Renaud Bourlès
email: renaud.bourles@univmed.fr

Bruno Ventelou
email: bruno.ventelou@inserm.fr

Maame Esi Woode
email: mwoode@gmail.com

Greqam

Child Income as an Insurance Mechanism Consequences for the Health-Education Relationship

Keywords: Health risk, Education, Insurance mechanism, Remittance. JEL Classification Numbers: I15, I25

This paper analyzes the relationships between HIV/AIDS and education taking into account the appropriative nature of child income. We first build a simple theoretical model linking parental health risk, educational choice and appropriation of future children's income. We show that considering (remittances from) child's income as an insurance asset can reverse the usual negative relationship between disease prevalence and educational investment. This prediction is tested on data compiled from the Demographic and Health Survey (DHS) database for 17 Sub-Sahara African (SSA) countries between the years 2003 to 2010 for children aged between 6 and 22-years-old. To account for the hierarchical nature of the data we employ a multilevel analysis. We find that, in general, the impact of community HIV prevalence on school enrollment is insignificant. Once the data is split to account for differences in appropriation, the effect of community prevalence becomes positive and sometimes significant for highly appropriable groups (rural, girls) and remains either negative for the rest.

Introduction

Education has been proven to be key in developping countries, regarding both macroeconomic (growth) and microeconomic (consumption smoothing, risk management) issues. Within risk-sharing systems, education can act as a way to smooth consumption and to reduce the risk of wealth loss induced by health risk. It is now well documented that health status and in particular HIV status has an important (negative) impact on child education. This is particularly true in developing countries where this negative relationship can lead to macroeconomic issues, such as poverty trap. However, if we focus on the impact of health risk (which can be proxied by prevalence) rather than health status, the impact on education seems less obvious. We therefore propose here to enrich the analysis of the link between HIV and education by focusing -both theoretically and empirically -on health risk (rather than status) and on the insurance role of education. This paper refers to three fields of economic literature: i) the relationship between epidemics and economic development at the micro level, i.e. how a health shock may jeopardize a household's future capacity to save, accumulate and generate future income (see Kawabata et There are two ways of insuring for old age: educating one's children or having a lot of children.

Investment in children, both quantitatively and qualitatively is an asset for parents, seen in developing countries as a substitute for social security (see for example [START_REF] Nugent | The Old Age Security Motive for Fertility[END_REF], Michel and Wigniole 2007 and Ehrlich and Lui 1991), especially in old age, which is often characterized by the absence of labor income and the presence of elevated health-risk. However, the fundamental nature of this insurance asset is that it is mobilized differently depending on the realization of a risk: actual 1 The welfare effects of HIV can be approximated through the share of the household budget absorbed by HIV related medical expenditures, (see for example [START_REF] Wagstaff | Catastrophe and Impoverishment in Paying for Health Care: With Applications to Vietnam 1993-1998[END_REF][START_REF] Wagstaff | Catastrophe and Impoverishment in Paying for Health Care: With Applications to Vietnam 1993-1998[END_REF][START_REF] Xu | Household Catastrophic Health Expenditure: A Multicountry Analysis[END_REF]. When health expenditures represent a large share of the healthcare the situation is defined as "catastrophic" as this implies either a substantial sacrifice of the household current consumption, possibly including basic needs, or the mobilization of other resources (such as savings, assets, credit and transfers from friends and family).

repayments are contingent on bad event. Starting from this point, we add to the idea of investment in children as an old-age insurance the notion of changing appropriation. This refers to the return on investment in a child's education being dependent on a variety of characteristics of the child and it's household. Moreover, the intensity with which parents invoke their "rights" on the income of their descendants varies depending on their health status. This is not only inspired by the need to draw a parallel with the classic insurance framework, but can be connected to the notion of "child income appropriation" (see [START_REF] Schoonbroodt | Who Owns Children and Does it Matter[END_REF]: in a lot of developing countries, the property rights on labor-income are shared between the different members of a household and, at least partially, submitted to discretionary re-appropriation decisions coming from the household community. [START_REF] Bazen | The Impact of Parental Health on Child Labor: Evidence from Bangladesh[END_REF] and [START_REF] Mcintyre | What are the Economic Consequences for Households of Illness and of Paying for Healthcare in Low-and middle-income Country Contexts?[END_REF] showed that "child-labor" and "premature removal from school" are connected with health shocks, suggesting that child income indeed plays the role of an insurance asset, with repayments contingent on a bad event. The recent paper of [START_REF] Maccini | Under the Weather: Health, Schooling, and Economic Consequences of Early-Life Rainfall[END_REF] clearly follows the same direction, but links schooling decisions to other types of shock (rainfall shocks) and focuses on the asymmetric effect between girls and boys in the household (as in [START_REF] Duflo | Child Health and Household Resources in South Africa: Evidence from the Old Age Pension Program[END_REF]. More generally, if we extend the analysis to fields outside schooling choice, numerous authors, as [START_REF] Fafchamps | Solidarity Networks in Preindustrial Societies: Rational Peasants with a Moral Economy[END_REF], [START_REF] Harrower | Consumption Smoothing in the Zone Lacustre, Mali[END_REF] and [START_REF] Park | Risk Pooling between Households and Risk Coping Measure in Developing Countries: Evidence from Rural Bangladesh[END_REF], consider that shocks are insured through risk sharing networks (not only household) and that remittances of labor-income act as contingent repayments in case of negative shocks (see [START_REF] Gertler | Insuring Consumption against Illness[END_REF]Gruber 2002, Conroy et al. 2007). These intra-community transfers are more or less compulsory, due to the need of reciprocity in the co-insurance system of the network [START_REF] Fafchamps | Solidarity Networks in Preindustrial Societies: Rational Peasants with a Moral Economy[END_REF], or due to social punishment [START_REF] Rapoport | The Economics of Migrants' Remittances, on the Economics of Giving, Reciprocity and Altruism[END_REF].

In the following, we first present a theoretical model to obtain testable predictions on the relationship between health shock (on adults) and child education considering child education as an insurance asset. We present the property of a "changing appropriation" in case of bad event. We reinterpret the impact of the health shock on educational choice of households in a totally new manner which allows to deal with an empirical "puzzle". Indeed, our empirical findings are not compatible with the existing view that a health shock necessarily has a negative effect on the education of children.

Theoretical Model

In this section we build a simple theoretical model to highlight the importance of child income appropriation for the relationship between education and disease prevalence. While the literature examines the quantitative choice (i.e. fertility) in respond to health shocks (see [START_REF] Becker | A Reformulation of the Economic Theory of Fertility[END_REF] for a survey), we focus on the qualitative decision on education enrollment, modeling here a two-person household. We consider the problem of a representative household composed of a head (the parent) and a child in a two-period model2 .

In the first period (activity period), the parent has a secure revenue, ω, and chooses the proportion of time they educate their child, t. When not enrolled in the educational system, that is in a proportion (1 -t) of the period, children earn for the household a wage, ω i , that can differ per child depending on a set of characteristics, for example the gender of the child. The time spent in school leads to an increase in the "basic" wage in the second period. We denote by ρ the return on each unit of time spent in school, so that choosing t in period 1 gives as child income (1 -t)ω i in period 1, and a period 2 income of (1 + ρt)ω i .

The health risk arises in first period after a decision on the education of the child has been made.

We model health risk as risk on wealth. With probability p, the parent falls ill in both periods (for the sake of simplicity -and consistent with the pattern of HIV/AIDS -we assume that no seropositivity is declared in the second period). The parent, if ill in the first period, loses an amount ε 1 of his revenue for that period and an amount ε 2 of his second period revenue.

The second period is modeled as a retirement period in which the parent does not personally earn money but rather relies on the appropriation of his child's income (in accordance with the literature on old-age security). We assume ε 1 > ε 2 , that is, the monetary cost of illness is higher in the first period than in the second. This is because while in the second period the loss of revenue is only due to health costs, in the first period this loss is not only due to health costs but also due to the loss of work time which translates into an income loss.

The key mechanism of our model is the assumption that the proportion of child income appropriated by the parent in the second period differs according to the parent's health status and that this proportion also differs across children. The variation may be explained for example, by gender (see [START_REF] Maccini | Under the Weather: Health, Schooling, and Economic Consequences of Early-Life Rainfall[END_REF]Yang 2009 or Duflo 2000). To be precise, we assume that the parent appropriates a proportion δ of his child's revenue if healthy and a proportion δ + ∆ i if unhealthy, where ∆ i is not necessarily homogeneous.

If we set β as the discount factor between the two periods, the problem of the parent in period 1 becomes:

max t p u C 1 + βu C 2 + (1 -p) u C 1 + βu C 2 (1)
where

C 1 = (1 -t)ω i + ω C 1 = (1 -t)ω i + ω -ε 1 C 2 = δ(1 + ρt)ω i C 2 = (δ + ∆ i )(1 + ρt)ω i -ε 2
For the model to remain tractable and realistic, we further assume that, if healthy, the parent is always richer in the first period than in the second period, that is C 1 > C 2 (which corresponds to ω > δ(1 + ρ)ω i ) and always better off healthy than unhealthy, that is

C 1 > C 1 and C 2 > C 2 (which is written as ∆ i (1 + ρ)ω i < ε 2 ).
To capture the role of "changing appropriation", we first analyze the baseline model where appropriation is the same irrespective of the health status of the parent, i.e. ∆ i = 0. We compare this to the case where, through appropriation, education can play an insurance role ∆ i > 0. We show that while optimal education is decreasing with disease prevalence in the baseline case, the asymmetric feature of appropriation can lead to a positive relationship.

2.1 The benchmark case: no insurance role for education (∆ i = 0)

Let us first assume that appropriation of child income by the parent does not increase in case of disease, that is ∆ i = 0. In such a case, the first order condition of (1) writes:

-pω i u C 1 + βpδρω i u C 2 -(1 -p)ω i u C 1 + β(1 -p)δρω i u C 2 = 0
That is:

E(u (C 1 )) E(u (C 2 )) = βδρ
To analyze the effect of disease prevalence, p, on the optimal time spent in school, t, we define

f (t, p) ≡ E(u (C 1 )) E(u (C 2 ))
-βδρ and use the Implicit Function Theorem:

∂t ∂p = - ∂f /∂p ∂f /∂t to obtain, ∂f ∂p = u C 1 -u C 1 E(u (C 2 )) -u C 2 -u C 2 E(u (C 1 )) [E(u (C 2 ))] 2 = u C 1 u C 2 -u C 1 u C 2 [E(u (C 2 ))] 2 (2)
When ∆ i = 0, C 2 = δ(1 + ρt)ω i -ε 2 and C 2 -C 2 = 2 for any t. As C 1 -C 1 = 1 ∀t, the income risk
faced by the parent is independent of the level of education he chooses for his child. Therefore, the educational choice reduces to a wealth transfer between the two periods. To abstract from the second order effect of wealth on risk aversion, we consider Constant Absolute Risk Aversion (CARA) preferences:

u(C) = - 1 α e -αC
It then turns out that:

Proposition 1. If the proportion of child revenue appropriated by the parent does not increase when ill, i.e. ∆ i = 0, the level of education, t, decreases with disease prevalence, p.

Proof. See Appendix A.

This theoretical result is as a direct consequence of the income effects and intertemporal smoothing decisions obtained in this framework. A health shock implies, first of all, household impoverishment, with income losses of 1 and 2 . For 1 > 2 , we have that the education decision is downsized in the face of diseases, to compensate potential income-loss in period 1. The slope between health risk, p, and the level of education, t, is then negative as in [START_REF] Fortson | Mortality Risk and Human Capital Investment: The Impact of HIV/AIDS in Sub-Saharan Africa[END_REF], who uses a different micro modeling technique (human capital decision and mortality risk).

Assuming an insurance role for education (∆ i > 0)

If we now assume that the part of child revenue appropriated in period 2 by the parent is higher in case of ill health, i.e. ∆ i > 0, the first order condition of program (1) becomes:

g(t, p) ≡ E(u (C 1 )) -pβ∆ i ρu (C 2 ) E(u (C 2 )) -βδρ = 0 with, ∂g ∂p = ∂f ∂p - β∆ i ρu (C 2 )E(u (C 2 )) -pβ∆ i ρu (C 2 )(u (C 2 ) -u (C 2 )) [E(u (C 2 ))] 2 = u C 1 u C 2 -u C 1 u C 2 -β∆ i ρu (C 2 )u (C 2 ) [E(u (C 2 ))] 2 (3)
Note here that two forces are at play when we compare (3) and ( 2): (i) an extra negative term appears in the numerator of ( 3) and (ii) ∆ i (positive) increases C 2 so that we now have

C 2 > C 2 -2 .
On the one hand, education has a higher return for the parent due to their ability to appropriate more, but on the other hand, the extra (expected) wealth in the second period reduces the need for education for intertemporal consumption smoothing.

Moreover

∂g ∂t = ∂f ∂t - pβ∆ i ρ(δ + ∆ i )ρω i u (C 2 )E(u (C 2 )) -pβ∆ i ρu (C 2 ) p(δ + ∆ i )ρω i u (C 2 ) + (1 -p)δρω i u (C 2 ) [E(u (C 2 ))] 2
Assuming a CARA utility function this simplifies to:

∂g ∂t = ∂f ∂t |∆ i =0 - p∆ i ρω i [E(u (C 2 ))] 2 u C 2 E u (C 1 ) + (1 -p)βρ∆ i u (C 2 )u (C 2 ) > 0
We therefore end up with,

∂t ∂p = 1 ω i u (C1)u (C2)-u (C2)[u (C1)+β∆ i ρu (C 2 )] E(u (C 1 ))E(u (C 2 ))+δρE(u (C 2 ))E(u (C 1 ))+p∆ i ρω i [u (C2)E(u (C 1 ))+(1-p)βρ∆ i u (C 2 )u (C 2 )] (4)
that leads to the following proposition Proposition 2. When the proportion of child revenue appropriated by the parent is higher in the case of ill health (∆ i > 0), the level of education decreases with disease prevalence (in the case of CARA preferences) if and only if agents are impatient enough, that is if and only if

β < 1 ∆ i ρ u (C 1 ) u (C 2 ) - u (C 1 ) u (C 2 ) ≡ β
In the other case, that is if β > β (with β > 0 whenever 1 > 2 ), the level of education increases in p.

Remark 1. The role of changing appropriation (that is the "insurance role" of education) can be inferred from this condition. When ∆ i is positive, the negative slope between prevalence and education (found when ∆ i = 0, see Proposition 1) is more difficult to obtain. For high level of β (β > β), we even find that the sign changes, with the slope becoming positive.

The intuition behind this result is quite simple. Assuming a change in the degree of appropriateness under bad health gives education an insurance role, calling for an increase in education when risk increases. The total effect of an increase in disease prevalence depends on whether this "insurance effect" dominates the consumption smoothing effect found in the previous section. According to proposition 2, the insurance effect dominates when the discount factor is large enough, that is, when the weight of the second period (the one in which education plays its insurance role) is large enough in the intertemporal utility function.

These theoretical results (Propositions 1 and 2) provides us with interesting testable predictions.

Indeed, data would confirm these mechanisms if, after proxying health risk, we find that (i) the relationship between health risk and education is generally negative, but (ii) can become positive when education plays an important insurance role, for example for groups that are highly appropriable. In the next sections, we test these predictions on data assuming that child or parent/household characteristics can lead to different values of ∆ i , that is, to heterogeneity in the insurance role of education.

Data

We We test for the impact of HIV prevalence at the community level on educational enrollment focusing on individuals between the ages of 6 and 22 inclusive. There are, for our analysis, 357873 individuals within 128575 households nested in 6814 communities which are in turn nested in 17 countries. We take a look at the relationship between the HIV prevalence rates (that proxies health risk) and enrollment rates at the community level, for each country of our sample. The HIV prevalence rates are calculated using individuals between the ages of 15 and 49 years inclusive. From Figure 1, there appears not to exist a priori a clear relationship between prevalence rates and enrollment rate.

Depending on the country, this relationship can be positive, negative, or even insignificant.

Insert Figure 1 here

Our theoretical model suggests that differences in appropriation of child income can be the source of this observed heterogeneity. A first way to examine this mechanism is to look at the differences in the bi-dimensional relationship between HIV prevalence rates and enrollment rates when differentiating urban and rural areas (appropriation being easier in rural areas according to [START_REF] Nugent | The Old Age Security Motive for Fertility[END_REF]. Figure 2 takes a first look at this relationship for Lesotho (one of the countries with the most striking figures 3 ). In Lesotho, the relation between community prevalence and educational enrollment is either positive or negative depending on the type of place of residence. In figure 1 we have that the overall effect is slightly positive (certainly insignificant) but when we take type of place of residence into account, we find that this effect can be decomposed into a negative relationship in urban areas and a positive one in rural areas where appropriation of child income is much easier.

Insert Figure 2 here This bi-dimensional analysis however needs to be confirmed by a deeper study, controlling for various confounding factors known to influence educational enrollment rate. Therefore, in the following, in addition to community prevalence rates and country enrollment rates, we control for child and household characteristics including

Empirical Model

We now analyze more precisely the effect of HIV risk, proxied by community level HIV prevalence, on education, i.e. the attendance status of children. Recall that our theoretical model predicts that this effect is generally negative but can turn out to be positive for households in which the part of child income appropriated increases in case of bad health (this intuition being that is it more likely among highly appropriable children). To study the impact of community HIV prevalence on the school attendance status of a child we use a multilevel model. The multilevel model allows us to take into account random components at the individual, household, community and country level.

As our data is hierarchical in nature, ignoring the clustering will imply that the independence condition is violated. In such a case, the standard errors are underestimated (see [START_REF] Guo | Multilevel Modeling for Binary Data[END_REF]; results from the standard binary model can be found in Appendix D). The hierarchical structure in our data comes from the fact that we pool our data across countries and also from the design of the DHS, where individuals are nested within households which are in turn nested within communities. Due to the complex nature of our analysis, that is the existence of 4 levels, we do a preliminary test of the significance of the levels. We are able to include contextual HIV/AIDS factors at the community level. We observe y ijkl , the enrollment status of individual i in household j in community k in country l where

y ijkl =     
1 if currently enrolled or was enrolled in current year 0 otherwise [START_REF] Bennell | The Impact of HIV/AIDS Epidemic on the Education Sector in Sub-Saharan Africa: A Synthesis of the Findings and Recommendations of Three Country Studies[END_REF] Let us denote by p ijlk the probability that y ijkl = 1 that is:

p ijkl = P r (y ijkl = 1)
Then by denoting π ijkl = ln p ijkl 1-p ijkl we have that

π ijkl = α 0 + αX ijkl + βZ jkl + γW kl + ηV l + θ ijkl + υ jkl + kl + φ l ( 6
)
where α 0 is the intercept, α is the vector of coefficients for the covariates X ijkl , β is the vector of coefficients associated with the covariates Z jkl , γ the vector of coefficients for covariates W kl and η the vector of coefficients for covariates V l . The covariates, X ijkl varies at the individual level, Z jkl varies at the household level, W kl at the community level and V l at the country level. The error terms at the individual, household, community and country levels are given as θ ijkl , υ jkl , kl and φ l respectively and have means 0 and variances, σ 2 in , σ 2 hh , σ 2 cm and σ 2 ct respectively, (see [START_REF] Goldstein | Nonlinear Multilevel Models with an Application to Discrete Response Data[END_REF]. We first calculate the intra-class correlations to look at the contribution of each level to the overall variance. This variable is calculated from the empty model:

π ijkl = α 0 + θ ijkl + υ jkl + kl + φ l (7)
Estimates of household, community and country level variances are used to calculate the intra-class correlation coefficients to determine the proportion of group-level variance compared to total variance, using the following formula:

ρ hh = σ 2 hh σ 2 hh + σ 2 cm + σ 2 ct + σ 2 in ( 8
)
ρ cm = σ 2 rg σ 2 hh + σ 2 cm + σ 2 ct + σ 2 in ( 9
)
and

ρ ct = σ 2 ct σ 2 hh + σ 2 cm + σ 2 ct + σ 2 in ( 10
)
where ρ hh , ρ cm and ρ ct are the intra-class correlations at the household, community and country levels and σ 2 in is the variance at the individual level, where for the case of a multilevel logistic regression is fixed at π 2 3 , i.e. [START_REF] Bazen | The Impact of Parental Health on Child Labor: Evidence from Bangladesh[END_REF].289 (see [START_REF] Hedeker | MIXOR: A Computer Programme for Mixed Effects Ordinal Regression Analysis[END_REF].

The empty model gives us as variances for the four levels σ 2 in = 3.2899, σ 2 hh = 0.7017, σ 2 cm = 0.9317 and σ 2 ct = 0.7487. From equation ( 8) we obtain as intra-class correlation rates ρ hh = 12.3713, ρ cm = 16.4263 and ρ ct = 13.1999 meaning that 12.3713% of the total variation is explained at the household level, with 16.4263% explained at the community level and 12.3713% at the country level.

Based on this result, we include the four levels in our multilevel model. We now discuss the results we obtain from our multilevel model.

Results

Running our empirical model on Demographic and Health Surveys, we confirm previous findings (see table [?]). We indeed find that (i) having an HIV+ household member implies a negative impact on the probability of enrolling in school, a result that can be reconciled with [START_REF] Fortson | Mortality Risk and Human Capital Investment: The Impact of HIV/AIDS in Sub-Saharan Africa[END_REF] (ii) living in rural areas also has a negative and significant effect on enrollment whereas (iii) being from a high income household has a positive and significant impact on the probability of being enrolled. Regarding, our variable of interest, we first find that, taking into account the whole sample, the probability of being enrolled in school is not significantly affected by the probability of adult household members becoming HIV+.

The results are nuanced if, as in [START_REF] Nugent | The Old Age Security Motive for Fertility[END_REF], and based on the second part of the theoretical model (allowing for heterogeneous appropriateness), we split our data based on gender and on type of place of residence of the household.

Insert Table 3 here An increase in HIV prevalence (the probability of adult household members becoming HIV+) leads to a reduction in male enrollment but an increase in female children enrollment, though the effect for females are insignificant. According to our theoretical, this would come from the fact that females are more appropriable than their male counterparts. The result can be related to the The same kind of results can be inferred from the split between urban and rural regions. As children are more easily appropriable in rural regions, we find that an increase in HIV-prevalence decreases the probability of enrollment in urban areas but increase it in rural areas. These two results on subsamples confirm the predictions of our theoretical model, as we find that for subgroup of children that are highly appropriable (female children and children in rural areas), the insurance role of education can lead to a positive (and sometimes significant) relationship between the probability of becoming HIV+ and the probability of being enrolled in school.

Insert Table 4 here

As a final check we split the data based on the age groups (see Table 4). We find that our key relationship (between HIV prevalence and educational enrollment) is positive and significant for children from 6 to 12, insignificant for children from 13 to 18 and negative and significant for children aged 19 to 22. This seems to confirm the insurance role plaid by children future income as the positive relationship highlighted in our theoretical model is mainly present for young children (who closely corresponds to the one modeled in the theoretical part).

The two last columns of Table 4 split data with respect to the degree of community HIV prevalence.

We then find that our positive relationship is restricted to areas where HIV prevalence remains low.

In our view, this underlines that the total effect consists of the insurance (positive) effect presented here and the usual negative effect (through household income effect and other macroeconomic mechanisms 4 ); and that the positive effects can only dominate in low prevalence community.

Discussion and Conclusion

We have analyzed the impact of health risk, and specifically HIV/AIDS risk, on education taking into account the fact that education of children can act as a form of insurance against shocks, and in this particular case, health shocks in old-age. We first built a theoretical model where active adults invest in the education of their children. We find that in the absence of extra appropriation in case of ill health, an increase in the probability of falling ill leads to a reduction in the time children spend in school. However when there is an opportunity for extra appropriation in bad health events, we have that health risk may in some cases lead to an increase in the amount of education, especially when family's response in terms of required remittance (the extra amount appropriable) is very high.

We test the idea that an increase in the probability of falling ill can lead to an increase in educational investments and find that (i) for the overall population of 6 to 22-year-olds the effect is negative but that (ii) once the data is split between males and females the effect becomes positive for females. When we also split the data between rural and urban households we find a positive and significant effect in the rural areas.

We therefore verify the conventional result (a negative relationship between education and HIV prevalence, see Fortson (2011)) for a subsample of children: male children and children living in urban areas. But we enrich it for children of another type: female children and children in rural areas for whom the relationship seems to be positive (a result already emphasized by some: see [START_REF] Bennell | The Impact of HIV/AIDS Epidemic on the Education Sector in Sub-Saharan Africa: A Synthesis of the Findings and Recommendations of Three Country Studies[END_REF] and [START_REF] Bennell | AIDS in Africa: Three Scenarios for the Education Sector[END_REF]). This empirical "puzzle", i.e. this surprising non-monotonicity, opens the door to a new theoretical reasoning different from the usual one of consumption smoothing. Our interpretation of these results is that, depending on how appropriable the income of ones child is, education acts as an insurance asset, providing protection in bad events. This attribute could reverse the sign of the relationship between disease prevalence and schooling. Our theoretical model provides one possible explanation of such a mechanism.

These results suggest that the role of remittance systems would be an interesting avenue to explore.

In our empirical setting, we have proxied differences in appropriateness by exogenous characteristics (gender, urban/rural areas) that have been proven in other papers to be related to appropriateness.

However, more attention needs to be paid to our key mechanism of "changing appropriateness", for example by analyzing to what extent (and based on which characteristics) remittances change depending on the health status of parents.

Last, regarding the policy implications of our work, it seems worth emphasizing that the general picture we obtain is that HIV/AIDS disease indeed has a negative impact on school enrollment.

First, the fact that family members are infected (individual HIV status) is always associated with a decrease in the probability of schooling. Second, when we obtain a positive effect of HIV risk rate (community prevalence), we must keep in mind that the estimated coefficient evaluates the actual consolidation of different forces, sometimes positive (insurance effect) and generally negative (income effect, supply side effect on the educational sector). This is clearly demonstrated by the various stratifications of the estimations, showing that the positive effect only dominates for young children, in rural areas, in countries of low prevalence. In brief, the findings of this paper clearly do not support a decrease in schooling support in Least Developed Countries(LDC). To the contrary, the lesson is that supply-oriented policies should be favored, as the demand for education can be partly self-sustained, given the insurance effect.

Insert Table 5 here Insert Table 6 here Insert Table 7 here

C Community Enrollment vs Prevalence Over Type of Place of Residence for All Countries

Insert Figure 3 here

D Results from the Binary Model

Insert Table 8 here

E Graphs and Figures

  al. 2002, Wagstaff and Doorslaer 2003, Xu et al. 2003 and Flores et al. 2008 1 ; Fortson (2011) already demonstrates the impact of HIV/AIDS on children's education), ii) child labor and child income appropriation by parents (for example Basu and Pham 1998 and Schoonbroodt and Tertilt 2010), iii) educational choice as old-age insurance in the absence of social security, the most common situation in a lot of African countries (Ehrlich and Lui 1991 for example studied the effect of changing mortality in an overlapping-generations model in which children provide old-age support for their parents).

  findings of Maccini and Yang (2009) or De la Brière et al. (2002) who advocate that girls are often used as insurance assets (in the case of rainfall shocks for Maccini and Yang (2009)).

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Community Enrollment vs Prevalence Per Country

  use the nationally representative cross-sectional data set, the Demographic and Health Survey (DHS), for 17 Sub-Saharan African countries, namely Burkina Faso (2003), Cameroon (2004),

	Congo Kinshasa (2007), Ethiopia (2005), Ghana (2003), Guinea (2005), Kenya (2008/09), Lesotho
	(2009), Liberia (2007), Malawi (2010), Mali (2006), Niger (2006), Senegal (2005), Sierra Leone

(2008)

, Swaziland (2006/07),

Zambia (2007)

and Zimbabwe

(2005/06)

. In this survey, data is collected at both the individual and household level. The DHS has been conducted in developing countries since 1984 with the aim of providing countries with data needed to monitor and evaluate population, health and nutrition programs on a regular basis. The data is collected usually every 5 years though a few are collected over shorter time intervals. It contains household data on basic characteristics of members of each household and also specific data on both male and female household members between the ages 15 and 49 inclusive. These surveys include testing for the HIV/AIDS status of members of the household 15 to 49 years of age. The sampling frame used in most DHS is, by definition, a list of non-overlapping area units with a majority of DHS sample designs clustered and stratified.

  85% of their children below the ages of 5. Approximately 69.07% of households are located in rural areas with 74.14% of the households being headed by males. About 43.93% of the household heads are uneducated, while 5.9801% of the households have an HIV+ member.

	Insert Table 1 here
	Regarding educational level (primary, secondary or university) we have:
	Insert Table 2 here

• Child characteristics: Age, gender, residency status and relation to head.

• Household characteristics: Proportion of children below 5, type of place of residence, income, gender of head, age of head, HIV status of the household and educational level of head.

Regarding these characteristics (see table

1

), our sample contains children of average age 12.85 years old with an almost equal number of males and females. Most of these children are usual residents of their households with a little over half being children of the head of the household. Households have about 16.

3 See Appendix 3 for results for all countries

Table 1 :

 1 Summary Statistics

	E.1 Tables

Table 2 :

 2 Highest Educational Level of Individuals Per Age Group

		Fixed Effects
	Variable	Total	Male	Female	Urban	Rural
	Intercept	-3.3371 * * * -2.4997 * * * -3.6439 * * * -1.7544 * * * -4.8271 * * *
		(0.1397)	(0.0853)	(0.2626)	(0.3045)	(0.1768)
	Age	1.1302 * * *	1.1362 * * *	1.1719 * * *	0.9639 * * *	1.2402 * * *
		(0.0069)	(0.0094)	(0.0105)	(0.0119)	(0.0086)
	Age-Squared	-0.0480 * * * -0.0464 * * * -0.0517 * * * -0.0410 * * * -0.0528 * * *
		(0.0003)	(0.0003)	(0.0004)	(0.0004)	(0.0003)
	Male	0.4530 * * *			0.5300 * * *	0.4173 * * *
	Usual Resident	0.1763 * * *	0.2892 * * *	0.0493	0.2249 * * *	0.0916 * *
		(0.0330)	(0.0486)	(0.0458)	(0.0582)	(0.0404)
	Child of Head	0.6871 * * *	0.4069 * * *	0.8397 * * *	0.8605 * * *	0.5921 * * *
		(0.0120)	(0.0166)	(0.0169)	(0.0206)	(0.0147)
	Household HIV+	-0.0487 *	-0.0683 *	-0.0281	0.0122	-0.0877 * * *
		(0.0274)	(0.0362)	(0.0368)	(0.0467)	(0.0337)
	Household HIV Unknown -0.0266 * *	-0.0063	-0.0388 * *	-0.0150	-0.0337 * *
		(0.0129)	(0.0164)	(0.0174)	(0.0244)	(0.0152)
	Rural	-0.8149 * * * -0.8188 * * * -0.8352 * * *
		(0.0312)	(0.0336)	(0.0363)
	Middle Income	0.3195 * * *	0.3155 * * *	0.3739 * * *	0.2657 * * *	0.3616 * * *
		(0.0171)	(0.0217)	(0.0229)	(0.0565)	(0.0180)
	High Income	0.7941 * * *	0.8002 * * *	0.8855 * * *	0.9762 * * *	0.7527 * * *
		(0.0192)	(0.0300)	(0.0257)	(0.0522)	(0.0211)
	Prop. of Child. < 5	-0.8896 * * *	-0.0335	-1.3894 * * * -1.5303 * * * -0.6006 * * *
		(0.0462)	(0.0629)	(0.0624)	(0.0876)	(0.0543)
	Head Uneducated	-0.7298 * * * -0.7720 * * * -0.7910 * * * -0.9386 * * * -0.6136 * * *
		(0.0158)	(0.0201)	(0.0210)	(0.0289)	(0.0188)
	Head 40-59	0.1765 * * *	0.1341 * * *	0.2147 * * *	0.2731 * * *	0.1235 * * *
		(0.0144)	(0.0190)	(0.0195)	(0.0262)	(0.0172)
	Head 60-79	0.2958 * * *	0.2051 * * *	0.3758 * * *	0.3917 * * *	0.2308 * * *
		(0.0184)	(0.0239)	(0.0249)	(0.0352)	(0.0217)
	Head > 80	0.4318 * * *	0.3225 * * *	0.5504 * * *	0.6588 * * *	0.3078 * * *
		(0.0426)	(0.0551)	(0.0568)	(0.0916)	(0.0482)
	Head Male	-0.2703 * * * -0.2368 * * * -0.3440 * * * -0.2600 * * * -0.2552 * * *
		(0.0146)	(0.0194)	(0.0195)	(0.0258)	(0.0172)
	Cluster HIV Prevalence	0.0009	-0.0035 * *	0.0017	-0.0051 * *	0.0108 * * *
		(0.0017)	(0.0016)	(0.0021)	(0.0024)	(0.0023)
	Country Enrollment Rate	0.0624 * * *	0.0570 * * *	0.0693 * * *	0.0271 * * *	0.0758 * * *
		(0.0023)	(0.0010)	(0.0044)	(0.00582)	(0.0029)
		Random Effects
	Household
	Int. Variance	1.0133	0.9574	0.9006	1.1560	0.9283
	Int. Std. Deviation	1.0066	0.9784	0.9490	1.0752	0.9635
	Community
	Int. Variance	0.8990	0.8651	1.0008	0.3874	0.9538
	Int. Std. Deviation	0.9481	0.9301	1.0004	0.6224	0.9766
	Country
	Int. Variance	0.0173	-	0.0744	0.0963	0.0290
	Int. Std. Deviation	0.1317	-	0.2728	0.3103	0.1703
	Number of Obs.	357873	178561	179332	105563	252310
	Groups: Household	128575	93002	101415	37060	91515
	Groups: Cluster	6814	6812	6810	2291	4523
	Groups: Country	17	17	17	17	17

Table 3 :

 3 Impact of HIV Prevalence

			Fixed Effects
	Variable	6 -12	13 -18	19 -22	Low Prevalence High Prevalence
	Intercept	-3.4162 * * * -3.2379 * * * -2.5942 * * *	-3.1362 * * *	-2.9248 * * *
		(0.6458)	(0.3262)	(0.6636)	(0.1321)	(0.2992)
	Age	2.9692 * * *	0.7783 * * * -2.5452 * * *	1.0722 * * *	1.2356 * * *
		(0.0391)	(0.1019)	(0.5104)	(0.0080)	(0.0140)
	Age-Squared	-0.1406 * * * -0.0395 * * * 0.0520 * * *	-0.0445 * * *	-0.0552 * * *
		(0.0022)	(0.0033)	(0.0125)	(0.0003)	(0.0005)
	Male	0.1929 * * *	0.6193 * * *	0.8772 * * *	0.5416 * * *	0.1860 * * *
		(0.0152)	(0.0170)	(0.0265)	(0.0114)	(0.0202)
	Usual Resident	0.1444 *	0.1801 * * * -0.3535 * * *	0.1085 * *	0.0856 *
		(0.0767)	(0.0491)	(0.0561)	(0.0476)	(0.0486)
	Child of Head		* * *	0.6162 * * *	0.6996 * * *	0.7241 * * *
		(0.0204)	(0.0198)	(0.0313)	(0.0142)	(0.0233)
	Household HIV+	-0.0501	-0.0806 * -0.2088 * * *	-0.0176	-0.0900 * * *
		(0.0468)	(0.0413)	(0.0556)	(0.0599)	(0.0340)
	Household HIV Unknown -0.0410 * * -0.1021 * * *	-0.0088	-0.0211	-0.0594 * *
		(0.0193)	(0.0198)	(0.0288)	(0.0147)	(0.0285)
	Rural	-1.1558 * * * -0.7229 * * * -0.6304 * * *	-1.0414 * * *	-0.2272 * * *
		(0.0450)	(0.0381)	(0.0423)	(0.0392)	(0.0498)
	Middle Income	0.4307 * * *	0.4437 * * *	0.1872 * * *	0.3289 * * *	0.3351 * * *
		(0.0248)	(0.0260)	(0.0421)	(0.0200)	(0.0340)
	High Income	1.0130 * * *	0.8784 * * *	0.8174 * * *	0.7943 * * *	0.8823 * * *
		(0.0287)	(0.0287)	(0.0410)	(0.0226)	(0.0378)
	Prop. of Child. < 5	-0.0785	-1.0899 * * * -2.6911 * * *	-0.6944 * * *	-1.4031 * * *
		(0.0710)	(0.0753)	(0.1069)	(0.0546)	(0.0907)
	Head Uneducated	-0.9036 * * * -0.8728 * * * -0.8385 * * *	-0.7500 * * *	-0.6491 * * *
		(0.0236)	(0.0236)	(0.0350)	(0.0185)	(0.0315)
	Head 40-59	0.0014	0.2395 * * *	0.6403 * * *	0.1878 * * *	0.1811 * * *
		(0.0214)	(0.0242)	(0.0375)	(0.0171)	(0.0282)
	Head 60-79	0.0676 * *	0.3239 * * *	0.6862 * * *	0.2537 * * *	0.4100 * * *
		(0.0289)	(0.0285)	(0.0432)	(0.0219)	(0.0362)
	Head > 80	0.2274 * * *	0.5024 * * *	0.6635 * * *	0.4326 * * *	0.4312 * * *
		(0.0670)	(0.0610)	(0.0958)	(0.0515)	(0.0790)
	Head Male	-0.2526 * * * -0.2967 * * * -0.2743 * * *	-0.2929 * * *	-0.2225 * * *
		(0.0235)	(0.0219)	(0.0302)	(0.0182)	(0.0259)
	Cluster HIV Prevalence	0.0121 * * *	-0.0031	-0.0065 * * *	0.0380 * * *	-0.0025
		(0.0027)	(0.0021)	(0.0023)	(0.0083)	(0.0022)
	Country Enrollment Rate	0.0892 * * *	0.0635 * * *	0.0186 *	0.0610 * * *	0.0567 * * *
		(0.0109)	(0.0055)	(0.0112)	(0.0021)	(0.0048)
			Random Effects
	Household
	Int. Variance	1.52714	0.7429	0.4828	1.0164	1.1390
	Int. Std. Deviation	1.2358	0.8619	0.6948	1.0082	1.0673
	Community
	Int. Variance	1.4231	0.9406	0.6329	0.9899	0.5351
	Int. Std. Deviation	1.1929	0.9698	0.7956	0.9950	0.7315
	Country
	Int. Variance	0.4925	0.1188	0.5212	0.0113	0.0765
	Int. Std. Deviation	0.7018	0.3447	0.7219	0.1064	0.2766
	Number of Obs.	184164	116685	57024	255785	102088
	Groups: Household	94988	73415	46124	88929	39646
	Groups: Cluster	6802	6802	6773	4607	2207
	Groups: Country	17	17	17	17	17

Table 4 :

 4 Impact of HIV Prevalence (Age and Prevalence)Community Enrollment Burkina Faso Cameroon D.R. Congo Ethiopia Ghana Guinea Kenya Lesotho Liberia Malawi Mali Niger Senegal Sierra Leone Swaziland Zambia Zimbabwe

	18.0582	9.3527	17.0611	398
	14.2148	10.2653	13.5156	319
	26.8747	12.2443	25.5852	275
	1.5185	3.5170	0	353
	10.7839 1.2155 1.0425 0.8874	11.6696 2.8586 2.8892 2.5364	7.84921 0 0 0	847 407 342 376
	1.6783	2.4476	0	298
	7.0473 22.2591	9.4101 12.6784	4.6556 21.7038	395 400
	1.5534	3.2699	0	292
	2.2586	3.8330	0	412
	2.0049	4.1870	0	535
	1.3277	2.4655	0	300
	5.7832	7.0585	4.2834	466
	1.8167	3.7134	0	400
	Mean	Std. Dev.	Median	Obs

Table 7 :

 7 Summary Statistics of Cluster Prevalence Rate (%)

	Low Prevalence High Prevalence	-1.9407 * * * -2.0589 * * *	(0.0489) (0.0738)	0.8297 * * * 0.9867 * * *	(0.0066) (0.0117)	-0.0344 * * * -0.0439 * * *	(0.0003) (0.0004)	0.3968 * * * 0.1457 * * *	(0.0095) (0.0170)	-0.0318 -0.0497	(0.0385) (0.0383)	0.4332 * * * 0.5008 * * *	(0.0109) (0.0185)	0.0044 -0.0549 *
	19 -22	-1.6266 * * *	(0.0827)	-1.7640 * * *	(0.4427)	0.0349 * *	(0.0108)	0.7057 * * *	(0.0229)	-0.2854 * * *	(0.0457)	0.4751 * * *	(0.0266)	-0.1962 * * *	(0.0466)
	13 -18	-1.9405 * * *	(0.0605)	0.7280 * * *	(0.0867)	-0.0343 * * *	(0.0028)	0.4715 * * *	(0.0142)	0.1296 * *	(0.0400)	0.5715 * * *	(0.0160)	-0.0824 *	(0.0330)
	6 -12	-1.8746 * * *	(0.0689)	1.9408 * * *	(0.0303)	-0.0917 * * *	(0.0017)	0.1102 * * *	(0.0117)	-0.1467 *	(0.0570)	0.1661 * * *	(0.0145)	0.0132	(0.0324)
	Rural	-3.0074 * * *	(0.0452)	0.9759 * * *	(0.0072)	-0.0415 * * *	(0.0003)	0.3113 * * *	(0.0099)	-0.0142	(0.0322)	0.3592 * * *	(0.0114)	-0.0558 *	(0.0243)
	Urban	-1.2963 * * *	(0.0703)	0.7618 * * *	(0.0100)	-0.0323 * * *	(0.0004)	0.4068 * * *	(0.0152)	0.0722	(0.0471)	0.5930 * * *	(0.0162)	-0.0044	(0.0329)
	Female	-2.1554 * * *	(0.0529)	0.9306 * * *	(0.0086)	-0.0409 * * *	(0.0003)			-0.1859 * * *	(0.0366)	0.5545 * * *	(0.0134)	-0.0265	(0.0282)
	Male	-1.6603 * * *	(0.0541)	0.8771 * * *	(0.0078)	-0.0358 * * *	(0.0003)			0.2271 * * *	(0.0390)	0.2549 * * *	(0.0131)	-0.0586 *	(0.0276)
	Total	-2.0635 * * *	(0.0377)	0.8894 * * *	(0.0057)	-0.0378 * * *	(0.0002)	0.3364 * * *	(0.0082)	0.0132	(0.0265)	0.4289 * * *	(0.0093)	-0.0380	(0.0196)
	Variable	Intercept		Age		Age-Squared		Male		Usual Resident		Child of Head		Household HIV+

Table 8 :

 8 Binary Model

The model is easily extended to situations where the revenue of the parent comes from their own parents (in terms of educational investments). Results are robust to an overlapping-generation modeling with three periods (childhood, working age and retirement) in which the child is born in good health and can only fall ill in the second period.

Unicef 2000[START_REF]The Progress of Nations[END_REF] underlines that the number of African children having lost their teachers because of AIDS was around 860 000 in 1999, It means that the replacement rate of teachers, due to the pandemic, is not large enough to provide a desirable level of education.

Table 6: Summary Statistics of Cluster Enrollment Rate (%)

A Proof of Proposition 1

We have:

Therefore, ∂f ∂p is of the sign of

, that is, in the case of ∆ i = 0, the sign of

Note that the previous lemma also hold for preferences exhibiting Increasing Absolute Risk Aversion and that the result is ambiguous for Decreasing absolute risk aversion As, moreover,


John Douglas
email: j.douglas@brgm.fr

Seismic network design to detect felt ground motions from induced seismicity

Keywords: seismic networks, geothermal power production, induced seismicity, spatial correlation, enhanced geothermal systems, earthquake ground motions, receiver operating characteristic

Human activities, such as fluid injection as part of the stimulation of an enhanced geothermal system (EGS) for heat and power production, can cause damaging earthquake ground motions. A difficulty in quickly settling or rejecting insurance claims to the policy of the operator of the EGS is the lack of ground truth on the observed shaking at sites of reported damage.

To overcome this problem a local seismic network could be installed prior to injection to constrain the ground-motion field at points of potential damage.

Since the installation and maintenance of seismometers is costly there is an incentive to keep the number of instruments to a minimum. In this short communication, ground-motion fields are simulated and receiver operating characteristic analysis conducted to guide decisions on the number of sensors required to obtain a certain confidence in the rate of false alarms and missed detections. For densities of 10-20 instruments per km 2 the ability to estimate potentially-damaging ground motions is reasonable but associated with a significant chance of missed detections and false alarms. If an EGS operator

Introduction

Heat and power production via enhanced geothermal systems (EGS) is an attractive low-carbon renewable energy source and a number of tests of this technology are operating (e.g. Soultz-sous-Forêts, France) or are in the planning stage (e.g. Newberry, USA). In an EGS the permeability of a geothermal reservoir is enhanced using hydraulic stimulation, which purposely induces micro-earthquakes but because of existing (tectonic) may also trigger larger shocks [START_REF] Majer | Induced seismicity associated with En-hanced Geothermal Systems[END_REF]. As discussed by Gardini [START_REF] Giardini | Geothermal quake risks must be faced[END_REF], during the Deep Heat Mining (Basel, Switzerland) project a mainshock of magnitude M L 3.4 was triggered along with thousands of smaller earthquakes. The shaking from these tremors were cause for concern to the local population and led to the shutdown of this project and insurance claims amounting to more than $9 million for minor building damage (e.g. non-structural cracking). A difficulty in resolving these insurance claims is knowing whether the observed building damage was actually due to shaking caused by earthquakes induced or triggered by the EGS or were actually pre-existing and only noticed after the tremors.

Resolution of this question requires information on two aspects: the earth-quake shaking that the local building stock was subjected to due to the EGS project, and the vulnerability of the buildings to this shaking, which could be modelled using appropriate fragility curves. The second of these topics is not discussed here, although fragility curves to predict cracking from highfrequency ground motions from shallow induced seismicity is a topic that has not be widely researched and hence is ripe for investigation. This short communication concerns the ability to recover the true earthquake shaking, characterized by a scalar intensity measure (e.g. peak ground velocity, PGV) that is useful for the prediction of building damage, based on measurements from a local seismic network.

The advantage of having a local seismic network in a population centre close to an area of geothermal power production is shown by a recent study concerning the Icelandic village of Hvergerdi, which is monitored by the dense network known as ICEARRAY [START_REF] Halldorsson | On the effects of induced earthquakes due to fluid injection at Hellisheidi geothermal power plant, Iceland[END_REF]. These instruments were used to demonstrate that the pseudo-spectra accelerations in the village due to some recent induced seismicity (including an M L 3.8 earthquake) from a nearby (at 11 km) geothermal power plant, Hellisheidi, were close to and sometimes surpassed the codified design spectral accelerations for the majority of the building stock.

The aim of this study is to provide guidance on the number of instruments required to obtain a certain detection rate of an intensity measure surpassing a given threshold. It is based on the simulation of local ground-motion fields consistent with those observed from induced seismicity. These simulations are discussed in the following section. The subsequent section presents the method (kriging) used to recover this ground-motion field based on obser-vations at a limited number of (simulated) instruments. The penultimate section analyzes the results of thousands of such simulations using the approach known as receiver operating characteristic (ROC) analysis to draw graphs allowing the number of required instrument density to obtain certain true and false positive rates to be estimated. This short communication ends with some brief conclusions and suggestions for future research.

Simulated ground-motion fields

The procedure presented by [START_REF] Strasser | Review: Strong ground motions -Have we seen the worst?[END_REF]Bommer [4, pp. 2625-2626] is used to simulated the spatially-correlated ground-motion fields taking into account the within(intra)-event variability. Between(inter)-event variability is considered in the ground-motion fields by the inclusion of event terms within the PGV computation computed using the between-event standard deviation (τ ) and the normal distribution. Similar approaches have been used in previous studies [e.g. 5]. Since the aleatory variabilities associated with the GMPE used here are very large the ground-motion fields show great differences and hence many runs are required to obtain stable results from this analysis.

The empirical ground-motion prediction equation (GMPE) derived by Douglas et al. [START_REF] Douglas | Predicting ground motion from induced earthquakes in geothermal areas[END_REF] for PGV based on thousands of records from microseismicity in geothermal zones is used here to obtain the deterministic groundmotion field assuming a point source, which is a common assumption for small earthquakes even though the causative rupture will have length of a few hundreds of metres. This equation, for a rock site based on a generic velocity profile, is (in terms of cm/s and hypocentral distance, r hyp ): log 10 y = -2.3426 + 0.8526M w -1.4048 log 10 ( r 2 hyp + 2.9330 2 ) -0.013r hyp [START_REF] Majer | Induced seismicity associated with En-hanced Geothermal Systems[END_REF] with a between-event standard deviation of τ = 0.6746 and a within-event standard deviation of φ = 0.4467. The impact of the high value of φ on the results was tested and within a reasonable range for this component of the aleatory variability it does not have a significant influence on the detection rates so it is not considered further.

This deterministic ground-motion field is perturbed by the addition of random field derived from a multivariate normal distribution based on an exponential correlation function with a correlation length h 0 (i.e. exp(-h/h 0 ), where h is the separation distance between points of interest) and a standard deviation equal to the within-event variability of the selected GMPE.

The exponential correlation function has been found to fit the observed spatial correlation of earthquake ground motions but h 0 appears to vary with the structural period, geographical region and earthquake [e.g. 5, 7]. To consider the observed range of h 0 two values are considered here: 5 and 20 km. In addition, two end-members: h 0 = 0.001 (no correlation between sites) and h 0 = 10000 km (complete correlation between sites) are simulated to understand the role of this key parameter, which is poorly known particularly for induced seismicity. The spatial correlation of ground motions from small earthquakes could be higher than for larger earthquakes since the point source assumption is more appropriate but on the other hand the short seismic wavelengths from small earthquakes could mean that the spatial variability is large because of scattering due to a heterogeneous crust.

A moment magnitude (M w ) of 3.5 and a focal depth of 3 km are assumed for the scenario earthquake because EGS operators would hope to contain the seismicity to smaller events and hence such a scenario could be considered a worst case for most projects. An area of 1 km × 1 km above the injection well is considered for these calculations with a resolution of 20 m (i.e. a grid of 50 × 50 = 2500 points) and the earthquake epicentre in the centre of this zone. A building density of 1 000 houses per km 2 , corresponding to a European suburban area, is assumed. The number of instruments co-located with the houses is varied between 10, roughly corresponding to the density of the ICEARRAY [START_REF] Halldorsson | ICEARRAY: The first small-aperture, strong-motion array in Iceland[END_REF], and 100, which is much higher than any existing strong-motion array [START_REF] Trifunac | Recording strong earthquake motion -Instruments, recording strategies and data processing[END_REF] and corresponds to an instrument every 100 m. A PGV threshold of 7.4 mm/s is considered because this corresponds to the 2nd-percentile (median minus two standard deviation) PGV for an Modified Mercalli intensity of V (slight cracking of plaster possible) according to the ground-motion intensity conversion equations (GMICEs) of Worden et al. [START_REF] Worden | Probabilistic relationships between ground-motion parameters and Modified Mercalli intensity in California[END_REF]. These GMICEs are used because they are the most recent ones available and they are based on a large database of observations (roughly 200 000 intensity-PGV pairs).

An example simulated ground-motion field with the associated houses and sensors is presented in Figure 1 for h 0 = 5 km, showing the considerable spatial variation to be expected in earthquake ground motions that is not explained by the distance from the source. It is this spatial variation in shaking that makes the estimation of ground motions at the location of a house based on measurements at surrounding instruments prone to large uncertainties. Simulated (dark grey) and recovered (light grey) contour of PGV threshold using the measurements at the 20 instruments and the relations between these thresholds at the 100 considered houses colour-coded thus: red, true positive; blue, false negative; green, false positive; and black (unfilled), true negative. Note that 100 rather than 1000 houses are assumed to more clearly show the results.

Recovery of observed ground motion at sites of interest

In this section it is assumed that all that is available to estimate the ground-motion field are the PGV observations at the simulated sensors. From these measurements the PGVs estimated at the considered houses will be computed and checked against the considered threshold. The geostatistical technique known as kriging with a trend is used for this purpose. Kriging is a commonly-used technique for the interpolation of measurements at geographically-distributed locations and it has been used in ground-motion estimation in a number of studies [e.g. [START_REF] Carr | Use of geostatistics for accurate mapping of earthquake ground motion[END_REF][START_REF] Ambraseys | Magnitude calibration for north Indian earthquakes[END_REF][START_REF] Douglas | Inferred ground motions on Guadeloupe during the 2004 Les Saintes earthquake[END_REF]. The trend model used as part of this procedure accounts for the overall distance decay that on average is present in earthquake ground motions -it is assumed here that the earthquake location is precisely known (this is reasonable given that the earthquakes are assumed to be induced in an EGS reservoir that is monitored by a local seismic network). This technique roughly corresponds to the inverse of the method used to simulate the ground-motion field because kriging makes use of a spatial correlation model, for which here an exponential model is assumed for consistency with the simulations. A comparison between the simulated and recovered ground-motion thresholds at the considered houses is shown in Figure 1. It should be noted that the same value of h 0 is assumed as for the simulations -the impact of this assumption is investigated below. In general this procedure leads to a high correlation between the simulated and recovered PGVs at all grid points, particularly when the density of instruments is large (Figure 2).

Receiver operating characteristic

In this study, the interest is in the ability to predict the surpassing or not of a ground-motion threshold indicating a possibly damaging level of shaking at houses within the epicentral zone. A useful tool in such a situation is ROC analysis in which the rates of the four possible outcomes of a prediction are considered. These four outcomes are: true positive (observed and predicted PGV both above threshold), false positive (false alarm, i.e. predicted PGV above threshold but observed PGV actually below threshold), true negative (observed and predicted PGV both below threshold) and false negative (missed detection, i.e. observed PGV above threshold but predicted PGV below threshold). A good local network would be one where the true positive and negative rates were maximized with respect to the false positives and negatives because this would allow, for example, the insurance claims for building damage to be quickly resolved.

To construct a ROC curve the true positive rate is plotted against the false positive rate. The better the prediction method the closer the ROC curve is to the upper left hand corner of the graph (0, 1), i.e.: a true positive rate of 100% and a false positive rate of 0% (no false alarms or missed detections).

The closer the ROC curve is to the 45 degree line the worse the detection technique. These curves, therefore, provide a useful way of comparing the ability of different densities of sensors to assess the PGVs at sites of interest.

The impact of increasing the density of instruments is shown in Figure 3, where it can be seen that increasing the number of sensors moves the results of the ROC analyses away from the 45 degree line and towards the topleft corner. Because of the large variation in the ground-motion fields and the random distribution of houses and instruments the results of the ROC analysis do not form a curve but show considerable dispersion. In the extreme case of a seismometer in every house the ROC curve would correspond to a point at (0, 1) indicating a perfect detection rate and no false alarms.

Modification of the size or depth of the scenario earthquake or the PGV threshold would increase or decrease the absolute number of samples above or below the considered limit without necessarily changing the ability of the network to discriminate between potential damaging amplitudes or not. For example, Figure 4 shows the results of a ROC analysis using the same procedure but assuming a smaller magnitude (M w 3) and a lower PGV threshold, and 100 within a 1 km 2 zone assuming a spatial correlation length h 0 = 5 km, M w 3 and a PGV threshold of 2.2 mm/s.. plaster possible) again using the same GMICEs as above [START_REF] Worden | Probabilistic relationships between ground-motion parameters and Modified Mercalli intensity in California[END_REF]. By comparing Figures 3 and4 it can be seen that the rates of true and false positives are almost unchanged by the modification of the scenario and the threshold.

Apart from the instrument density, a key parameter affecting the ROC results is the correlation length h 0 . In Figure 5 the impact of varying this parameter is shown assuming 20 instruments. As expected, as the value of h 0 increases (PGVs at neighbouring points are increasing correlated) the ROC results tend to the top left corner, i.e. better detection rate with fewer false alarms. When there is no correlation between neighbouring points (h 0 very small) the ground-motion field cannot be recovered from the available measurements and the question of whether PGV surpassed the considered threshold becomes a matter of complete chance. This is indicated by the ROC curve falling along the 45 degree line in Figure 5. When ground motions are perfectly correlated spatially (h 0 is very large) the ground-motion field can be recovered at all points based on only a few measurements that enable the decay with distance to be estimated. This leads to a ROC curve that falls in the top left hand corner of the graph (Figure 5). The reason why there appears to be far fewer points for h 0 = 10000 km than for the other three cases is that almost all of the points overlap since the ground-motion fields only differ due to the between-event variability. There is generally a small difference in the results when h 0 = 5 or h 0 = 20 km.

One assumption made above is that the same h 0 is assumed to simulate the ground-motion field and to reconstruct the field based on PGVs at the hypothetical instruments. In reality, however, h 0 is uncertain. To investigate the impact of not knowing h 0 an additional test is conducted. ROC analyses are conducted assuming 20 instruments and h 0 = 5 km for the simulations but inversion assuming h 0 = 20 km or h 0 = 1 km, which can then be compared to the results when h 0 is assumed to equal 5 km (Figure 6). This graph supports the observation made previously that for h 0 within the range 5 (or even 1) to 20 km, which corresponds to the values observed for moderate to large earthquakes in active regions, the exact value used for this parameter is not critical.

Conclusions

In this brief article, the benefits of installing a (very) dense seismic network in the vicinity of a source of induced seismicity are assessed from the point of view of detecting potentially damaging motions at local buildings.

The considered scenarios are limited to those that are realistic from the point of view of current monitoring technology, induced ground motions and the latest research on spatial correlations. It is found that installing a local network of a density similar to some recent arrays (e.g. ICEARRAY), i.e. 10 to 20 instruments per km 2 would lead to a reasonable detection rate but with the risk of a considerable number of false alarms and missed detections. As the number of sensors increases to densities not yet seen in engineering seismology (50 to 100 instruments per km 2 ) the number of missed detections and false alarms decreases significantly. The density of instruments installed is a decision for the EGS operator and the regulatory authority since it is related to the chance of correctly recovering potentially-damaging ground motions that is accepted by these organizations.

The type of study presented here could form part of a wider cost-benefit analysis that seeks to find the optimum number of instruments to install in the vicinity of a project that could induce earthquakes (e.g. an EGS) based on the trade-off between the cost of the instruments and their installation and maintenance and their benefit in terms of the ability to accurately estimate the ground motions at points of interest and also, if the instruments could also be used to locate earthquakes, the improvement in reservoir monitoring. Such an analysis would require, however, estimates on the likely cost of resolving insurance claims, required length of monitoring and so forth, which are difficult aspects to define a priori. For the location of a future project where the locations of the buildings, their overall vulnerabilities to earthquakes and a local ground-motion model are known, a site-specific analysis could be conducted following a similar approach to give more precise guid-ance on the installation of a local monitoring network.

The results obtained here are dependent on the value assumed for the spatial correlation length h 0 . As demonstrated by Jayaram and Baker [START_REF] Jayaram | Correlation model for spatially distributed ground-motion intensities[END_REF] and others h 0 appears to show a dependency on structural period, earthquake magnitude and geographical region. All previous studies on this parameter focus on its value for moderate and large tectonic earthquakes (M w > 5) in active regions with high instrument densities (predominantly California, Taiwan and Japan) and, therefore, there is uncertainty on the values to use for this key parameter for small induced earthquakes, particularly those occurring in areas with low instrument densities. Therefore, there is a need for additional research on spatial correlation of ground motions from small (induced) earthquakes to better constrain spatial correlation models.

Acknowledgments

Figure 1 :

 1 Figure 1: Left: Example ground-motion field simulated for this study, with the simulated locations of 100 houses (unfilled circles) and 20 instruments (filled circles). Right:

Figure 2 :

 2 Figure 2: Top: Simulated versus recovered PGVs at the 2 500 grid points for an example ground-motion field assuming 200 instruments. Bottom: Simulated versus recovered PGVs at the 2 500 grid points for an example ground-motion field assuming only 20 instruments.

Figure 3 :

 3 Figure 3: Results of ROC analyses for four different densities of instruments: 10, 20, 50 and 100 within a 1 km 2 zone assuming a spatial correlation length h 0 = 5 km, M w 3.5 and a PGV threshold of 7.4 mm/s.

2. 2 Figure 4 :

 24 Figure 4: Results of ROC analyses for four different densities of instruments: 10, 20, 50

Figure 5 :

 5 Figure 5: Results of ROC analyses for four different values of h 0 : 0 (no spatial correlation), 5, 20 and 10000 km (all points perfectly correlated) assuming 20 instruments.

Figure 6 :

 6 Figure 6: Results of ROC analyses for assuming 20 instruments and using h 0 = 5 km for the simulations and the same value for the inversion or h 0 = 20 km or h 0 = 1 km for the inversion.

  This study was funded by GEISER (Geothermal Engineering Integrating Mitigation of Induced Seismicity in Reservoirs) project funded under contract 241321 of the EC-Research Seventh Framework Programme (FP7). I thank Stefan Wiemer, Falko Bethmann, Dirk Kraaijpoel, Vincenzo Convertito and Ben Edwards for the discussions that led to this study, and their comments on an earlier draft.


Cédric Alexis

Clarence Charnay
email: clarence.charnay@univ-montp2.fr

Vincent Lapinte
email: vincent.lapinte@univ-montp2.fr

Jean-Jacques Robin

Hydrophilization coating of silylated polyoxazoline using sol-gel process

Hybrid films prepared from TEOS and polyoxazoline cross-linkers(Si-POx-Si) were coated on different substrates in order to modify their surface properties. The film cohesion and adhesion on substrates were expected through the hydrogen bonding of the polyoxazoline crosslinked network. Low molecular-weight α,-unsaturated polyoxazolines (DA-PMOx)s were synthesized by a one step cationic ring-opening polymerization (CROP) of 2-methyl-2-oxazoline (MOx) with a good control over the molecular weight. Based on double thiol-ene coupling (d-TEC), a postfunctionalization of end chains DA-PMOx gave in good yield polyoxazoline cross-linker (Si-POx-Si). Glass and various polymer substrates (PP, PEI, POM...) were spin coated by the organic-inorganic hybrid films through sol-gel process. AFM, SEM, visible reflectance spectroscopy and contact angle experiments allowed the full characterization of targeted surfaces and demonstrated the efficiency of the polyoxazoline coating.

Introduction

Nowadays, sol-gel process was known to be one of the mainly practical methods for substrate coating. Nevertheless, the achievement of crack-free hydrophilic films on organic polymer substrates still reminded challenging. Indeed a main limitation in using coated plastics was the weak interfacial adhesion of the inorganic thin film on the polymeric substrate resulting in cracking upon drying process [1]. In order to prevent this drawback,surface activation methods such as plasma have been developed, to activate the surface and improve the wettability by bonding between the coating and the substrate [2]. Moreover, glycidoxypropyltrimethoxysilane (GOTMS) was used to enhance the properties ofthe deposited film [3]. In this way, our efforts were concentrated on the elaboration of crack-free coating for polymer supports using hybrid solgel technique with a new polyoxazoline based precursor. The modified polyoxazoline acted as hydrophilic cross-linker of the polymeric network where its flexible chains avoided cracks upon drying process.

Over the last few years, polyoxazolines have received significant attention, due to their high potential for applications in technical matters [3,4] and biomedical applications [5][6][7][8]. They have allowed the synthesis of a wide range of functionalized polymers [9][10][11][12]. Functionalities can be introduced into the polymer through the initiation [13,14] and/or the termination steps [15] giving head and/or end groups. Poly(2-oxazoline)s with methyl (PMOx) and ethyl side groups are water-soluble polymers. These candidates are applied as compatibilizer between reinforcing agent and the polymer matrix in composites [16], biocompatible agent in the fabrication of tip for scanning probe nanolithography [17] and protein-repellent agent in surface coatings [18].

Various hybrid polyoxazoline coatings were already explored with colloidal polymer hybrids where the C=O bonds interacted by hydrogen bonds with Si-OH of silicate network [20][21][22]. The grafting of PMOx on silicone was another polyoxazoline coating method [23]. Finally the sol-gel process between an inorganic cross-linker such as tetraethyl orthosilicate (TEOS) and PMOx bearing an alkoxysilane coupling agent was investigated. The sol-gel reactive group was introduced either in pendant chain of oxazoline monomer [24,25] or in the polyoxazoline endchain [26]. In the latter case, hydrosilylation reaction was used to modify the αor -end group of polyoxazoline [23,[27][28][29][30]. Nevertheless, this method suffers from low yield.

In this contribution, a new synthetic pathway was investigated to convert by one step process α,-allylic PMOx (DA-PMOx) into α,-alkoxysilane PMOx (Si-PMOx-Si) through a double thiol-ene coupling (d-TEC). The TEC reaction has been extensively exploited in polymer chemistry [31,32] and in biomaterials for application in medicine, especially dentistry [33], taking advantage of the absence of toxic transition-metal catalyst. However, only few examples of thiol-ene coupling from polyoxazoline were described in the literature [34][35][36][37][38] and never a one step double thiol-ene coupling.

In a second part of the study, this new sol-gel precursor, Si-PMOx-Si, was used to achieve an improved crack-free inorganic-organic hybrid coating on raw polymeric substrates through a sol-gel route and a deposition carried out by spin-coating technique. Then, the surface of the coated substrates was observed on SEM images and the roughness was measured by AFM spectroscopy. The modification of the surface was finally evaluated by contact angle measurement as well as the transparency was measured by visible reflectance spectroscopy.

Experimental section

Materials and chemicals

Allylic alcohol, tosyl chloride, allylamine, 3-mercaptopropyl-triethoxysilane (TEOSithiol), tetraethyl orthosilicate (TEOS), 3-glycidoxypropyltrimethoxysilane (GOTMS), methanol, ethanol, H 2 SO 4 , HCl, diethyl ether, KOH and CaH 2 were purchased from ACROS and were used as received. Acetonitrile was dried and distilled according to standard procedures [START_REF] Perrin | Purification of laboratory chemicals[END_REF]. 2-Methyl-2-oxazoline (MOx) was dried, distilled from CaH 2 and stored under a dry nitrogen atmosphere.

Acetate butyrate cellulose (CAB), Polystyrene (PS), Polyoxomethylene (POM), Polyetherimide (PEI) and Polypropylene (PP) substrates were purchased from Goodfellow and were used as received. Glass substrates (microscope slides from Marienfeld) were carefully washed with a solution of H 2 SO 4 /H 2 O 2 and then intensively rinsed with MilliQ water, and dried. Deuterated solvent (CDCl 3 ) was purchased from SDS and were used without further purification.

Instrumentation and characterization

1 H and 13 C NMR spectra were recorded using a Bruker AC 200 with CDCl 3 as solvent. In 1 H NMR chemical shifts were referenced to the peak of residual CHCl 3 at 7.26 ppm. In 13 C NMR chemical shifts were referenced to CDCl 3 at 77 ppm. Size exclusion chromatography (SEC) was performed on a PL-GPC 50 Plus equipped with an RI refractive index detector. Three PL aquagel-OH columns (25, 7.5 and 4.6 mm ID) were used at 40°C with a 0.8 mL•min -1 flow rate of H 2 O/CH 3 OH: 7/3 (0.1M LiNO 3 ), calibrated using PEO standards.Contact angles were determined with a Digidrop camera contact-angle meter, Fast/60 from GBX, with an automatic liquid dispenser. Experiments were carried out at room temperature. A liquid droplet (4 µL) was carefully placed on the sample surface and imaged with a coupling device camera. The contact angle was determined mathematically through the fitting of the Young-Laplace equation. The displayed values are the average value of 3 measurements with a dispersion of measured data lesser than 5 %.Scanning electron microscopy (SEM) images were obtained on a Hitachi S2600N scanning electron microscope. AFM images were obtained with an AFM Digital D 3100 (Digital Instruments) in tapping mode and phase. Visible reflectance spectroscopy was carried out using a reflectance fiber optic probe FCR-7IR200-2 from Avantes, a tungsten halogen light source (150W) and an AvaSpec-2048TEC spectrometer.

Preparation of Allylic p-toluene sulfonate (ATs)

. KOH (15 g, 267 mmol, 2 eq) was dissolved in 160 mL of ethyl ether solution containing allylic alcohol (2.5 eq). At 5°C,tosyl chloride (25 g, 131 mmol, 1 eq) was added dropwise .The resultingmixture was stirred for 6 h at room temperature and filtered and finallyconcentrated under pressure. The product ATs was isolated in 89% yield.

Typical polymerization of MOx using ATs (DA-PMOx).

All reactions were carried out under a dry nitrogen atmosphere. ATs and MOx were dissolved in dry acetonitrile (4 M). The solution was vigorously stirred at 80°C. The reaction product was quenched by addition of an adequate amount of allylamine (10 eq). The flask was maintained for 24 h at 50°C. After cooling, the polymer was isolated by slow precipitation from cold diethyl ether. DA-PMOx was isolated in 93% yield. 1 H NMR (CDCl 3 ) δ (ppm): 5.8 (m, 1H, CH=), 5.2 (m, 2H, CH 2 =), 3.9-3.3 (m, 4nH, CH 2 of POx), 2.8 (m, 4H, =CH-CH 2 -N), 2.0 (s, 3nH, CH 3 of POx). 13 C NMR (CDCl 3 ) δ (ppm): 171.2-170.6 (C=O of POx), 132.2 (CH=), 118.1-117.1 (CH 2 =), 51.1-43.2 (CH 2 -N and CH 2 n of POx), 20.9 (CH 3 of POx).

Typical di-thiol-ene coupling (d-TEC) (Si-PMOx-Si).

In a 50 mL round flask DA-PMOx, TEOSi-thiol (3 eq per C=C group) and AIBN (0.12 eq) were dissolved in CHCl 3 solution (0.3 M).

Nitrogen was bubbled in the solution for 30 min. The reaction mixture was heated at 60°C for 72 hours under a nitrogen atmosphere. The reaction mixture was poured in a large excess of cold ethyl ether. Si-PMOx-Si was isolated in 86% yield after precipitation. 1 H NMR (CDCl 3 ) δ (ppm):

Film elaboration.

The reactive polyoxazoline Si-PMOx-Si was co-hydrolyzed and cocondensed with TEOS and GOTMS with HCl as acidic catalyst to yield a series of hybrid sol-gel sols. The solutions stirred for 2 hours were then spincoated at 1800 rpm, for 20s on different inorganic or polymer substrates, glass, POM, PS, CAB, PEI and PP. Every coating procedure was performed around 30°C. In the preparation of the sol, Si-PMOx-Si was first diluted in absolute ethanol to obtain a 10wt% solution. The weight ratio between Si-PMOx-Si solution and silica precursors was optimized and the following composition of the hybrid sols 1:0.3:0.2 (Si-PMOx-Si:GOTMS:TEOS weight ratio) displayed coated films with no macroscopic defects.

Results and discussion

Synthesis of polyoxazoline cross-linker

Hybrid sol-gel coatings were realized from PMOx bearing alkoxysilane end groups, Si-PMOx-Si, as shown in Scheme 1. The previous stage was the synthesis of α,-telechelic polyoxazoline DA-PMOx bearing allylic end-groupsobtainedby both the addition of allyl tosylate initiator ATs and allylamine as termination agent.

Scheme 1.Synthesis of Si-PMOx-Si.

Low molecular weight polyoxazolines were further used as sol-gel cross-linkers. As a consequence, the control over molecular weight was investigated using SEC and NMR techniques. The experimental number-average molecular weights, Mn,SEC and the theoretical values (calculated from the molar ratio of introduced monomer MOx, DPn,th) versus conversion werereported in Supporting Information. The Mn,SEC values wereslightly higher than the theoretical ones. M n, SEC values increased linearly until approximately 80% whichindicatedthat the transfer reactions were negligible in this range. Above this limit the decrease of the molecular weight and the enhancement of polydispersity indicated the presence of transfer reactions. In conclusion,,-allylic polyoxazolines DA-PMOx could be prepared with a good molecular weight control under relatively high conversion (<80%) and alow polydispersity (M w /M n < 1.3).

The number average polymerization degree was obtained by SEC analysis and NMR spectroscopy as summarized in Figure 1. A narrow polydispersity ranged from 1.2 to 1.3 and the unimodal SEC traces were detected. The peaks shifted towards higher molecular weights with the same distribution of molecular weight. It wasimportant to notice that polymerization reactions were stopped at about 80% conversion in order to minimize transfer and control the end-chain functionalities. Whatever the DP n , DP n, SEC and DP n, NMR ,the values were consistent with the theoretical ones. For instance,a target degree of polymerization of 20 actually producedan experimental degree of polymerization of 23.0 and 21.5. The efficiency of the end functionalization using allylamine was evidenced by NMR spectroscopy.

Post-functionalization based on double thiol-ene coupling and sol gel condensation

To date, the synthesis of polyoxazoline bearing alkoxysilane head-and end-groups for further sol-gel condensation used hydrosilylation reaction. Owing to low yield of polyoxazoline modification by hydrosilylation, another synthetic strategy of polyoxazoline postfunctionnalization was explored. One reaction that is emerging as an attractive process is the century-old addition of thiols to alkenes, which is currently called thiol-ene coupling (TEC). The TEC reaction has been extensively exploited in polymer chemistry and more specifically in polyoxazoline chemistry on functional monomer or polymer. In our case, the TEC reaction was employed as post-functionalization of polyoxazoline carrying α,-unsaturated end groups DA-PMOx. The TEC reaction transformed allylic into alkoxysilane end groups. The thiol-ene coupling from DA-PMOx (DP 15, exp ) was realized in excess of mercaptan agent: TEOSi-thiol considering the typical thiol-disulfide interchange reaction [START_REF] Kiran | [END_REF]. Under free radical condition, the double end group modification (d-TEC) of DA-PMOx into alkoxysilane groups gave Si-PMOx-Si (Scheme 1). The conversion was monitored by 1 H NMR analysis using the integration of theethylenic signals of DA-PMOx. The ethylenic signals at 5.2 and 5.7 ppm decreased during the TEC reaction. On the other hand, after purification of Si-PMOx-Si, the integration of CH 3 signal of CH 3 -CH 2 -O-Si at 1.2 ppm confirmed the reaction of TEOSi-thiol on DA-PMOx (Figure 2).

Our first experiments using 0.06 and 0.12 equivalent of AIBN in 0.1M yielded in minor amount the target compound: 10 and 15% conversion, respectively (Supporting Information). This result prompted us to investigate the course of the reaction more thoroughly. An additional amount of AIBN improved the conversion until 45%. Parameter such as the concentration of the reaction mixture was varied. As expected, higher concentration until 0.3M concentration favored the formation of thioether compound with an optimum 86% yield. The sol-gel process was carried out through hydrolysis and condensation of Si-PMOx-Si with a mixture of tetraethoxysilane (TEOS) and (3-glycidoxypropyl)trimethoxysilane (GOTMS) (Scheme 2). The latter was used to develop linkage, on one hand with silica network structure using alkoxy groups and on the other hand with polyoxazoline chains through interchain hydrogen bonding [41]. Thanks to the flexibility of PMOx chains the composition of the sol-gel precursor solution was optimized to obtain crack-free film onto substrates. Scheme 2. Elaboration of polyoxazoline coating by sol-gel process.

Characterization of polyoxazoline coating on polymer and glass materials

Contact angle measurements

The efficiency of the hybrid sol-gel coating was studied on various polymer supports ranged from PP, CAB, PS, POM to PEI and glass. To get more information on the surface energy of the hydrophilic coating, the contact angle measurements were carried out using probe liquids (water and diiodomethane) as summarized in Table 1. The obtained surface energy values for the raw substrates are in agreement with those previously reported [42]. The experimental values for hybrid coating on substrates are around 63±2 mJ.m -2 , exceptfor PP. Indeed, the surface energy value is lower as a result of the film cracks. As polyoxazoline is a hydrophilic polymer, [43] it is observed an increase of the surface energy of the substrate related to the hydrophilicity of the coated film. Whatever the substrates, the constant value of the surface energy of the composite materials gave evidence the homogeneity of the hybrid film. The adhesion of hybrid polyoxazoline film on the materials was also studied. The value of interfacial energy is an important parameter to determine the adhesion between two components in a composite; it is also a description of the thermodynamic state of the interface [44]. The work of adhesion W 12 measures the attraction between two different phases and W 12 =       for the separation process. Therefore the adhesion increases when the interfacial energy decreased. Interfacial energy between substrates (1) and hybrid film (2) in the composite has been determined with:

𝛾 12 = 𝛾 1 + 𝛾 2 -2 𝛾 1 × 𝛾 2
Assuming an average value of 63 mJ.m -2 for   

The low values of the interfacial energy are in good agreement with the observed adhesion of the polyoxazoline hybrid film coating on the different substrates with surface energy higher than 30 mJ.m -2 .

SEM images

The top view SEM images of the hybrid silica film deposited on different substrates (Glass, POM, PEI and PP) exhibit a very homogeneous coating with very few defects on the substrates (glass, POM and PEI on Figure 3). The uniformity of the films indicates the effectiveness of the coating resulting from the efficient crosslinking of polyoxazoline with alkoxysilane end groups Si-PMOx-Si. On the other hand, the coating on PP substrate shows important cracks that reveal a low adherence of the film as expected the high interfacial energy (Table 1). The coated film thickness, determined from the cross-sectional image (Figure 3), is around 5 µm. the root mean square of the amplitude was used as a feedback value to maintain the oscillation amplitude constant. In order to determine the nature of changes that took place in the surface morphology of substrate after hybrid silica coating, AFM was carried out and the results are depicted in Figure 4 for PEI substrate. The observations of other composite materials provide very similar results. No significant alteration of the surface morphology was detected, with a minor reduction in roughness, found to be equal to 0.364 nm on the raw substrate and 0.268 nm after coating. The result could be explained by the use of flexible polyoxazoline segments allowing a perfect fit of the surface structure between two crosslink nodes.

Optical reflectance measurements

The reflectance measurements show similar spectra for the raw and coated substrates except for the glass (Figure 5) where the reflectivity of the material decreased with the coating. This result indicates that the coated film is efficient to reduce the surface reflection of transparent materials as required for different applications to improve the light transmission [45]. In the case of PEI, the polymer presents a light absorption in the blue region and exhibits a yellow color. The presence of the coated film decreases the absorbance leading to a slight increase of the reflectivity in this region. As a result, the reflectivity measurements indicate that the coated hybrid film does not significantly modify the optical properties of the substrate because of its transparency.

Conclusions

Hybrid sol-gel coatings were successfully realized without any preliminary surface activation from PMOx onto various polymer substrates as well as glass support. The crack-free hydrophilic films resulted from a crosslinked network between polyoxazoline flexible chains and inorganic cross-linkers. The α,-functionalized polyoxazoline was reached by an original and efficient synthetic route using an one pot head and end-allylation chains followed by the thiol-ene reaction which was a powerful alternative route to silylation for the sol-gel condensation. Finally the reflectivity measurements indicated that the coated hybrid films do not significantly modify the optical properties of the substrates and no defect was detected in AFM and SEM images of the coatings.

3 (

 3 1 H NMR (CDCl 3 ) δ (ppm): 7.8 (d, 2H, H aromatic ), 7.4 (d, 2H, H aromatic ), 5.7-5.6 (m, 1H, CH=), 5.3 (m , 2H, CH 2 =), 4.6 (d, 2H, =CH-CH 2 -), 2.5 (s, 3H, CH 3 ). 13 C NMR (CDCl 3 ) δ (ppm): 145.0 (C aromatic -CH 3 ), 133.2 (C aromatic -S), 130.3 and 127.9 (CH aromatic ), 120.CH 2 =), CH= no visible, 70.9 (CH 2 -O), 21.7 (CH 3 ).

Fig. 1 .

 1 Fig. 1. Molecular weight, polydispersity and SEC trace of DA-PMOxs.

Fig. 2 .

 2 Fig. 2. 1 H NMR spectra a) before (DA-PMOx) and b) after thiol-ene reaction (Si-PMOx-Si).

Fig. 3 . 3 . 3 . 3

 3333 Fig. 3. Top view SEM images and cross sectional image of the hybrid film coated on different

Fig. 4 .

 4 Fig. 4.AFM surface image (1*1 μm2) of uncoated PEI (Rq=0.268 nm, on the left) and coated PEI

Fig. 5 .

 5 Fig. 5. Reflectance spectra of some raw and coated substrates: a: glass microscope slide,

  3.8-3.3 (m, (4n+12) H, CH 2 POx and CH 3 -CH 2 of OEt), 3.2-2.8 (s, 4H, CH 2 -N), 2.5 (m, 8H, CH 2 -S), 2.1 (s, 3nH, CH 3 of POx), 1.8-1.6 (m, 8H, CH 2 -CH 2 -Si and CH 2 -CH 2 -N), 1.2 (t, 18H, CH 2 -CH 3 of OEt), 0.7 (t, 4H, CH 2 -Si). 13 C NMR (CDCl 3 ) δ (ppm): 170.1 (C=O of POx), 57.8 (CH 2 of O-Et), 51.4-42.6 (CH 2 of POx, CH 2 -CH 2 -S and CH 2 -N), 34.6 (CH 2 -S side end group), 28.1 (CH 2 -S side POx), 20.5 (CH 3 of POx), 17.7 (CH 3 of O-Et), 9.3 (CH 2 -Si).

Table 1 .

 1 Surface energy of raw and coated substrates.

		Raw substrate	Coated substrate
	substrate	contact angle	Surface energy  1

/mJ.m -2 contact angle Surface energy  2 /mJ.m -2 interfacial energy  12 /mJ.m -2 water CH 3 I water CH 3 I PP

		88 64.5	28	63.8 45.2	43	7
	POM	76.7 40.3	40	32.7 40.9	63	2.6
	CAB	77.4 37.1	42	30.7 40.9	65	2.1
	PS	96.0 34.2	42	29.3 40.1	65	2.1
	PEI	70.6 31.8	45	19.7 37.1	64	1.5
	glass	16.9 42.3	72	35 39.8	62	0.3