V D Nguyen

O Fruchart

S Pizzini

J Vogel

J.-C Toussaint

N Rougemaille
email: nicolas.rougemaille@neel.cnrs.fr

Third type of domain wall in soft magnetic nanostrips

Magnetic domain walls (DWs) in nanostructures are low-dimensional objects that separate regions with uniform magnetisation. Since they can have different shapes and widths, DWs are an exciting playground for fundamental research, and became in the past years the subject of intense works, mainly focused on controlling, manipulating, and moving their internal magnetic configuration. In nanostrips with in-plane magnetisation, two DWs have been identified: in thin and narrow strips, transverse walls are energetically favored, while in thicker and wider strips vortex walls have lower energy. The associated phase diagram is now well established and often used to predict the low-energy magnetic configuration in a given magnetic nanostructure. However, besides the transverse and vortex walls, we find numerically that another type of wall exists in permalloy nanostrips. This third type of DW is characterised by a three-dimensional, flux closure micromagnetic structure with an unusual length and three internal degrees of freedom. Magnetic imaging on lithographically-patterned permalloy nanostrips confirms these predictions and shows that these DWs can be moved with an external magnetic field of about 1 mT. An extended phase diagram describing the regions of stability of all known types of DWs in permalloy nanostrips is provided.

.

.

. For flat strips of soft magnetic materials with in-plane magnetisation, like permalloy (Py), simulations showed the existence of two type of domain walls, the so-called transverse (TW) and vortex (VW) domain walls. A phase diagram was determined, predicting the regions of stability of

both walls in a typical thickness range of 0-50 nm and widths 100-600 nm [START_REF] Mcmichael | Head to head domain wall structures in thin magnetic strips[END_REF][START_REF] Nakatani | Head-to-head domain walls in soft nano-strips: a refined phase diagram[END_REF]. This phase diagram was recently extended to much wider strips and double / triple vortex configurations were found to be low-energy states [START_REF] Estévez | Head-to-Head Domain Wall Structures in Wide Permalloy Strips[END_REF].

Using micromagnetic simulations and Magnetic Force Microscopy (MFM) measurements, we investigate the magnetic properties of domain walls in the case of Py strips thicker than 50 nm, for which the vortex wall is expected to be energetically favored on the basis of the existing phase diagram [START_REF] Mcmichael | Head to head domain wall structures in thin magnetic strips[END_REF][START_REF] Nakatani | Head-to-head domain walls in soft nano-strips: a refined phase diagram[END_REF][START_REF] Thiaville | Micromagnetic understanding of currentdriven domain wall motion in patterned nanowires[END_REF]. However, above a critical thickness t C ≈ 60 nm, and almost independently of the width of the strip, we find that the vortex wall transforms into a flux closure configuration stretched along the length of the strip, with in-plane magnetisation curling around a Bloch wall of finite length. This prediction is confirmed experimentally by MFM images obtained on a series of 80 nm-thick Py strips of various widths. The observed domain walls show two unusual features: i) contrary to usual TWs and VWs, they are significantly longer than the width of the strip, ii) they are characterised by three internal degrees of freedom.

RESULTS

We first consider, numerically, a 200 nm wide, 60 nm-thick Py strip. Two configurations can be obtained, as shown in Fig. 1: a vortex wall [Figs. 1(a-d The overall micromagnetic configuration is similar to a rectangular Landau flux-closure pattern [START_REF] Arrott | Micromagnetics and hysteresis as prototypes for complex systems[END_REF][START_REF] Hertel | Computation of the magnetic domain structure in bulk permalloy[END_REF][START_REF] Masseboeuf | Dimensionality Crossover in Magnetism: From Domain Walls (2D) to Vortices (1D)[END_REF]. In the following, we therefore refer to the Landau domain wall when describing this type of DW.

Investigating the influence of the width w of the strip on the critical thickness t C at which the VW / Landau DW transition occurs, we find that t C is essentially independent of w and of the order of 55 nm. This observation, together with the value found for t C , suggests that this transition may be linked to the Néel -Bloch wall transition observed in continuous films, which is known to occur at a film thickness of a few tens of nanometers, around 40 nm in Py [START_REF] Middelhoek | Domain walls in thin Ni-Fe films[END_REF][START_REF] Torok | Transition between Bloch and Néel Walls[END_REF][START_REF] Rivkin | Continuous Néel-to-Bloch transition in strips whose thickness increases: Statics and dynamics[END_REF]. Besides, the tilted feature of the Bloch part of the Landau wall is consistent with the known zigzag structure of head-to-head domain walls in thin films with in-plane uniaxial anisotropy [START_REF] Hubert | Magnetic Domains: The Analysis of Magnetic Microstructures[END_REF][START_REF] Hubert | Charged walls in thin magnetic-films[END_REF]. In both systems, the tilt allows spreading the magnetostatic charges over a larger area, and thus decreasing the magnetostatic energy. In strips, the anisotropy is of dipolar origin (shape anisotropy), while it is of magnetocrystalline origin in films.

To check these predictions, we fabricated a series of 40 nm and 80 nm-thick Py strips with different widths, ranging from 200 to 1200 nm. The strips are made with a curved shape to easily set a domain wall in the bent region after the application of an external magnetic field, transverse to the strip [START_REF] Taniyama | Resistivity due to Domain Walls in Co Zigzag Wires[END_REF][START_REF] Kläui | Direct observation of domain-wall configurations transformed by spin currents[END_REF]. The resulting magnetic configuration is then imaged after moving the DW in the straight region of the strip. Consistent with micromagnetic simulations, we always observe Landau DWs in the thickest (80 nm) strips and conventional VWs in the thinnest ones (40 nm). An example of a MFM image for a 1 µm-wide Py strip is reported in Fig. 2(a). Note, in that image, the black magnetic contrast atop the Bloch wall crossing the entire DW and the unusual length of the overall micromagnetic configuration.

DISCUSSION

When increasing the thickness of the nanostrip, we observe numerically and experimentally an increase of the domain wall width. This result is expected as the amount of volume magnetic charges within the DW increases with the thickness of the strip. Using the MFM images, we measure the angle θ made by the long axis of the strip and the line passing through the points where the magnetic flux, transverse to the strip, enters and exits the Landau DW [see Fig. 2(a)]. Comparison between the predicted DW angle θ and the experimental findings, is reported in Fig. 2(b). Although our experimental measurements are scattered, a fairly good quantitative agreement is found between the angle values deduced from the MFM images and the predictions from micromagnetic simulations. The scattering of the experimental data points is likely due to imperfections in the strips, such as edge / interface roughness or pinning sites that could slightly deform the DWs, or can be linked to the way the DWs have been prepared and moved prior to their imaging. However, we want to stress that, above a certain critical thickness (about 50 nm experimentally), we always observe Landau domain walls, independently of the width of the strip (at least in the 200 -1200 nm range probed in this work). There is no exception and experiments have been reproduced several times on different samples (see Fig. 1 and Fig. 2 in supplementary materials). We also observe discrepancies for the widest strips (for 1000 and 1200 nm-wide strips).

We emphasize though that micromagnetic simulations are hardly achievable for such large widths and thicknesses, and understanding if this non-monotonic behavior is an experimen- Interestingly, when the Landau DW is found numerically, it always has a lower energy than the VW. In other words, within the set of parameters we use in the micromagnetic simulations, we never find the Landau DW as a metastable state. For example, a VW is always found if the thickness t of the strip is set to 50 nm or less, even if the initial condition in the micromagnetic simulation is a Landau DW. These results indicate that Landau DWs may not persist as a metastable state and suggest a second-order transition, consistent with the existence of a breaking of symmetry. We emphasize though that characterizing the order of the VW / Landau DW phase transition deserves more investigations, beyond the scope of the present work.

Inspection of the dependence of the DW width δ as a function of both the strip width w and strip thickness t is also useful to compare quantitatively experiments and micromagnetic predictions. In the following, we define the DW width δ = w/ tan θ [Fig. 2 Néel caps (magenta arrows in Fig. 4). Similar to what is observed in thick, self-assembled Fe micro-dots [START_REF] Yan | Calculations of three-dimensional magnetic normal modes in mesoscopic permalloy prisms with vortex structure[END_REF][START_REF] Cheynis | Controlled Switching of Néel Caps in Flux-Closure Magnetic Dots[END_REF][START_REF] Fruchart | Asymmetric Hysteresis of Neel Caps in Flux-Closure Magnetic Dots[END_REF], the two Néel caps have their magnetisation pointing in opposite directions to minimize the magnetostatic energy of the asymmetric Bloch wall.

Finally, this entire, three-dimensional micromagnetic configuration can be moved under an applied magnetic field. This is illustrated in Fig. 5, where a field of 1 mT was applied

METHODS

Numerically, our approach is based on the finite difference OOMMF code [START_REF] Donahue | The Object Oriented MicroMagnetic Framework (OOMMF) project at ITL/NIST[END_REF]. Material parameters are µ 0 M Py = 1.0053 T for the spontaneous magnetisation, A Py = 10 pJ/m for the exchange stiffness, and zero magnetocrystalline anisotropy. This implies a dipolar exchange length ∆ d = 2A/µ 0 M Py 2 ≈ 5 nm. Although directly applicable to Py, our simulations may be applied to any other magnetic material with no magnetocrystalline anisotropy, scaling lengths with ∆ d . Cell size in all simulations presented here was set to 4 × 4 × 5 nm 3 . Using a cell size of 4 × 4 × 2.5 nm 3 yields only negligible differences. Damping is set to 1 to speed up convergence, with no consequence on the result as we are interested in equilibrium states. Magnetic moments at the two extremities of the strips are fixed to avoid non-uniform magnetisation distributions at the edges, and the length of the strips is chosen such that the aspect ratio is at least larger than 10 to limit finite size effects.

Experimentally, the internal micromagnetic configuration of the domain walls gives rise essentially to a monopolar contrast. This is typical for Bloch walls or asymmetric Néel walls with a significant out-of-plane magnetisation within the core, while Néel walls would mainly lead to a dipolar contrast. The MFM contrast may be dark or bright in Fig. 2(a), consistent with the possibility to have either up or down perpendicular component of the magnetisation within the wall [see Fig. 1(a)].

To determine the width δ of the Landau domain walls, we use the following model to calculate the dipolar and exchange energies. Dipolar energy of a head-to-head DW (whatever its exact shape) is estimated from a 2D slab carrying the surface magnetic charge σ = 2twM Py /(wδ) = 2tM Py /δ. The resulting dipolar energy integrated in space is (1/2)µ 0 H 2 d , with H d of the order of σ/2 at each surface of the strip and significant only above the wall over a distance w. The total dipolar energy is then of the order of

E d ≈ (w 2 t 2 /δ)µ 0 M Py 2 .
The Landau DW being bounded by two segments of wall, nearly perpendicular to the strip whatever the thickness and width (see Fig. 1), we neglect their contribution in the energy minimization. Most of the remaining energy is then associated with the tilted Bloch wall.

The Bloch wall energy per unit length is mostly of exchange origin because of the rather good closure of the flux inside Bloch walls. For the thicknesses considered here, Bloch walls have a close-to-circular cross-section [START_REF] Hubert | Stray-field-free magnetisation configurations[END_REF][START_REF] Labonte | Two-dimensional Bloch-type domain walls in ferromagnetic thin films[END_REF] and thus have an area of the order of t 2 . This leads to an energy per unit length of A(π/t) 2 t 2 = Aπ 2 . Finally, the Bloch wall is assumed to extend over the diagonal of the rectangular shape of the Landau wall, and its energy reads:

E W = Aπ 2 √ δ 2 + w 2 .
Minimization of the total energy versus δ provides the equilibrium value: δ eq ∼ ( √ 2/π)wt/∆ d . To obtain this value, we dropped the term related to w 2 in the length of the Bloch wall, quickly being negligible above the transition thickness. As often for scaling laws, the numerical factor is not directly applicable, due to the crude approximations made in the model: (2)/π ≈ 1/2, against 1/4 in the phenomenological law extracted from the simulations.

  )] and another flux closure magnetic state [Figs. 1(e-h)]. Contrary to the vortex wall in which the magnetisation distribution can be essentially considered as uniform within the thickness of the strip, the magnetisation distribution in this second DW varies significantly across the thickness. Figs. 1(d,h) are cross section views illustrating this distribution: the micromagnetic configuration reveals that the DW is essentially a vortex with a core stretched along the length of the strip, i.e. a flux closure magnetic configuration curling around a Bloch wall instead of a vortex core. The magnetic flux is entering and exiting this Bloch wall at different locations of the domain wall, introducing an asymmetry compared to what is observed in a vortex core [see Figs. 1(d,h)].

  tal artefact or if it has a physical origin remains an open question. However, micromagnetic simulations performed for Py strips with widths and thicknesses in the range of 200-600 nm and 40-70 nm, respectively, allow us to provide an extended phase diagram, describing the regions of stability of the TW, VW and Landau DW [Fig. 2(c)].

  (a)]. Figs. 3(ab) show δ and δ/w as a function of the strip thickness t. The latter clearly reveals two phenomenological scaling laws: δ ≈ w for t < t C and δ ≈ w(t -6∆ d )/(4∆ d ) for t > t C , with ∆ d = 2A/µ 0 M Py 2 the dipolar exchange length. The first scaling law simply reflects the double triangular shape of the VW, rather rigidly kept due to the internal 90 • walls. The second one may be explained with a crude model accounting for the competition between dipolar and exchange energies, and assuming that the Bloch wall extends along the diagonal of the rectangular shape of the Landau DW. This model predicts δ ∼ wt (see Method section), thus describing accurately the slope of the scaling law deduced from the simulations. The absence in the model of the horizontal shift 6∆ d is thought to be linked to the length of the Bloch wall [see Fig. 2(a)], which is smaller than the full diagonal of the supposed rectangular shape of the Landau pattern. Also, the deviation at large t for narrow strips results from the ersatz used in the model to estimate the dipolar energy, assuming a thin strip. Qualitatively, the total length of the wall measured experimentally is well reproduced: while transverse and vortex walls have a typical size comparable to the width of the strip [1, 7], this Landau DW is significantly larger, two -three times the strip width, consistent with the simulations [Fig. 4(b)]. Landau DWs have additional interesting properties. Like (asymmetric) transverse and vortex walls, they have two internal degrees of freedom: a chirality (TW) or circulation (VW and Landau DW), associated with the clockwise or anticlockwise circulation of the in-plane magnetisation, and a polarity, associated with the out-of-plane direction of the magnetisation within the Bloch wall. But unlike the vortex and transverse walls, the Landau DW has a third internal degree of freedom associated with the in-plane direction of the top and bottom

  along the strip length to move the DWs from the straight zone to the bent region of the strip, where they were first nucleated. Similar to what is often observed in thinner Py strips, all Landau DWs could be moved with a relatively small field, 1-2 mT typically. A closer inspection of the MFM images, before and after the application of the external magnetic field, reveals some changes in the internal micromagnetic configurations of the Landau DW. These changes are likely due to the domain wall dynamics during their motion. Understanding the magnetisation dynamics of Landau DWs deserves however more investigations.CONCLUSIONIn summary, we have demonstrated experimentally and numerically the existence of a third type of magnetic domain wall in thick Py nanostrips. This wall is characterised by a three-dimensional micromagnetic structure with an in-plane flux closure configuration curling around a Bloch wall of finite length, similar to a Landau flux-closure pattern. The width of this Landau DW is unusually large and can significantly exceed a few microns in Py strips (see Fig.2and Fig.5). The Landau DW has also interesting properties: besides the chirality of the flux closure configuration and the polarity of the Bloch wall, the wall has well-defined directions for the Néel caps. The static and dynamic properties of this wall could open new prospects in nanomagnetism and field-or current-induced domain wall motion.

FIG. 1 .

 1 FIG. 1. (Color online) Micromagnetic configuration of a head-to-head domain wall in a 200 nm wide, 60 nm thick Py strip. (a-d) Vortex domain wall. (e-h) Landau domain wall. In all images, X, Y and Z stand for the length, width and thickness of the strip, respectively. The red / blue color code gives the amplitude of the magnetisation component along X (a,e), Y (b, f) and Z (c, d, g, h). Black arrows represent the local direction of magnetisation. (a-c, e-g) XY cross section views of the domain walls taken at mid-height (z=30 nm). (d,h) Cross section views of the same domain walls along the dashed white line shown in (c) and (g).

12

 12

FIG. 2 . 14 FIG. 4 .

 2144 FIG. 2. (Color online) (a) MFM image of a head-to-head domain wall in a 1 µm wide, 80 nm thick Py strip. (b) Predicted angle θ for different width of a 80 nm thick Py strip (yellow dots). The dashed line is an extrapolation of the θ decrease as the width of the strip becomes larger. The purple squares are experimental data points deduced from the MFM images of 80 nm thick Py strips. (c) Extended phase diagram of Py strips showing the regions where the TW, asymmetric TW (ATW), VW and Landau DW have the lowest energy. The red and black data points are reproduced from Ref.7.

ACKNOWLEDGMENTS

The authors acknowledge the support of the Nanofab team at the Institut Néel and also thank S. Le-Denmat for technical help with AFM/MFM measurements.

ADDITIONAL INFORMATION

Competing financial interests: The authors declare no competing financial interests.


Damien Givry

Wolff-Michael Roth

Gestures: A Mode of Conceptualization in Science 1 Gestures: A Mode of Conceptualization in Science

Problem

. We reconstruct a student's ideas on the basis of his/her actions, especially, verbal productions, communicative gestures, and manipulations of experiences.

.

Background

Communicative gestures have been classified into different types including those of deictic (pointing), iconic, metaphorical, and beat nature [START_REF] Kendon | Some uses of gesture[END_REF][START_REF] Mcneil | Hand and Mind: what gestures reveal about thought[END_REF][START_REF] Scherer | Les fonctions des signes non verbaux dans la conversation. Dans La communication non verbale[END_REF]. Deictic gestures "point" out some aspect of the context. Iconic gestures and the entities they stand for are characterized by mapping relationships such that both be understood in terms of the same topological features. For an example, a scientist might outline a graph by following it using her finger, and thereby highlights the shape of its line. In a similar way, metaphorical gestures provide a visual expression of a metaphor, even though the concept or idea may be abstract. Finally symbolic gestures function as independent signs and obtain their sense through shared social conventions (raising of middle finger to signal an obscenity) [START_REF] Roth | From gesture to scientific langage[END_REF]. In this study, we will focus only on iconic and metaphoric gestures, especially their synchronization with speech, because together, speech and gesture constitute a good indicator of knowledge [START_REF] Goldin-Meadow | When gestures and words speak differently[END_REF].

Our general methodology for reconstructing student's ideas is based on the following theoretical framework. Vygotsky (1985) considers that the study of language could give us access to how people think. But students' discourse is not simply a "window on the mind", but a "dirty window", because the language used and produced depends on context [START_REF] Edwards | But What Do Children Really Think?: Discourse Analysis and Conceptual Content. In Children's Talk[END_REF]. We used the French concept of contexte pertinent (relevant, salient context;[START_REF] Kerbrat-Orecchioni | La conversation[END_REF] to define the context. Relevant context included the following elements: time of student's actions, task (the question that the students are doing in the sequence on gas), and description of the different actions that the students are doing: talking, reading, writing, handling. Furthermore communication has to be understood as occurring in multiple channels at once [START_REF] Lemke | Multiplying meaning: Visual and verbal semiotics in scientific text[END_REF] and as being interpretively flexible (many possibilities of sense). We therefore transcribed the verbal productions, communicative gestures, and manipulating gestures.

Procedure

In this study, an intensive in-depth case study approach was selected. Eight fifteen-year old French students were followed during a teaching sequence on gases, which had been designed by a team including teachers and researchers (this sequence could be downloaded on the site pegase http://nte-serveur.univ-lyon1.fr/pegase/). The design is based on a socio-constructivist approach with respect to three main dimensions.

1-The modelling activity [START_REF] Tiberghien | Designing teaching situations in the secondary school[END_REF]: students establish links between the world of objects/events and that one of theory/model favours students' learning. 2-The semiotic registers [START_REF] Duval | Sémiosis et pensée humaine, registres sémiotiques et apprentissage intellectuels [Semiosis and human thought, semiotic registers and intellectual leaning[END_REF]: students represent the same concept in various semiotic registers (natural language, diagrams, graphs...) and establishing relations between these registers favours students' learning (e.g., force). 3-The students' conceptions of gases [START_REF] Benson | Students' preconceptions of the nature of gases[END_REF][START_REF] Méheut | Les limites de l'atomisme enfantin : expérimentation d'une démarche d'élaboration d'un modèle particulaire par des élèves de collège[END_REF][START_REF] Séré | analyse des conceptions de l'état gazeux qu'ont les enfants de 11 à 13 ans en liaison avec la notion de pression, et propositions de stratégies pédagogiques pour en faciliter l'évolution[END_REF][START_REF] Stavy | Chidren's conception of gas[END_REF]: students construct new knowledge from initial knowledge. Studying the evolution of student's ideas in a fine-grained manner involves analysing the students' learning processes during a real teaching sequence. We therefore filmed the eight students continuously and collected their written productions-there were three experimental sessions of 1h 30 and three working sessions of 1h. On the whole, the video corpus duration is 8 hours and the written production consists of 10 pages per student approximately. "Syringe" "we close with the finger to enclose air" Data Analyses and Findings. The analysis of our data video show how synchronization between gestures and speech could be a pertinent indicatory of student's conceptualization. Gestures are interpretively flexible 3 and the same gesture can be used to refer to different entities and have therefore different sense. This sense depends on context. For example, the closing of the hand (Figure 1) in itself does not "mean"; it is together with speech and context that the topic of communication is established.

Example of transcription with the elements of context (

Figure 1: Gesture of bringing together her finger

We give an example of this gesture in a different context during the second working session (1h). In this session students are working in small group of four persons.

The first example concerns the measure of pressure in a syringe when the student was decreasing the volume of the syringe. In this extract, student A explains the question to another student M. It is the beginning of the working session (Time = 10:53 m) 01 A: and after it's when we push when we ma-when we make smaller the volume (gesture of bringing together her finger) 02 M: when we make smaller (M is mocking of A's pronunciation) 03 A: when we decrease the volume

In line 1, student A is searching for words and finally she uses the verb "rappetissir" (make smaller). In this case gesture and speech occurred simultaneously. We hypothesize that student A use this gesture because it allows her to specify the sense of the word. And this hypothesis should be confirm by the following fact: when she employed the correct physics word "decrease" (line 3), she did not employ a concurrent gesture. Afterwards, she never used gestures again when uttering sentences involving the "decrease of volume".

The second example involves the description of actions of a gas in the syringe while the student was decreasing the volume of the syringe. This example is during the middle of the working session (Time = 31:36 m). In this small extract, student A and E are talking about the behavior of molecules.

01 E: what are you writing? 02 A: that the that the (gesture of bringing together her finger) 03 Student A uses the same gesture to do a description of the molecules behavior. But in this case, she has not specifics words. Afterwards, she searches a word to describe this phenomenon.

This research of good word is illustrating by the last extract ( This extract shows that student A had developed an idea about molecules behavior but she did not have a word to describe. She used only the same metaphoric gestures.

These tree examples illustrate how synchronization between gesture and speech is a good criterion for establishing a student's current conceptualizations :

• When a gesture appears before speech (example 2 & 3), we a student is at the beginning of the conceptualization. Because he/she has an idea about the phenomenon but he/she hasn't any word to express it. • When gesture and speech are simultaneous (example 1), the meaning of the word is not enough precise and gesture should be like a kind of "complement" of meaning. • When speech is used without a gesture (example 1), we consider that meaning of word has overtake some stability. Our data illustrate the fact that the same gesture should be have different meaning and associate to different concepts (volume (example 1) and molecules (example 2 & 3)).

Contribution.

In the past, research on conceptions and conceptual change has relied solely on students' utterances to specify their knowledge and understanding. This study shows the significant contribution to the specification of conceptions and conceptual change that are made by the study of gestures. This study allowed us to specify criteria for following the evolution of students' learning and development with respect to their conceptual knowledge.

Table 1

 1

	):


Laetitia Laguzet
email: laguzet@ceremade.dauphine.fr

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-10 Influenza A

Keywords: individual vaccination, mean field games, vaccine scares, SIR model, epidemic control, Nash equilibrium MSC: 49, 62P10, 92B05

H1N1) epidemic in France

Introduction

The vaccination, when available, is one of the most classical defense against an evolving epidemic. Theoretical works have been proposed to describe the optimal vaccination policy as function of the vaccine cost and epidemic dynamics and severity, see for instance [START_REF] Hethcote | Optimal vaccination schedules in a deterministic epidemic model[END_REF][START_REF] Abakuks | Optimal immunisation policies for epidemics[END_REF][START_REF] Morton | On the optimal control of a deterministic epidemic[END_REF][START_REF] Sethi | Optimal control of some simple deterministic epidemic models[END_REF][START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF][START_REF] Laguzet | Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis[END_REF]. The point of view of these initial studies is that of a benevolent planner that optimizes the overall societal welfare. However in most situations vaccination is not compulsory and people have the choice to vaccinate or not. For a number of childhood diseases (with no compulsory vaccination) a decay in the vaccination coverage has been observed in developed countries and debates over the danger of vaccines and their usefulness emerged. This suggested that the individual point of view has also to be taken into account: any individual may choose to vaccinate or not and the epidemic dynamics is influenced by all these individual choices. The individual defines a cost r V incurred if he takes the vaccine and a cost r I incurred if infected; the costs can be expressed in terms of money, medical side effects or general morbidity. For a discussion on this topic refer to [START_REF] Zeckhauser | Where now for saving lives? Law and Contemporary Problems[END_REF], [START_REF] Anand | Disability-adjusted life years: a critical review[END_REF], [START_REF] Sassi | Calculating QALYs, comparing QALY and DALY calculations[END_REF] and related literature on QALY/DALY measuring scales. The cost r I is to be weighted by the probability to be infected; it is immediate that when the epidemic is near extinction there is very low incentive to vaccinate (because the probability to be infected is small). Thus people stop vaccinating before the epidemic stops, which can cause in fact the epidemic to start again. In theory the situation can be highly unstable oscillating between no vaccination, followed by an epidemic outburst, then massive vaccination followed by epidemic near extinction and so on. An important question is the existence of a stable equilibrium and the impact on the vaccination coverage.

Several works on this topic appeared as early as [START_REF] Fine | Individual versus public priorities in the determination of optimal vaccination policies[END_REF], [START_REF] Brito | Externalities and compulsary vaccinations[END_REF] and [START_REF] Geoffard | Disease eradication: Private versus public vaccination[END_REF] and ask the question of disease eradication, market equilibrium and externalities in relationship with vaccination; recently [START_REF] Bauch | Group interest versus self-interest in smallpox vaccination policy[END_REF] (using a SEIR model) contributed to the renewal of interest on vaccination policies and individual decisions. In a subsequent paper, [START_REF] Bauch | Vaccination and the theory of games[END_REF] use a SIR model with vital dynamics to describe the propagation of a childhood disease; the individual choices converge to a sub-optimal vaccine coverage. The vaccination strategy of the individual is of all-or-nothing type and time-independent. Furthermore, [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] proposes a learning process based on a "rule of thumb" to explain why and how the people vaccinate, resulting in a time-dependent vaccination strategy. The presence of a time-dependent optimal strategy is more realistic and an advance over the previous work but the model is dependent on the "rule of thumb" chosen a priori. This study was completed by [START_REF] Shim | The influence of altruism on influenza vaccination decisions[END_REF] where the depart from an 100% egocentric decision is investigated. [START_REF] Coelho | Dynamic modeling of vaccinating behavior as a function of individual beliefs[END_REF] and [START_REF] Codeço | Vaccinating in diseasefree regions: a vaccine model with application to yellow fever[END_REF] also model the vaccination behavior and apply the results to the yellow fever scare in Brazil. [START_REF] Buonomo | Global stability of an SIR epidemic model with information dependent vaccination[END_REF] also introduce a feed-back mechanism but for a SEIR system.

In a very elegant approach, [START_REF] Francis | Optimal tax/subsidy combinations for the flu season[END_REF] studied a situation that match very well our setting: the equilibrium in the SIR model. They study in addition the impact of taxes and subsidies and propose revenue-neutral health policies to encourage vaccination. Although the techniques are very intuitive, they do not explicitly introduce a cost functional for the individual and as such this approach has yet to be extended to general settings (for instance when using discounting). We present in Section 3 additional comments comparing the two approaches.

In another work [START_REF] Reluga | Evolving public perceptions and stability in vaccine uptake[END_REF] studied the impact of the perceptions of the relative risk between vaccine and epidemic; they used a SIR model with vital dynamics and also discussed the imitation dynamics. [START_REF] Galvani | Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum[END_REF] consider a double SIR periodic model of influenza with vaccination and two age groups (more or less than 65 years). Vaccination is separated from dynamics and arrives once at the beginning of each season. The effect of the group dependent vaccination is analyzed. [START_REF] Cojocaru | Dynamics of vaccination strategies via projected dynamical systems[END_REF] and [START_REF] Cojocaru | Dynamic equilibria of group vaccination strategies in a heterogeneous population[END_REF], [START_REF] Chen | A susceptible-infected epidemic model with voluntary vaccinations[END_REF] also consider the mathematical questions related to the presence of several groups having distinct epidemic characteristics.

Further contributions for models with vital dynamics include d 'Onofrio et al. (2007), d 'Onofrio et al. (2008), [START_REF] Reluga | A general approach for population games with application to vaccination[END_REF].

A review on the relationship between human behavior and epidemic dynamics is proposed by [START_REF] Funk | Modelling the influence of human behaviour on the spread of infectious diseases: a review[END_REF]. In particular they discuss the relationship between the timescales of the vaccination and epidemic dynamics, which is also the object of several contributions by [START_REF] Vardavas | Can influenza epidemics be prevented by voluntary vaccination?[END_REF], [START_REF] Breban | Mean-field analysis of an inductive reasoning game: Application to influenza vaccination[END_REF]. [START_REF] Chen | Modeling the effect of information quality on risk behavior change and the transmission of infectious diseases[END_REF] discusses how the available information change the decisions made by an individual.

Even if realistic vaccination behavior is very likely to depend on the imitation and altruistic dynamics, there is a need to separate this part from the optimization of the cost-benefit ratio at the individual level.

A recent example of low individual vaccination is the 2009-10 influenza A (H1N1) epidemic. In this case the vaccine is only relevant for one epidemic season and therefore previous models do not describe accurately the dynamics. We compare in Table 1 the difference between the target coverage of the vaccines, as defined by the sanitary authorities, and the effective rate of vaccination obtained. In many European countries one order of magnitude separates the two.

But today there is still need for theoretical guidance to explain this data with existing models. In particular we expect the optimal vaccination strategy to be time-dependent. We also want to identify the vaccination dynamics and propose a model that only takes into account the effect of the individual cost optimization; such a result can then help identify the impact of other effects: imitation, altruism.

We prove the existence of an equilibrium between individual vaccination and an epidemic propagation described by a deterministic SIR-model (with or without discounting). We compare the theoretical result with the optimal pol- (2010), [START_REF] Mereckiene | and the VENICE project gatekeepers group. Influenza A(H1N1)pdm09 vaccination policies and coverage in Europe[END_REF], [START_REF] Walter | Monitoring pandemic influenza A(H1N1) vaccination coverage in Germany 2009/10results from thirteen consecutive cross-sectional surveys[END_REF][START_REF] Schwarzinger | Positive attitudes of french general practitioners towards A/H1N1 influenzapandemic vaccination: A missed opportunity to increase vaccination uptakes in the general public?[END_REF], [START_REF] Brien | The determinants of 2009 pandemic A/H1N1 influenza vaccination: A systematic review[END_REF], (Door, 2010, page 157).

icy at the societal level and describe the differences between the two. Finally we apply the theoretical result to the Influenza A (H1N1) 2009-10 epidemic season in France and observe that people were not homogeneous in their perception of the r V /r I quotient: while some saw the vaccine as harmless, a non-negligible portion of the population saw the vaccine as potentially risky.

The model

We consider an epidemic spreading in a non-immune population; the dynamics of the epidemic follows a SIR model [START_REF] Anderson | Infectious Diseases of Humans Dynamics and Control[END_REF]May, 1992, Diekmann and[START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF]. The epidemic is supposed to take place at a rapid time frame (several months up to 1-2 years) when compared with the demographic dynamics (births, deaths) and therefore the model does not take into account any vital dynamics.

The costs incurred by an infected individual (either in terms of monetary value or of medical condition) are the same for any individual and are denoted by r I . We also suppose that a vaccine giving lifelong immunity exists. Its cost r V is considered known by all the individuals and takes into account not only the economic price but also all possible side-effects of the vaccine. The people in the Susceptible class can choose to vaccinate or not (people in other classes cannot vaccinate).

The mathematical equation describing the SIR model with vaccination is:

dS(t) = -βS(t)I(t)dt -dU (t), S(0 -) = S 0 -, dI(t) = βS(t)I(t) -γI(t) dt, I(0 -) = I 0 -. (1)
Here dU (t) is the vaccination rate, S(t) is the proportion of individuals in the Susceptible class and I(t) is the proportion in the Infected class. The proportion of people in the Recovered class is t 0 -γI(τ )dτ and V (t) := t 0 -dU (τ ) is the proportion of people that vaccinated by the time t. The parameters β, γ and the initial conditions S 0 -and I 0 -are supposed known. Note that when vaccination is instantaneous the function S(t) may be discontinuous and its derivative is a measure because dU (t) only exists as a measure. We refer to the Appendix A for the mathematical details.

Equations (1) only represents the overall, societal dynamics. At the microscopic level, the individual dynamics is modeled by a continuous time Markov chain with individual jumping between the Susceptible, Infected, Recovered and Vaccinated classes, as illustrated in Figure 1. The jump from the Susceptible to the Vaccinated class depends on the willingness of the individual to vaccinate. We suppose that each individual is aware of the propagation dynamics at the societal level (1) but cannot influence it. His only concern is whether it is useful for himself to vaccinate or not and when to do it. This decision is called a strategy. A first interpretation of the strategy is the instant t ∈ [0, ∞[ when the individual vaccinates (if he is still in the Susceptible class at time t). Such a strategy is denoted Π t ; if the individual never vaccinates his strategy is Π ∞ . However it turns out (see discussion in Appendix B) that it is better to represent the individual strategies as mixed strategies which are probability laws on the set of all pure strategies. We represent such a probability law by its CDF (cumulative distribution function) ϕ V with ϕ V (0 -) = 0; then ϕ V (t) represents the probability to choose a pure strategy in the interval [0, t]. When ϕ V (∞) = 1 this means that with probability 1ϕ V (∞) the individual never vaccinates. When ϕ V (0) > 0 then the individual vaccinates immediately with probability ϕ V (0)ϕ V (0 -) = ϕ V (0). When the vaccination capacity is limited by a constant u max > 0, the CDF ϕ V has to comply with the constraints given in equation ( 19).

Another useful mathematical object is the cumulative probability of infection in [0, t], denoted ϕ I (t):

ϕ I (t) = 1 -e -t 0 -βI(τ )dτ . (2)
Note that ϕ I (t) depends on S 0 -, I 0 -and U ; however to ease notations this dependence will not be written explicitly.

Given the epidemic dynamics (fully determined by S 0 -, I 0 -and U ) the individual can associate a cost, denoted J pure (•), to each pure strategy. For instance J pure (Π 0 ) = r V . For general t ∈ [0, ∞[ the cost of Π t is the sum of the cost of infection r I weighted by the probability ϕ I (t) that infection occurs before time t plus the cost of vaccination weighted by the probability 1ϕ I (t) that infection did not occurred before time t:

J pure (Π t ) = r I ϕ I (t) + r V (1 -ϕ I (t)).
(3)

The non-vaccination strategy Π ∞ costs r I ϕ I (∞). Summing up the terms, the cost of the mixed strategy with CDF ϕ V (t), denoted J indi (ϕ V ), is:

J indi (ϕ V ) = (1 -ϕ V (∞))J pure (Π ∞ ) + ∞ 0 - J pure (Π t )dϕ V (t) = r I ϕ I (∞) + ∞ 0 - r V -r I ϕ I (∞) + (r I -r V )ϕ I (t) dϕ V (t). ( 4
)
Alternative equivalent definitions have been proposed in the general context of timing games, see (Fudenberg and Tirole, 1991, page 118).

When a discount factor D > 0 is introduced, the discounted risk of infection is:

Φ I (t) = t 0 - e -Dτ dϕ I (τ ), Φ I (0 -) = 0.
(5)

The cost of pure strategies is for t ∈ [0, ∞[:

J D pure (Π t ) = r I Φ I (t) + e -Dt r V (1 -ϕ I (t)). (6)
In this case

J D pure (Π ∞ ) = r I Φ I (∞).
The the cost of the mixed strategy with CDF ϕ V (t) is:

J D indi (ϕ V ) = r I Φ I (∞) + ∞ 0 - r I (Φ I (t) -Φ I (∞)) + r V e -Dt (1 -ϕ I (t)) dϕ V (t).
(7) We work under the constraints (S, I) ∈ Ω, where

Ω = {(S, I) ∈ R 2 | S, I > 0, S + I < 1}. ( 8
)
Recall that the global dynamics is an aggregation of individual dynamics. Therefore I(t) in Equation ( 1) is the same as in ( 16) and a compatibility relation has to exist between dU (t) (societal vaccination) and ϕ V (individual vaccination).

To make explicit this last compatibility requirement one has to investigate the relationship between the Markov chain of one individual and the evolution dynamics of the population; such an endeavor is beyond the scope of this work but has been explored in several papers starting with [START_REF] Kurtz | Solutions of ordinary differential equations as limits of pure jump Markov processes[END_REF], see also Sandholm (2010, Theorem 10.2.3 pages 373-374) for an application to population dynamics. Further results very close to the setting of this work are given by [START_REF] Gomes | Continuous time finite state mean field games[END_REF], [START_REF] Guéant | Existence and uniqueness result for mean field games with congestion effect on graphs[END_REF]. The conclusion is that the societal dynamics is compatible with the individual dynamics when:

dU (t) = dϕ V (t) 1 -ϕ V (t) S(t). (9)
An intuitive understanding of this formula is the following: for a given individual, the probability that it is vaccinated at time t + ∆t provided that it was not vaccinated at time t equals ϕ

V (t+∆t)-ϕ V (t) 1-ϕ V (t)
+ o(∆t) (see Equation ( 15) for a similar computation). At the same time, since individuals are all the same, the same probability has to be U (t+∆t)-U (t)

S(t)

+ o(∆t). Comparing the two in the limit ∆t → 0 we obtain the Equation (9).

Questions regarding the equilibrium

When everybody is vaccinating this collective behavior stops the epidemic. For a given individual that still has to make its own choice, the perceived benefit of vaccination is low because the risk associated with the epidemic is very low. Why should he vaccinate then ? The individual will therefore not act as everybody else.

On the contrary, when nobody vaccinates and the individual foresees a severe epidemic (because of lack of vaccination for instance), the individual is lead to vaccinate, therefore will not act as everybody else.

In both situations there is an incoherence between the overall, societal, dynamics and the individual self-interest. A legitimate question is whether an equilibrium scenario exists where the best vaccination policy of the individual and the vaccination policy of everybody else agrees.

To illustrate this question consider the Figure 2 corresponding to the model in Section 1.1. Vaccination is represented by a parameter indicated the time during which vaccination takes place. A low time indicates low vaccination level and a large time a high vaccination level. We plot a three-dimensional surface of the cost J indi (θ I ) incurred by an individual which has the vaccination time θ I in a circumstance when the societal vaccination time is θ G . Note that, with an abuse of notation, we write

θ G instead of dU = 1 [0,θ G ] u max dt and θ I instead of the solution of dϕ V (t) = (1 -ϕ V (t))1 [0,θ I ] u max /S(t)dt, see Appendices A and B.
For any societal vaccination strategy represented by the parameter θ G the individual minimizes θ I → J indi (θ I ). The optimal value θ opt I of θ I as a function of θ G is represented as the solid bottom curve in the Figure 2. The existence and uniqueness of the function θ opt I (θ G ) (for general societal vaccination strategies dU , not necessarily given in terms of vaccination time) is discussed in Section 2.1 and Appendix C for D = 0 and Section 3 and Appendix G for D > 0. As expected, for low θ G the curve θ opt I (θ G ) takes a large value while for large values of θ G the curve θ opt I (θ G ) takes small values. The equilibrium is when θ opt I (θ G ) = θ G . We therefore ask for the existence of a common point of the solid and dashed curves. The existence and uniqueness of an equilibrium is discussed in Section 2.2 and Appendix D for D = 0 and Section 3 and Appendix G for D > 0. Such questions have been given a firm mathematical ground since the introduction of the "Mean Field Games" theory in the pioneering works of Lasry and Lions (2006a,b), [START_REF] Huang | Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the nash certainty equivalence principle[END_REF][START_REF] Huang | Nash equilibria for large-population linear stochastic systems of weakly coupled agents[END_REF] (see also [START_REF] Gomes | Continuous time finite state mean field games[END_REF], [START_REF] Guéant | Existence and uniqueness result for mean field games with congestion effect on graphs[END_REF] for related aplications).

Illustration of the theoretical results: the undiscounted case

We consider in this Section the undiscounted case corresponding to D = 0.

Individual optimal strategy for arbitrary epidemic propagation

If the societal vaccination strategy dU is given, there exists an individual strategy ϕ V that minimizes the individual cost (4). The strategy is unique (except degenerate cases). The technical details concerning the existence, uniqueness and other properties of the optimal individual strategy are proved in the Appendix C. Here we only illustrate these results in a particular case.

The theoretical results tell us that, in order to find the optimal decision at time t, an individual has to calculate the quantity ϕ I (∞)-ϕ

I (t) 1-ϕ I (t) and compare it with r V /r I . While ϕ I (∞)-ϕ I (t) 1-ϕ I (t) ≥ r V /r I it is optimal for the individual to vaccinate, otherwise he should not vaccinate. Note that ϕ I (∞)-ϕ I (t) 1-ϕ I (t)
can only decrease during the course of an epidemic.

The Figure 3 gives an example of such a situation with U = 0. Until the time 0.06, ϕ I (∞)-ϕ I (t) 1-ϕ I (t) ≥ r V /r I and the individual vaccinates. After this time individual vaccination stops. The probability to be vaccinated is constant equal to 0.07% after time 0.06. 19). Left: The trajectory (S(t), I(t)) of the system (1) with U = 0. Right: The decreasing dotted curve is t →

ϕ I (∞)-ϕ I (t) 1-ϕ I (t)
, the constant densely dashed line is the level r V /r I and the increasing dashed-dotted curve is the probability of vaccination over time.

Equilibrium strategies

In classical settings that disregard individual choices (see (Abakuks, 1974, Laguzet and[START_REF] Laguzet | Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis[END_REF]) the cost for the society defined as:

J soc (S 0 -, I 0 -, U ) = ∞ 0 - r I S(t) dϕ I (t) 1 -ϕ I (t) + r V dU (t), (10)
is the only quantity of interest and is minimized. An optimal societal strategy can be found, which will be denoted from now on (OS). The strategy (OS) partitions the domain Ω into a region of vaccination and a region without vaccination. The frontier of the two regions is the curve {(S,

I) ∈ Ω |∂ S ζ(S, I) = r V /r I }.
Remark 1 We recall that for any (S, I) ∈ Ω one can introduce the number ζ(S, I) of infected people (in absence of vaccination) for a trajectory starting in (S, I). In particular ζ is the solution of the following equation (see [START_REF] Abakuks | Optimal immunisation policies for epidemics[END_REF][START_REF] Abakuks | Some optimal isolation and immunisation policies for epidemics[END_REF][START_REF] Laguzet | Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis[END_REF]):

1 -ζ(S, I)/S = e -β γ (I+ζ(S,I)) . (11)
In our setting, the situation is not a simple optimization. Any individual optimizes its cost but the coherence of the model requires that the aggregation of individual optimal policies ϕ V result in the global vaccination policy dU . Therefore the situation is more adequately described by an equilibrium in the form of a fixed point property: a global vaccination dU gives optimal individual responses ϕ V which sum up to form a societal response dU . The equilibrium is reached only when dU = dU .

The first good news is that the results in Appendix D show that an equilibrium always exists, i.e., even if the individuals are only driven by self-interest some vaccination level is conserved (and does not drop to zero). This is completely coherent with other results from the literature, see [START_REF] Bauch | Vaccination and the theory of games[END_REF] and related works.

The couple of optimal individual strategy and resulting societal strategy in equilibrium will be denoted from now on (EIS). The equilibrium (EIS) is also described as a partition of Ω into a region of vaccination and a region without vaccination. Individuals will vaccinate in the first region and stop vaccination upon reaching the boundary of the second region. The line that delimits the two regions is of equation:

{(S, I) ∈ Ω |I + (r V /r I )S + (γ/β) ln(1 -r V /r I ) = 0 } . ( 12
)
The Proposition 1 shows that the vaccination region of the (OS) strategy includes strictly the vaccination region of the (EIS) strategy.

The Figure 4 presents the comparison of the regions in Ω. The response of an individual to a given state (S, I) of the epidemic dynamics depends on the location of (S, I) in Ω. In the gray region the individual (in the (EIS) strategy) will vaccinate and also the (OG) strategy is to vaccinate. In the hashed region the individual (in the (EIS) strategy) will not vaccinate but the (OG) strategy is to vaccinate. In the white region the individual (in the (EIS) strategy) will not vaccinate and the (OS) strategy is to not vaccinate.

Therefore only the hashed region, delimited by the curve ∂ S ζ(S, I) = r V /r I and the line I + (r V /r I )S + (γ/β) ln(1r V /r I ) = 0, is conflictual, in the sense that the individual does not have incentive to vaccinate but the societal best decision is to vaccinate. In this region the individual will tend to free-ride on the vaccination of the rest of the society. We illustrate in Figure 5 the societal trajectories for three possible strategies in equilibrium, that is, satisfying the constraint (9) (but not all optimal at the individual level). The dashed curve with no label corresponds to no vaccination and the individual cost is 0.65. The second dashed curve corresponds to the (EIS) strategy: between point (S 0 -, I 0 -) and A, the individual vaccinates (dϕ V = u max /S(t)(1ϕ V (t))) and after point A, there is no vaccination (dϕ V = 0). The individual cost is 0.55. The third curve corresponds to the trajectory with vaccination coherent with the (OS) societal strategy: vaccination occurs between point (S 0 -, I 0 -) and B. The individual cost is 0.53.

A very counter-intuitive fact is that the individual cost for the (OS) strategy is lower than the cost of the (EIS) strategy. Is this not in contradiction with the fact that the (EIS) strategy is the optimal strategy for an individual ? How can be something better than the optimal ? In fact the equilibrium is a Nash equilibrium which is not globally optimal. Although everybody would be better off adopting the (OS) strategy this choice is not stable at the individual level. A new individual added to such a population has the incentive to vaccinate less, driving the equilibrium towards the (EIS) stable point. See also Appendix E for the relation between the two costs. In game theory such a circumstance is termed "price of anarchy".

Illustration of the theoretical results: the discounted case

We consider in this Section D > 0. Such a situation appears when individuals favor the present more than the future: when faced with two identical events, one in t > 0 years and one in the present, the event in the future is seen as less severe (or less beneficial) than the event in the future. The attenuation factor is by definition e -Dt . There is no general prescription on the precise numerical value to use for D because it is specific to the subjective behavior of the individuals in relationship with a given epidemic. Discounting has been also discussed in the context of the QALY/DALY scales (see [START_REF] Zeckhauser | Where now for saving lives? Law and Contemporary Problems[END_REF], [START_REF] Anand | Disability-adjusted life years: a critical review[END_REF], [START_REF] Sassi | Calculating QALYs, comparing QALY and DALY calculations[END_REF]).

This situation features several important novelties. First, the optimal individual strategies are not based on the comparison between the infection risk and the vaccine cost. This approach, used profitably by [START_REF] Francis | Optimal tax/subsidy combinations for the flu season[END_REF] (see also [START_REF] Francis | Dynamic epidemiology and the market for vaccinations[END_REF]) is only valid when there are precisely two regions (a vaccination and a non-vaccination region) and the only pure strategies ever used are Π 0 and Π ∞ . Or, in general, it is impossible to know in advance the structure of the solution and therefore their approach needs to be extended, as shown in this work.

When D > 0, vaccinating in the future makes sense when I 0 -is small and S 0 -large because the infection risk is temporarily attenuated by the discount factor. The criterion is the difference between the quantity r V (1e -D∆t ) (gained from postponing vaccination ∆t units of time) and the increase in the risk of infection during the ∆t units of time. The correct solution requires proper consideration of the probability law ϕ V , see Appendix G for details. A graphical illustration of this situation is given in Figure 6.

When D > 0 the equilibrium features three types of regions: a no-vaccination region Ω n , a instantaneous vaccination region Ω i and a delayed vaccination region Ω d . The zone Ω d does not exist when D = 0 or D > 0 and r V /r I is large; however Ω d appears in the realistic situation when D > 0 and r V /r I is small. The Figure 7 illustrates this partition. r V e -Dt (1-ϕ I (t)) ; the initial value is 0.3897. The vaccination does not occur when the risk of infection reaches r V for the first time (at time t = 0.0048) nor at the maximum value of the infection risk (time t = 0.0425).

When the initial point (S 0 -, I 0 -) ∈ Ω d the equilibrium is such that vaccination is not stable and will not be requested until the infection level I = I * has been reached (see Appendix F for the definition of I * ). The equilibrium dynamics is as follows: if (S 0 -, I 0 -) ∈ Ωn it will always remains there; if (S 0 -, I 0 -) ∈ Ω i partial vaccination will bring it to Ωn and total vaccination to the boundary S = 0; if (S 0 -, I 0 -) ∈ Ω d it will evolve to some point (S, I * ), then vaccinate till the point (S * , I * ) on the boundary of all three regions, then will evolve in the region Ωn. If on the contrary r V /r I = 0.6 we have Ω d = ∅; this situation is not illustrated here as it is similar to Figure 4).

An application to the Influenza A (H1N1) 2009/2010 epidemic in France

We apply in this Section the previous theoretical results to a practical situation. Recall that in our model individuals are all the same and are rational optimizing agents with perfect knowledge and foresight. Although this is never true in reality, the model may still prove useful in order to understand in what regards the reality is different from the model. In particular we will show here that it is unlikely that all individuals shared the same r V in the specific example chosen in this Section.

Epidemic context

The 2009/2010 Influenza A (H1N1) epidemic in France is a recent example of vaccination campaign that displays a large difference between the target, planned, vaccination coverage and the effective coverage obtained at the end of the campaign (see also [START_REF] Schwarzinger | Positive attitudes of french general practitioners towards A/H1N1 influenzapandemic vaccination: A missed opportunity to increase vaccination uptakes in the general public?[END_REF]).

The 2009/2010 worldwide H1N1 epidemic spread through 213 countries and has been attributed 18 156 deaths by 15/06/2010 (when epidemic was declared over by the WHO). In France the first cases appeared in May 2009 and 1334 severe forms were declared out of, approximately, 7.7 to 14.7 Millions people infected. The vaccination campaign in France was costly (around 500M EUR, although cost estimation vary) for a low efficiency of 8% coverage (to be compared with 24% in the US or 74% in Canada) although the target was set to 75%. During the French vaccination campaign some undesired neurological side-effects of the adjuvant present in the vaccines were under debate and known to the public.

The campaign was designed in waves (people were called to the vaccine centers according to their risk status) and all vaccination centers were not 100% operational at once. Moreover the government mobilized additional medical personnel (military, medicine interns,...) latter during the epidemic propagation. The joint combination of these effects is that the maximum vaccination capacity was non-constant increasing from zero to some peak obtained by the end of 2009.

Data sources

The information concerning the vaccination capacities were available in the general news but no quantitative estimation is, up to our knowledge, available. However the actual cumulative vaccination curve was reported by [START_REF] Guthmann | Insufficient influenza A(H1N1) 2009 vaccination coverage in the global population and high risk groups during the 2009-2010 pandemic in France[END_REF] (up to the maximum coverage of 7.9%). The vaccine effectiveness coefficient was estimated by [START_REF] Valenciano | Estimates of pandemic influenza vaccine effectiveness in Europe, 2009-2010: Results of influenza monitoring vaccine effectiveness in Europe (I-MOVE) multicentre case-control study[END_REF] to be 71.9%. We set U in the model (1) to fit this data, see Figure 8. In what concerns the size of the Infected class, the French "Sentinel" network (cf. [START_REF] Flahault | Virtual surveillance of communicable diseases: a 20-year experience in France[END_REF]) reports the estimation of the number of infections per week starting from the beginning of the epidemic. Using the same data Valleron and Guidet (2010) compared the epidemic with seasonal epidemic dynamics. Finally [START_REF] Bone | Incidence of H1N1 2009 virus infection through the analysis of paired plasma specimens among blood donors, France[END_REF] measure seroconversion and obtain results on the number of people immunized at the end of the epidemics. All this data was used to estimate the curve I(t) in the model (1). The data obtained by the Sentinel network takes into account the number of consultations in medical offices. However some infected persons are asymptomatic (but contribute to the propagation) and thus only some percentage of the infected individuals will consult a medical doctor. This percentage is difficult to evaluate; we follow the European Centre for Disease Prevention and Control (ECDC) (2009) who estimated this rate at 50%; thus to estimate the number of infections we set I(t) two times larger than the curve reported by the Sentinel network.

W 4 3 Y 0 9 W 4 6 Y 0 9 W 4 9 Y 0 9 W 5 2 Y 0 9 W 0 2 Y 1 0 W 0 5 Y 1 0 W 0 8 Y 1 0 W 1 1 Y 1 0 W 1 4 Y 1 0 W 1 7 Y 1 0 0 2 4 6 •10 -2 Data Model W 4 3 Y 0 9 W 4 6 Y 0 9 W 4 9 Y 0 9 W 5 2 Y 0 9 W 0 2 Y 1 0 W 0 5 Y 1 0 W 0 8 Y 1 0 W 1 1 Y 1 0 W 1 4 Y 1 0 W 1
The propagation parameters β and γ were chosen consistent with ranges from the literature (see [START_REF] Boëlle | Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review[END_REF]) although large confidence intervals were present, see for instance [START_REF] Carrat | Planning for the next influenza H1N1 season: a modelling study[END_REF]. We took R 0 = β/γ = 1.35. We fit γ in a very limited range, S 0 -in the range 0.8 to 0.95 (compatible with estimation concerning already immunized persons) and I 0 -. The parameters were optimized in order to reproduce the cumulative number of infections ∞ 0 -βS(t)I(t)dt and the peak of the epidemic. We obtained γ = 365/3.2, S 0 -= 0.84 and I 0 -= 2 × 10 -6 . The overall fit obtained is described in Figures 8 and9. Given the large uncertainties surrounding the propagation parameters and the number of infections the fit is considered very satisfactory. Given the short time of propagation of the epidemic we took

D = 0. W 1 9 Y 0 9 W 2 3 Y 0 9 W 2 7 Y 0 9 W 3 1 Y 0 9 W 3 5 Y 0 9 W 3 9 Y 0 9 W 4 3 Y 0 9 W 4 7 Y 0 9 W 5 1 Y 0 9 W 0 2 Y 1 0 W 0 6 Y 1 0 W 1 0 Y 1 0 W 1 4 Y 1 0 W 1 8 Y 1 0 0 0.5 1 1.5 •10 -2 Data Model W 1 9 Y 0 9 W 2 3 Y 0 9 W 2 7 Y 0 9 W 3 1 Y 0 9 W 3 5 Y 0 9 W 3 9 Y 0 9 W 4 3 Y 0 9 W 4 7 Y 0 9 W 5 1 Y 0 9 W 0 2 Y 1 0 W 0 6 Y 1 0 W 1 0 Y 1 0 W 1 4 Y 1 0 W 1 8

Methods

Once the overall dynamics ( 1) is given, there is still the quotient r V /r I to be estimated. Such an endeavor is very difficult because this is depending on the individuals' perception and very few data is available on the heterogeneity of this parameter within the population. Rather than using very uncertain data we preferred to see what the model says about the quotient r V /r I .

On the vaccination intensity Figure 8 (left) one can note that vaccination had a sharp increase up to week 50 -51 of 2009 followed by a sharp decrease. The decrease was not due to the capacity of vaccination centers. We interpret it as reflecting a perception of a low infection risk coupled with high vaccination risk, i.e., a high quotient r V /r I . We can compute the precise value of the quotient r V /r I compatible with our model. In Figure 10 we plot three risk indicators: first of them is ϕ I (∞)-ϕ I (t) 1-ϕ I (t) , the second ζ(S(t),I(t))

S(t)

; the third is the quotient between the number of infections from t to ∞ divided by the number of susceptible people in t (which is the initial value S 0 -minus the total infections and vaccinations between 0 and t). Note that this last criterion is model free and can be computed from the vaccination coverage report and the Sentinel network data (weighted by the asymptomatic infected individuals rate).

We looked therefore at the level of the three criterions that were attained at the time when vaccination decreased. According to our model this level is an indication of the quotient r V /r I .

Results

An individual that stops vaccinating at the peak of the curve in Figure 8 (left) will have r V /r I in the range 5%-10%. This is a huge value meaning that 1 out of 10 vaccinated people will develop side-effects as severe as the influenza A H1N1 itself. Such situations reflect probably a communication failure around the epidemic and the vaccines (infection risk, severity, vaccine side effects, need for a mass vaccination campaign, ...) rather than specific medical data about the risks involved. See also [START_REF] Brien | The determinants of 2009 pandemic A/H1N1 influenza vaccination: A systematic review[END_REF] for a discussion on the determinants of individual vaccination and [START_REF] Basu | Integrating epidemiology, psychology, and economics to achieve hpv vaccination targets[END_REF] for a related discussion concerning the controversial Human papillomavirus (HPV) vaccines.

Note also that vaccination did not stop completely at week 50 -51 of 2009 but continued at detectable levels up to week 10 of 2010. An individual that stops vaccinating at week 10/2010 time has r V /r I < 1% (probably much less as the model cannot be more precise with available data). This is a more common accepted figure.

Therefore at least two groups with very heterogeneous perceptions of the vaccination risk were present: a first group very worried about vaccine risk and another group less pessimistic.

We want to stress again that these results are highly dependent on the quality of the model and should be interpreted with care. Moreover the robustness of the results with respect to the input parameters γ, β, dU have also to be assessed in practice.

Discussion and conclusions

In the context of individual vaccination (as opposed to global, compulsory vaccination) we develop an equilibrium model that allows to quantify the relationship between the individual perception of vaccine side-effects and of the epidemic morbidity.

The first important result is that, even when individuals are only driven by self interest, such an equilibrium exists. Secondly we are able to characterize the individual decision. When there is no discounting the individual will vaccinate or not depending on his estimations of the infection risk compared with the cost of the vaccine. When discounting is taken into account the individual may choose to wait until epidemic risk rises and only then vaccinate.

The equilibrium is a Nash equilibrium with an infinity of players (as pioneered by Lasry and Lions (2006a,b)), which means in particular that it can have (and has) a non-null "price of anarchy". This means that self-interest, although stable from both individual-societal points of view, is not the best solution that can be obtained.

Finally, we apply the results to the Influenza A 2009/2010 vaccination campaign in France and see that, under the assumptions of the model, it is hinted that at least two distinct groups existed, one very pessimistic about vaccine side-effects and the other rather optimistic. Fig. 10 Using the dynamics in (1), fitted with our data, we plot three criterions:

W 1 9 Y 0 9 W 2 3 Y 0 9 W 2 7 Y 0 9 W 3 1 Y 0 9 W 3 5 Y 0 9 W 3 9 Y 0 9 W 4 3 Y 0 9 W 4 7 Y 0 9 W 5 1 Y 0 9 W 0 2 Y 1 0 W 0 6 Y 1 0 W 1 0 Y 1 0 W 1 4 Y 1 0 W 1 8 Y 1 0 0 0.
ϕ I (∞)-ϕ I (t) 1-ϕ I (t) , ζ(S(t),I(t)) S(t)
and the quotient between the infections in [t, ∞[ and the susceptibles in t (see text). By the time the first group of people stopped vaccination (week 50 -51 of 2009) there was still 5 -10% chance for an individual to contract the Influenza A. The individual that do not vaccinate at this point estimates the side effects of the vaccine to occur with frequency of about 5 -10%, which is considered a very large and pessimistic value. Individuals that stopped last (week 10 of 2010) had this figure down to less than 1% (probably much less given data uncertainties). Although the precise values may depend on the parameters we noticed that two distinct groups were found for a wide domain of parameters (compatible with data and with dU ). This work has several limitations which open the way for future developments; the parameters in Section 4 were calibrated in the following way: the reproduction number γ/β was taken in an interval centered around the parameters of seasonal influenza; on the contrary the initial number of Susceptibles (S 0 -) and Infected (I 0 -) was calibrated from data specific to this season. Parameters were optimized to fit the final number of Infected, Vaccinated and give a correct epidemic peak. However it is not certain that only one set of parameters is compatible with data and the robustness has to be investigated, in particular for epidemics which are different from historical data.

In addition, our hypothesis that r V is known may be a limitation in general. Nevertheless, although vaccines are specific to each Influenza season, their risks are generally well understood and can be transferred from one season to another (but the vaccine immunity does not). For the specific case of Influenza A (H1N1) 2009-10 season in France it turns our that the vaccination arrived after several other European countries already started their vaccination programs; data from these earlier programs were largely discussed and even sparkled important controversies in France concerning the vaccine sideeffects. In general, when the risk r V is not known the model could be extended as in [START_REF] Bhattacharyya | Wait and see" vaccinating behaviour during a pandemic: A game theoretic analysis[END_REF] where the authors discuss the learning of r V parameter during the Influenza A (H1N1) 2009-10 epidemic season in North America. Another limitation arises from the schedule of the vaccination campaign: people were assigned to risk groups and in principle, cannot vaccinate without a written invitation received by mail (at a time corresponding to the risk group they belonged). However many centers were under-utilized by mid December 2009, see (Cour des Comptes, page 26) and on the other hand almost 2.7 Millons on-the-spot invitations were issued to people that presented at their initiative for vaccination, see (Door, 2010, page 95). We conclude that absence of invitations did not prevented, by itself, voluntary vaccination at the end of 2009.

A On the societal and individual SIR model with vaccination

The mathematical description of the SIR model with vaccination has to take into account the possibility of instantaneous vaccination, which means that a non-negligible proportion of the total population can be vaccinated instantaneously at some time t. A first consequence is that S(0) can be strictly less than S 0 -(when some vaccination occurs at time 0). Recall that V (t) represents the proportion of people vaccinated by the time t. If, for instance, nobody vaccinates before time t * = 0.25 years (3 months) and 30% of the population vaccinates at time t * this means that V (t) is discontinuous at this point (see Figure 11 for an illustration). In particular it is not derivable and neither will be S(t); as such it is not possible to use in Equation (1) the derivative dS(t)/dt. This explains why the equation is only written in the differential form: dS(t) = -βS(t)I(t)dt-dU (t). In this writing all objects have a well defined mathematical meaning: dU (t) is a positive measure which, for our example will be the Dirac mass 0.3δ t * . We refer to [START_REF] Bressan | Impulsive control systems with commutative vector fields[END_REF], Dal [START_REF] Maso | On systems of ordinary differential equations with measures as controls[END_REF], [START_REF] Miller | The generalized solutions of nonlinear optimization problems with impulse control[END_REF], [START_REF] Silva | Necessary conditions for optimal impulsive control problems[END_REF] for the mathematical properties of the solutions to such evolution equations. In the particular situation when U (t) has a jump at 0 we will have S 0 --S(0) = U (0) -U (0 -) = U (0).

In this dynamics all individuals are the same. Each individual is following a continuous time Markov dynamics jumping between the states Susceptible, Infected, Recovered and Vaccinated. Let Mt be the state of one such individual at time t. The time of the jump from the Infected to the Recovered class is a exponential random variable of mean 1/γ. We write in terms of probabilities:

P M t+∆t = Recovered Mt = Inf ected = γ∆t + o(∆t). ( 13
)
For a given individual in the Susceptible class at time t, the probability to be infected during the time interval [t, t + ∆t] is βI(t)∆t + o(∆t), therefore

P M t+∆t = Inf ected Mt = Susceptible = βI(t)∆t + o(∆t). ( 14
)
0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 t Fig. 12 The individual cumulative probability of vaccination ϕ V (t) is an increasing, right continuous with left limits (càdlàg) function with ϕ V (0 -) = 0, ϕ V (∞) ≤ 1.

On the other hand, denote by ϕ I (t) the probability of infection (in the absence of vaccination) during the time interval [0, t]. Then

P M t+∆t = Inf ected Mt = Susceptible = P (M t+∆t = Inf ected, Mt = Susceptible) P (Mt = Susceptible) = ϕ I (t + ∆t) -ϕ I (t) 1 -ϕ I (t) = 1 1 -ϕ I (t) ϕ I (t)∆t + o(∆t). (15)
Passing to the limit ∆t → 0 in Equations ( 14) and ( 15) we obtain:

ϕ I (t) = βI(t)(1 -ϕ I (t)), ϕ I (0 -) = 0. ( 16
)
therefore ϕ I is given by formula (2). Note that since I(t) is continuous ϕ I is differentiable everywhere.

In particular, for an individual in the Susceptible class at time τ that does not vaccinate any more from that time on, the probability of infection after time τ is:

ϕ I (∞) -ϕ I (τ ) 1 -ϕ I (τ ) . (17)
We can prove, by direct computations, that dU = 0 on [τ, ∞[ implies:

ϕ I (∞) -ϕ I (τ ) 1 -ϕ I (τ ) = ζ(S(τ ), I(τ )) S(τ ) . ( 18
)
Remark 2 Since ϕ I (∞) < 1 the cost Jpure(Π∞) is not the limit of the costs Jpure(Πt) (for t → ∞). On the contrary J D pure (Π∞) = limt→∞ J D pure (Πt).

B Individual strategies

The simplest individual strategy is to vaccinate or not at some given time (provided he is still susceptible). Such a strategy is called a pure strategy. However pure strategies do not always have good theoretical properties and in his Nobel award winning work John Forbes Nash proved that on the contrary, any finite game admits equilibrium if mixed strategies are allowed. A mixed strategy is a probability law on the set of all pure strategies. With the notations of Section 1.1 the mixed strategy are probability laws on [0, ∞] with special meaning of values 0 (immediate vaccination) and ∞ (no vaccination). The CDF function ϕ V (t) is such that ϕ V (0 -) = 0 (no vaccination before time 0). Note that ϕ V is not necessary a continuous function (the discussion is very much similar to the one in Appendix A). In particular when the individual chooses the pure strategy Π t * then ϕ V is the Heaviside function H(• -t * ). See the illustration in Figure 12. It may also be necessary to impose some constraints. Suppose that global vaccination (at the society level) can only happen with the maximal rate of umax percent of population in a unit time. Then, with the notations in Equation (1): U (t + ∆t) -U (t) ≤ umax∆t.

Case a/ If r I ≤ r V , since ϕ I is an increasing function, the quantity in ( 24) is the sum of positive terms. The minimum attainable value is therefore zero and it obtained when dϕ V (t) = 0 ∀t. Or ϕ V (0 -) = 0 thus ϕ V (t) = 0 ∀t ≥ 0. Cases b/ and c/ If r I ≥ r V let us compute

d dt [r V -r I ϕ I (∞) + (r I -r V )ϕ I (t)] = (r I -r V ) dϕ I (t) dt ≥ 0, (25)
(recall that ϕ I (t) is increasing) and moreover r V -r I ϕ I (∞)+(r I -r V )ϕ I (0) = r V -r I ϕ I (∞) is positive as soon as ϕ I (∞) ≤ r V /r I . To minimize the cost, vaccination should not occur when the term to integrate against dϕ V (t) is positive therefore there is no vaccination if ϕ I (∞) ≤ r V /r I . If ϕ I (∞) > r V /r I , vaccination occurs for all t such that r V -r I ϕ I (∞)+(r I -r V )ϕ I (t) ≤ 0, or equivalently

ϕ I (∞)-ϕ I (t) 1-ϕ I (t) ≥ r V r I . Moreover, d dt ϕ I (∞) -ϕ I (t) 1 -ϕ I (t) = - [1 -ϕ I (∞)] [1 -ϕ I (t)] 2 dϕ I (t) dt < 0. ( 26
) Then t → ϕ I (∞)-ϕ I (t) 1-ϕ I (t)
is a decreasing, continuous function from ϕ I (∞) to zero. To minimize the cost, the vaccination should appear right at the beginning and last until the boundary of the domain Ω is attained by the dynamics (1) or until time θ I , the unique solution of

ϕ I (∞)-ϕ I (θ I ) 1-ϕ I (θ I ) = r V r I
. Therefore in order to minimize the integral one has to set dϕ V = 0 on [θ I , ∞[ and vaccinate on [0, θ I [ with maximal values coming first. Taking into account the constraint ( 21) we obtain equation ( 22

) (the constraint umax < ∞ implies that if Z S ≤ θ I then Z S 0 umax S(τ ) dτ = ∞ thus ϕ V (Z - S ) = 1). Remark 3 There is no individual vaccination if r V /r I > 1 -e -β/γ .
The next results applies when the individual vaccination can be unbounded, i.e., dϕ V can contain Dirac masses.

Theorem 2 (Case umax = ∞) The individual strategy ϕ * V that minimizes the cost (4) with the system (1) is:

a/ if r I ϕ I (∞) < r V then ϕ * V ≡ 0, b/ if r I ϕ I (∞) > r V then dϕ * V = δ 0 , c/ if r I ϕ I (∞) = r V then dϕ * V = αδ 0 with α arbitrary in [0, 1]. Proof Case a/ If ∞ 0 -r V -r I ϕ I (∞) + (r I -r V )ϕ I (t)dϕ V (t) is positive, then the mini- mum is greater than r I ϕ I (∞). This value is attained only if ∞ 0 -r V -r I ϕ I (∞) + (r I - r V )ϕ I (t)dϕ V (t) = 0 so ϕ V (t) = 0 for all t. Case b/ The strict monotony of the integrand r V -r I ϕ I (∞) + (r I -r V )ϕ I (t) allows to write: ∞ 0 - r V -r I ϕ I (∞) + (r I -r V )ϕ I (t)dϕ V (t) ≥ ∞ 0 - [r V -r I ϕ I (∞) + (r I -r V )ϕ I (t)] t=0 -dϕ V (t) = [r V -r I ϕ I (∞)] ∞ 0 - dϕ V (t) ≥ r V -r I ϕ I (∞),
where we used ϕ V (∞) ≤ 1 and r V -r I ϕ I (∞) ≤ 0. This gives a lower bound for the minimum. The bound is attained when both inequalities become equalities that is dϕ V (t) is the Dirac mass in 0. Case c/ The difference with the previous case is that the last term r V -r I ϕ I (∞) is 0. Therefore the last inequality is always satisfied. We obtain the conclusion.

Theorem 4 (Case umax = ∞) The equilibrium is: a/ If r V ≥ r I or r V < r I and I 0 -+ (r V /r I )S 0 -+ (γ/β) ln(1 -r V /r I ) ≤ 0 then the unique equilibrium is dU = 0 and dϕ V = 0 (no vaccination). b/ If I 0 -+ (γ/β) ln(1 -r V /r I ) > 0 then the unique equilibrium is dU = δ 0 S 0 -and dϕ V = δ 0 (total instantaneous vaccination). c/ Otherwise the unique equilibrium is dU = α * δ 0 and dϕ V = (α * /S 0 -)δ 0 (partial instantaneous vaccination) with α * = r V ((I 0 -+ r V /r I S 0 -+ γ/β) ln(1 -r V /r I ))/r I .

Proof Previous results indicate that in the unbounded case the individual optimal strategies are of the form dϕ V = ηδ 0 thus dU = ηS 0 -δ 0 . Moreover, let α = ηS 0 -and recall that S(0) = S(0 -) -α = S 0 --α; moreover:

d dα ζ(S 0 --α, I 0 -) S 0 --α = 1 (S 0 --α) 2 ∂ξ ∂S (-1)(S 0 --α) -ξ(-1) = ζ(ζ -S 0 -+ α) (S 0 --α) 2 (γ/β + ζ -S 0 -+ α) < 0, so the function α → ζ(S 0 --α,I 0 -) S 0 --α is decreasing. Recall that ζ(S 0 --α, I 0 -)/(S 0 --α) = r V /r I is the same as I 0 -+ r V /r I (S 0 --α) + γ/β ln(1 -r V /r I ) = 0.
Similar arguments as in the proof of the Theorem 3 apply and allow to reach the conclusion.

Proposition 1 The vaccination region of the (OS) strategy contains the vaccination region of the (EIS) strategy.

Proof In the (OS) strategy, the vaccination only stops if

∂ S ζ ≤ r V r I
(see [START_REF] Laguzet | Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis[END_REF] for details). Or

∂ S ζ = ζ S 1 1+ β γ (ζ-S) and ζ -S ≤ 0 so 1 + β γ (ζ -S) ≤ 1. Then ζ S ≤ ∂ S ζ ≤ r V r I
and the conclusion follows.

E Relation between global and individual cost.

Lemma 1 Let ϕ V and U satisfy Equation (9). Then the individual cost is the average of the global cost, that is:

1 S 0 - Jsoc(S 0 -, I 0 -, U ) = J indi (ϕ V ). ( 33
)
Proof We can write:

∞ 0 - r I (1 -ϕ V (t))dϕ I (t) = ∞ 0 - r I (1 -ϕ V (t))(1 -ϕ I (t)) - dS(t) S(t) - dU (t) S(t) (34)
because from (1) for S(t) = 0:

dϕ I (t) 1-ϕ I (t) = βI(t) = - dS(t) S(t) - dU (t) S(t) . Furthermore (9) implies: ∞ 0 - r V (1 -ϕ I (t))dϕ V (t) = ∞ 0 - r V (1 -ϕ V (t))(1 -ϕ I (t)) dU (t) S(t) . (35)
By summing the Equations ( 34) and ( 35) we get:

J indi (ϕ V ) = ∞ 0 - (1 -ϕ V (t))(1 -ϕ I (t)) -r I dS(t) S(t) -r I dU (t) S(t) + r V dU (t) S(t) = ∞ 0 - S(t) S 0 - -r I dS(t) S(t) -r I dU (t) S(t) + r V dU (t) S(t) = 1 S 0 - ∞ 0 - [-r I dS(t) -r I dU (t) + r V dU (t)] , (36)
where we used (1 -ϕ V (t))(1 -ϕ I (t)) = S(t)/S 0 -. Using Equation ( 16) and the definition of Jsoc(S 0 -, I 0 -, U ) in Equation ( 10), we obtain the result.

Corollary 1 The average cost per person with the (OS) strategy is lower than the average cost per person with the (EIS) strategy.

Proof Denote by U G the (OG) strategy and ϕ G V its individual counterpart. Also denote by ϕ I

V the individual strategy in the (EIS) equilibrium and U I its global counterpart. By the definition of the optimality of U G : Jsoc(S 0 -, I 0 -, U G ) ≤ Jsoc(S 0 -, I 0 -, U I ). From Lemma 1 after simplification by S 0 -we obtain:

J indi (ϕ G V ) ≤ J indi (ϕ G V )
where, in the last inequality, we used that t 0 -e -γu du ≤ t. Item 2: It is enough to show that the tangent in (S * , I * ) to C r V /r I coincides with the direction (-βS * I * , βS * I * -γI * ) (the dynamics of the system without vaccination). It is standard to prove (see [START_REF] Laguzet | Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis[END_REF]) that ζ d satisfies the following equation: G Individual and equilibrium strategy with a discount factor Let g be the following function: g(t) = r I (Φ I (t) -Φ I (∞)) + r V e -Dt (1 -ϕ I (t)). ( 40)

The individual cost functional can then be written as:

J D indi (ϕ V ) = r I Φ I (∞) + ∞ 0 - g(t)dϕ V (t). ( 41
)
Since I(t) is continuous with continuous derivative the same is true for g and

g (t) = e -Dt (1 -ϕ I ) (r I -r V )βI -r V D . ( 42
)
Therefore the function g is increasing when I < I * and decreasing otherwise. On the other hand I decreases to zero at ∞ and may remain superior to I * on a bounded time interval.

As such, depending on (S 0 -, I 0 -) and U , the following possible behaviors can occur:

-g(t) is decreasing from g(0) > 0 to g(∞) = 0 -g(t) is decreasing from g(0) to some value g(t 1 ), increases from g(t 1 ) to g(t 2 ) and then decreases from g(t 2 ) to g(∞) = 0.

The minimum of J D indi (ϕ V ) is realized as following:

A/ If g(t) > 0 for all t then ϕ V = 0 (never vaccinate). B/ If inf t≥0 g(t) < 0 then the optimum is realized when dϕ V is a Dirac mass placed at the (unique) time τ * such that g(τ * ) = inf t≥0 g(t). In particular τ * < ∞. C/ Otherwise there exists an unique τ * such that g(τ * ) = 0 and the optimal strategies are dϕ V = αδτ * (δτ * being the Dirac mass in τ * ), α ∈ [0, 1] arbitrary.

Remark 4 For the individual strategy described in item B/ it is optimal to delay vaccination to a latter time. Such a strategy is never encountered when D = 0, where vaccination occurs either at t = 0 or never.

Finally, as in Equation ( 18) we can prove that if dU = 0 on [τ, ∞[, for any t ≥ τ the value g(t) is positive / negative / null if and only if ζ d (S(t), I(t)) is less / larger / equal to (r V /r I )S(t).

Consider the following notations:

Fig. 2 J

 2 Fig. 2 J indi (θ I ) for parameters β = 73, γ = 36.5, umax = 10, r I = 1, r V = 0.5 and D = 0. The solid bottom curve represent the optimal individual choice θ opt I (θ G ) and the dashed bottom curve is θ G = θ I .

Fig. 3

 3 Fig.3The parameters used are (S 0 -, I 0 -) = (0.75, 0.1), β = 73, γ = 36.5, r I = 1, r V = 0.5, umax = 10 and constraint in Equation (19). Left: The trajectory (S(t), I(t)) of the system (1) with U = 0. Right: The decreasing dotted curve is t →

Fig. 4

 4 Fig.4The representation of the domain Ω and its regions. Gray region: vaccination in the (OG) strategy and in the (EIS) strategy. Hashed region: vaccination in the (OG) strategy but not in the (EIS) strategy. White region: no vaccination in the (OG) strategy and no vaccination in the (EIS) strategy.

Fig. 5

 5 Fig. 5 The dashed curve (without the point A) is with no vaccination (dU = 0) and has individual cost 0.68; the solid curve is with (OS) strategy and has individual cost 0.51; the dashed curve (with point A) is with the (EIS) strategy and has individual cost 0.54. The parameters used for the three trajectories are β = 73, γ = 36.5, umax = 10, r V = 0.5, r I = 1, S 0 -= 0.75, I 0 -= 0.1.

Fig. 6

 6 Fig. 6 The parameters used are (S 0 -, I 0 -) = (0.9, 0.04), β = 73, γ = 36.5, r V = 0.4, r I = 1 and D = 10. Left: Evolution of the function g (see Appendix G). The optimal individual vaccination occurs at time τ * = 0.0329 (minimum of the function). Right: The risk of infection t → r I (Φ I (∞)-Φ I (t))

Fig. 7

 7 Fig.7The partition of the domain Ω in three regions: Ω i (dotted, instantaneous vaccination), Ωn (white, no vaccination), Ω d (gray, delayed vaccination) for γ = 36.5,β = 73, r V /r I = 0.4 and D = 10. In this case Ω d = ∅. The equilibrium dynamics is as follows: if (S 0 -, I 0 -) ∈ Ωn it will always remains there; if (S 0 -, I 0 -) ∈ Ω i partial vaccination will bring it to Ωn and total vaccination to the boundary S = 0; if (S 0 -, I 0 -) ∈ Ω d it will evolve to some point (S, I * ), then vaccinate till the point (S * , I * ) on the boundary of all three regions, then will evolve in the region Ωn. If on the contrary r V /r I = 0.6 we have Ω d = ∅; this situation is not illustrated here as it is similar to Figure4).

Fig. 8

 8 Fig. 8 Left: Fit of the cumulative vaccination in percents weighted by the effectiveness coefficient. The curve U (labeled "Model") and the curve in Guthmann et al. (2010) (labeled "Data") cannot be distinguished. Right: Instantaneous vaccination dU , weighted by the effectiveness coefficient.

Fig. 9

 9 Fig. 9 Comparison between the Infected class as obtained from the Sentinel network and the one from the model in (1). Left: Infected class as function of time. The peak of the epidemic is well reproduced. Right: cumulative number of Infections. The overall number of infection (at the end of the epidemic) is well reproduced.

Fig. 11

 11 Fig. 11 Illustration of instantaneous vaccination of 30% percent of the population at time t * = 0.25. The function V (t) = t * 0 -dU (τ ) is plotted. It has a discontinuity at time t * and is equal to H(• -t * ) with H(•) the Heaviside function.

F

  which is the conclusion. Some properties of the discounted number of infectedIn this Section and in Appendix G we consider D > 0 and umax = ∞. DefineI * = (r V D)/((r I -r V )β) and ζ d (S 0 -, I 0 -) = ∞0 -e -Dt βS(t)I(t)dt where (S(t), I(t)) is a nonvaccinating dynamics starting in (S 0 -, I 0 -). Denote alsoC r V /r I = {(S, I) ∈ Ω|ζ d (S, I) = (r V /r I )S}.Lemma 2 1. The point (0, I * ) is always below the curve C r V /r I . In particular either C r V /r I intersects the line I = I * in a point (S * , I * ) ∈ Ω or C r V /r I is completely above the line I = I * . 2. When C r V /r I ∩ {(S, I) ∈ Ω|I = I * } = ∅ , the non-vaccination dynamics starting from (S, I) ∈ C r V /r I is entering the domain {(S, I) ∈ Ω|ζ d (S, I) ≤ (r V /r I )S} if and only if I ≥ I * . Proof Item 1: It is enough to prove that lim →0 ζ d ( , I * )/ ≤ r V /r I . Let (S(t), I(t)) be the evolution of the system without vaccination starting from the point ( , I * ). From I (t) = (βS(t) -γ)I(t) we obtain I(t) = I * e t 0 -(βS(u)-γ)du ∈ [I * e -γt , I * e (β -γ)t ] and from S (t) = (-βI(t))S(t) we obtain S(t) = e t 0 --βI(u)du ∈ [ e t 0 --βI * e (β -γ)u du , e t 0 --βI * e -γu du ]. For → 0-βI * t dt = r V /r I ,

∂

  S ζ d (S, I)(-βSI) + ∂ I ζ d (S, I)(βSI -γI) -Dζ d (S, I) + βSI = 0. (38) On the other hand the normal to the curve C r V /r I in (S * , I * ) is (∂ S ζ d (S * , I * )-r V /r I , ∂ I ζ d (S * , I * )). A simple computation shows that the condition (∂ S ζ d (S * , I * ) -r V /r I , ∂ I ζ d (S * , I * )) ⊥ (-βS * I * , βS * I * -γI * ) (39) reduces to (38) when I = I * .

I

  ) When, with the notations of Lemma 2, a point (S * , I * ) ∈ C r V /r I exists let S * * be the unique solution in [γ/β, 1] of the equation: x -(γ/β)ln(x) = I * + S * -(γ/β)ln(S * ).Then the domain Ω is divided in three subdomains:Ω i = {(S, I) ∈ Ω|ζ d (S, I) > (r V /r I )S, I > I * }, Ω d = {(S, I) ∈ Ω|I < I * , I + S -(γ/β)ln(S) > I * + S * -(γ/β)ln(S * )}, Ωn = Ω \ Ω i ∪ Ω d . II) When C r V /r I is above the line I = I * , Ω i = {(S, I) ∈ Ω|ζ d (S, I) > (r V /r I )S}, Ωn = {(S, I) ∈ Ω|ζ d (S, I) < (r V /r I )S}.Theorem 5 Let (S 0 -, I 0 -) ∈ Ω; the unique equilibrium of the individual-societal vaccination is the following:a/ If (S 0 -, I 0 -) ∈ Ωn then dU = 0 and dϕ V = 0 (no vaccination). b/ If (S 0 -, I 0 -) ∈ Ω i let α ∈ [0, 1] be the maximum value such that (S 0 --α, I 0 -) ∈ Ω i .Then dU = αδ 0 and dϕ V = (α/S 0 -)δ 0 (partial or total instantaneous vaccination). c/ If (S 0 -, I 0 -) ∈ Ω d let τ * be the time at which the system (without vaccination) evolving from (S 0 -, I 0 -) reaches the line I = I * . Then dU = (S(τ - * ) -S * )δτ * and dϕ V = (S(τ - * )/S * -1)δτ * (vaccination after waiting the time τ * ).

Table 1

 1 Vaccination Coverage expected and realized in different countries as percentage of population during the 2009-10 Influenza A (H1N1) epidemic. Sources: Guthmann et al.

	Country	Target	Effective rate
		coverage	of vaccination
	Germany 30 -100% Spain 40% France 70 -75% Italy 40%	8 -10% 4 -27.1% 7.9 -8.5% 1.4 -4%

  Individual dynamics: continuous time Markov jumps between Susceptible, Infected, Recovered and Vaccinated classes.

	rate βI	rate γ
	Susceptible	Infected	Recovered
	...
	Vaccinated
	Fig. 1

Suppose now that all individuals want to vaccinate at the same time with the same ϕ V then the constraint above, coupled with (9), implies that ϕ V is differentiable in any t with S(t) > 0 and denoting f V (t) = umax S(t) :

∀t ≥ 0 with S(t) > 0 :

Another interpretation of the constraint is the following: when the number of people that want to vaccinate exceeds the capacity of the vaccination centers people will have to wait. In this model all individuals are the same (that is, have the same characteristics therefore same strategies) then the probability for a given individual to obtain vaccination in a time interval [t, t + ∆t] is umax∆t S(t) . The probability to be not yet vaccinated at time t but be vaccinated by time t + ∆t is on the one hand

and on the other hand is umax∆t S(t)

+ o(∆t); for ∆t → 0 we obtain the constraint

C Proof of the properties of the individual optimal strategy: the undiscounted case

We set D = 0.

Theorem 1 (Case umax < ∞) Let U be a given societal policy in (1), increasing, U (t) ≤ 1, ∀t ≥ 0 and f V (t) = umax S(t) . Suppose that the set of admissible individual strategies is composed of all increasing functions ϕ V differentiable in any t with S(t) > 0 such that:

Then the individual optimal strategy ϕ * V that minimizes the cost in Equation (4) (with the system (1)) is:

with the notation

The parameter θ I is the unique solution of the equation:

Proof We use individual cost in the form in Equation ( 4). It may be noted that:

D Equilibrium strategy: the undiscounted case

We set D = 0.

Theorem 3 (Case umax < ∞) Consider admissible individual strategies as in Theorem 1. Then:

starting from S 0 -, I 0 -with dU = umaxdt at all times until S(t) = 0. Let θ umax (S 0 -, I 0 -) be the first time when this dynamics touches the ensemble of curves:

Then the unique equilibrium is:

Proof Case a/ If r V > r I then any optimal individual strategy is ϕ V = 0 and the compatibility relation ( 9) imply U = 0. Therefore the only possible equilibrium in this case is

It is easy to see that this is effectively an equilibrium. If r V ≤ r I and ζ(S 0 -, I 0 -) ≤ (r V /r I )S 0 -, from Equations ( 18) and ( 26) it follows that an individual strategy that vaccinates cannot be optimal, thus ϕ V = 0. As before we obtain the unique equilibrium U = 0 = ϕ V . Case b/ Since ζ(S 0 -, I 0 -) > r V /r I S 0 -any optimal individual strategy must satisfy ( 22). Therefore, from the compatibility relation ( 9) we obtain dU (t) = umax1 [0,θ] (t) for some θ ≥ 0. We have to find θ such that θ is solution of (23) for dU (t) = umax1 [0,θ] (t). This is a fixed point equation.

Let (S, I) be the solution of the system (1) with dU (t) = umax1 [0,θ] (t). Since dU = 0 on [θ, ∞] by Equation ( 18), any θ solution of ( 23) is also solution of

But (S,

Therefore, any θ that represents an equilibrium is also solution of the equation:

But, [START_REF] Laguzet | Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis[END_REF] for details and the expressions of the partial derivative of ζ).

Thus if (32) has a solution then this solution is unique. Therefore in any case at most one equilibrium exists. Under the hypothesis ζ(S 0 -, I 0 -) < r V r I S 0 -Equation (32) does have a solution so θ is also solution of Equation (30) and this is the equilibrium.

Proof We will consider only the more involved situation when a point (S * , I * ) ∈ C r V /r I exists.

Recall that the evolution of the system without vaccination satisfies I +S -(γ/β)ln(S) = cst. Therefore the frontier of Ω d and Ωn is a trajectory of the system without vaccination. Let us consider the global vaccination strategy U given in this Theorem. This strategy does not vaccinate in domain Ωn, vaccinates (instantaneously) in domain Ω i and when the evolution starts in domain Ω d it waits to reach the line I = I * ; at that time it vaccinates until reaching the point (S * , I * ). This dynamics is illustrated in Figure 7. In order to prove that this is effectively an equilibrium we still have to prove that the best individual policies ϕ V are coherent with U .

But the properties of the function g(t) show that g(t) is positive in Ωn: therefore for a starting point (S 0 -, I 0 -) ∈ Ωn the best individual strategy is to never vaccinate.

When the starting point (S 0 -, I 0 -) ∈ Ω i , the instantaneous vaccination makes it arrive at time 0 on the boundary Ωn and Ω i ; therefore g(0 -) = g(0) = 0. Optimal individual strategies are ηδ 0 with η ∈ [0, 1]; among those only one is coherent with U (the one described in item b/).

Finally, when the starting point (S 0 -, I 0 -) ∈ Ω d , the free (non-vaccination) evolution makes it arrive at some point (S(τ - * ), I * ). Then at time τ * it reaches the point (S * , I * ) thus g(τ * ) = g(τ - * ) = 0. But, before τ * the coordinate I was inferior to I * thus g(t) was decreasing to zero during this time. Then all ηδτ * , η ∈ [0, 1] will be optimal strategies. Among those, only one value of η is compatible with U . Thus the strategies proposed in this Theorem are equilibrium strategies. Uniqueness is proved as in Theorems 1 and 2.

Remark 5 For D > 0 and umax < ∞ we were not able to find an analytic expression for the domains Ω i , Ωn and Ω d . The individuals, being aware of the shortage of vaccines, will wait in line in advance to be vaccinated by the optimal time; it is possible to obtain the corresponding Hamilton-Jacobi-Bellmann equilibrium equation but the equation has to be solved numerically.