is an eigenvalue if there exists x ∈ E such that T x = λx and x = 0.

If E is also a functional space, for g ∈ E, we denote by M g the multiplication (possibly unbounded) operator defined by M g (h) = gh for all h ∈ E.

Kernel operators.

We define a kernel (resp. signed kernel ) on Ω as a R + -valued (resp. R-valued) measurable function defined on (Ω 2 , F ⊗2 ). For f, g two non-negative measurable functions defined on Ω and k a kernel on Ω, we denote by f kg the kernel defined by: ( 9)

f kg : (x, y) → f (x) k(x, y)g(y).
When γ is a positive measurable function defined on Ω, we write k/γ for kγ -1 , and remark that it may differ from γ -1 k.

For p ∈ (1, +∞), we define the double norm of a signed kernel k by: (10)

k p,q = Ω Ω |k(x, y)| q µ(dy) p/q µ(dx) 1/p
with q given by 1 p

+ 1 q = 1.
Assumption 1 (On the kernel model [(Ω, F , µ), k]). Let (Ω, F , µ) be a probability space. The kernel k on Ω has a finite double-norm, that is, k p,q < +∞ for some p ∈ (1, +∞).

To a kernel k such that k p,q < +∞, we associate the positive integral operator T k on L p defined by:

(11) T k (g)(x) = Ω k(x, y)g(y) µ(dy) for g ∈ L p and x ∈ Ω.
According to [22, p. 293], operator T k is compact. It is well known and easy to check that:

(12) T k L p ≤ k p,q .
For η ∈ ∆, the kernel kη has also a finite double norm on L p and the operator M η is bounded, so that the operator T kη = T k M η is compact. We can define the effective spectrum function Spec[k] from ∆ to K by:

(13) Spec[k](η) = Spec(T kη ), the effective reproduction number function R e [k] = rad • Spec[k] from ∆ to R + by: (14) R e [k](η) = rad(Spec(T kη )) = ρ(T kη ),
and the corresponding reproduction number :

(15) R 0 [k] = R e [k](1) = ρ(T k ).
When there is no ambiguity, we simply write R e for R e [k] and R 0 for R 0

[k]. We say a vaccination strategy η ∈ ∆ is critical if R e (η) = 1.
Following the framework of [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], for q ∈ (1, +∞), we also consider the following norm for the kernel k:

k ∞,q = sup x∈Ω Ω k(x, y) q µ(dy) 1/q
.

Clearly, we have that k ∞,q finite implies that k p,q is also finite, with p such that 1/p+1/q = 1. When k ∞,q < +∞, the corresponding positive bounded linear integral operator T k on L ∞ is similarly defined by:

(16) T k (g)(x) = Ω k(x, y)g(y) µ(dy) for g ∈ L ∞ and x ∈ Ω.
Notice that the integral operators T k and T k corresponds respectively to the operators T k and Tk in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]. According to [7, Lemma 3.7], the operator T 2 k on L ∞ is compact and T k has the same spectral radius as T k :

(17) ρ(T k ) = ρ(T k ).

2.3.

Dynamics for the SIS model and equilibria. In accordance with [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], we consider the following assumption. Recall that k/γ = kγ -1 .

Assumption 2 (On the SIS model [(Ω, F , µ), k, γ]). Let (Ω, F , µ) be a probability space. The recovery rate function γ is a function which belongs to L ∞ + and the transmission rate kernel k on Ω 2 is such that k/γ ∞,q < +∞ for some q ∈ (1, +∞).

Assumption 2 implies Assumption 1 for the kernel k = k/γ. Under Assumption 2, we also consider the bounded operators T k/γ on L ∞ , as well as T k/γ on L p , which are the so called next-generation operator. The SIS dynamics considered in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] (under Assumption 2) follows the vector field F defined on L ∞ by: ( 18)

F (g) = (1 -g)T k (g) -γg.
More precisely, we consider u = (u t , t ∈ R), where u t ∈ ∆ for all t ∈ R + such that:

(19) ∂ t u t = F (u t ) for t ∈ R + ,
with initial condition u 0 ∈ ∆. The value u t (x) models the probability that an individual of feature x is infected at time t; it is proved in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] that such a solution u exists and is unique.

An equilibrium of ( 19) is a function g ∈ ∆ such that F (g) = 0. According to [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF], there exists a maximal equilibrium g, i.e., an equilibrium such that all other equilibria h ∈ ∆ are dominated by g: h ≤ g. The reproduction number R 0 associated to the SIS model given by [START_REF] Fine | Herd Immunity: A Rough Guide[END_REF] is the spectral radius of the next-generation operator, so that using the definition of the effective reproduction number ( 14), ( 15) and [START_REF] Feng | Evaluating targeted interventions via meta-population models with multi-level mixing[END_REF], this amounts to:

(20) R 0 = ρ(T k/γ ) = R 0 [k/γ] = R e [k/γ](1).
If R 0 ≤ 1 (sub-critical and critical case), then u t converges pointwise to 0 when t → ∞. In particular, the maximal equilibrium g is equal to 0 everywhere. If R 0 > 1 (super-critical case), then 0 is still an equilibrium but different from the maximal equilibrium g, as Ω g dµ > 0.

Vaccination strategies.

A vaccination strategy η of a vaccine with perfect efficiency is an element of ∆, where η(x) represents the proportion of non-vaccinated individuals with feature x. Notice that η dµ corresponds in a sense to the effective population.

Recall the definition of the kernel f kg from [START_REF] Delmas | Optimal vaccination for a 2 sub-populations SIS model[END_REF]. For η ∈ ∆, the kernels kη/γ and kη have finite norm • ∞,q under Assumption 2, so we can consider the bounded positive operators T kη/γ and T kη on L ∞ . According to [7, Section 5.3.], the SIS equation with vaccination strategy η is given by [START_REF] Fine | Herd Immunity: A Rough Guide[END_REF], where F is replaced by F η defined by:

(21) F η (g) = (1 -g)T kη (g) -γg.
We denote by u η = (u η t , t ≥ 0) the corresponding solution with initial condition u η 0 ∈ ∆. We recall that u η t (x) represents the probability for an non-vaccinated individual of feature x to be infected at time t. Since the effective reproduction number is the spectral radius of T kη/γ , we recover [START_REF] Duijzer | The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect[END_REF]

as ρ(T kη/γ ) = ρ(T kη/γ ) = R e [k/γ](η) with k = k/γ.
We denote by g η the corresponding maximal equilibrium (so that g = g 1 ). In particular, we have: [START_REF] Grobler | Compactness conditions for integral operators in Banach function spaces[END_REF] F η (g η ) = 0.

We will denote by I the fraction of infected individuals at equilibrium. Since the probability for an individual with feature x to be infected in the stationary regime is g η (x) η(x), this fraction is given by the following formula:

(23)

I(η) = Ω g η η dµ = Ω g η (x) η(x) µ(dx).
We deduce from ( 21) and ( 22) that g η η = 0 µ-almost surely is equivalent to g η = 0. Applying the results of [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] to the kernel kη, we deduce that:

(24)

I(η) > 0 ⇐⇒ R e [k/γ](η) > 1.
We conclude this section with a result on the maximal equilibrium g which is a direct consequence of Proposition 8.2 proved in Section 8.1. This result completes what is known from [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]. Notice that, if R 0 > 1, then Property (ii) implies that the strategy 1g is critical. (i) For any h ∈ ∆, h = g if and only if

F (h) = 0 and R e (1 -h) ≤ 1. (ii) If g = 0, then R e (1 -g) = 1.
3. P r e l i m i n a ry t o p o l o g i c a l r e s u lt s 3.1. On the weak topology. We first recall briefly some properties we shall use frequently. We can see ∆ as a subset of L 1 , and consider the corresponding weak topology: a sequence (g n , n ∈ N) of elements of ∆ converges weakly to g if for all h ∈ L ∞ we have:

(25) lim n→∞ Ω hg n dµ = Ω hg dµ.
Notice that [START_REF] Hill | The critical vaccination fraction for heterogeneous epidemic models[END_REF] can easily be extended to any function h ∈ L q for any q ∈ (1, +∞); so that the weak-topology on ∆, seen as a subset of L p with 1/p + 1/q = 1, can be seen as the trace on ∆ of the weak topology on L p . The main advantage of this topology is the following compactness result.

Lemma 3.1 (Topological properties of ∆). We have that: (i) The set ∆ endowed with the weak topology is compact and sequentially compact.

(ii) A function from ∆ (endowed with the weak topology) to a metric space (endowed with its metric topology) is continuous if and only if it is sequentially continuous.

Proof. Let p ∈ (1, +∞), and consider the weak topology on ∆ as the trace on ∆ of the weak topology on L p . We first prove (i). Since L p is reflexive, by the Banach-Alaoglu theorem [6, Theorem V.4.2], its unit ball is weakly compact. The set ∆ is closed and convex, therefore it is weakly closed; see [6, Corollary V.1.5]. Thus, ∆ is weakly compact as a weakly closed subset of the weakly compact unit ball. By the Eberlein-Šmulian theorem [6, Theorem V.13.1], ∆ is also weakly sequentially compact.

We now prove (ii). A continuous function is sequentially continuous. Conversely, the inverse image of a closed set by a sequentially continuous function is sequentially closed. Besides, a sequentially closed subset of a sequentially compact set is sequentially compact. Using the Eberlein-Šmulian theorem, we deduce that the inverse images of closed sets are compact. In particular, they are closed which proves a sequentially continuous function is continuous.

3.2.

Invariance and continuity of the spectrum for compact operators. We recall a few facts on operators. Let (E, • ) be a Banach space. Let A ∈ L(E). We denote by A the adjoint of

A. A sequence (A n , n ∈ N) of elements of L(E) converges strongly to A ∈ L(E) if lim n→∞ A n x -Ax = 0 for all x ∈ E. Following [1], a set of operators A ⊂ L(E) is collectively compact if the set {Ax : A ∈ A , x ≤ 1} is relatively compact.
We collect some known results on the spectrum of to compact operators. Recall that the spectrum of a compact operator is finite or countable and has at most one accumulation point, which is 0. Furthermore, 0 belongs to the spectrum of compact operators in infinite dimension. Lemma 3.2. Let A, B be elements of L(E).

(i) If A, B and A -B are positive operators, then we have:

(26) ρ(A) ≥ ρ(B).
(ii) If A is compact, then we have:

Spec(A) = Spec(A ) (27)
Spec(AB) = Spec(BA) [START_REF] Kallenberg | Foundations of modern probability[END_REF] and in particular: [START_REF] Kato | Perturbation theory for linear operators[END_REF] ρ(AB) = ρ(BA).

(iii) Let (E , • ) be a Banach space such that E is continuously and densely embedded in E. Assume that A(E ) ⊂ E , and denote by A the restriction of A to E seen as an operator on E . If A and A are compact, then we have:

(30) Spec(A) = Spec(A ).
(iv) Let (A n , n ∈ N) be a collectively compact sequence which converges strongly to A. Then, we have

lim n→∞ Spec(A n ) = Spec(A) in (K , d H ), and lim n→ ρ(T n ) = ρ(T ).
Proof. Property (i) can be found in [START_REF] Marek | Frobenius theory of positive operators: comparison theorems and applications[END_REF]Theorem 4.2]. Equation [START_REF] Janson | Graphons, cut norm and distance, couplings and rearrangements. NYJM Monographs[END_REF] from Property (ii) can be deduced from the [START_REF] König | Eigenvalue distribution of compact operators[END_REF]Theorem page 20]. Using the [32, Proposition page 25], we get that Spec(AB) ∩ C * = Spec(BA) ∩ C * , and thus [START_REF] Kato | Perturbation theory for linear operators[END_REF]. As A is compact we get that AB and BA are compact, thus 0 belongs to their spectrum in infinite dimension. Whereas in finite dimension, as det(AB) = det(A)det(B) = det(BA) (where A and B denote also the matrix of the corresponding operator in a given base), we get that 0 belongs to the spectrum of AB if and only if it belongs to the spectrum of BA. This gives [START_REF] Kallenberg | Foundations of modern probability[END_REF].

Property (iii) follows from [23, Corollary 1 and Section 6]. We eventually check Property (iv). We deduce from [1, Theorems 4.8 and 4.16] (see also (d) and (e) in [2, Section 3]) that lim n→∞ Spec(T n ) = Spec(T ). Then use that the function rad is continuous to deduce the convergence of the spectral radius from the convergence of the spectra (see also (f) in [2, Section 3]). We generalize a continuity property on the spectral radius originally stated in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] by weakening the topology. Let us remark the proof holds even if k takes negative values.

F i r s t p ro p e rt i e s o f t h e f u n c t i o n s R

(i) R e (η 1 ) = R e (η 2 ) if η 1 = η 2 µ-almost surely. (ii) R e (0) = 0 and R e (1) = R 0 . (iii) R e (η 1 ) ≤ R e (η 2 ) if η 1 ≤ η 2 µ-almost surely. (iv) R e (λη) = λR e (η) for all λ ∈ [0, 1]. Proof. If η 1 = η 2 µ-
Proof. Let B denote the unit ball in L p , with p ∈ (1, +∞) from Assumption 1. Since the operator T k is compact, the set T k (B) is relatively compact. For all η ∈ ∆, set ηB = {ηg : g ∈ B}. As ηB ⊂ B, we deduce that T kη (B) = T k (ηB) ⊂ T k (B). This implies that the family (T kη , η ∈ ∆) is collectively compact.

Let (η n , n ∈ N) be a sequence in ∆ converging weakly to some η ∈ ∆. Let g ∈ L p . The weak convergence of η n to η implies that (T kηn (g), n ∈ N) converges µ-almost surely to T kη (g). Consider the function:

K(x) = Ω k(x, y) q µ(dy) 1/q
, which belongs to L p , thanks to [START_REF] Delmas | Optimal vaccination: various (counter) intuitive examples[END_REF]. Since for all x,

|T kηn (g)(x)| ≤ T k (|η n g|)(x) ≤ K(x) η n g p ≤ K(x) g p ,
we deduce, by dominated convergence, that the convergence holds also in L p : [START_REF] Kloeckner | Effective perturbation theory for simple isolated eigenvalues of linear operators[END_REF] lim n→∞ T kηn (g) -T kη (g) p = 0, so that T kηn converges strongly to T kη . Using Lemma 3.2 (iv) (with T n = T kηn and T = T kη ) on the continuity of the spectrum, we get that We give a stability property of the spectrum and spectral radius with respect to the kernel k. ). Let p ∈ (1, +∞). Let (k n , n ∈ N) and k be kernels on Ω with finite double norms on L p . If lim n→∞ k nk p,q = 0, then we have:

lim n→∞ Spec[k](η n ) = Spec[k](η).
(32) lim n→∞ sup η∈∆ R e [k n ](η) -R e [k](η) = 0 and lim n→∞ sup η∈∆ d H Spec[k n ](η), Spec[k](η) = 0.
Proof. We first prove that

lim n→∞ Spec[k n ](η n ) = Spec[k](η)
, where the sequence (η n , n ∈ N) is any sequence in ∆ which converges weakly to η ∈ ∆.

The operators A = {T k } ∪ {T kn : n ∈ N} are compact, and we deduce from (12) that:

lim n→∞ T kn -T k L p = 0.
The family A is then easily seen to be collectively compact. (Indeed, let

(y n = T k in (x n ), n ∈ N) be a sequence with x n ≤ 1, i n ∈ N ∪ {∞} and the convention k in = k if i n = ∞.
Up to taking a sub-sequence, we can assume that either the sequence

(i n , n ∈ N) is constant and (T k i 0 (x n ), n ∈ N) is convergent (as T k i 0 is compact) or that the sequence (i n , n ∈ N) is increasing and the sequence (T k (x n ), n ∈ N)
is convergent (as T k is compact) towards a limit, say y. In the former case, clearly the sequence (y n , n ∈ N) converges. In the latter case, we have:

T k in (x n ) -y p ≤ T k in -T k L p + T k (x n ) -y p ,
which readily implies that the sequence (y n , n ∈ N) converges towards y. This proves that the family A is collectively compact.) This implies, see [1, Proposition 4.1(2)] for details, that the family A = {T M η : , T ∈ A and η ∈ ∆} is collectively compact. We deduce that the sequence

(T n = T knηn = T kn M ηn , n ∈ N) of elements of A is collectively compact and that T = T kη = T k M η is compact. Let g ∈ L p .
We have:

T n (g) -T (g) p ≤ T kn -T k L p g p + T kηn (g) -T kη (g) p .
Using lim n→∞ T kn -T k L p = 0 and (31), we get that lim n→∞ T n (g)-T (g) p , thus (T n , n ∈ N) converges strongly to T . With Lemma 3.2 (iv), we get that

lim n→∞ Spec(T n ) = Spec(T ), that is lim n→∞ Spec[k n ](η n ) = Spec[k](η).
Then, as the function η → d H Spec[k n ](η), Spec[k](η) is continuous on the compact set ∆, thanks to Theorem 4.2, it reaches its maximum say at η n ∈ ∆ for n ∈ N. As ∆ is compact, consider a sub-sequence which converges weakly to a limit say η. Since

sup η∈∆ d H Spec[k n ](η), Spec[k](η) = d H Spec[k n ](η n ), Spec[k](η n ) ≤ d H Spec[k n ](η n ), Spec[k](η) + d H Spec[k](η n ), Spec[k](η) ,
using the continuity of Spec[k], we deduce that along this sub-sequence the right hand side converges to 0. Since this result holds for any converging sub-sequence, we get the second part of [START_REF] König | Eigenvalue distribution of compact operators[END_REF]. The first part then follows from the definition [START_REF] Delmas | Effective reproduction number: convexity, invariance and cordons sanitaires[END_REF] of R e as a composition, and the Lipschitz continuity of the function rad. Proof. According to [7, Proposition 2.10], the solution u t of the SIS model with vaccination ∂ t u t = F η (u t ) and initial condition u 0 = g is non-decreasing since F η (g) ≥ 0. According to [7, Proposition 2.13], the pointwise limit of u t is an equilibrium. As this limit is dominated by the maximal equilibrium g η and since u t is non-decreasing, this proves that g ≤ g η .

We may now state the main properties of the function I. Proof. If η 1 = η 2 µ-almost surely, then the operators T kη 1 and T kη 2 are equal. Thus, the equilibria g η 1 and g η 2 are also equal which in turns implies that I(η 1 ) = I(η 2 ). Point (ii) is already stated in Equation [START_REF] Hethcote | Gonorrhea Transmission Dynamics and Control[END_REF].

To prove the monotonicity (Point (iii)), consider

η 1 ≤ η 2 . Since T kη 1 ≤ T kη 2 , we get F η 1 (g) ≤ F η 2 (g) for all g ∈ ∆.
In particular, taking g = g η 1 and using [START_REF] Grobler | Compactness conditions for integral operators in Banach function spaces[END_REF], we get F η 2 (g η 1 ) ≥ 0. By Lemma 4.4 this implies g η 1 ≤ g η 2 . To sum up, we get:

(33) η 1 ≤ η 2 =⇒ g η 1 ≤ g η 2 .
This readily implies that I(η 1 ) = Ω g η 1 η 1 dµ ≤ Ω g η 2 η 2 dµ = I(η 2 ). We conclude using Point (i).

We now consider Point (iv). Since λ ∈ [0, 1], we deduce from (33) that g λη ≤ g η . This implies that I(λη) = Ω g λη λη dµ ≤ λ Ω g η η dµ = λI(η).

The proof of the following continuity results are both postponed to Section 8.1. n k n , n ∈ N) have finite double norm in L p and that lim n→∞ k nk p ,q = 0. Then we have:

(34) lim n→∞ sup η∈∆ I[k n , γ n ](η) -I[k, γ](η) = 0.

5.

Pa r e t o a n d a n t i -Pa r e t o f ro n t i e r s 5.1. The setting. To any vaccination strategy η ∈ ∆, we associate a cost and a loss.

• The cost function. The cost C(η) measures all the costs of the vaccination strategy (production and diffusion). The cost is expected to be a decreasing function of η, since η encodes the non-vaccinated population. Since doing nothing costs nothing, we also expect C(1) = 0, see Assumptions 3 below. We shall also consider natural hypothesis on C, see Assumptions 4 and 6. A simple cost model is the affine cost given by:

(35) C aff (η) = Ω (1 -η(x)) c aff (x) µ(dx),
where c aff (x) is the cost of vaccination of population of feature x, with c aff ∈ L 1 positive.

The particular case c aff = 1 is the uniform cost C = C uni :

(36) C uni (η) = Ω (1 -η) dµ.
The real cost of the vaccination may be a more complicated function ψ(C aff (η)) of the affine cost, for example if the marginal cost of producing a vaccine depends on the quantity already produced. However, as long as ψ is strictly increasing, this will not affect the optimal strategies. • The loss function. The loss L(η) measures the (non)-efficiency of the vaccination strategy η. Different choices are possible here. We prove in this section general results that only depend on a few natural hypothesis for L; see Assumptions 3, 5 and 7. These hypothesis are in particular satisfied if the loss is the effective reproduction number R e (kernel and SIS models), or the asymptotic proportion of infected individuals I (SIS model); more precisely see Lemmas 5.6, 5.11 and 5.12. We shall consider cost and loss functions with some regularities. Definition 5.1. We say that a real-valued function H defined on ∆ endowed with the weak topology is:

• Continuous: if H is continuous with respect to the weak topology on ∆.

• Non-decreasing:

if for any η 1 , η 2 ∈ ∆ such that η 1 ≤ η 2 , we have H(η 1 ) ≤ H(η 2 ). • Decreasing: if for any η 1 , η 2 ∈ ∆ such that η 1 ≤ η 2 and Ω η 1 dµ < Ω η 2 dµ, we have H(η 1 ) > H(η 2 ). • Sub-homogeneous: if H(λη) ≤ λH(η) for all η ∈ ∆ and λ ∈ [0, 1].
The definition of non-increasing function and increasing function are similar.

Assumption 3 (On the cost function and loss function). The loss function L : ∆ → R is non-decreasing and continuous with L(0) = 0. The cost function C : ∆ → R is non-increasing and continuous with C(1) = 0. We also have:

max := max ∆ L > 0 and c max := max ∆ C > 0.
Assumption 3 will always hold. In particular, the loss and the cost functions are non-negative and non-constant.

We will consider the multi-objective minimization and maximization problems: Multi-objective problems are in a sense ill-defined because in most cases, it is impossible to find a single solution that would be optimal to all objectives simultaneously. Hence, we recall the concept of Pareto optimality. Since the minimization problem is crucial for vaccination, we shall define Pareto optimality for the bi-objective minimization problem. A strategy η ∈ ∆ is said to be Pareto optimal for the minimization problem in [START_REF] Newburgh | The variation of spectra[END_REF] if any improvement of one objective leads to a deterioration of the other, for η ∈ ∆:

(38) C(η) < C(η ) =⇒ L(η) > L(η ) and L(η) < L(η ) =⇒ C(η) > C(η ).
Similarly, a strategy η ∈ ∆ is anti-Pareto optimal if it is Pareto optimal for the bi-objective maximization problem in [START_REF] Newburgh | The variation of spectra[END_REF]. Intuitively, the "best" vaccination strategies are the Pareto optima and the "worst" vaccination strategies are the anti-Pareto optima.

We define the feasible region as all possible outcomes:

F = {(C(η), L(η)), η ∈ ∆}.
Then, we first consider the minimization problem for the "best" strategies. The set of Pareto optimal strategies will be denoted by P L , and the Pareto frontier is defined as the set of Pareto optimal outcomes:

F L = {(C(η), L(η)) : η Pareto optimal}.
We consider the minimization problems related to the "best" vaccination strategies, with ∈ [0, max ] and c ∈ [0, c max ]:

Minimize: L(η) (39a) subject to: η ∈ ∆, C(η) ≤ c, (39b)
as well as

Minimize: C(η) (40a) subject to: η ∈ ∆, L(η) ≤ . (40b)
We denote the values of Problems ( 39) and ( 40) by:

L (c) = inf{L(η) : η ∈ ∆ and C(η) ≤ c} for c ∈ [0, c max ], C ( ) = inf{C(η) : η ∈ ∆ and L(η) ≤ } for ∈ [0, max ].
We now consider the maximization problem related to the "worst" vaccination strategies, with ∈ [0, max ] and c ∈ [0, c max ]:

Maximize: L(η) (41a) subject to: η ∈ ∆, C(η) ≥ c, (41b)
as well as

Maximize:

C(η) (42a)

subject to: η ∈ ∆, L(η) ≥ . (42b)
We denote the values of Problems ( 41) and ( 42) by:

L (c) = sup{L(η) : η ∈ ∆ and C(η) ≥ c} for c ∈ [0, c max ], C ( ) = sup{C(η) : η ∈ ∆ and L(η) ≥ } for ∈ [0, max ].
We denote by P Anti L the set of anti-Pareto optimal strategies, and by F Anti L its frontier:

F Anti L = {(C(η), L(η)) : η anti-Pareto optimal}.
If necessary, we may write C ,L and C ,L to stress the dependence of the function C and C in the loss function L.

Under Assumption 3, as the loss and the cost functions are continuous on the compact set ∆, the infima in the definitions of the value functions C and L are minima; and the suprema in the definition of the value functions C and L are maxima. Since ∆ in endowed with the weak topology, we will consider the set of Pareto and anti-Pareto optimal vaccination modulo µ-almost sure equality.

See Figure 2 Outline of the section. It turns out that the anti-Pareto optimization problem can be recast as a Pareto optimization problem by changing signs and exchanging the cost and loss functions. In order to make use of this property for the kernel and SIS models, we study the Pareto problem under assumptions on the cost that are general enough to cover the choices C uni and -L, and assumptions on the loss that cover the choices R e , I and -C uni .

The main result of this section states that all the solutions of the optimization Problems [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] or [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF] are Pareto optimal, and gives a description of the Pareto frontier F L as a graph in Section 5.2, and similarly for the anti-Pareto frontier in Section 5.3. Surprisingly, the problem is not completely symmetric, compare Lemma 5.6 used for the Pareto frontier and Lemmas 5.11 and 5.12 used for the anti-Pareto frontier. In the latter lemmas, notice the kernel considered is quasi-irreducible, whereas this condition is not needed for the Pareto frontier. 5.2. On the Pareto frontier. We first check that Problems [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] and ( 40) have solutions.

Proposition 5.3 (Optimal solutions for fixed cost or fixed loss). Suppose that Assumption 3 holds. For any cost c ∈ [0, c max ], there exists a minimizer of the loss under the cost constraint C(•) ≤ c, that is, a solution to Problem [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF]. Similarly, for any loss ∈ [0, max ], there exists a minimizer of the cost under the loss constraint L(•) ≤ , that is a solution to Problem [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF].

Proof. Let c ∈ [0, c max ].
The set {η ∈ ∆ : C(η) ≤ c} is non-empty as it contains 11 since C(1) = 0. It is also compact as C is continuous on the compact set ∆ (for the weak topology). Therefore, since the loss function L is continuous (for the weak topology), we get that L restricted to this compact set reaches its minimum. Thus, Problem (39) has a solution. The proof is similar for the existence of a solution to Problem [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF].

We start by a general result concerning the links between the three problems. Proposition 5.4 (Single-objective and bi-objective problems). Suppose Assumption 3 holds.

(i) If η is Pareto optimal, then η is a solution of (39) for the cost c = C(η ), and a solution of (40) for the loss = L(η ). Conversely, if η is a solution to both problems (39) and (40) for some values c and , then η is Pareto optimal. (ii) The Pareto frontier is the intersection of the graphs of C and L : Proof. Let us prove (i). If η is Pareto optimal, then for any strategy η, if C(η) ≤ C(η ) then L(η) ≥ L(η ) by taking the contraposition in [START_REF] Poghotanyan | Constrained minimization problems for the reproduction number in meta-population models[END_REF], and η is indeed a solution of Problem [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] with c = C(η ). Similarly η is a solution of Problem [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF].

F L = {(c, ) ∈ [0, c max ] × [0,
For the converse statement, let η be a solution of [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] for some c and of (40) for some . It is also a solution of (39) with c = C(η ). In particular, we get that for η ∈ ∆, L(η) < L(η ) implies that C(η) > c = C(η ), which is the second part of [START_REF] Poghotanyan | Constrained minimization problems for the reproduction number in meta-population models[END_REF]. Similarly, use that η is a solution to [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF], to get that the first part of (38) also holds. Thus the strategy η is Pareto optimal.

To prove Point (ii), we first prove that F L is a subset of {(c, ) : c = C ( ) and = L (c)}. A point in F L may be written as (C(η ), L(η )) for some Pareto optimal strategy η . By Point (i), η solves Problem [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] for the cost C(η ), so L (C(η )) = L(η ). Similarly, we have C (L(η )) = C(η ), as claimed. We now prove the reverse inclusion. Assume that c = C ( ) and = L (c), and consider η a solution of Problem [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF] for the loss : L(η) ≤ and C(η) = C ( ) = c. Then η is admissible for Problem [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] with cost c = C ( ), so L(η) ≥ L (C ( )) = L (c) = . Therefore, we get L(η) = L (c), and η is also a solution of Problem [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF]. By Point (i), η is Pareto optimal, so (C(η), L(η)) = (c, ) ∈ F L , and the reverse inclusion is proved.

Finally we prove Point (iii). We have C (0) = min{C(η) : η ∈ ∆ and L(η) = 0} ∈ [0, c max ]. Let η ∈ ∆ such that L(η) = 0 and C(η) = C (0). We deduce that L (C (0)) ≤ L(η) = 0 and thus L (C (0)) = 0 as L is non-negative. We deduce from (ii) that (C (0), 0) belongs to F L . Since C is non-increasing, we also get that C = 0 on [C (0), c max ]. The other properties of (iii) are proved similarly.

The next two hypotheses on C and L will imply that the Pareto frontier is connected. Under these hypotheses, the picture becomes much nicer, see Figure 2( d ), where the only flat parts of the graphs of C and L occur at zero cost or zero loss. Proposition 5.5. Under Assumption 3 and 4 the following properties hold:

(i) The optimal cost C is decreasing on [0, L (0)].

(ii) If η solves Problem [START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF] for the loss ∈ [0, L (0)], then L(η) = (that is, the constraint is binding). Moreover η is Pareto optimal, and:

(43) L (C ( )) = .
(iii) The Pareto frontier is the graph of C :

(44) F L = {(C ( ), ) : ∈ [0, L (0)]}.
Similarly, under Assumptions 3 and 5, the following properties hold: (iv) The optimal loss L is decreasing on

F L = {(c, L (c)) : c ∈ [0, C (0)]}.
Finally, if Assumptions 4 and 5 hold, then L is a continuous decreasing bijection of [0, C (0)] onto [0, L (0)] and C is the inverse bijection, and the Pareto frontier is compact and connected. Proof. We prove (i). Let 0 ≤ < ≤ L (0), and let η be a solution of Problem ( 40 We now prove (ii). If the inequality in [START_REF] Van Mieghem | Decreasing the spectral radius of a graph by link removals[END_REF] was strict, that is L(η ) < , then we would get a contradiction as C(η ) ≥ C (L(η )) > C ( ) = C(η ). Therefore any solution η of (40) satisfies L(η ) = , and in particular C (L(η )) = C ( ) = C(η ). This implies in turn that η also solves [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF]: if η satisfies L(η) < L(η ), then using the definition of C , the fact that it decreases, and the definition of η , we get:

C(η) ≥ C (L(η)) > C (L(η )) = C(η ).
By contraposition, we have L(η) ≥ L(η ) for any η such that C(η) ≤ C(η ), proving that η is also a solution of [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] with c = C(η * ). By Point (i) of Proposition 5.4, η is Pareto optimal. Therefore (C(η ), L(η )) = (C ( ), ) belongs to the Pareto frontier. Using Point (ii) of Proposition 5.4, we deduce that = L (C ( )).

To prove Point (iii), note that Equation [START_REF] Smith | Concepts of herd protection and immunity[END_REF] shows that, if c = C ( ) for ∈ [0, L (0)], then = L (c). Use Point (ii) and (iii) of Proposition 5.4, to get that

F L = {(c, ) : c = C ( ), ∈ [0, L (0)]}.
The claims (iv), (v) and (vi) are proved in the same way, exchanging the roles of L and C.

To conclude the proof, it remains to check that C and L are continuous under Assumptions 3, 4 and 5. We deduce from Point (ii) and Proposition 5.3 that [0, L (0)] is in the range of L .

Since L is decreasing, thanks to Point (iv) and L (C (0)) = 0, see Proposition 5.4 (iii), we get that L is continuous and decreasing on [0, L (0)], and thus one-to-one from [0, C (0)] onto [0, L (0)]. Then use [START_REF] Smith | Concepts of herd protection and immunity[END_REF] to get that C is its inverse bijection. The continuity of L and [START_REF] Stevanović | On spectral radius and energy of complete multipartite graphs[END_REF] implies that F L is compact and connected.

Finally, let us check that Assumptions 4 and 5 hold under very simple assumptions, which are in particular satisfied by the cost functions C uni and C aff and the loss functions R e and I (recall from Propositions 4.1 and 4.5 that R e and I are sub-homogeneous). Proof. Let η ∈ ∆. If C has a local minimum at η, then, as C is non-increasing, for ε > 0 small enough, we get that

C(η) ≥ C(η + ε(1 -η)) ≥ C(η). If C is decreasing, this is only possible if η = 1 almost surely, so that η is a global minimum of C. This also gives L (0) = max . Similarly if L has a local minimum at η, then for ε > 0 small enough L(η) ≤ L((1 -ε)η) ≤ (1 -ε)L(η), so L(η) = 0 and η is a global minimum of L.
Corollary 5.7. Suppose that Assumptions 3, 4 and 5 hold. The set of Pareto optimal strategies P L is compact (for the weak topology).

Proof. Since L is continuous thanks to Proposition 5.5, we deduce that F L , which is given by [START_REF] Stevanović | On spectral radius and energy of complete multipartite graphs[END_REF], is compact and thus closed. Since P L = f -1 (F L ), where the function f = (C, L) defined on ∆ is continuous, we deduce that P L is closed and thus compact as ∆ is compact. (i) If η is anti-Pareto optimal, then η is a solution of (41) for the cost c = C(η ), and a solution of (42) for the loss = L(η ). Conversely, if η is a solution to both problems [START_REF] Schaefer | Banach lattices and positive operators[END_REF] and [START_REF] Smith | Prospects for the control of infectious disease[END_REF] for some values c and , then η is anti-Pareto optimal. (ii) The anti-Pareto frontier is the intersection of the graphs of C and L :

F Anti L = {(c, ) ∈ [0, c max ] × [0, max ] : c = C ( ) and = L (c)}.
(48) F Anti L = {(c, L (c)) : c ∈ [C ( max ), c max ]}.
Finally, if Assumptions 3, 6 and 7 hold, then L is a continuous decreasing bijection of [C ( max ), c max ] onto [L (c max ), max ], C is the inverse bijection, and the anti-Pareto frontier is compact and connected. Furthermore, the set of anti-Pareto optimal strategies P Anti L is compact (for the weak topology).

The following result is similar to the first part of Lemma 5.6.

Lemma 5.10. Suppose Assumption 3 holds. If the cost function C is decreasing, then Assumption 6 holds and L (c max ) = 0.

Proof. Let η ∈ ∆ and ε ∈ (0, 1). Since C is decreasing, C((1ε)η) ≥ C(η), with equality if and only if η = 0 µ-almost surely. Therefore the only local maximum of C is η = 0, and it is a global maximum. Since C(η) = c max implies that η = 0 µ-almost surely, we also get that L (c max ) = L(0) = 0.

5.3.2.

The particular case of the kernel and SIS models. We show that, under an irreducibility hypothesis on the kernel, Assumption 7 holds for the loss functions R e and J. The reducible case is more delicate and it is studied in more details in [START_REF] Delmas | Effective reproduction number: convexity, invariance and cordons sanitaires[END_REF] for the loss function L = R e ; in particular Assumption 7 may not hold and the anti-Pareto frontier may not be connected.

Let us recall some notation. Let k be a kernel with finite double norm. For A, B ∈ F , we write A ⊂ B a.s. if µ(B ∩ A c ) = 0 and A = B a.s. if A ⊂ B a.s. and B ⊂ A a.s. For A, B ∈ F , x ∈ Ω and a kernel k, we simply write k(x, A) = A k(x, y) µ(dy), k(B, x) = B k(z, x) µ(dz) and: A,µ) is an invariant closed subspace for T k , seen as an operator on L p (Ω, µ).)

k(B, A) = B×A k(z, y) µ(dz)µ(dy). A set A ⊂ F is k-invariant if k(A c , A) = 0. (Notice that if A is k-invariant, then L p (
A kernel k is irreducible (or connected) if any k-invariant set A is such that a.s. A = ∅ or a.s. A = Ω. Define {k ≡ 0} as {x ∈ Ω : k(x, Ω) + k(Ω, x) = 0}, so that k(A, Ω) + k(Ω, A) = 0 implies that a.s. A ⊂ {k ≡ 0}. A kernel k is quasi-irreducible if the restriction of k to {k ≡ 0} c is irreducible, that is if any k-invariant set A is such that A ⊂ {k ≡ 0} a.
C ( max ) = C(1 {k≡0} c ) (which is 0 if k is irreducible).
Proof. The quasi-irreducible case can easily be deduced from the irreducible case, so we assume that k is irreducible. In particular, we have k(Ω, y) > 0 almost surely. Let η ∈ ∆ be a local maximum of R e on ∆; we want to show that it is also a global maximum.

Suppose first that inf η > 0. Then kη is irreducible with finite double norm. According [41, Theorem V.6.6 and Example V.6.5.b], the eigenspace of T kη associated to R e (η) is one-dimensional and it is spanned by a vector v d such that v d > 0 almost surely, and the corresponding left eigenvector associated to R e (η), say v g , can be chosen such that v g , v d = 1 and v g > 0 almost surely. According to [START_REF] Kloeckner | Effective perturbation theory for simple isolated eigenvalues of linear operators[END_REF]Theorem 2.6], applied to L 0 = T kη and L = T k(η+ε(1-η)) , we have, using that L 0 -L = O(ε) thanks to [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF]:

R e (η + ε(1 -η)) = R e (η) + ε v g , T k(1-η) v d + O(ε 2 ).
Since R e has a local maximum at η, the first order term on the right hand side vanishes, so v g (x)k(x, y)(1η(y))v d (y) = 0 for µ almost all x and y. Since v g and v d are positive almost surely and k is irreducible, we get that k(Ω, y)(1η(y)) = 0 almost surely and thus η(y) = 1 almost surely. Therefore η = 1, which is a global maximum for R e .

Finally, suppose that inf η = 0. Let O be an open subset of ∆ on which R e ≤ R e (η) and with η ∈ O. For ε > 0 small enough, the strategy

η ε = η + ε(1 -η) belongs to O and satisfies R e (η) ≤ R e (η ε ) ≤ R e (η)
(where the first inequality comes from the fact that R e is non-decreasing). Therefore η ε is a local maximum, and thus η ε = 1 almost surely. This readily implies that η = 1 almost surely.

We deduce that if η is a local maximum, then η = 1 almost surely. Thus η is a global maximum and C ( max ) = C(1) = 0. This ends the proof.

( max ) = C(1 {k≡0} c ) (which is 0 if k is irreducible).
Proof. The quasi-irreducible case can easily be deduced from the irreducible case, so we assume that k is irreducible.

Set k = k/γ. Suppose that I has a local maximum at some η ∈ ∆. For ε ∈ (0, 1), the kernel kη ε , with η ε = η + ε(1η), is irreducible (with finite double norm) since k is irreducible and γ is positive and bounded. We have that for ε > 0 small enough:

I(η) ≥ I(η ε ) = Ω g ηε η ε dµ ≥ Ω g ηε η dµ ≥ Ω g η η dµ = I(η),
where we used that η ≤ η ε and 0 ≤ g η ≤ g ηε , see [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF]. Therefore all these quantities are equal. Since the equilibrium g ηε is µ-a.e. positive thanks to [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]Remark 4.11] as kη ε is irreducible, we must have η ε = η a.s, which is only possible if η = 1 almost surely.

Since I(1) > I(η) for any η = 1, we also get C ( max ) = 0, with max = I(1) = Ω g dµ.

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We prove results concerning the feasible region, the stability of the Pareto frontier and its geometry.

6.1.

No holes in the feasible region. We check there is no hole in the feasible region. Proposition 6.1. Suppose that Assumption 3 holds. The feasible region F is compact, path connected, and its complement is connected in R 2 . It is the whole region between the graphs of the one-dimensional value functions:

F = {(c, ) ∈ R 2 : 0 ≤ c ≤ c max , L (c) ≤ ≤ L (c)} = {(c, ) ∈ R 2 : 0 ≤ ≤ max , C ( ) ≤ c ≤ C ( )}.
Proof. The region F is compact and path-connected as a continuous image by (C, L) of the compact, path-connected set ∆.

By symmetry, it is enough to prove that F is equal to

F 1 = {(c, ) ∈ R 2 : 0 ≤ c ≤ c max , L (c) ≤ ≤ L (c)}. Let (c, ) ∈ F and η ∈ ∆ be such that (c, ) = (C(η), L(η))
. By definition of L and L , we have:

L (c) = L (C(η)) ≤ L(η) ≤ L (C(η)) = L (c). We deduce that (c, ) ∈ F 1 .
Let us now prove that F 1 ⊂ F. Let us first consider a point of the form (c, L (c)), where 0 ≤ c ≤ c max . By definition, there exists η such that C(η) ≤ c and L(η) = L (c). Let η t = tη. The map t → C(η t ) is continuous from [0, 1] to [C(η), c max ], and c ∈ [C(η), c max ], so there exists s such that C(η s ) = c. Since L is non-decreasing, L(η s ) ≤ L(η). By definition of L (c), L(η s ) ≥ L (c). Therefore (c, L (c)) = (C(η s ), L(η s )) belongs to F. Similarly the graphs of C , C and L are also included in F.

So, it is enough to check that, if A = (c, ) is in F 1 , with c ∈ (0, c max ) and ∈ (L (c), L (c)), then A belongs to F. We shall assume that A ∈ F and derive a contradiction by building a loop in F that encloses A and which can be continuously contracted into a point in F.

Since L (c) < < L (c), there exist η SO and η NE such that:

C(η SO ) ≤ c, L(η SO ) < , C(η NE ) ≥ c and L(η NE ) > .
We concatenate the four paths defined for u ∈ [0, 1]:

u → uη SO , u → (1 -u)η SO + u, u → (1 -u) + uη NE and u → (1 -u)η NE ,
to obtain a continuous loop (η t , t ∈ [0, 4]) from [0, 4] to ∆, such that:

η 0 = η 4 = 0, η 1 = η SO , η 2 = 1 and η 3 = η NE .
We now define a continuous family of loops (γ s , s

∈ [0, 1]) in R 2 by γ s (t) = (C(sη t ), L(sη t ), t ∈ [0, 4]).
By definition, for all s ∈ [0, 1], γ s is a continuous loop in F . Since A = (c, ) / ∈ F, the loops γ s do not contain A, so the winding number W (γ s , A) is well-defined (see for example [START_REF] Huggett | A topological aperitif[END_REF]Definition 6.1]). As A ∈ F, we get that γ s is a continuous deformation in R 2 \ {A} from γ 1 to γ 0 . Thanks to [START_REF] Huggett | A topological aperitif[END_REF]Theorem 6.5], this implies that W (γ s , A) does not depend on s ∈ [0, 1].

For s = 0, the loop degenerates to the single point (C(0), 0) so the winding number is 0. For s = 1, let us check that the winding number is 1, which will provide the contradiction. To do this, we compare γ 1 with a simpler loop δ defined by:

δ(0) = δ(4) = (c max , 0), δ(1) = (0, 0), δ(2) = (0, max ) and δ(3) = (c max , max ),
and by linear interpolation for non integer values of t: in other words, δ runs around the perimeter of the axis-aligned rectangle with corners (0, 0) and (c max , max ). Clearly, we have

W (δ, A) = 1.
Let M t , N t denote γ 1 (t) and δ(t) respectively. For t ∈ [0, 1], we have N t = ((1t)c max , 0), so the second coordinate of --→ AN t is non-positive. On the other hand L(tη SO ) ≤ L(η SO ) < , so the second coordinate of --→ AM t is negative. Therefore the two vectors --→ AN t and --→ AM t cannot point in opposite directions. Similar considerations for the other values of t ∈ [START_REF] Anselone | Collectively compact operator approximation theory and applications to integral equations[END_REF][START_REF] Britton | A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2[END_REF] show that --→ AN t and --→ AM t never point in opposite directions. By [26, Theorem 6.1], the winding numbers W (γ 1 , A) and W (δ, A) are equal, and thus W (γ 1 , A) = 1.

This gives that A ∈ F by contradiction, and thus F 1 ⊂ F.

Finally, it is easy to check that F 1 has a connected complement, because F 1 is bounded, and all the points in F c 1 can reach infinity by a straight line: for example, if > L (c), then the half-line

{(c, ), ≥ } is in F c 1 . 0 1/2 3/4 1 0 1/2 3/4 1 x y ( a ) Grayplot of the kernel k ε , with Ω = [0, 1]
and µ the Lebesgue measure (k ε is equal to the constant κ > 0 on the black zone and to ε on the gray zone, with ε > 0 small). Let η ∈ ∆ be the weak limit of a sequence (η n , n ∈ N) of Pareto optima, that is η n ∈ P L (n) for all n ∈ N. If C(η) ≤ C (0), then we have η ∈ P L . Remark 6.3 (On the continuity of the Pareto Frontier). It might happen that some elements of P L are not weak limit of sequence of elements of P L (n) ; see [START_REF] Delmas | Optimal vaccination for a 2 sub-populations SIS model[END_REF] for such discontinuity. It might also happen that a sequence (η n , n ∈ N) such that η n ∈ P L (n) and L (n) (η n ) > 0 converges to some η that does not belong to P L if L(η) = 0. In particular, in this case, C ,L (n) (0) does not converge to C ,L (0), where C ,L is the value function C associated to the loss L . This situation is represented in Figure 3. In Figure 3( a ), we have plotted a perturbation

k ε = k + ε n∈N 1 In×In of the multipartite kernel k defined in Example 1.7 for ε > 0 small. According to Proposition 4.3, R e [k ε ] converges uniformly to R e [k]
when ε vanishes. However, the Pareto optimal strategies for k ε that cost more than 1/2 do not converge to some Pareto optimal strategies for k. This can be seen in Figure 3( b ), where the Pareto frontier of k ε (in blue) corresponding to costs larger than 1/2 does not have a counterpart in the Pareto frontier of k (in red). Since C is affine, we have:

C(θη) = θC(η) + (1 -θ)c max ≤ θc + (1 -θ)c max .
Therefore, θη is admissible for Problem [START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] with cost constraint C(•) ≤ θc + (1θ)c max . This implies that L (θc + (1θ)c max ) ≤ L(θη) ≤ θL (c), thanks to the sub-homogeneity of the loss function L.

In some case, we shall prove that the considered loss function is convex (which in turn implies Assumption 5). In this case, choosing a convex cost function implies that Assumption 4 holds and the Pareto frontier is convex. A similar result holds in the concave case. We provide a short proof of this result. Proof. Let 0 , 1 ∈ [0, max ]. By Proposition 5.3, there exist η 0 , η 1 such that L(η i ) ≤ i and

C(η i ) = C ( i ) for i ∈ {0, 1}. For θ ∈ [0, 1], let = (1 -θ) 0 + θ 1 .
Since C and L are assumed to be convex, η = (1θ)η 0 + θη 1 satisfies:

C(η) ≤ (1 -θ)C ( 0 ) + θC ( 1 ) and L(η) ≤ (1 -θ) 0 + θ 1 . Therefore, we get that C ((1 -θ) 0 + θ 1 ) ≤ C(η) ≤ (1 -θ)C ( 0 ) + θC ( 1 )
, and C is convex. The proof of the convexity of L is similar. The concave case is also similar.

E q u i va l e n c e o f m o d e l s b y c o u p l i n g

Even if in full generality, the cost function could also be treated as a parameter, we shall for simplicity consider only the uniform cost C uni given by [START_REF] Miettinen | Nonlinear multiobjective optimization[END_REF] in this section. (The interested reader can use Remark 5.2 for a first generalization to the affine cost function given by (35).) 7.1. Motivation. The aim of this section is to provide examples of different set of parameters for which two kernel or SIS models are "equivalent", in the intuitive sense that their Pareto frontiers are the same (as subsets of R 2 + ), and it is possible to map nicely the Pareto optima from one model to the another. In Section 7.4, we present an example where discrete models can be represented as a continuous models and an example based on measure preserving transformation in the spirit of the graphon theory. We shall consider the two families of models:

• where for the last equality the left hand-side refers to the SIS model and the right hand-side refers to the kernel model (where Assumption 1 holds as a consequence of Assumption 2). Using [START_REF] Kato | Perturbation theory for linear operators[END_REF] if inf γ > 0 (see [START_REF] Delmas | Effective reproduction number: convexity, invariance and cordons sanitaires[END_REF]Section 3] for details and more general results), then we also have R e [(Ω, F , µ), k/γ] = R e [(Ω, F , µ), γ -1 k].

7.2. On measurability. Let us recall some well-known facts on measurability. Let (E, E ) and (E , E ) be two measurable spaces. If E = R, then we take E = B(R) the Borel σ-field. Let f be a function from E to E . We denote by σ(f ) = {f -1 (A) : A ∈ E } the σ-field generated by f . In particular f is measurable from (E, E ) to (E , E ) if and only if σ(f ) ⊂ E . Let ϕ be a measurable function from (E, E ) to (E , E ). For ν a measure on (E, E ), we write ϕ # ν for the for the push-forward measure on (E , E ) of the measure ν by the function ϕ (that is ϕ # ν(A) = ν(ϕ -1 (A)) for all A ∈ E ). By definition of ϕ # ν, for a non-negative measurable function g defined from (E , E ) to (R, B(R)), we have:

(49) E g dϕ # ν = E g • ϕ dν.
Let f be a measurable function from (E, E ) to (R, B(R)). We recall that:

(50) σ(f ) ⊂ σ(ϕ) =⇒ f = g • ϕ,
for some measurable function g from (E , E ) to (R, B(R)).

The random variables we consider are defined on a probability space, say (Ω 0 , F 0 , P).

7.3. Coupled models. We refer the reader to [START_REF] Janson | Graphons, cut norm and distance, couplings and rearrangements. NYJM Monographs[END_REF] for a similar development in the graphon setting. We first define coupled models in the next definition and state in Proposition 7.3 that coupled models have related (anti-)Pareto optima and the same (anti-)Pareto frontiers.

In the kernel model, we consider the models A measure π on (Ω 1 × Ω 2 , F 1 ⊗ F 2 ) is a coupling if its marginals are µ 1 and µ 2 .

Param i = [(Ω i , F i , µ i ), k i ] for i ∈ {1, 2},
Definition 7.1 (Coupled models). The models Param 1 and Param 2 are coupled if there exists two independent Ω 1 × Ω 2 -valued random vectors (X 1 , X 2 ) and (Y 1 , Y 2 ) (defined on a probability space (Ω 0 , F 0 , P)) with the same distribution given by a coupling ( i.e. X i and Y i have distribution µ i ) such that, P-almost surely:

Kernel model: k 1 (X 1 , Y 1 ) = k 2 (X 2 , Y 2 ), SIS model: γ 1 (X 1 ) = γ 2 (X 2 ) and k 1 (X 1 , Y 1 ) = k 2 (X 2 , Y 2 ).
In this case, two real-valued measurable functions v 1 and v 2 defined respectively on Ω 1 and Ω 2 are coupled (through V ) if there exists a real-valued σ(X 1 , X 2 )-measurable integrable random variable V such that P-almost surely:

E [V | X i ] = v i (X i ) for i ∈ {1, 2}.
Remark 7.2. We keep notation from Definition 7.1 (i) Since V is real-valued and σ(X 1 , X 2 )-measurable, we deduce from (50) that there exits a measurable function v defined on Ω 1 ×Ω 2 such that V = v(X 1 , X 2 ), thus the following equality holds P-almost surely:

E [v(Y 1 , Y 2 )| Y i ] = v i (Y i ) for i ∈ {1, 2}. (ii) If W is a real-valued integrable σ(X 1 )∩σ(X 2 )-measurable random variable, then setting v i (X i ) = E [W |X i ] = W , the equality v 1 (X 1 ) = v 2 (X 2 )
holds almost surely, and we get that v 1 and v 2 are coupled (through W ).

measure µ d into a continuous part, or, conversely, merge all points that behave similarly for k c and γ c into an atom, without altering the Pareto frontier.

Example 7.5. We consider the so called stochastic block model, with 2 populations for simplicity, in the setting of the SIS model, and give in this elementary case the corresponding discrete and continuous models. Then, we explicit the relation with the formalism of the same model developed in [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF] by Lajmanovich and Yorke.

The discrete SIS model is defined on Ω d = {1, 2} with the probability measure µ d defined by µ d ({1}) = 1µ d ({2}) = p with p ∈ (0, 1), and a kernel k d and recovery function γ d given by the matrix and the vector:

k d = k 11 k 12 k 21 k 22 and γ d = γ 1 γ 2 .
Notice p is the relative size of population 1. The corresponding discrete model is

Param d = [({1, 2}, F d , µ d ), k d , γ d ]; see Figure 4( b ).
The continuous model is defined on the state space Ω c = [0, 1) is endowed with its Borel σfield, F c , and the Lebesgue measure µ c = Leb. The segment [0, 1) is partitioned into two intervals B 1 = [0, p) and B 2 = [p, 1), the transmission kernel k c and recovery rate γ c are given by: k c (x, y) = k ij and γ c (x) = γ i for x ∈ B i , y ∈ B j , and i, j ∈ {1, 2}.

The corresponding continuous model is Param c = [([0, 1), F c , Leb), k c , γ c ]; see Figure 4( a ). By the general discussion above, the discrete and continuous models are coupled, and in particular they have the same Pareto and anti-Pareto frontiers. Furthermore, in this simple example, it is easily checked that a discrete vaccination η d = (η 1 , η 2 ) and a continuous vaccination η c = (η c (x), x ∈ [0, 1)) are coupled if and only if there exists a function η defined on Ω c × Ω d = [0, 1) × {1, 2} such that:

   η i = 1 Leb(B i ) B i η(x, i) dx, i ∈ {1, 2}, η c (x) = η(x, 1)1 B 1 (x) + η(x, 2)1 B 2 (x), Leb-a.s.,
which occurs if and only if:

η i = 1 Leb(B i ) η c (x)1 B i (x) dx, i ∈ {1, 2}.
Therefore, in this case, the optimal strategies of the continuous model are easily deduced from the optimal strategies of the discrete model.

To conclude this example, we rewrite, using the formalism of the discrete model Param d , the next-generation matrix K in the setting of [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF], and the effective next-generation matrix K e (η) when the vaccination strategy η is in force (recall η i is the proportion of population with feature i which is not vaccinated): Let (Ω, F , µ) be a measurable space. We say a measurable function ϕ from (Ω, F ) to itself is measure preserving if µ = ϕ # µ. For example the function ϕ : x → 2x mod (1) defined on the probability space ([0, 1], B([0, 1], Leb) is measure preserving. Let ϕ be measure preserving function on Ω. Let k 1 be a kernel and γ 1 a function on Ω such that the model Param 1 = [(Ω, F , µ), k 1 , γ 1 ] satisfies Assumption 2. Let X 1 be a random variable with probability distribution µ and let X 2 = ϕ(X 1 ), so that (X 1 , X 2 ) is a coupling of (Ω, F , µ) with itself. Then for the kernel k 2 and the function γ 2 defined by:

K = k 11 p k 12 (1 -p) k 21 p k 22 (1 -p) and K e (η) = k 11 p η 1 k 12 (1 -p) η 2 k 21 p η 1 k 22 (1 -p) η 2 , with p = µ d ({1}), 1 -p = µ d ({2}) and k d = k d /γ d , that is: k d = k 11 k 12 k 21 k 22 = k 11 /γ 1 k 12 /γ 2 k 21 /γ 1 k 22 /γ 2 .
1 = [(Ω 1 , F 1 , µ 1 ), k 1 , γ 1 ]. Let ϕ be a measurable function from (Ω 1 , F 1 ) to (Ω 2 , F 2 ). Assume that: σ(γ 1 ) ⊂ σ(ϕ) and σ(k 1 ) ⊂ σ(ϕ) ⊗ σ(ϕ).
We can then build an elementary coupling. Let X 1 and Y 1 be independent µ 1 distributed random elements of Ω 1 , and set

(X 2 , Y 2 ) = (ϕ(X 1 ), ϕ(Y 1 )). Since σ(γ 1 ) ⊂ σ(ϕ) and σ(k 1 ) ⊂ σ(ϕ) ⊗ σ(ϕ), we get that γ 1 (X 1 ) is σ(X 2 )-measurable and k 1 (X 1 , Y 1 ) is σ(X 2 , Y 2 )-measurable.
According to (50), there exists two measurable functions γ

2 : Ω 2 → R and k 2 : Ω 2 × Ω 2 → R such that γ 1 = γ 2 • ϕ and k 1 = k 2 (ϕ ⊗ ϕ) that is almost surely: γ 1 (X 1 ) = γ 2 (X 2 ) and k 1 (X 1 , Y 1 ) = k 2 (X 2 , Y 2 ).
Let µ 2 = ϕ # µ 1 be the push-forward measure of µ 1 by ϕ. Using (49) it is easy to check that the integrability condition from Assumption 2 is fulfilled, so we can consider the reduced model

Param 2 = [(Ω 2 , F 2 , µ 2 ), k 2 , γ 2 ]
. By Definition 7.1, Param 1 is coupled with Param 2 through the (deterministic) coupling π given by the distribution of (X 1 , ϕ(X 1 )).

Eventually, we get from Corollary 7.4 with G = σ(ϕ), that η 1 ∈ ∆ 1 is Pareto optimal if and only if E 1 [η 1 | ϕ] is Pareto optimal (for the model Param 1 ), where E 1 correspond to the expectation with respect to the probability measure µ 1 on (Ω 1 , F 1 ).

T e c h n i c a l p ro o f s

8.1. The SIS model: properties of I and of the maximal equilibrium. We prove here Theorem 4.6 and Proposition 4.7, and properties of the maximal equilibrium. For the convenience of the reader, we only use references to the results recalled in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF] for positive operators on Banach spaces. For an operator A, we denote by A its adjoint. We first give a preliminary lemma. Lemma 8.1. Suppose Assumption 2 holds, and consider the positive bounded linear integral operator T k/γ on L ∞ . If there exists g ∈ L ∞ + , with Ω g dµ > 0 and λ > 0 satisfying: T k/γ (g)(x) > λg(x), for all x such that g(x) > 0, then we have ρ(T k/γ ) > λ.

Proof. Set T = T k/γ . Let A = { g > 0 } be the support of the function g. Let T be the bounded operator defined by T

(f ) = 1 A T (1 A f ). Since T (g) = 1 A T (1 A g) = 1 A T (g) > λg,
we deduce from the Collatz-Wielandt formula, see [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]Proposition 3.6], that ρ(T ) ≥ λ > 0. According to [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]Lemma 3.7 (v)], there exists v ∈ L q + \ {0}, seen as an element of the topological dual of L ∞ , a left Perron eigenfunction of T , that is such that (T ) (v) = ρ(T )v. In particular, we have v = 1 A v and thus A v dµ > 0 and Ω vg dµ > 0. We obtain:

(ρ(T ) -λ) v, g = v, T (g) -λg > 0.
This implies that ρ(T ) > λ. Since T -T is a positive operator, we deduce from (26) that ρ(T ) ≥ ρ(T ) > λ.

We now state an interesting result on the characterization of the maximal equilibrium g. We keep notations from Sections 2.3 and 2.4 and write R e for R e [k/γ]. Recall that R 0 = R e (1) and F defined by [START_REF] Feng | An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing[END_REF]. Let DF [h] denote the bounded linear operator on L ∞ of the derivative of the map f → F (f ) defined on L ∞ at point h:

DF [h](g) = (1 -h)T k (g) -(γ + T k (h))g for h, g ∈ L ∞ .
Let s(A) denote the spectral bound of the bounded operator A, see [START_REF] Lajmanovich | A deterministic model for gonorrhea in a nonhomogeneous population[END_REF] in [START_REF] Delmas | An Infinite-Dimensional SIS Model[END_REF]. Proposition 8.2. Suppose Assumption 2 holds and write R e for R e [k/γ]. Let h in ∆ be an equilibrium, that is F (h) = 0. The following properties are equivalent:

(

i) h = g, (ii) s(DF [h]) ≤ 0, (iii) R e ((1 -h) 2 ) ≤ 1. (iv) R e (1 -h) ≤ 1. We also have: g = 0 ⇐⇒ R 0 ≤ 1; as well as: g = 0 =⇒ R e (1 -g) = 1. Proof. Let h ∈ ∆ be an equilibrium, that is F (h) = 0.
Let us show the equivalence between (ii) and (iii). According to [7, Proposition 4.2], s(DF [h]) ≤ 0 if and only if:

ρ (T k ) ≤ 1 with k(x, y) = (1 -h(x)) k(x, y) γ(y) + T k (h)(y) • Since F (h) = 0, we have (1 -h)/γ = 1/(γ + T k (h)). This gives: (52) k(x, y) = (1 -h(x)) k(x, y)(1 -h(y)) γ(y) and thus T k = M 1-h T k/γ M 1-h
, where M f is the multiplication operator by f . Recall the definition [START_REF] Duijzer | The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect[END_REF] of R e . According to (29), we have:

(53) ρ (T k ) = ρ T k/γ M (1-h) 2 = R e ((1 -h) 2 ).
This gives the equivalence between (ii) and (iii).

We prove that (i) implies (iv). Suppose that R e (1h) > 1. Thanks to (29), we have

ρ(M 1-h T k/γ ) = ρ(T k/γ M 1-h ) = R e (1 -h) > 1. According to [7, Lemma 3.7 (v)], there exists v ∈ L q + \ {0} a left Perron eigenfunction of T (1-h)k/γ , that is T (1-h)k/γ (v) = R e (1 -h)v.
Using F (h) = 0, and thus (1h)T k (h) = γh, for the last equality, we have:

R e (1 -h) v, γh = v, (1 -h)T k/γ (γh) = v, γh .
We get v, γh = 0 and thus v,

1 A = 0, where A = { h > 0 } denote the support of the function h. Since T (1-h)k/γ (v) = R e (1 -h)v and setting v = (1 -h)v (so that v = v µ-almost surely on A c
), we deduce that:

T k /γ (v ) = R e (1 -h)v , where k = 1 A c k 1 A c . This implies that ρ(T k /γ ) ≥ R e (1 -h). Since k = (1 -h)k and T k/γ -T k /γ
is a positive operator as kk ≥ 0, we get, using [START_REF] Huggett | A topological aperitif[END_REF] for the inequality, that

ρ(T k /γ ) = ρ(M 1-h T k /γ ) ≤ ρ(M 1-h T k/γ ) = R e (1-h). Thus, the spectral radius of T k /γ is equal to R e (1 -h). According to [7, Proposition 4.2], since ρ(T k /γ ) > 1, there exists w ∈ L ∞ + \ {0} and λ > 0 such that:
T k (w)γw = λw. This also implies that w = 0 on A = { h > 0 }, that is wh = 0 and thus wT k (h) = 0 as T k (h) = γh/(1h). Using that F (h) = 0, T k (w) = T k (w) = (γ + λ)w and hT k (w) = 0, we obtain:

F (h + w) = w(λ -T k (w)). Taking ε > 0 small enough so that εT k (w) ≤ λ/2 and εw ≤ 1, we get h + εw ∈ ∆ and F (h + εw) ≥ 0. Then use Lemma 4.4 to deduce that h + εw ≤ g and thus h = g.

To see that (iv) implies (iii), notice that (1h) 2 ≤ (1h), and then deduce from Proposition 4.1 (iii) that R e ((1h) 2 ) ≤ R e (1h).

We prove that (iii) implies (i). Notice that F (g) = 0 and g ∈ ∆ implies that g < 1. Assume that h = g. Notice that γ/(1

-h) = γ + T k (h), so that γ(g -h)/(1 -h) ∈ L ∞
+ . An elementary computation, using F (h) = F (g) = 0 and k defined in (52), gives:

T k γ g -h 1 -h = (1 -h)T k (g -h) = γ g -h 1 -g = 1 -h 1 -g γ g -h 1 -h • Since h = g and h ≤ g, we deduce that (1-h)/(1-g) ≥ 1, with strict inequality on { g -h > 0 }
which is a set of positive measure. We deduce from Lemma 8.1 (with k replaced by kγ) that ρ (T k ) > 1. Then use (53) to conclude.

To conclude notice that g = 0 ⇐⇒ R 0 ≤ 1 is a consequence of the equivalence between (i) and (iv) with h = 0 and R 0 = R e [START_REF] Anselone | Collectively compact operator approximation theory and applications to integral equations[END_REF].

Using that F (g) = 0, we get T k (g) = γg/(1g). We deduce that T k(1-g)/γ (T k (g)) = T k (g). If g = 0, we get T k (g) = 0 (on a set of positive µ-measure). This implies that R e (1g) ≥ 1. Then use (iv) to deduce that R e (1g) = 1 if g = 0.

In the SIS model, in order to stress, if necessary, the dependence of a quantity H, such as F η , R e or g η , in the parameters k and γ (which satisfy Assumption 2) of the model, we shall write H[k, γ]. Recall that if k and γ satisfy Assumption 2, then the kernel k/γ has a finite double norm on L p for some p ∈ (1, +∞). We now consider the continuity property of the maps η → g η [k, γ] and (k, γ, η) → g η [k, γ]. Lemma 8.3. Let ((k n , γ n ), n ∈ N) and (k, γ) be kernels and functions satisfying Assumption 2 and (η n , n ∈ N) be a sequence of elements of ∆ converging weakly to η.

(i) We have lim n→∞ g ηn [k, γ] = g η [k, γ] µ-almost surely.

(ii) Assume furthermore there exists p ∈ (1, +∞) such that k = γ -1 k and (k n = γ -1 n k n , n ∈ N) have finite double norm on L p and that lim n→∞ k nk p ,q = 0. Then, we have lim n→∞ g ηn [k n , γ n ] = g η [k, γ] µ-almost surely.

Proof. The proof of (i) and (ii) being rather similar, we only provide the latter and indicate the difference when necessary. To simplify, we write g n = g ηn [k n , γ n ]. We set h n = η n g n ∈ ∆ for n ∈ N. Since ∆ is sequentially weakly compact, up to extracting a subsequence, we can assume that h n converges weakly to a limit h ∈ ∆. Since F ηn [k n , γ n ](g n ) = 0 for all n ∈ N, see [START_REF] Grobler | Compactness conditions for integral operators in Banach function spaces[END_REF], we have:

(54)

g n = T kn (η n g n ) 1 + T kn (η n g n ) = T kn (h n ) 1 + T kn (h n ) •
We set g = T k (h)/(1 + T k (h)). Notice that T kn (h n ) = (T kn -T k )(h n ) + T k (h n ). We have lim n→∞ T k (h n ) = T k (h) pointwise. Since (T kn -T k )(h n ) p ≤ k nk p ,q , up to taking a sub-sequence, we deduce that lim n→∞ (T kn -T k )(h n ) = 0 almost surely. (Notice the previous step is not used in the proof of (i) as k n = k and lim n→∞ T k (h n ) = T k (h) pointwise.) This implies that g n converges almost surely to g. By the dominated convergence theorem, we deduce that g n converges also in L p to g. This proves that h = ηg almost surely. We get g = T k (ηg)/(1 + T k (ηg)) and thus The random variables we consider, are defined on a probability space, say (Ω 0 , F 0 , P). We recall an elementary result on conditional independence. Let A , B and H be σ-fields subsets of F 0 , such that H ⊂ A ∩ B. Then, according to [START_REF] Kallenberg | Foundations of modern probability[END_REF]Theorem 8.9], we have that for any integrable real-valued random variable X which is B-measurable: We now state two technical lemmas.

Lemma 8.4 (Measurability). Let Param 1 and Param 2 be coupled models with independent coupling (X 1 , X 2 ) and (Y 1 , Y 2 ). Then the random variable γ 1 (X 1 ) is σ(X 1 ) ∩ σ(X 2 )-measurable.

For any measurable function v :

Ω 1 × Ω 2 → R, such that k 1 (X 1 , Y 1 )v(Y 1 , Y 2 ) is integrable, the random variable E [k 1 (X 1 , Y 1 )v(Y 1 , Y 2 )|X 1 ]
is also σ(X 1 ) ∩ σ(X 2 )-measurable.

Proof. The σ(X 1 ) ∩ σ(X 2 )-measurability of γ 1 (X 1 ) is an immediate consequence of the almostsure equality γ 1 (X 1 ) = γ 2 (X 2 ). Since E [k(X 1 , Y 1 )v(Y 1 , Y 2 )|X 1 ] is σ(X 1 )-measurable, it remains to prove that it is also σ(X 2 )-measurable. Since (X 1 , X 2 ) is independent from (Y 1 , Y 2 ), the σfields A = σ(X 1 , X 2 ) and B = σ(X 1 , Y 1 , Y 2 ) are conditionally independent given H = σ(X 1 ). Using (55), we deduce that:

E [k 1 (X 1 , Y 1 )v(Y 1 , Y 2 )|X 1 ] = E [k 1 (X 1 , Y 1 )v(Y 1 , Y 2 )|X 1 , X 2 ] .
Since k 1 (X 1 , Y 1 ) = k 2 (X 2 , Y 2 ) P-almost surely, we get:

E [k 1 (X 1 , Y 1 )v(Y 1 , Y 2 )|X 1 ] = E [k 2 (X 2 , Y 2 )v(Y 1 , Y 2 )|X 1 , X 2 ] = E [k 2 (X 2 , Y 2 )v(Y 1 , Y 2 )|X 2 ] ,
where the last equality follows from another application of (55) with A = σ(X 1 , X 2 ), B = σ(X 2 , Y 1 , Y 2 ) which are conditionally independent given H = σ(X 2 ). The last expression is σ(X 2 ) measurable, so the proof is complete.

In the following key lemma, we simply write H i for H[Param i ] for H the loss functions R e and I, the cost function C = C uni and the spectrum Spec. Proof. Let (X 1 , X 2 ) and (Y 1 , Y 2 ) be two independent couplings, and assume that η 1 and η 2 are coupled through the function η, see Remark 7.2 (i):

(57) E [η(X 1 , X 2 )| X i ] = η i (X i ) for i ∈ {1, 2}.

Step 1: The cost function (H = C uni ). We directly have:

C 1 (η 1 ) = 1 -E [η 1 (X 1 )] = 1 -E [η(X 1 , X 2 )] = 1 -E [η 2 (X 2 )] = C 2 (η 2 ).
Step 2: The spectrum and the effective reproduction function (H = R e ). Set k i = k i /γ i for i ∈ {1, 2}. Let λ be a non-zero eigenvalue of T k 1 η 1 associated with an eigenvector v 1 . Notice that k(X 1 , Y 1 )η 1 (Y 1 )v(Y 1 ) is integrable thanks to the integrability condition from Assumption 2. By definition of eigenvectors, v 1 (X 1 ) is a version of the conditional expectation:

λ -1 E [k 1 (X 1 , Y 1 ) η 1 (Y 1 )v 1 (Y 1 )|X 1 ] .
By Lemma 8.4 applied to the function v(y 1 , y 2 ) = (v 1 η 1 /γ 1 )(y 1 ), the real-valued random variable v 1 (X 1 ) is σ(X 1 ) ∩ σ(X 2 )-measurable and thus σ(X 2 )-measurable. Thanks to (50), there exists v 2 such that v 2 (X 2 ) = v 1 (X 1 ) almost surely. Since (Y 1 , Y 2 ) is distributed as (X 1 , X 2 ), we deduce that (57) holds also with (X 1 , X 2 ) replaced by (Y 1 , Y 2 ) and that v 2 (Y 2 ) = v 1 (Y 1 ) almost surely. Recall that k i = k i /γ i , so that k 1 (X 1 , Y 1 ) = k 2 (X 2 , Y 2 ) almost surely. We may now compute:

(58)

λv 2 (X 2 ) = λv 1 (X 1 ) = E [k 1 (X 1 , Y 1 ) η 1 (Y 1 )v 1 (Y 1 )|X 1 ] = E [k 1 (X 1 , Y 1 ) η(Y 1 , Y 2 )v 1 (Y 1 )|X 1 ] (de-conditioning on (Y 1 , X 1 )) = E [k 1 (X 1 , Y 1 ) η(Y 1 , Y 2 )v 1 (Y 1 )|X 2 ] (Lemma 8.4) = E [k 2 (X 2 , Y 2 ) η(Y 1 , Y 2 )v 2 (Y 2 )|X 2 ] (a.s. equality) = E [k 2 (X 2 , Y 2 ) η 2 (Y 2 )v 2 (Y 2 )|X 2 ]
(conditioning on (Y 2 , X 2 )) = T k 2 η 2 v 2 (X 2 ).

Since the distribution of X 2 is µ 2 , we have λv 2 = T k 2 η 2 v 2 µ 2 -almost surely. Therefore λ is also an eigenvalue for T k 2 η 2 . By symmetry we deduce that the spectrum up to {0} of T k 1 η 1 and T k 2 η 2 coincide, that is Spec 1 (η 1 ) ∪ {0} = Spec 2 (η 2 ) ∪ {0}, and in particular the spectral radius coincide.

Step 3: The total proportion of infected population function (H = I). We assume without loss of generality that ρ(T k 1 /γ 1 ) > 1, which is equivalent to ρ(T k 2 /γ 2 ) > 1, thanks to (56) with H = R e and η 1 = η 2 = 1. Let g 1 = g η 1 be the maximal equilibrium for the model Param 1 . Since F η 1 (g 1 ) = 0, see [START_REF] Grobler | Compactness conditions for integral operators in Banach function spaces[END_REF], we have:

(59) g 1 = T k 1 (η 1 g 1 ) γ 1 + T k 1 (η 1 g 1 )
•

By Lemma 8.4, this implies that g 1 (X 1 ) is σ(X 1 ) ∩ σ(X 2 ) measurable. Thus, there exists g 2 such that g 2 (X 2 ) = g 1 (X 1 ) P-almost surely.. Therefore, by the same computation as in (58):

T k 1 (η 1 g 1 )(X 1 ) = T k 2 (η 2 g 2 )(X 2 ) Pa.s.

We set:

(60)

g 2 = T k 2 (η 2 g 2 ) γ 2 + T k 2 (η 2 g 2 )
•

Then, we deduce from (59) that g 2 (X 2 ) = g 2 (X 2 ) P-almost surely, that is g 2 = g 2 µ 2 -almost surely. Thus (60) holds with g 2 replaced by g 2 . In other words, g 2 satisfies (22): it is an equilibrium for the model given by Param 2 .

Let us now prove that g 2 is in fact the maximal equilibrium. Since g 2 (X 2 ) = g 1 (X 1 ) P-almost surely and g 1 (X 1 ) is σ(X 1 ) ∩ σ(X 2 )-measurable, we deduce from Remark 7.2 (ii), that (1g 1 ) and (1-g 2 ) are coupled, so R e [Param 1 ](1-g 1 ) = R e [Param 2 ](1-g 2 ), by Property (56) applied to H = R e . Since R 0 > 1 and g 1 is the maximal equilibrium for Param 1 , we deduce from Proposition 8.2 that R e [Param 1 ](1g 1 ) = 1. Using again Proposition 8.2, this gives that g 2 is the maximal equilibrium for Param 2 .

We may now compute:

I 1 (η 1 ) = E [η 1 (X 1 ) g 1 (X 1 )] = E [η(X 1 , X 2 ) g 1 (X 1 )] (deconditioning on X 1 ) = E [η(X 1 , X 2 ) g 2 (X 2 )] ( a.s. equality) = E [η 2 (X 2 ) g 2 (X 2 )]
(conditioning on X 2 ) = I 2 (η 2 ), thus (56) holds for H = I, and the proof is complete.

We now give the proof of Proposition 7.3. Its first part is an elementary consequence of the Lemma 8.5; and the second part is a direct consequence of Remark 7.2 (iii).

  of vaccinating the whole population, 0 = L(0) ≤ L(η) ≤ L(1) = max = loss incurred in the absence of vaccination.

  6 states that the Pareto frontier is non empty and has a continuous parametrization for the cost C = C uni and the loss L = R e or L = I; see Figure 1( b ) below for a visualization of the Pareto frontier.Theorem 1.4 (Properties of the Pareto frontier). For the kernel model with loss function L = R e or the SIS model with L ∈ {R e , I}, and the uniform cost function C = C uni , the function C ,L is continuous and decreasing on [0, max ], the function L is continuous on [0, c max ] decreasing on [0, C ,L (0)] and zero on [C ,L (0), c max ]; furthermore the Pareto frontier is connected and:

  Grayplot of the kernel k, with Ω = [0, 1] and µ the Lebesgue measure (k is equal to the constant κ > 0 on the black zone and to 0 on the white zone). The Pareto frontier in solid red line compared to the cost and loss of the uniform vaccinations in dotted line and the worst vaccination strategy in red dashed line.

F i g u r e 1 .

 1 Example of optimization with L = R e .

Proposition 2 . 1 (

 21 On the maximal equilibrium). Suppose Assumption 2 holds and write R e for R e [k/γ].

e a n d I 4 . 1 .

 41 The effective reproduction number R e . We consider the kernel model [(Ω, F, µ), k] under Assumption 1, so that k is a kernel on Ω with finite double norm. Recall the effective reproduction number function R e [k] defined on ∆ by (14): R e [k](η) = ρ(T k M η ) and the reproduction number R 0 [k] = ρ(T k ). We simply write R e and R 0 for R e [k] and R 0 [k] respectively when no confusion on the kernel can arise.

Proposition 4 . 1 (

 41 Basic properties of R e ). Suppose Assumption 1 holds. Let η, η 1 , η 2 ∈ ∆. The function R e = R e [k] satisfies the following properties:

Theorem 4 . 2 (

 42 Continuity of R e [k] and Spec[k]). Suppose Assumption 1 holds. Then, the functions Spec[k] and R e [k] are continuous functions from ∆ (endowed with the weak-topology) respectively to K (endowed with the Hausdorff distance) and to R + (endowed with the usual Euclidean distance).

  The function Spec[k] is thus sequentially continuous, and, thanks to Lemma 3.1, it is continuous from ∆ endowed with the weak topology to the metric space K endowed with the Hausdorff distance. The continuity of R e [k] then follows from its definition[START_REF] Delmas | Effective reproduction number: convexity, invariance and cordons sanitaires[END_REF] as the composition of the continuous functions rad and Spec[k].

Proposition 4 . 3 (

 43 Stability of R e [k] and Spec[k]

4. 2 .

 2 The asymptotic proportion of infected individuals I. We consider the SIS model [(Ω, F , µ), k, γ] under Assumption 2. Recall from (23) that the asymptotic proportion of infected individuals I is given on ∆ by I(η) = Ω g η η dµ, where g η is the maximal solution in ∆ of the equation F η (h) = 0. We first give a preliminary result. Lemma 4.4. Let η, g ∈ ∆. If F η (g) ≥ 0, then we have g ≤ g η .

Proposition 4 . 5 (

 45 Basic properties of I). Suppose that Assumption 2 holds. Let η, η 1 , η 2 ∈ ∆. The function I has the following properties:(i)I(η 1 ) = I(η 2 ) if η 1 = η 2 µ-almost surely. (ii) I(η) = 0 if and only if R e [k/γ](η) ≤ 1. (iii) I(η 1 ) ≤ I(η 2 ) if η 1 ≤ η 2 µ-almost surely.(iv) I(λη) ≤ λI(η) for all λ ∈ [0, 1].

Theorem 4 . 6 (

 46 Continuity of I). Suppose that Assumption 2 holds. The function I defined on ∆ is continuous with respect to the weak topology.We write I[k, γ] for I to stress the dependence on the parameters k, γ of the SIS model.

Proposition 4 . 7 (

 47 Stability of I). Let ((k n , γ n ), n ∈ N) and (k, γ) be a sequence of kernels and functions satisfying Assumption 2. Assume furthermore that there exists p ∈ (1, +∞) such that k = γ -1 k and (k n = γ -1

  Before going further, let us remark that for the reproduction number optimization in the vaccination context, one can without loss of generality consider the uniform cost instead of the affine cost. Remark 5.2. Consider the kernel model Param = [(Ω, F , µ), k] with the affine cost function C aff and the loss R e . Furthermore, if we assume that c aff is bounded and bounded away from 0 (that is c aff and 1/c aff belongs to L ∞ + ), and without loss of generality, that c aff dµ = 1, then we can consider the weighted kernel model Param 0 = [(Ω, F , µ 0 ), k 0 ] with measure µ 0 (dx) = c aff (x) µ(dx) and kernel k 0 = k/c aff . (Notice that if Assumption 2 holds for the model Param, then it also holds for the model Param 0 .) Consider the loss L = R e . Then for a strategy η ∈ ∆, we get that (C aff (η), L(η)) for the model Param is equal to (C uni (η), L(η)) for the model Param 0 . Therefore, for the loss function L = R e , instead of the affine cost C aff , one can consider without any real loss of generality the uniform cost. (This holds also for the SIS model.) However, this is no longer the case for the loss function L = I in the SIS model.

  for a typical representation of the possible aspects of the feasible region F (in light blue), the value functions and the Pareto and anti-Pareto frontiers under the general Assumption 3, and the connected Pareto and anti-Pareto frontiers under further regularity on the cost and loss functions (see Assumption 4-7 below) in Figure 2( d ). In Figure 1( b ), we have plotted in solid red line the Pareto frontier and in dashed red line the anti-Pareto frontier from Example 1.7.

  max ] : c = C ( ) and = L (c)}. (iii) The points (0, L (0)) and (C (0), 0) both belong to the Pareto frontier, and we have C (L (0)) = L (C (0)) = 0. Moreover, we also have C ( ) = 0 for ∈ [L (0), max ], and L (c) = 0 for c ∈ [C (0), c max ].

  Value functions for Problems[START_REF] Restrepo | Characterizing the Dynamical Importance of Network Nodes and Links[END_REF] and[START_REF] Schaefer | Banach lattices and positive operators[END_REF]. Value functions for Problems[START_REF] Saha | Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread[END_REF] and[START_REF] Smith | Prospects for the control of infectious disease[END_REF]. Pareto and anti-Pareto frontier under additional regularity Assumptions 4-7.

F i g u r e 2 .

 2 An example of the possible aspects of the feasible region F (in light blue), the value functions L , L , C , C , and the Pareto and anti-Pareto frontier (in red) under Assumption 3.

Assumption 4 .Assumption 5 .

 45 If the cost C has a local minimum (for the weak topology) at η, then C(η) = 0 and η is a global minimum of C. If the loss L has a local minimum (for the weak topology) at η, then L(η) = 0 and η is a global minimum of L.

  [0, C (0)]. (v) If η solves Problem (39) for the cost c ∈ [0, C (0)], then C(η) = c. Moreover η is Pareto optimal, and C (L (c)) = c. (vi) The Pareto frontier is the graph of L : (45)

  ) = C ( ) and L(η ) ≤ . The set O = {η : L(η) < } is open and contains η . Since L(η ) < L (0), we get C(η ) > 0, so η is not a global minimum for C. By Assumption 4, it cannot be a local minimum for C, so O contains at least one point η for which C(η ) < C(η ). Since η ∈ O, we get L(η ) ≤ , so that C ( ) ≤ C(η ) < C(η ) = C ( ). Since < are arbitrary, C is decreasing on [0, L (0)].

Lemma 5 . 6 .

 56 Suppose Assumption 3 holds. If the cost function C is decreasing, then Assumption 4 holds and L (0) = max . If the loss function L is sub-homogeneous, then Assumption 5 holds.

5. 3 .

 3 On the anti-Pareto frontier. 5.3.1. The general setting. Letting C (η) = max -L(η) and L (η) = c max -C(η), it is easy to see that: C (c) = max -L (c maxc) and L ( ) = c max -C ( max -), so that Proposition 5.5 may be applied to the cost function C and the loss function L to yield the following result. Proposition 5.8 (Single-objective and bi-objective problems for the anti-Pareto strategies). Suppose Assumption 3 holds.

(Assumption 6 .

 6 iii) The points (C ( max ), max ) and (c max , L (c max )) both belong to the anti-Pareto frontier, and we have C (L (c max )) = c max and L (C ( max )) = max . Moreover, we also haveC ( ) = c max for ∈ [0, L (c max )], and L (c) = max for c ∈ [0, C ( max )].The following additional hypotheses rule out the occurrence of flat parts in the anti-Pareto frontier. If the cost C has a local maximum at η (for the weak topology), then C(η) = c max and η is a global maximum of C. Assumption 7. If the loss L has a local maximum at η (for the weak topology), then L(η) = max and η is a global maximum of L. The following result is now a consequence of Proposition 5.5 and Corollary 5.7 applied to the loss function L and cost function C . Proposition 5.9. Under Assumption 3 and 6 the following properties hold: (i) The optimal cost C is decreasing on [C ( max ), c max ]. (ii) If η solves Problem (42) for the loss ∈ [L (c max ), max ], then L(η) = (that is, the constraint is binding). Moreover η is anti-Pareto optimal, and L (C ( )) = . (iii) The anti-Pareto frontier is the graph of C : C ( ), ) : ∈ [L (c max ), max ]}. Similarly, under Assumptions 3 and 7, the following properties hold: (iv) The optimal loss L is decreasing on [L (c max ), max ]. (v) If η solves Problem (41) for the cost c ∈ [C ( max ), c max ], then C(η) = c. Moreover η is anti-Pareto optimal, and C (L (c)) = c. (vi) The anti-Pareto frontier is the graph of L :

Lemma 5 . 11 .

 511 s. or A c ⊂ {k ≡ 0} a.s. Notice the definition of the quasi-irreducibility from [3, Definition 2.11] is slightly stronger as it uses a topology on Ω. Consider the kernel model Param = [(Ω, F , µ), k] under Assumption 1. If k is quasi-irreducible, then Assumption 7 holds for L = R e [k], and

Lemma 5 . 12 .

 512 Consider the SIS model Param = [(Ω, F , µ), k, γ] under Assumption 2. If k is quasi-irreducible, then Assumption 7 holds for L = I and C

  In red, the Pareto frontier of the kernel k represented in Figure1( a ) compared to the Pareto frontier of the kernel k ε in blue.

F i g u r e 3 .Proposition 6 . 2 .

 362 On the stability of the Pareto frontier 6.2. Stability. We can consider the stability of the Pareto frontier and the set of Pareto optima. Recall that, thanks to[START_REF] Stevanović | On spectral radius and energy of complete multipartite graphs[END_REF], the graph {(c, L (c)) : c ∈ [0, c max ]} of L is the union of the Pareto frontier and the straight line joining (0, C (0)) to (0, c max ) and can thus be seen as an extended Pareto frontier. The proof of the following proposition is immediate. It implies in particular the convergence of the extended Pareto frontier. This result can also easily be adapted to the anti-Pareto frontier. Let C be a cost function and (L(n) , n ∈ N) a sequence of loss functions converging uniformly on ∆ to a loss function L. Assume that Assumptions 3, 4 and 5 hold for the cost C and the loss functions L (n) , n ∈ N, and L. Then L (n) converges uniformly to L .

6. 3 .

 3 Geometric properties. If the cost function is affine, then there is a nice geometric property of the Pareto frontier. Lemma 6.4. Suppose that Assumption 3 holds, the cost function is affine (i.e., C = C aff given by (35)) and the loss function L is sub-homogeneous. Then, we have L (θc+(1-θ)c max ) ≤ θL (c) for all c ∈ [0, c max ] and θ ∈ [0, 1]. Remark 6.5. Geometrically, Lemma 6.4 means that the graph of the loss L : [0, c max ] → [0, max ] is below its chords with end point (1, L (1)) = (1, 0). See Figures 1( b ) for a typical representation of the Pareto frontier (red solid line). Proof. Let c ∈ [0, c max ] and θ ∈ [0, 1]. Thanks to Lemma 5.6, Assumption 5 holds. Thus, thanks to Proposition 5.5 (iv), there exists η ∈ P L with cost C(η) = c and thus L(η) = L (c).

Proposition 6 . 6 .

 66 Suppose that Assumption 3 holds. If the cost function C and the loss function L are convex, then the functions C and L are convex. If the cost function C and the loss function L are concave, then the functions C and L are convex.

  the kernel model characterized by Param = [(Ω, F , µ), k], with Assumption 1 fulfilled, and loss function L = R e ; • the SIS model characterized by Param = [(Ω, F , µ), k, γ], with Assumption 2 fulfilled, and loss function L ∈ {R e , I}; where (Ω, F , µ) is a probability space, k and k are non-negative kernels on Ω and γ is a non-negative function on Ω. In order to emphasize the dependence of a quantity H on the parameters Param of the model, we shall write H[Param] for H. For example we write: ∆[Param] for the set of functions {η ∈ L ∞ (Ω, F ) : 1 ≥ η ≥ 0}, which clearly depends on the parameters Param; and the effective reproduction function R e [Param]. For example, under Assumption 2, we have the equality of the following functions: R e [(Ω, F , µ), k, γ] = R e [(Ω, F , µ), k/γ, 1] = R e [(Ω, F , µ), k/γ],

  where Assumption 1 holds for each model; in the SIS model, we consider the modelsParam i = [(Ω i , F i , µ i ), k i , γ i ] for i ∈ {1,2}, where Assumption 2 holds for each model. In what follows, we simply write ∆ i the set of functions ∆ for the model Param i .

  Continuous model: kernel k c on Ω c = [0, 1) with the Lebesgue measure. Discrete model: kernel k d on Ω d = {1, 2} with the measure pδ 1 + (1p)δ 2 .

F i g u r e 4 .

 4 Coupled continuous model (left) and discrete model (right). 7.4.2. Measure preserving function. This section is motivated by the theory of graphons, which are indistinguishable by measure preserving transformation, see [34, Sections 7.3 and 10.7].

k 2

 2 (x, y) = k 1 (ϕ(x), ϕ(y)) and γ 2 (x) = γ 1 (ϕ(x)), the models Param 1 and Param 2 = [(Ω, F , µ), k 2 , γ 2 ] are coupled. Roughly speaking, we can give different labels to the features of the population without altering the Pareto and anti-Pareto frontiers. 7.4.3. Model reduction using deterministic coupling. This example is in the spirit of Section 7.4.1, where one merges individual with identical behavior. We consider a SIS model Param

1 .( 8 . 2 .

 182 F η [k, γ](g) = 0: g is an equilibrium for F η [k, γ]. We recall from [8, Section 3] the functional equality R e [k h] = R e [hk ],where k is a kernel, h a nonnegative functions such that the kernels k h and hk have some finite double norm. We deduce from the weak-continuity and the stability of R e , see Theorem 4.2 and Proposition 4.3, that:R e [k/γ](η(1g)) = R e [k](η(1g)) = lim n→∞ R e [k n ](η n (1g n )) = lim n→∞ R e [k n /γ n ](η n (1g n )) ≤ Only the weak-continuity of η → R e [k/γ](η ) is used in the proof of (i) to get R e [k/γ](η(1g)) ≤ 1.) We deduce that property (iv) of Proposition 8.2 holds with k replaced by kη, and thus property (i) therein implies that g = g η [k, γ].Proofs of Theorem 4.6 and Proposition 4.7. Under the assumptions of Lemma 8.3, taking the pair (k n , γ n ) equal to (k, γ) in the case (i) therein, we deduce that (η n g ηn [k n , γ n ], n ∈ N) converges weakly to η g η[k, γ]. This implies that:lim n→∞ I[k n , γ n ](η n ) = lim n→∞ Ω η n g η [k n , γ n ] dµ = Ω η g η [k, γ] dµ = I[k, γ](η).Taking (k n , γ n ) = (k, γ) provides the continuity of I[k, γ] and thus Theorem 4.6. Then, arguing as in the end of the proof of Proposition 4.3, we get Proposition 4.7. Coupling and Pareto optimality. We prove here Proposition 7.3. We only consider the SIS model Param = [(Ω, F , µ), k, γ], as the kernel model can be handled similarly. We suppose throughout this section that Assumption 2 holds.

  (55)A and B are conditionally independent givenH =⇒ E [X| A ] = E [X| H ] .

Lemma 8 . 5 .

 85 If Param 1 and Param 2 are coupled models, and if the functions η 1 ∈ ∆ 1 and η 2 ∈ ∆ 2 are coupled, then Spec 1 (η 1 )∪{0} = Spec 2 (η 2 )∪{0} and for H any one of the mappings C uni , R e or I: (56) H 1 (η 1 ) = H 2 (η 2 ).

  almost surely, then we have that T kη 1 = T kη 2 , and thus R e (η 1 ) = R e (η 2 ). This gives Point (i). Point (ii) is a direct consequence of the definition of R

e . Since for any fixed λ ∈ C and any operator A, the spectrum of λA is equal to {λs, s ∈ Spec(A)}, Point (iv) is clear. Finally, note that if η 1 ≤ η 2 µ-almost everywhere, then the operator T kη 2 -T kη 1 is positive. According to

[START_REF] Huggett | A topological aperitif[END_REF]

, we get that ρ(T kη 1 ) ≤ ρ(T kη 2 ). This concludes the proof of Point (iii).

This work is partially supported by Labex Bézout reference ANR-10-LABX-58.

(iii) Let η 1 ∈ ∆ 1 . According to (50), there exists η 2 ∈ ∆ 2 such that E [η 1 (X 1 )|X 2 ] = η 2 (X 2 ).

Thus, by definition η 1 and η 2 are coupled (through V = η 1 (X 1 )).

The main result of this section, whose proof is given in Section 8.2, states that coupled models have coupled Pareto optimal strategies, and thus the same (anti-)Pareto frontier.

and X = X can be seen as the identity map on Ω. The conclusion then follows from Proposition 7.3.

Examples of couplings.

In this section, we consider the SIS model as the kernel model can be handled in the same way. We denote by Leb the Lebesgue measure.