Alexandra Tebkieta
email: tebkieta_alexandra.tapsoba@etu.udamail.fr

PhD Tapsoba

Grégoire Rota

Catherine Araujo Bonjean

Keywords: Poverty, Remittances, Natural Disasters, Resilience, Burkina Faso. JEL Codes I32, F24, Q54, N57

Using a multi-topic household panel survey conducted in Burkina Faso by the "Institut national de la statistique et de la démographie" (INSD), in 2014, this paper assesses the impact of remittances on poverty in Burkina Faso. To do so, a poverty index is computed using household's housing characteristics. Propensity score matching method is used as an empirical strategy, and results show that remittances have a negative impact on poverty. Another important result is that remittances have a higher impact on the resilience of households, when they have experienced disasters in the past. Therefore, when it comes to natural disasters, these inflows act as an important tool for populations to be more resilient.

Introduction

Poverty is an old, but much up to date topic. As a testimony, the first goal of the post 2015 development agenda is aiming to: « End poverty in all its forms everywhere » (UNDP, 2015). Remittances also constitute one of the main concerns of this agenda. In respect of the goal 10, which aims to reduce inequalities within and among countries, one of the targets is to "reduce to less than 3% the transaction costs of migrant remittances and eliminate remittance corridors with costs higher than 5%" (UNDP, 2015). This goal emphasises the importance of remittances especially for developing countries. In fact, they have increased at an impressive rate since the beginning of the 2000s and represent now a major source of foreign inflows in developing countries [START_REF] Yang | Are remittances insurance? Evidence from rainfall shocks in the philippines[END_REF]. According to the (World-Bank), the amount of remittances towards developing countries was around $430 billion in 2014 and $431.6 billion in 2014.

The Sustainable Development Goals then underlines the importance of remittances especially for developing countries, which can be seen as a hedge against poverty.

Remittances and poverty have been broadly discussed in the literature, and most of the studies find that remittances reduce poverty in the developing part of the world (Adams, R. & Page, J., 2005) [START_REF] Acosta | What is the impact of international remittances and poverty and inequality in Latin America?[END_REF], [START_REF] Gupta | Effect of remittances on poverty and financial development in Sub-Saharan Africa[END_REF].

Using household data of 71 developing countries, (Adams & Page, J, 2005), show that remittances lead to a decrease of 3.5% in the share of poor people. This result is shared by [START_REF] Acosta | Remittances and development in Latin America[END_REF], for international remittances, with the result that a increase by 1 point in the ratio of remittances to GDP lead to a decrease of 0.4% of poverty rates. [START_REF] Moises | How do international remittances affect poverty in developing countries? A quantile regression analysis[END_REF], using a panel of 66 developing countries between 1981 and 2011 show that international remittances reduce poverty, especially for the 10% most poor countries.

On the micro level, [START_REF] Lopez-Cordova | Globalization, migration and development: The role of Mexican migrant remittances[END_REF], in Mexico, [START_REF] Lokshin | Work-related migration and poverty reduction in Nepal[END_REF] in Nepal, [START_REF] Gubert | Do remittances affect poverty and inequality: evidence from Mali[END_REF] in Mali, and [START_REF] Adams | The impact of remittances on Investment and poverty in Ghana[END_REF] in Ghana, found that remittances reduce poverty.

Focusing on Burkina Faso, and using the last multi-topic household's survey conducted in the country in 2014, this paper aims to understand the linkages and particularly assess the impact of remittances on poverty. To our knowledge, this is the first econometrical study applied to this survey.

Despite the fact that this topic has been broadly discussed in the macroeconomics, and microeconomics literature, this paper brings another light on the topic, as we present a new way of measuring poverty.

In the case of Burkina Faso, [START_REF] Bambio | Impact of remittances on poverty and inequality in rural Burkina Faso[END_REF] found that remittances reduce poverty using low skilled worker emigration in Ivory Coast, and rural-rural move data between 2004 and 2006. As for [START_REF] Lachaud | Envois de fonds, inégalités et pauvreté au Burkina Faso[END_REF], using a national survey on Burkina Faso, remittances tend to reduce the incidence of poverty by 7.2% for rural households, and by 3.2% for urban ones. As majority of papers on poverty, these authors used monetary measures such as income per capita, usually necessary to insure a certain amount of calories per day, expenditures per capita, and so on.

We then chose to compute a poverty index, using housing variables. This index will contribute to the assessment of households' resilience to natural disasters and climate shocks that some developing countries have to face currently. As remittances and poverty, climate issues occupy an important place in the post 2015 agenda. Hence, the goal 13 aims to "take urgent action to combat climate change and its impacts" (UNDP, 2015). One of the target being to strengthen resilience and adaptive capacity to climate related hazards and natural disasters in all countries.

The rest of the paper is organised as follows: the section 1 consists in addressing the definitions of poverty, and presenting the index computed in this paper; Section 2 presents the scope of the study, some descriptive statistics and the empirical construction of the index; Section 3 is dedicated to the presentation of the methodology, and lastly, Section 4 presents the results and discuss of a potential learning process of households regarding previous shocks.

What is poverty?

1.1 : Poverty in the literature As stated earlier, poverty is an old scourge, fought worldwide, but yet to be solved, especially in developing countries. For [START_REF] Hartwell | Consequences of the industrial revolution in England for the poor[END_REF], "Economics is, in essence, the study of poverty". It cannot be restrained to one definition, but historically, and on the empirical level, poverty has been most commonly defined as a lack of a given amount of income [START_REF] Ipc | Poverty in Focus[END_REF].

According to the work of (Ravaillon, 1992), some questions are important when it comes to assessing poverty: "How do we assess individual well-being or "welfare?" "At what level of measured well-being do we say that a person is not poor", and "how do we aggregate individual indicators of well-being into a poverty measure?" For the author, the two first questions are referring to an identification problem, of which persons are considered poor, and how poor they are, when the last question is referring to an aggregation problem, and lead to answering of the question how much poverty is there?

When it comes to the measurement of poverty, one common practice is to count the number of people who fall below a poverty line defined in an arbitrary way [START_REF] Bardhan | Development Microeconomics[END_REF].

According to [START_REF] Streeten | First things first: meeting basic needs in developing countries[END_REF], poverty entails the non-achievement of basic needs such as "….human needs in terms of health, food, education, water, shelter, transport."

In this work we chose to compute an index based on the shelter part of these basic needs, in order to measure household's poverty status, as well to assess their resilience of households when receiving remittances, in the context of climate shocks.

1.2: The poverty index

We based our reflexion on the fact that monetary variables in surveys are usually underestimating poverty due to several factors, such as the fact that household might not include in their response, their annexe activities from informal sector for example. (Filmer, D & Pritchett, L.H) proposed the construction of an index using PCA (Principal Component Analysis) or FA (Factor Analysis) as a proxy of the household's wealth. This index is an alternative measure to the monetary one, which can easily be biased and non-objective for developing countries [START_REF] Briand | La demande de raccordement des ménages auprès des petits opérateurs privés d'eau potable: le cas des quartiers de Maputo[END_REF].

In our index, we then focused on housing data, which include the following variables:

-The presence of a roof in metal sheet -The presence of a concrete ground -The presence of drainage system -The presence of a lavatory -The presence of concrete walls around the house -The household own a mobile phone in operating condition -The fact that the household lives in a parcelled area The variables included in this index are referring to the standard of living dimension of the multidimensional poverty index define by UNDP. Multidimensional poverty index is an index of poverty that reflects the deprivation in very rudimentary services and core human functions (Alkire & Santos, M.E, 2010). This index accounts for health, education and standard of living, and was implemented as a result of the millennium objectives indicators [START_REF] Owusu | Non monetary poverty in Ghana . Ghana statistical Service[END_REF][START_REF] Owusu | Non monetary poverty in Ghana . Ghana statistical Service[END_REF].

The standard of living dimension is referring to aspects such as electricity (household considered deprived if there is no electricity), drinking water (household considered deprived if there is no access to clean drinking water, or clean water is more than 30 minutes of walk away from home), sanitation (deprived if no access to improved toilet), flooring (considered deprived if there is dirt, sand…), cooking fuel (deprived if the household cook with wood, charcoal), and assets (household considered deprived if it does not own more than one radio, TV, telephone, bike, motorbike, …) (Alkire & Santos, ME, 2010).

The variable of "living in a parcelled area" is included to take into account a growing phenomenon in the country, which is the anarchic and illegal occupation of spaces. This phenomenon can be seen as a result of the growing population in the cities, which inevitably lead to a competition and rise of parcelled areas' prices.

According to [START_REF] Vallée | Stratification de la ville de Ouagadougou (Burkina Faso) à partir d'une image panchromatique Spot 5: Une première étape à la mise en place d'une enquête de santé[END_REF], outparcelled areas were representing half of sections of Ouagadougou the city capital. They are mainly characterized by barely existence of basic public services and water evacuation systems increasing the risk of being exposed to floods.

Burkina Faso is characterized by Soudano-Sahelian weather with an average annual rainfall between less than 600mm and more than 900mm1. During the past 10 years, floods have been recurrent in the Niger bassin, consequently in Burkina Faso. For the year 2012 for example, in the provinces of Soum, Oudalan, et Yagha, rainfall recorded for the 30th of September exceed 750 mm where the mean rainfall barely exceeds 600mm a year in the region (sahelian region) [START_REF] Sighomnou | Crue exceptionnelle et inondations au cours des mois d'août et septembre 2012 dans le Niger Moyen et Inférieur[END_REF]. This combined with the fact that the soils where already saturated with water after the rain season beginning in early July-late June. [START_REF] Hangnon | Précipitations extrêmes et inondations à Ouagadougou: Quand le développement urbain est mal maîtrisé[END_REF] work on extreme precipitations and floods in the main city of Burkina Faso, Ouagadougou for the last ten years show interesting results. In fact, they found that the floods are more the result of unplanned urban growth than any change in the frequency or intensity of extreme rainfall.

We then include this variable in our poverty index, as it also captures the ability of people to leave in safer and therefore parcelled areas.

Ultimately, we think that this index, when computed will give information about the poverty status of the household, and their resilience to natural disasters recorded in the country such as floods and violent wind.

Remittances have been found in the work of [START_REF] Mohapatra | Remittances and Natural Disasters: ex-post response and contribution to ex ante preparedness[END_REF] to help household in their preparedness to natural disasters. They show in the case of Ghana, that remittances receiving households tend to live in concrete houses, have electricity, and mobile phones, that play a major part in the preparedness of households to natural disasters. For Burkina Faso, they found that households that receive remittances ten to own concrete houses. We believe that this work can contribute to the debate, by apprehending the resilience of households to natural disasters.

2-The scope of the study / A new poverty indicator

2.1-An overview of the database

This paper took advantage of the most recent household survey conducted in 2014 in Burkina Faso. Burkina Faso is located in sub-Saharan Africa, an area where mass poverty in the world is the most geographically concentrated [START_REF] Bardhan | Development Microeconomics[END_REF].

The data are from the "Enquête Multisectorielle Continue" (EMC) of the "Institut National de la Statistique et de la Démographie", and is a rich database of over 10000 (10800 precisely) households in the 13 regions and the 45 provinces. This survey aimed to understand the characteristics of the households, their employment status, housing, health, education and other socio-economic conditions. expenses does not equal this number. On this basis, on the national level, 40.1% of people are determined as poor.

The least poor region is the Central-East region with 9.2% considered as monetary poor. This region happens to be the most migration intensive region of the country, that lead to the intuition that migration may be as the literature largely prove a hedge against poverty.

Descriptive analysis of the database shows that 26.92% of the households are remittance receiving ones, regardless of the origin of the transfers. 86% of the households are lead by men, 27% are educated, and the majority of them, more than 80% are employed. Households are composed by approximately 7 persons, and live in majority in rural areas. which are Ouagadougou and Bobo-Dioulasso (Central -Hauts Bassins-Cascades). This can be explained by the development of the region and the fact that there are more transaction services such as western union and MoneyGram for examples, the main channel of remittances sending.

Region

2.2-Computing the poverty index

For this analysis, we use MCA (Multiple Correspondence Analysis)2, as we only consider here binary variables. The analysis codes the data by creating tables with binary columns for each of the variables, where the columns can only get the value 1 once at the time.

According to [START_REF] Hervé | Multiple correspondance analysis[END_REF], due to the fact that this coding adds additional dimension, causing the variance to be artificially inflated. As a consequence, the percentage of inertia explained by the first dimension of the analysis is underestimated.

Two corrections are often used, [START_REF] Benzécri | Sur le calcul des taux d'inertie dans l'analyse d'un questionnaire[END_REF] and [START_REF] Greenace | Correspondace Analysis in practice[END_REF]. We use the Benzecri correction and the results can be consulted in the appendix.

We then normalized the first axis scores in order to have an index between 0 and 1. Table 3 shows correlation coefficient for our poverty index and remittances, as well as variables that can help us understand the poverty status of the household. These correlations will help confirm that we are indeed capturing the poverty status of households.

Quart

Regarding remittances, coefficients are negative. This gives a first view of the relationship between poverty and remittances, which might be negatively impacting each other.

The coefficients show that larger households are most likely to be poorer; living in a urban area is negatively correlated with poverty, which is a non surprising result. Total expenditures of the household per capita, which is one of the most common measure of poverty in some papers is negatively correlated to the poverty index. This means that the more the household spend, the richer they are, which is an intuitive fact. Lastly, and regarding household promiscuity, which the number of people sharing a same room3 is positively correlated to the index, which means that the poorest households are the ones with more promiscuity.

In the recent literature, some papers are using luminosity in order to determine the development of a region. The utilization of this type of data is to say that the most developed regions are the ones with more luminosity, and then this could have a negative impact on poverty. However, the development of the region is captured by the variable "parceled area", as in Burkina Faso, only households situated in a parceled area are the one that can join the electrification network, as well as the sanitation, and water network.

However, and in order to validate even more the fact that our index is capturing household's poverty status, we tried to see the correlation of the index with the luminosity. Coefficients show that nighttime luminosity and poverty are negatively correlated. This confirms the intuition that the development of the region is negatively correlated to our poverty index.

3-Methodology

3.1-Propensity score matching framework

In this study, we are trying to compare the poverty and preparedness status of remittances receiving households, and those who are non-recipients. To this end, we use 3 Variable previously present in the database.

different propensity score matching methods to assess the average treatment effect of the remittances on our poverty index, which is the average treatment effect on the treated (ATET).

We define in this equation Remi as the treatment status dummy that take 1 if the household receive remittances, and 0 otherwise.

The value of the observed outcome, which is the score of our poverty index when the household receive remittances, Remi=1 is equal to Yi1(Rem=1), and Yi0(Rem=1) is the potential outcome of the same household, if it did not receive remittances.

The average treatment effect on the treated is defined as follows:

ATET=E[(Y i1 -Y i0 ) |Rem i =1]=E[Y i1 | Rem i =1]-E[Y i0 |Rem i =1] (1)
The ATET measures the difference in poverty scores as a result of the treatment for an household. In practice, the observational rule for Yi interferes with the estimation of the ATET as the outcome Yi0(Rem=1) cannot be observed. In an experimental scenario,

E[Yi0 | Remi = 1] = E[Yi0 | Remi = 0]
, so that the observed outcomes for the untreated observations can replace those of the treated observations as a counterfactual.

This however does not hold as the assignment into the treatment can be influenced by factors affecting the outcome as well. In our case, receiving remittances can be affected by variables that affect poverty as well.

Matching estimators, hence assume that there is a set of observable characteristics Xi such that the outcomes are independent of assignment to treatment conditional on Xi. This is known as the conditional independence assumption, define by the following equation:

Y i ⊥ Remi | Xi (2)
Xi here is the vector of variables that jointly influence the fact that households receive remittances. Hence, once controlling for these observables covariates, differences in the outcome of interest between treated and control households with the same values of covariates Xi can be attribute to the remittances receiving status [START_REF] Caliendo | Some practical guidance for implemation of propensity score matching[END_REF].

If we consider f(x) as the probability of assignment to treatment, which is also called the propensity score. (Rosenbaum, P.R & Rubin, D.B) show that if f(x) € 0,1, and equation ( 2) holds, and we have:

Yi ⊥ Remi | f(x) (3)
The outcome Y0 is independent from the assignment to treatment conditional upon f(x), as f(x) represents a balancing score that ensures that Xi ⊥ z|f(x) (Rosenbaum, P.R & Rubin, D.B).

Hence, the expected value of the unobserved outcome for treated households conditional upon f(x) coincides with the expected value of the observed outcome Y0 for untreated households:

ET [Yi0 | Remi= 1, f(x)=p] = EU [Yi0 | Remi=0, f(x) =p] (4)
Several propensity matching method exists to estimate equation 5. Firstly we have Nnearest neighbour matching which consists of matching each remittances receiving household with the N-untreated households with the closest propensity score, but does not receive remittances. In our study, we consider the nearest (n=1), the second nearest (n=2) and the third nearest (n=3).

Secondly, we use the radius matching method of [START_REF] Dehejia | Propensity score matching methods for nin experimental causal studies[END_REF], which consists in matching each treated household with an untreated one located at some distance in terms of propensity score. Hence, we consider (r=0.001), (r=0.01) and (r=0.05) radius.

Lastly, we use the kernel matching by (Heckman, J, Ichimura, H, & Todd, P, 1998), which matches each treated household with distribution of untreated households in the common support, with weights that are inversely proportional to the gap with respect to the propensity score of each treated.

After implementing these methods, we analyse the robustness and sensitivity of our results with respect to the two main assumptions of the PSM, which are the common support and the conditional independence.

Regarding the first assumption, one commonly used way to test the first assumptions is a t-test of the null-hypothesis of the mean between treated and untreated, conditional to the estimated propensity score. Hence, after controlling for the propensity scores, there should not be any significant difference between treated and untreated households in the common support area.

This approach can be criticised as the t-test is made on observations from the common support, and therefore, the result will be sensitive to the matching algorithm [START_REF] Lee | Propensity score matching and variations on the balancing test[END_REF]. This approach should be implemented on the total sample characteristics according to (Imai, king, G, & Stuart, E.A, 2008). We then follow [START_REF] Sianesi | An evaluation of the swedish system of active labor market programs in the 1990's[END_REF] an reestimate the propensity scores on matched units using a Probit model.

For each estimated ATET, we report the pseudo R2, which is define as the difference between the pseudo R2 for the matched sample, and the unmatched sample. We are then looking for a small pseudo R2, signal that propensity scores can be used as balanced scores.

Regarding the second assumption, the matching estimations validity is related to the potential influence of non-observable variables. To test if our results are sensible to unobserved variables, the method suggested by [START_REF] Mantel | Statistical aspects of the analysis od data from retrospective studies[END_REF]) is often used. This test evaluates how unobserved variables contribution can bias our results by testing the null hypothesis that the effect of receiving remittances on poverty is zero.

Études et Documents n° 8, CERDI, 2017

Overestimation and underestimation of the effect is often reported in studies, with statistics for respectively, the upper bound and the lower bound. However, the MH bounds are used when the outcome variable is a binary one. As our outcome variable is continuous, we use the Rosenbaum bounds sensitivity test.

3.2-Selection of the covariates / Propensity scores

We estimate the propensity scores by using a probit model. Several determinants have been found in the literature to explain the probability of receiving remittances such as migrant's characteristics (age, academic level, migration duration…) household's characteristics such as the household's size, and household's head characteristics (age, education, occupation…). In this paper we choose to use a set of variables that have been found in the micro economic empirical field to explain the probability for the migrants to remit.

First we chose to use the gender, education and age of the household's head. [START_REF] Randazzo | Remittances and Household Expenditure Behaviour in Senegal[END_REF] , found on the work on Senegal that having a female household head, an household head with a secondary education level, and the household head age, have a positive impact on receiving remittances.

Regarding the working status of the household's head, [START_REF] Durand | International migration and development in Mexican communities[END_REF] ( [START_REF] Holst | Migration and money-What determines remittances? Evidence from Germany[END_REF], [START_REF] Craciun | Migration and Remittances in the Republic of Moldova: Empirical Evidence at Micro Level[END_REF] show that household wealth have a negative impact on the probability of receiving remittances. Making the hypothesis that a working household head will contribute to the wealth status of the household, we think that having a working household head will have a negative impact on receiving remittances.

We lastly use the distance to the registry office and the distance to a practicable road, and expect them to have a positive impact of the receiving status of remittances. For the former, one can think that the registry office is the state representation that can provide for traveling documents, in order to migrate. When people migrate, there is therefore a high probability of receiving remittances. Using the same corridor of explanation, we can think that the proximity of a road to the villages, can have a positive impact on migration, and therefore on remittances. The results in table 4 show the results for the determinants of receiving remittances.

Having a household head that can read and write in French have a positive impact on receiving remittances. This can be explained as the fact that at least literate household head will have fewer difficulties to fill forms, when receiving remittances for example.

Having a male household head have a negative impact, which goes in the same direction of the literature. In fact, having a female as a HHH can entail that her husband went abroad, but still is the breadwinner of the household, and as a migrant, will remit more.

The age of the HHH rise with the probability of receiving remittances, going in the same direction of the literature as well ( [START_REF] Randazzo | Remittances and Household Expenditure Behaviour in Senegal[END_REF] [START_REF] Richard | Remittances and poverty in Ghana[END_REF].

Having a working household head, the distance to the civil services, or registry office and the practicable road, have, as expected, a negative impact on receiving remittances. 5 show that remittance-receiving households are less poor than non-receiving households. This result is robust to multiple matching techniques, which are the nearest neighbor matching, the kernel matching, as well as the radius matching with different caliper numbers.

Results of the PSM in table

Table 5: Impact of remittances on the poverty index

Results in table 5 also present the pseudo R2 as well as the rosenbaum bounds, in order to check for the validity of matching assumptions. As we stated earlier, a small pseudo R2 signals that propensity scores can be used as balanced scores. In our case, we see that the pseudo R2 is close to zero, varying between 0.001 and 0.003. These results imply that our matching allows obtaining balanced scores in order to estimate the effect of receiving remittances on poverty.

As we have a continuous outcome variable, we use here rbounds other than mhbounds.

Results show that reducing effect of remittances on poverty is highly robust to overestimation of the impact of unobservable on our estimations, as the odds ratio found is greater than 2. This result is in line with the findings of (Roth,, V and al, 2014) who also find that the reducing effect of remittances on poverty in robust to unobservable variables odds ratio greater than 24.

(Roth, V…) also found that yet for underestimation, the numbers are low, around 1.1.

In our case, the level to which our estimations might be questionable varies weather we look at the Wilcoxon results or the Hodges-Lehman point estimates. The results suggest that unobservables characteristics will make our results questionable around an odds ratio between 1.02 and 1.12. These results can be considered sensible to unobserved characteristics when considering the work of Duvendack and Palmer-Jones (2011), who stated that odds ratio below 2 indicate high sensitivity to unobservable. However, this statement can be considered pessimistic (Clement Matthieu & al 2012) as the field of this work cannot be compared to other sciences results such as medicine, where the numbers often exceed 5. Aakvik (2001) also argued that sensitivity analysis describes a 4 We made our calculations for a gamma between (1 (0.01) 2 ) in accordance to the literature. Hence, (Aakvik, 2001)show that an odds ratio of 2 is a very large number. This implies that the estimations will be biased only if unobservables cause the odds to differ between the treated and the non-treated households by 100%.

"worst case scenario" as it portrays how a given treatment effect can alter results and estimations in case of hidden bias, not indicating, then, that such bias do exists.

4.2-Robustness Checks a-International remittances

Receiving international remittances VS not receiving any remittances We then chose to check our results by focusing only on international remittances. Therefore, we are trying here to understand if the result is consistent. Table 6 shows that our results are consistent, and the impact is stronger. In fact when comparing the coefficients of the ATT in Table 6 andin Table 5, one can see that they are much higher Targeting households that receive the largest amounts of remittances, we found with table 8 that the results still robust, even if the coefficient of the ATT slightly changes.

Compared to the baseline results, the impact of remittances on poverty is higher for this subpopulation.

This result shows that when comparing the most receiving households, to those who do not receive remittances, the former population is much less poor than the latter. This result can be explained by the fact that households who receive remittances can be those who where able to send a member abroad, and usually are the richest ones.

Therefore, when receiving remittances, their living standards will inevitably grow faster.

3.3-Remittances and previous disasters

Table 8: Results for the upper quarter of households

Another interpretation of the computed index lies in the preparedness of households in case of disasters. The work of [START_REF] Mohapatra | Remittances and Natural Disasters: ex-post response and contribution to ex ante preparedness[END_REF], found that household in Burkina Faso, that receive remittances tend to live in concrete houses, making them more resilient to disasters. Our index can be seen as a extension of this idea, as it captures some aspects of ex ante dispositions to face natural disasters, such as living in a concrete roof, concrete walls, owning a mobile phone, having a drainage system, and living in a parcelled area, factors that contribute to considerably reduce the probability of experiencing floods, or be more resilient in case of natural disaster.

In this part, we use data from the CONASUR (Conseil National de Secours d'Urgence et de Réhabilitation) of Burkina Faso, which is in charge of rescuing population in case of all nature emergencies and disasters. This database gives the number of disasters that occurred in the country since 2007. We focused on the data of 2013 and 2012, and generate a variable that take the value 1 if the household have experienced floods or violent wind, and zero otherwise5. 5 A simple correlation test shows that the poverty index and the disaster variable are not significantly correlated, as the p-value equals 0.1540. The disaster variable can therefore be considered exogenous. Results in table 9 show that for households that have experienced disasters in the past, remittances acts in a more efficient way. Hence, receiving remittances is lowering more the poverty scores for these types of households, than for those that have not experienced disasters in the past.

Matching

Another interpretation is that households, who experienced floods and violent wind, use remittances in a most efficient way when it comes to housing.

Therefore, remittances and past disasters can be seen as important factors in the resilience aspects of Burkina Faso's households to potential future disasters. However it is difficult to state between an ex ante preparation of households6, or an ex post reaction.

6 [START_REF] Mohapatra | Remittances and Natural Disasters: ex-post response and contribution to ex ante preparedness[END_REF]

Conclusion

Migration and climate issues constitue some of the trending topics in international discussions today as they play an important role in the development of countries, especially most vulnerable ones such as those in sub-saharan Africa.

We contribute to the existing debate by investigating the impact of remittances on poverty in Burkina Faso.

The paper is based on a multi-topic household panel survey conducted in Burkina Faso by the "Institut National de la Statistique et de la Démographie" (INSD), in 2014.

An index of poverty, taking into account variables that reflect household's deprivation in housing commodities, helped us to understand the linkages between remittances and poverty. An alternative interpretation of this index allows us to understand the impact of remittances on household's preparedness to natural disasters.

We used propensity score matching techniques to compare households' poverty levels when they receive or do not receive remittances.

Our findings go along with the ones in the literature, concluding that remittances reduce poverty. The results are consistent when using different matching techniques, and subsampling.

Using data on the occurrence of floods and violent wind in the country for 2013 and 2012, we found those households that receive remittances, and have been hit by such disaster tend to be less poor, and more resilient.

An important conclusion stems from this result, as we can say that there is a learning process of households when it comes to natural disasters, and that remittances can be an instrument of preparedness for households by making resources available for investments in houses improvements, in order to increase their resilience to disasters [START_REF] Mohapatra | Remittances and Natural Disasters: ex-post response and contribution to ex ante preparedness[END_REF].

Rather than seeing remittances as any other income, they should be seen as a contribution that can help reinforcing the efforts made by governments and organizations regarding household's resilience to natural disasters.

  The EMC survey have been implemented to help the institute and the country in updating the annual indicators of the "Stratégie de Croissance Accélélérée et de Développement Durable (SCADD), the 2015 Development Agenda, and today, for the Sustainable Development Goals. Restitutions of the institute regarding this survey concerning poverty is mainly based on monetary poverty and highlight the global improvement of the country in this area. Hence, they tried to explain the incidence, severity and depth of the poverty. A food basket has been defined using 30 products representing more than 80% of the total annual consumption in the 13 regions. The cost of this basket is approximately 153530 FCFA per adult person, and a person in poor if he Études et Documents n° 8, CERDI, 2017

Table 1

 1

		Remittance
		receiving	Percentage
		households
	Hauts Bassins	246	9%
	Boucle du Mouhoun	265	9%
	Sahel	183	6%
	Est	176	6%
	Sud-Ouest 124	4%
	Centre-Nord	313	11%
	Centre-Ouest	283	10%
	Plateau Central	177	6%
	Nord	230	8%
	Centre-Est 201	7%
	Centre	284	10%
	Cascades	259	9%
	Centre-Sud 114	4%

give a picture of the distribution of remittances across the country. We can see that the most receiving households are those around the main cities of the countries,

Table 1 :

 1 Remittances by regions

Études et Documents n° 8, CERDI, 2017

Table n°3 :

 n°3 most of the papers in the literature, we can see with table 2 that receiving remittances decrease with the poverty index, which means that the richest household of the sample are the ones receiving more remittances. This is a common result in the literature as it is proved that middle classes are the ones that can afford migration, and then are the ones that are more susceptible to receive remittances. In this perspective, Correlation between poverty index and other variables.

	(Le De, Gaillard, & Friesen, 2015) show in the case of post cyclone in Samoa that poor
	households, that have little to no access to remittances increase inequalities and
	vulnerabilities in the community of migrant's origin.
	Also, according to (Gubert, Mesplés-Somps, S, & Lassourd, T, 2010), migration cost
	constitutes a huge barrier for the poor households. Hence, migration occurs more in the
	lower or middle-income countries, and therefore, they are most susceptible to gain from
	it.
	Variable		Correlation
	Remittances		-0.0611
	Household size	0.0877
	Living in a urban area	-0.6288
	Household Promiscuity	0.1801
	Total per capita expenditures -0.4013
	Luminosity		-0.3525
		Remittances Percentiles
	1 (Most
	rich
	according	916	0,32084063
	to the
	index)
	2	835	0,292469352
	3	583	0,204203152
	4 (Most poor)	521	0,182486865
	Table2: Repartition of remittances receiving households by poverty scores

2 MCA can be seen as a FA with multiple qualitative data based on a Burt matrix.

As

Table 4 :

 4 Probit regression for remittancesNotes: Rosenbaum bounds are calculated using the command rbounds in Stata. Wilcoxon Γ and Hodges-Lehman Γ are the values above which unobservable information bias our results

	4-Results
	4.1-Preliminary Results
				Overestimation Γ	Underestimation
		ATT	Pseudo-R2		Wilcoxon Γ	Hodges-Lehman Γ
	VARIABLES
	n1	-0.0162**	0.003	>2	1.02	1.1
		(0.00802)
	n2	-0.0180*	0.002	>2	1.04	1.11
		(0.00923)
	n3	-0.0196**	0.002	>2	1.06	1.12
		(0.00804)
	Kernel	-0.0157***	0.001	>2	1.05	1.11
		(0.00591)
	radius (0.001)	-0.0154*	0.001	>2	1.04	1.1
		(0.00889)
	radius (0.01)	-0.0136*	0.001	>2	1.03	1.09
		(0.00819)
	radius (0.05)	-0.0158**	0.001	>2	1.05	1.12
		(0.00671)
	Observations	10,592
	Standard errors in parentheses
	*** p<0.01, ** p<0.05, *
	p<0.1

Table 6 :

 6 Impact of international remittances on the poverty index

c-25% Most receiving households.

				Overestimation	Underestimation
		ATT	Pseudo-R2	95%	95%
	VARIABLES
	n1	-0.0863***	0.009	>2	1.51
		(0.0192)
	n2	-0.0959***	0.004	>2	1.76
		(0.0149)
	n3	-0.0910***	0.003	>2	1.75
		(0.0130)
	Kernel	-0.108***	0.004	>2	>2
		(0.0126)
	radius
	(0.001)	-0.0891***	0.002	>2	1.84
		(0.0135)
	radius (0.01) -0.0934***	0.003	>2	>2
		(0.0109)
	radius (0.05)	-0.109***	0.005	>2	>2
		(0.0139)
	Observations	8330
	Standard errors in parentheses
	*** p<0.01, ** p<0.05, * p<0.1

Table 9 :

 9 Impact of remittances on poverty in the context of past disastersWe saw in previous results that remittance-receiving households are less poor and more prepared to potential disasters. Results in table 9 now show that households that receive remittances and have previously experienced natural disasters are even less poor than households that did not experienced disasters in the past.

Impact of internal remittances on poverty

  Where p represents the number of active variables 7 , and μi is the ni eigen value. This calculation is made only for the Eigen values greater that 1/p. Notes: Rosenbaum bounds are calculated using the command rbounds in Stata.

	Variable			Total		Internal	International	Nb of	Unit	Data Source
				Survey	Remittances	Remittances	HH
							Receiving	Receiving
	HH Gender Original Values Axis Eigen HH Educate Value 1 0.3052	86,27% Percentage 27% explained 30.53%	HH 74,60% Histogram 31,98%	HH 78,77% 21,56 % Benzecri correction 10806 Household Head is a man Eigen Value Percentage 10809 Household Head Educate explained 0.035911 93.66%	Enquête Multisectorielle continue (2014) Enquête Multisectorielle continue (2014)
	HH Work 2	0.1851	81,09% 18.51%		70,75%	71,62%	10809 Household Head Working 0,002433 6.34%	Enquête Multisectorielle
	3	0.1354	13.55%			continue (2014)
	Urban 4	0.1157	39,43% 11.58%		45,55%	30,91%	10852 Household living in an urban area	Enquête Multisectorielle continue (2014)
	HH Size 5	0.1064	7,27 10.65%		6,42	7,90	10849 Household's size	Enquête Multisectorielle
	6	0.0613	9.05%			continue (2014)
	HH Head Age			46			48	51	10806 Household's Head Age	Enquête Multisectorielle
								continue (2014)
	Distance to civil service Appendix 5 : Overstimation 3,49 3,22	3,65 Underestimation 10796 Distance between the household and the registry office measured in minutes	Enquête Multisectorielle continue (2014)
	Distance to practicable road VARIABLES	ATT	2,26 Pseudo-R2	2,03 Wilcoxon Γ	2,37 Wilcoxon Γ	10798 Distance between the household and the nearest practicable road measured Hodges-Lehman Γ	Enquête Multisectorielle continue (2014)
	n1	-0.0281**	0.001			>2	1.07	in minutes 1.16
			(0.0125)
	Remittances n2		26.92% -0.0182		17.92%	8.38%	1	10852 Remittances receiving HH 1	Enquête Multisectorielle
	Disasters n3 Kernel Households radius	(0.0118) 31.39% -0.0218** 0.001 (0.00905) -0.0150** 0.001 27.05% (0.00756)		29.82% 21.18%	>2 >2	27.39% 1.05 1.01 26.18%	10859 Household's hit by floods or violent wind during the past months 1.13 1.09 10806 Equals one if more than 3	continue (2014) Conseil National de Secours d'Urgence et de Réhabilitation. Enquête Multisectorielle
	(0.001)		-0.0129
			(0.0110)
	radius (0.01)		-0.0147*	0.001			>2	1.02	1.09
		(0.00877)
	radius (0.05)		-0.0148
		(0.00905)
	Observations		8627
	Standard errors in parentheses
	*** p<0.01, ** p<0.05, * p<0.1

7

Variables used to construct the index

Direction of Meteorology.Études et Documents n° 8, CERDI,

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). http://www.cerdi.org/

for international remittances. Households that receive international remittances seem to be even less poor than those who do not receive remittances. This result can be explained by the fact that international remittances amounts are higher than internal remittances, despite the fact that households mostly receive the latter. This leads us to comparing households based on the origin of remittances that they receive.

b-International remittances vs internal remittances

Now we try to test the results when comparing international remittances and internal remittances. Households that receive international remittances will be attributed 1 and 0 when they receive internal remittances. Results in table 7 show here that receiving remittances from abroad reduce poverty even deeper than when receiving internal remittances. Appendix 3: Multiple Correspondence Analysis variable loadings.

These graphs have been obtained with the software TANAGRA. The graph here shows another interpretation of the index, via an index of wealth, rather than poverty. The figures are the same, but this presents another way of seeing the index.

Appendix 4: The Benzecri correction

The Benzecri correction equals to: Tμi = (p/1-p) (μi-1/p)


Emma Hubert

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

NASH-MFG EQUILIBRIUM IN A SIR MODEL WITH TIME DEPENDENT NEWBORN VACCINATION

Keywords: April 24, 2017 vaccination, Mean Field Games, SIR model, individual vaccination, vaccination games, vaccination efficacy, Nash equilibrium

Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination

Emma Hubert, Gabriel Turinici

Introduction

To analyze the impact of societal vaccination policies on epidemic dynamics, one has to distinguish two situations. First, when the vaccination is compulsory or when the public adheres to a very large extent to the official recommendations, the vaccination policy can be supposed implementable in full and the problem can take the form of an optimization problem of costs with respect to benefits (either in terms of money or of medical states, or of both), 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] Suresh | 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].

On the contrary, when vaccination is not compulsory or the official recommendations are followed to a lesser extent, the individual choices have to be taken into account. Works in this direction include [START_REF] Fine | Individual versus public priorities in the determination of optimal vaccination policies[END_REF][START_REF] Dagobert | Externalities and compulsary vaccinations[END_REF][START_REF] Geoffard | Disease eradication: Private versus public vaccination[END_REF][START_REF] Bauch | Group interest versus self-interest in smallpox vaccination policy[END_REF][START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF][START_REF] Shim | The influence of altruism on influenza vaccination decisions[END_REF][START_REF] Flávio | Dynamic modeling of vaccinating behavior as a function of individual beliefs[END_REF][START_REF] Codeço | Vaccinating in disease-free regions: a vaccine model with application to yellow fever[END_REF][START_REF] Reluga | Evolving public perceptions and stability in vaccine uptake[END_REF][START_REF] Galvani | Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum[END_REF][START_REF] Cojocaru | Dynamics of vaccination strategies via projected dynamical systems[END_REF][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][START_REF] Buonomo | Global stability of an {SIR} epidemic model with information dependent vaccination[END_REF][START_REF] Alberto D'onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF][START_REF] Alberto D'onofrio | Fatal SIR diseases and rational exemption to vaccination[END_REF][START_REF] Reluga | A general approach for population games with application to vaccination[END_REF][START_REF] Funk | Modelling the influence of human behaviour on the spread of infectious diseases: a review[END_REF][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][START_REF] Müller | Optimal vaccination strategies-for whom?[END_REF][START_REF] Fukuda | Risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks[END_REF][START_REF] Bai | Uniqueness of nash equilibrium in vaccination games[END_REF] (see also the recent review [START_REF] Wang | Statistical physics of vaccination[END_REF]). It was shown that this decision depends on the present and past states of the epidemic (see [START_REF] Manfredi | Modeling the interplay between human behavior and the spread of infectious diseases[END_REF]), sometimes on the prediction of its future course (see [START_REF] Reluga | A general approach for population games with application to vaccination[END_REF]) and on the individual perception of two costs: a cost r V incurred if the individual vaccinates and a cost r I incurred if the individual is infected; again, the costs are not necessarily expressed in terms of money but can also be medical side effects or general morbidity (see [START_REF] Zeckhauser | Where now for saving lives?[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] for an introduction on QALY/DALY).

So, on the one hand, each individual chooses according to his/her own perception and interests. His/Her decision, as single individual, cannot influence the societal epidemic dynamics but is influenced by it. Such a situation is called in physical sciences a "mean field" that mediates the interaction between agents. At its turn, the mean field is the aggregation of the decisions of all individuals. An important question is the existence of a stable equilibrium between the individual decisions and the overall observed dynamics. From the mathematical point of view, the formalism is that of a Nash equilibrium in a game with an infinity of players. Recently, such approaches witnessed an important development with the introduction of the Mean Field Games (henceforth MFG) techniques in the works of Lasry and Lions [START_REF] Lasry | Jeux à champ moyen. II: Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Jeux à champ moyen. I: Le cas stationnaire[END_REF][START_REF] Lasry | Mean field games[END_REF] and of Huang, Malhamé and Caines [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] Benoussan | Mean Field Games and Mean Field Type Control Theory[END_REF] and [START_REF] Guéant | Mean field games and applications[END_REF] for entry points to the litterature).

Within this framework, the contribution presented here is more specifically targeted to analyze newborn vaccination in a SIR model with vital dynamics. Such a model has already been considered in [START_REF] Bauch | Vaccination and the theory of games[END_REF] which is the main inspiration and starting point of our work; however in [START_REF] Bauch | Vaccination and the theory of games[END_REF] the newborn vaccination rate is supposed constant in time. Our goal is [START_REF] Abakuks | Optimal immunisation policies for epidemics[END_REF] to show that an equilibrium exists when also considering time depending vaccination rate; (2) that this is the generic situation;

(3) and to propose a numerical algorithm to compute the equilibrium dynamics (because now the unknown is not just a real number but a real valued function defined on R + ).

As a technical remark, our analysis follows the line of thought formalized in [START_REF] Peter | Optimal tax/subsidy combinations for the flu season[END_REF][START_REF] Laguzet | Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-2010 influenza A (H1N1) epidemic in France[END_REF][START_REF] Reluga | A general approach for population games with application to vaccination[END_REF] with the distinction that here we have no analytic solution and have to resort to more general techniques.

The balance of the paper is the following: in section 2 we introduce the model and see that keeping the vaccination rate constant may not be realistic: several hundreds of years may be required to reach the constant-rate equilibrium, which is not compatible with the individual time span. Nevertheless the long time limit is right and one has only to study the transitory regime. This is done in section 3, where we prove that the model with a time-varying newborn vaccination rate has an equilibrium. The computation of the equilibrium is then explained in section 4 and numerical simulations are presented to illustrate the results.

Motivation and model notations

2.1. The model and previous works. We consider the SIR model with vital dynamics and newborn vaccination (see [START_REF] Anderson | Infectious Diseases of Humans Dynamics and Control[END_REF][START_REF] Bacaër | A short history of mathematical population dynamics[END_REF][START_REF] Diekmann | Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation[END_REF]) :

S (t) = µ(1 -p) -βSI -µS (2.1)
I (t) = βSI -γI -µI (2.2) R (t) = γI -µR (2.3) V (t) = µp -µV, (2.4)
where µ is the death / birth rate, β is the transmission rate, 1/γ is the infectious period and p(t) is the vaccine uptake level for the individuals born at time t (assuming, for simplicity, that infection cannot occur before vaccination). In this model S represents the susceptible individuals, I the infected ones, R the recovered and V the vaccinated. The initial conditions are set once for all:

S(0) = S 0 , I(0) = I 0 , R(0) = 1 -S(0) -I(0), V (0) = 0. (2.5)
Remark 2.1. The population is supposed at equilibrium i.e., the birth rate equals the death rate; the model can be adapted in a straightforward manner to treat the general case.

The model has already been considered in [START_REF] Bauch | Vaccination and the theory of games[END_REF], but there, p is assumed constant. It was proved that, if the individuals act according to their own self-interest (see later in section 2.4 the precise definition of the individual decision criterion), then the overall dynamics will converge, in some unknown time, to an endemic equilibrium characterized by some level of susceptibles S ∞ , of infected I ∞ and an equilibrium vaccination rate (stable Nash equilibrium), denoted p csne . Let us set R 0 = β/(γ + µ) and r = r V /r I (the quotient of the costs of the vaccine and of the illness).

In order to see a numerical illustration, we take the relevant situation when R 0 (1-r) > 1; then the equilibrium vaccination is

p csne = 1 - 1 R 0 (1 -r)
and the system converges to the endemic equilibrium:

S ∞ = 1/R 0 , I ∞ = µ γ + µ r R 0 (1 -r) , R ∞ = γ µ I ∞ , V ∞ = p csne . (2.6)
For the parameters in [START_REF] Bauch | Vaccination and the theory of games[END_REF], even if we start close to the theoretical equilibrium, the model may take long to converge (see figure 1). Therefore, even if the long time individual vaccination behavior is known, this still does not provide information on the transitory dynamics i.e., on the evolution towards this equilibrium, because the individuals are not expected to plan over such a long horizon. On the other hand, a time dependent vaccination rate results in faster dynamics and lowers the time-horizon of the individual decision.. Of course, the precise values of the time horizon depends on the specific parameters of the model. See also section 4.3 and figure 5 for additional advantages of a model with time dependent vaccination rate. [START_REF] Bauch | Vaccination and the theory of games[END_REF] with parameters : µ = 1/80, R 0 = 15, γ/µ = 1000, β = R 0 * (γ + µ), r = 0.1, T = 80(years) and initial parameters S 0 = 6.53%, I 0 = 7.4e -6. For these parameters p csne = 0.9259 is a convergently stable Nash equilibrium (within class of the time independent vaccination rates) with limit state S ∞ = 6.6%, I ∞ = 7.4e -6. More than a hundred years is required in order for the long time limit to be reached. Solid line: the evolution of the susceptibles and infected classes in the time dependent vaccination rate model (2.8)-(2.11) for a total time T = 5; faster convergence towards (S ∞ , I ∞ ) is observed.

2.2. First notations. We denote the set of all admissible vaccination strategies as:

P = { p : R + → [0, 1] , p measurable} . (2.7)
The interpretation is that the individual born at t will be vaccinated with probability p(t); note that this implies that we are dealing with mixed strategies (see [START_REF] Fudenberg | Game theory[END_REF] for some discussions on mixed versus pure strategies).

As a technical remark, note that the system (2.1)-(2.4) does not satisfy the usual hypothesis for the existence of a solution because p may have no regularity at all (other than being bounded and measurable). To show that a solution exists, one has to write the system in the form:

S (t) = µ(1 -u) -βSI -µS
(2.8)

I (t) = βSI -γI -µI (2.9) R (t) = γI -µR (2.10) V (t) = µ u -µV, (2.11)
where we used the cummulative vaccination function u(t) = t 0 p(τ )dτ which belongs to the set:

C = {u : [0, ∞[→ R|u(0) = 0, ∀b ≥ a ≥ 0 : |u(a) -u(b)| ≤ |a -b|, u(b) ≥ u(a)}. (2.12)
Here, we replaced p by u which is a Clarke generalized gradient of u (see appendix A and [START_REF] Frank | Generalized gradients and applications[END_REF][START_REF] Clarke | Functional analysis, calculus of variations and optimal control[END_REF]; we also denote p ∈ ∂u or p ∈ u). In this form, the results from [START_REF] Bressan | Impulsive control systems with commutative vector fields[END_REF] apply and allow to estimate the solution in terms of the L 1 norm of u. Remark 2.2. The model can accommodate the situation when vaccination may occur later during the lifetime of the individual (i.e., not necessarily at birth). In this case the term µp in equations (2.1) and (2.4) is to be replaced by the vaccination rate and more importantly the sets P and C are to be adapted in order to ensure S(t) ≥ 0 at all times.

Individuals seek to optimize an individual cost function. In order to define it, we have to consider the individual dynamics (see figure 2 for an illustration). It takes the form of a controlled Markov chain with several states, Susceptible (S), Vaccinated (V), Recovered (R), Infected (I), Death without any infection (D 1 ) and Death after infection (D 2 ).

The Markov chain of the individual born in t ≥ 0, whose state at time τ ≥ t is denoted M t τ , is described in terms of the passage probabilities :

P M t τ +∆τ = S | M t τ = S = 1 -βI(τ )∆τ -µ∆τ + o(∆τ ) (2.13) P M t τ +∆τ = I | M t τ = S = βI(τ )∆τ + o(∆τ ) (2.14) P M t τ +∆τ = D 1 | M t τ = S = µ∆τ + o(∆τ ) (2.15) P M t τ +∆τ = I | M t τ = I = 1 -γ∆τ -µ∆τ + o(∆τ ) (2.16) P M t τ +∆τ = R | M t τ = I = γ∆τ + o(∆τ ) (2.17) P M t τ +∆τ = D 2 | M t τ = I = µ∆τ + o(∆τ ) (2.18) P M t τ +∆τ = R | M t τ = R = 1 -µ∆τ + o(∆τ ) (2.19) P M t τ +∆τ = D 2 | M t τ = R = µ∆τ + o(∆τ ) (2.20) P M t τ +∆τ = V | M t τ = V = 1 -µ∆τ + o(∆τ ) (2.21) P M t τ +∆τ = D 1 | M t τ = V = µ∆τ + o(∆τ ) (2.22) P M t τ +∆τ = D 1 | M t τ = D 1 = 1 (2.23) P M t τ +∆τ = D 2 | M t τ = D 2 = 1 (2.24)
Since the individual born in t vaccinates with probability p (t), the initial state M t t follows the Bernoulli distribution :

M t t = S with probability 1 -p (t) V
with probability p (t) .

The probability of infection.

In order to explain the choice of an individual (to vaccinate or not and with which probability), we have to compute the probability of being infected. Let φ u I (.) be a function from R + to [0, 1] such that, for any t ∈ R + , φ u I (t) is the probability of infection during the life of an individual, born in t and not vaccinated, when the rest of the population follows the vaccination function p ∈ u. In mathematical terms, for any individual born in t ≥ 0,

φ u I (t) = P ∃τ ≥ t such that M t τ = I | M t t = S . (2.25)
We can prove (see Section B) that:

φ u I (t) = 1 - +∞ t µ exp - τ t (µ + βI u (s)) ds dτ. (2.26) Moreover φ u I (t) = 1 - F u I (∞) -F u I (t) f u I (t) , φ u I (0) = 1 -F u I (∞) , (2.27)
where the functions f u I (t) and F u I (t) satisfy, for t ≥ 0, the following differential equations:

d dt f u I (t) = -(µ + βI u (t)) f u I (t) , (2.28)
d dt F u I (t) = µf u I (t) , (2.29)
with initial conditions:

f u I (0) = 1, F u I (0) = 0. (2.30)
2.4. The individual cost. Recall that r I is the individual cost of the illness and r V the vaccination cost. We suppose that r I > r V and work with r = r V /r I . Suppose that the entire population follows the vaccination schedule u, the expected cost for any individual born in t with vaccination probability p I (t) is:

p I (t)r V + (1 -p I (t))r I φ u I (t) = p I (t)r V -p I (t)r I φ u I (t) + r I φ u I (t) . (2.31)
We consider that all individual are rational and minimize their cost. Given the vaccination u of the population, the minimization problem for an individual born in t is:

min p I (t)∈[0,1] {p I (t) (r V -r I φ u I (t))}. ( 2

Existence of an equilibrium

Set L = C 0 ([0, +∞[ , R
), the set of continuous functions in [0, +∞[ with real values. For any p ∈ P, the function t → u (t) = t 0 p (s) ds belongs to the set C ⊂ L defined by equation (2.12).

Conversely, for any u ∈ C, the set u of its Clarke generalized gradients [START_REF] Frank | Generalized gradients and applications[END_REF] is a subset of P. In particular ∀τ ≥ 0 : u(τ ) = τ 0 u(s)ds. Finding a Nash equilibrium in our model is equivalent to finding a fixed point of the following multivalued correspondence T : u ∈ C → T (u) ∈ P(C) (here P(C) is the ensemble of subsets of C) where

T (u) = {v ∈ C|∃p ∈ v : p (τ ) (r V -r I φ u I (τ )) ≤ q (τ ) (r V -r I φ u I (τ )) , ∀q ∈ P, ∀τ ≥ 0} . (3.1)
One of the main results of this work is the proof of existence of a Nash-MFG equilibrium and is formalized in the following Theorem 3.1. The model (2.1)-(2.4) admits at least one Nash equilibrium.

Proof We use the following fixed-point theorem (see [START_REF] Glicksberg | A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points[END_REF][START_REF] Ky | Fixed-point and minimax theorems in locally convex topological linear spaces[END_REF]): Theorem 3.2 (Kakutani-Fan-Glicksberg). Let L be a locally convex topological linear space and C a compact convex set in L. Let R(C) be the family of all closed convex (non empty) subsets of C. Then, for any upper semi-continuous point to set correspondence T from C to R(C), there exists a point x 0 ∈ C such that x 0 ∈ T (x 0 ). The assumptions of the theorem will be checked one by one in the rest of this Section.

3.1. The space L. In order to apply the previous theorem, we need to define a topology on L. Let K n = [0, n] be a sequence of compact sets in R + , we consider the countable family of seminorms η n (f ) = sup x∈Kn |f (x)| for f ∈ L. This family of seminorms is well defined, separated and induce the compact-open topology. The set L is completely metrizable with the distance d, which induce the same topology :

d (f, g) = n∈N * η n (f -g) 2 n (1 + η n (f -g))
.

The set L equipped with this family of seminorms is a locally convex topological vector space. Hence, L satisfies the assumptions of the theorem.

3.2. The subset C. We have to prove that C, defined by (2.12), is a compact convex set in L. Straightforward computations show that C is convex. In order to prove that C is a compact set, we will use the following result, which is a form of [39, Theorem 17, page 233] :

Theorem 3.3 (Arzela-Ascoli).
Let X be a separated topological space and Y be a metric space. Let C be a subset of C 0 (X, Y ) such that :

• C is equicontinuous; • C is closed; • for all x in X, the set C(x) = { u(x) | u ∈ C } is relatively compact in Y .
Then the subset C is compact in C 0 (X, Y ) for the compact-open topology.

First of all, X = R + is a separated topological space and Y = R is a metric space. For the first point, the functions in C are Lipschitz functions (with Lipschitz constant equal to 1), hence the space C is equicontinuous.

In order to prove the second assumption on C, let (u k ) k≥1 ⊂ C be a sequence converging to a limit u in the topology of L. First, u is an uniform limit (on compact sets) of a sequence of continuous functions, hence u is continuous. Secondly, u(0) = lim k→∞ u k (0) = 0 because (u k ) k≥1 is converging uniformly on compact sets to u, hence is also simply converging. Finally, u is a Lipschitz continuous function : indeed, let a, b ∈ R + , we have :

| u(a) -u(b) | = | u(a) -u k (a) + u k (a) -u k (b) + u k (b) -u(b) | ≤ | u(a) -u k (a) | + | u k (a) -u k (b) | + | u k (b) -u(b) |
The first and the last members are converging to zero when k is converging to +∞. The second member can be bounded by | a -b | because for all k, u k ∈ C, hence u k is Lipschitz. Thus, the limit u of the sequence (u k ) is in the subset C, hence C is closed.

Finally, for all x in R + , we have to show that the set

C(x) = { u(x) | u ∈ C } is relatively compact in R. As C(x) is a subset of R, we just have to prove that C(x) is bounded. Let u(x) ∈ C(x) : |u(x)| = |u(x) -u(0)| ≤ |x -0| = |x|.
Hence, for all x ∈ R + , the set C(x) = {u(x)|u ∈ C} is bounded, thus relatively compact, in R. By applying the previous theorem, the subset C is compact in L = C 0 ([0, +∞[ , R) for the compact-open topology. For given u ∈ C, and v ∈ T (u), then there exists p ∈ P ∩ v such that, for any t ≥ 0:

   p (t) = 0 if φ u I (t) < r p (t) arbitrary in [0, 1] if φ u I (t) = r p (t) = 1 if φ u I (t) > r
We need to show that T is with values in R(C), the family of all closed convex (non empty) subsets of C, and that T is an upper semi-continuous point to set correspondence.

3.3.1.

The set T (u) is closed. Let u in C and (v n ) n≥1 be a sequence in T (u) such that v n converges in L to a limit v. The sequence (v n ) n≥1 ⊂ T (u) is included in the compact set C, hence v ∈ C. Moreover, for all n ∈ N, there exists p n associated to v n in the sense of relation (3.1). Let us consider two cases :

(1) If φ u I (t 0 ) < r for t 0 ≥ 0 : since φ u I (•) is continuous, there exists > 0 such that φ u I (•) < r on V t 0 , = ]t 0 -, t 0 + [. Hence, on V t 0 , , p n = 0, i.e. v n = cst. Since v n converges uniformly to v on V t 0 , , v = cst on V t 0 , . Hence, for any p ∈ v we have p = 0 on V t 0 , and in particular p(t 0 ) = 0.

(2) If φ u I (t 0 ) > r for t 0 ≥ 0 : in this case, we can construct a neighborhood V t 0 , with p n = 1 on V t 0 , i.e., for all t ∈ V t 0 , , v n (t) -v n (t 0 -) = t -(t 0 -). By uniform convergence on compact sets, v (t) -v (t 0 -) = t -(t 0 -), hence, for any p ∈ v we have p = 1 on V t 0 , . In both cases, we showed that for any p ∈ v we have p = 0 when φ u I (•) < r and p = 1 when φ u I (•) > r. Recall that φ u I (t 0 ) = r for t 0 ≥ 0 does not impose and constraint on p which can be chosen arbitrary in [0, 1]. Hence, p and v satisfy the relation (3.1) and thus v ∈ T (u). (1) If φ u I (t 0 ) < r for t 0 ≥ 0 : since φ u I (•) is continuous, there exist such that φ u I (.) < r on V t, = ]t -, t + [. Hence, on V t, , p v = p w = 0, thus v and w are constant on V t, . This implies that z is also constant. Hence, p z = αp v + (1 -α)p w ∈ ż is such that p z = 0 on V t, .

T has values in R(C)

(2) If φ u I (t 0 ) > r for t 0 ≥ 0 : in this case, on some V t, we have p v = p w = 1, i.e. for all τ ∈ V t, :

v(τ ) -v(t -) = τ -(t -), w(τ ) -w(t -) = τ -(t -). (3.2)
Hence z(τ ) -z(t -) = τ -(t -). Thus, p z = 1 on V t, . Using the same arguments than those given in the Section 3.3.1, z ∈ T (u).

3.3.3.

T is upper semi-continuous. Since L is metrizable and T (u) is a compact set (closed and included in the compact set C), in order to prove that the mapping T is upper semi-continuous, we can use the definition in terms of limits of sequences.

Let (u n ) n≥1 be a sequence in C converging to a limit u and (v n ) n≥1 a sequence such that for all n, v n ∈ T (u n ) and converging to a limit v. We define, for all n ∈ N, p n associated to v n in the sense of relation (3.1). We need to prove that v ∈ T (u).

Consider the system (2.8)-(2.11) to which we adjoin (2.28)-(2.29) with initial conditions (2.5) and (2.30), written for the u n :

S n (t) = µ(1 -un (t)) -βS n (t)I n (t) -µS n (t), S n (0) = S 0 (3.3) I n (t) = (βS n (t) -γ -µ) I n (t), I n (0) = I 0 (3.4) R n (t) = γI n (t) -µR n (t), R n (0) = 1 -S n (0) -I n (0) (3.5) V n (t) = µ un (t) -µV n (t), V n (0) = 0 (3.6)
f un I (t) = -(µ + βI n (t)) f un I (t) , f un I (0) = 1 (3.7)
F un I (t) = µf un I (t) , F un I (0) = 0. (3.8)
This system is of the form Ẋ(t) = A(t) + u(t)B(t) and has been studied in [START_REF] Bressan | Impulsive control systems with commutative vector fields[END_REF] where it is shown that is has a well defined meaning for u n , u ∈ L 1 loc (R + ). In particular for any finite horizon T when (u n ) n≥1 converges in L 1 ([0, T ]) to u then the solution S n , ..., F un I (•) (corresponding to u n ) converges pointwise to the solution S, ..., F u I (•) (corresponding to u).

On the other hand, one can see that F u I (τ ) converges when τ → ∞ to F u I (∞) uniformly with respect to u (their difference is bounded by e -T µ ).

In conclusion since (u n ) n≥1 converges to u for the uniform convergence on compacts, we obtain by equation (2.27) that φ un I (t) is converging to φ u I (t) for any fixed t. Let us consider two cases :

(1) If φ u I (t) < r, then for some > 0 small enough there exists n such that φ un I (t) < r -for all n ≥ n > 0. Remark now that φ un I (•) are Lipschitz (with constant C L independent of u n ). Then on V = ]t -/(2C L ), t + /(2C L )[ we have φ un I (•) < r -/2 < r. In this neighborhood, and for n ≥ n η , p n (•) = 0, which implies that v n (•) = cst, thus v(•) = cst. Hence, v = 0 in the neighborhood V η and in particular v(t) = 0.

(2) We can prove, in the same way, that if φ u I (t) > r, then v(t) = 1.

Hence v ∈ T (u).

Remark 3.4. The Theorem 3.1 only guarantees the existence of an equilibrium and not its uniqueness. Proving uniqueness requires in general more information on the problem (for instance convexity / monotony as in [START_REF] Lasry | Jeux à champ moyen. II: Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Jeux à champ moyen. I: Le cas stationnaire[END_REF][START_REF] Diogo | Continuous time finite state mean field games[END_REF], which is not true in our cases, or analytic formulas as in [START_REF] Laguzet | Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-2010 influenza A (H1N1) epidemic in France[END_REF]) and can break down in general (see [START_REF] Lachapelle | Computation of mean field equilibria in economics[END_REF]). Nevertheless in all situations we tested numerically, only one equilibrium was found.

Numerical simulations

In order to find the Nash equilibrium, we need to minimize the cost of any individual; thus we need to simulate the model (2.1)-(2.4) but also the probability of infection φ u I (•).

4.1. Approximation for a finite horizon time T . Note that, in the numerical approach, the time horizon, denoted T , is finite. Hence, in order to compute F u I (∞), we need to make an approximation; here we suppose that the population of infected is constant after T , that is for any t ≥ T , I u (t) I u (T ). Under this assumption, we can estimate F u I (∞) in the following way :

F u I (∞) = ∞ 0 µ exp - τ 0 (µ + βI u (s)) ds dτ T 0 µ exp - τ 0 (µ + βI u (s)) ds dτ + ∞ T µ exp - T 0 (µ + βI u (s)) ds × exp - τ T (µ + βI u (T )) ds dτ = F u I (T ) + exp - T 0 (µ + βI u (s)) ds × ∞ T µe -(µ+βI u (T ))(τ -T ) dτ = F u I (T ) + µf u I (T ) × ∞ T e -(µ+βI u (T ))(τ -T ) dτ = F u I (T ) + µf u I (T ) × ∞ 0 e -(µ+βI u (T ))s ds = F u I (T ) + µf u I (T ) µ + βI u (T )
.

Hence we set :

φ u I (t) = 1 - 1 f u I (t) (F u I (∞) -F u I (t)) 1 - 1 f u I (t) × F u I (T ) + µf u I (T ) µ + βI u (T ) -F u I (t) . (4.1)
Thus, to simulate the model (2.1)-(2.4) and the probability infection, we solve numerically the system (3.3)-(3.8) and apply (4.1). Moreover in order to improve the precision of the numerical solver, we sometimes rewrite the system (3.3)-(3.8) by replacing I(t) by log(I(t)) and solve the corresponding equivalent ODE.

4.2.

Finding the fixed point. The result of the Section 3 guarantees the existence of at least one equilibrium. But this method is not constructive and has to be supplemented by a procedure to find the fixed point. The presentation of this procedure is the object of this section.

The procedure is iterative, denote by n the iteration counter. For n = 0, any p ∈ P can be chosen; for our numerical tests, we take p 0 (t) = p csne . Suppose now that we are at the iteration n. To p n we associate u n (•) = . 0 p n (s)ds and the probability of infection φ u n I (•) is calculated as indicated in section 4.1. As shown in the equation (2.32), the minimization problem of the cost of an individual born in t is equivalent to:

min p(t)∈[0,1] p (t) r V -r I φ u n I (t) . (4.2)
The specific of our procedure is that we do not take as next iterate the minimizer in the above term. Instead, in a way similar to gradient flows as implemented in [START_REF] Salvarani | Optimal individual strategies for influenza vaccines with imperfect efficacy and limited persistence[END_REF], we ask that the transition from p n to p n+1 be smooth; the intuitive interpretation is that the individual wants to minimize his/her cost but the new vaccination strategy need to be near the previous one. Hence, we minimize for all t and for given parameter δτ > 0 :

min p(t)∈[0,1] (p (t) -p n (t)) 2 2δτ + p (t) r V -r I φ u n I (t) . (4.3)
The constant δτ plays the role of a pseudo-time; we denote τ n = nδτ . If one forgets for a moment the restriction that p ∈ P, when δτ → 0 the sequence of p n will approach a solution of the following equation set in the space of curves ∂ ∂τ p τ (t)+r V -r I φ uτ (•) I

(t) = 0 (here u τ is the primitive of p τ ), see [START_REF] Turinici | Equilibrium flows on metric spaces[END_REF] for details.

The next iterate p n+1 is the solution of the minimization problem (4.3):

p n+1 (t) = proj [0,1] p n (t) -δτ r V -r I φ u n I (t)
, where proj [0,1] (x) = max (min (x, 1) , 0). Hence, we have a new vaccination function and the procedure iterates until a stopping criterion is met.

The stopping criterion is related to the incentive to vaccinate: given p n and u n , one may ask whether a Nash equilibrium has been reached. Recall that a Nash equilibrium is a strategy such that there is no gain in changing it unilateraly. We define the best obtainable cost at any time t by: min

p∈[0,1] pr V + (1 -p)r I φ u n I (t) = min r V , r I φ u n I (t) . (4.4)
In other terms, the Nash equilibrium is reached when, for the given u n and any time t, the best obtainable cost defined in equation (4.4) equals the cost obtained with p n (t).

Since the second cost is always larger than the first, a good stopping criterion is:

E p n (t) := p n (t)r V + (1 -p n (t))r I φ u n I (t) -min r V , r I φ u n I (t) . (4.5)
This will also be called the incentive to change strategy.

4.3.

Results. We consider, in this section, an epidemic with the following characteristics: µ = 1/80, R 0 = 15, γ = 1000µ, β = R 0 * (γ + µ), r = 0.1, T = 80, S 0 = 6.53%, I 0 = 7.4e -6.

In this case the long-term, time independent, convergently stable Nash equilibrium is attained at a level of p csne = 1 -1/(R 0 * (1 -r)) = 0.9259. We implement the procedure described in Section 4.2 (with δτ = 0.1) which converges as shown in figure 3, where the incentive to change strategies decreases by several orders of magnitude. We observe thus that a good solution has been obtained. The probability of vaccination p(t), which is the main result of the simulation and which completely characterize the equilibrium, is given in figure 4. Note that in order for the numerical simulation to complete, we had to choose a finite time horizon. So we have also checked that the solution is robust with respect to T . The comparison between two solutions, one with time horizon T = 20 and another with T = 80, is also presented in figure 4. Further numerical tests for T = 800, not shown here, display similar behavior. The similarity between the solutions for T = 20 and larger values of T shows that the main dynamics happens already in the T = 20 time horizon, which is more compatible with the time span of the individuals than the time-independent p csne vaccination strategy (see section 2). We refer to figure 1 for the curve t → (S(t), I(t)) . Finally, we compare in figure 5 the quality of two equilibrium strategies: the p(t) = p csne vaccination rate, which is an equilibrium within the class of time independent strategies (but not necessarily so in the class of time dependent vaccination rates) and the equilibrium strategy in figure 4 within the class of time dependent vaccination rates. The quality of each equilibrium is illustrated by plotting, for an individual born at t = 0, the two terms of the incentive to change strategies (see equation (4.5)): in grey the cost p(t)r V +(1-p(t))r I φ u I (t) experienced by the individual following the equilibrium strategy and in white the minimum possible cost min {r V , r I φ u I (t)}. The time dependent optimal strategy has lower incentive to change, resulting in a higher quality equilibrium. 1. The grey bar is always superior to the white bar and their difference is related to the quality of the equilibrium: smaller the difference higher the quality.

Note that f o (x, v) ≤ L v for any v ∈ X; moreover, as function of v, the directional derivative

f o (x, v) is subadditive i.e. f o (x, v + w) ≤ f o (x, v) + f o (x, w), ∀v, w ∈ X.
In particular it can be lower bounded by a linear functional in X * . The (Clarke) generalized gradient of f at x denoted ∂f (x) or ḟ (x) is the set of all such linear functionals; the formal definition is the following:

∂f (x) = {ξ ∈ X * |f o (x, v) ≥ v, ξ , ∀v ∈ X}. (A.2)
It can be shown that the Clarke generalized gradient is a non empty, convex, (weakly- * ) compact subset of X * . In particular when X = R k for some k ∈ N * , ∂f (x) is the convex hull of the set {lim →∞ ∇f (x )} for any sequence x converging to x such that: -∇f (x ) exists ∀ (recall that since f is Lipschitz it is differentiable a.e.) and -the limit lim →∞ ∇f (x ) exists.

Appendix B. Technical details concerning the probability of infection

Recall that φ u I (.) is a function from R + to [0, 1] such that, for any t ∈ R + , φ u I (t) is the probability of infection during the life of an individual, born in t and not vaccinated, when the population follows the vaccination strategy u. In mathematical terms, for any individual born in t ≥ 0,

φ u I (t) = P ∃τ ≥ t such that M t τ = I | M t t = S . (B.1)
In order to compute φ u I (t) we introduce the probability ϕ u,t I (.) of infection before τ :

ϕ u,t I (τ ) = P ∃s ∈ [t, τ ] such that M t s = I | M t t = S . (B.2)
Hence, we have : Hence, the probability of staying susceptible between t and τ is : r u,t (τ ) = e -µ(τ -t) exp - By an integration by parts, we obtain :

ϕ u,t I (τ + ∆τ ) = P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t t = S = P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = S × P M t τ = S | M t t = S + P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = I × P M t τ = I | M t t = S + P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = R × P M t τ = R | M t t = S + P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = D 1 × P M t τ = D 1 | M t t = S + P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = D 2 × P M t τ = D 2 | M t t = S , with P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = D 1 = 0 P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = I = 1 P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = R = 1 P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = D 2 = 1 P M t τ +∆τ ∈ I ∪ R ∪ D 2 | M t τ = S =
φ u I (t) = 1 - +∞ t µ exp - τ t
(µ + βI u (s)) ds dτ.

In fact, this probability is solution of the differential equation : Hence, for all t ≥ 0, we obtain :

dφ u I (t) dt = µ exp -
φ u I (t) = 1 - F u I (∞) -F u I (t) f u I (t)
, φ u I (0) = 1 -F u I (∞) .

1 ,

 1 I 1 ) S(t),I(t) for time-independent model S(t),I(t) for time-dependent model

Figure 1 .

 1 Figure1. Dotted line: evolution of the susceptibles and infected classes in the time independent vaccination rate model[START_REF] Bauch | Vaccination and the theory of games[END_REF] with parameters : µ = 1/80, R 0 = 15, γ/µ = 1000, β = R 0 * (γ + µ), r = 0.1, T = 80(years) and initial parameters S 0 = 6.53%, I 0 = 7.4e -6. For these parameters p csne = 0.9259 is a convergently stable Nash equilibrium (within class of the time independent vaccination rates) with limit state S ∞ = 6.6%, I ∞ = 7.4e -6. More than a hundred years is required in order for the long time limit to be reached. Solid line: the evolution of the susceptibles and infected classes in the time dependent vaccination rate model (2.8)-(2.11) for a total time T = 5; faster convergence towards (S ∞ , I ∞ ) is observed.

Figure 2 .

 2 Figure 2. Individual model is a continuous time non-homogeneous Markov chain.

3. 3 .

 3 The mapping T . Recall that the mapping T : u ∈ C → T (u) ∈ P(C) is such that T (u) is defined by equation 3.1.

  Let u in C, we need to show that T (u) is a closed convex (non empty) subset of C. By the Section 3.3.1, T (u) is a closed subset of C. For the convexity, let v, w ∈ T (u) and α ∈ [0, 1]. There exist p v and p w associated respectively to v and w, in the sense of relation (3.1). Denote z = αv + (1 -α)w. Let us consider two cases :

Figure 3 .

 3 Figure3. Decrease, with respect to τ , of the incentive to change strategy as defined in equation(4.5). The maximum mismatch between the current cost and optimal cost decreases by several orders of magnitude and as such one can consider the solution attained.

Figure 4 .

 4 Figure 4. The probability of vaccination (for T = 20 and T = 80).

Figure 5 .

 5 Figure 5. The comparison of the incentive to change the equilibrium strategy for the time dependent / independent classes. The parameters are as in figure1. The grey bar is always superior to the white bar and their difference is related to the quality of the equilibrium: smaller the difference higher the quality.

  βI u (τ ) ∆τ + o(∆τ ). Hence,ϕ u,t I (τ + ∆τ ) = βI u (τ ) ∆τ × P M t τ = S | M t t = S + P M t τ ∈ I ∪ R ∪ D 2 | M t t = S + o(∆τ ) = βI u (τ ) ∆τ × P M t τ = S | M t t = S + ϕ u,t I (τ ) + o(∆τ ). We denote r u,t (τ ) = P M t τ = S | M t t =S the probability of staying susceptible between t and τ . We compute this probability : r u,t (τ + ∆τ ) = P M t τ +∆τ = S | M t τ = S × P M t τ = S | M t t = S = (1 -βI u (τ ) ∆τ -µ∆τ ) r u,t (τ ) + o(∆τ ).

  τ t βI u (s) ds .

  ∆τ ) = βI u (τ ) ∆τ e -µ(τ -t) exp -τ t βI u (s) ds + ϕ u,t I (τ ) + o(∆τ ),which leads todϕ u,t I (τ ) dτ = βI u (τ ) e -µ(τ -t) expτ t βI u (s) ds = e -µ(τ -t) exp t 0 βI u (s) ds × βI u (τ ) expτ 0 βI u (s) ds = -e -µ(τ -t) exp t 0 βI u (s) ds d expτ 0 βI u (s) ds dτ = -e -µ(τ -t) exp t 0 βI u (s) ds dψ u I (τ ) dτ ,by setting ψ u I (τ ) = expτ 0 βI u (s) ds . Finally, we just have to compute the probability of infection during the life of an individual born in t who is not vaccinated, which is : (τ -t) + t 0 βI u (s) ds [ψ u I (τ )] dτ = -exp µt + t 0 βI u (s) ds +∞ t e -µτ [ψ u I (τ )] dτ.

  βI u (s)) ds -βI u (s)) ds dτ= µ -µ +∞ t (µ + βI u (t)) expτ t (µ + βI u (s)) ds dτ = µ -(µ + βI u (t)) (1 -φ u I (t)) = (µ + βI u (t)) φ u I (t) -βI u (t) .In order to get an explicit form for φ u I (0), we define, for all t ≥ 0 : ) = 1 -φ u I (t) . The last function satisfies the following differential equation :[F u I (t)] = -µ + (µ + βI u (t)) F u I (t) ) × f u I (t)] = [F u I (t)] -F u I (t) (µ + βI u (t)) × f u I (t) = -µf u I

  .32) 2.5. Nash equilibrium. A set of strategies is a Nash equilibrium if no individual can do strictly better by unilaterally changing his/her strategy. The goal is to prove that, in our model, there exists a Nash equilibrium. It is equivalent to finding a fixed point of the function which associate to any element in P all strategies in P which minimize the cost of an individual (supposing everybody else uses as strategy the argument of the function). See section 3.3 for a precise mathematical definition.

Acknowledgements G.T. acknowledges support from the Agence Nationale de la Recherche (ANR), projects EMAQS (ANR-2011-BS01-017-01), CINE-PARA and MFG (ANR-16-CE40-0015-01 ).

Appendix A. Clake generalized gradients

To recall the definition of the Clarke generalized gradients we follow the presentation in [16, section 10.1 page 194] and [START_REF] Frank | Generalized gradients and applications[END_REF]. Let X be a Banach space, X * its dual and x ∈ X; also take f : X → R to be a functional which is Lipschitz with constant L > 0 in a neighborhood of x, that is, for some > 0, we have f (y) -f (z) ≤ L y -z for all y, z in the ball of center x and radius . The generalized directional derivative of f at x in the direction v, denoted f o (x; v), is defined as