Johanna Hippolyte D'albis

Josselin Etner
email: jetner@parisnanterre.fr

Thuilliez

Hippolyte D'albis

Johanna Etner

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

Vaccination under pessimistic expectations in clinical trials and immunization campaigns

Keywords: Decision Theory, Vaccination, Clinical Trials, Immunization Campaigns Decision Theory, Vaccination, Clinical Trials, Immunization Campaigns JEL: D7, D81, I12

We provide one of the first formalizations of a vaccination campaign in a decisiontheoretic framework. We analyse a model where an ambiguity-averse individual must decide how much effort to invest into prevention in the context of a rampant disease.

We study how ambiguity aversion affects the effort and the estimation of the vaccine efficacy in clinical trials and immunization campaigns. We find that the behaviours of individuals participating in a clinical trial differ from individuals not participating. Individuals who are more optimistic toward vaccination participate more in trials. Their behaviours and efforts are also affected. As a result, because vaccine efficacy depends on unobserved behaviours and efforts, the biological effect of the vaccine becomes difficult to evaluate. During the scale-up phase of a vaccination campaign, provided that vaccine efficacy is established, we show that vaccine hesitancy may still be rational.

Introduction

The coronavirus vaccines approved in Europe are considered as remarkably effective. Several studies undertaken in real-world conditions indicate that overall, COVID-19 vaccination has reduced the number of new SARS-CoV-2 infections, with the largest benefit received after two vaccinations and for symptomatic and high viral burden infections [START_REF] Chodick | The effectiveness of the TWO-DOSE BNT162b2 vaccine: analysis of real-world data[END_REF][START_REF] Haas | Impact and effectiveness of mRNA BNT162b2 vaccine against SARS-CoV-2 infections and COVID-19 cases, hospitalisations, and deaths following a nationwide vaccination campaign in Israel: an observational study using national surveillance data[END_REF][START_REF] Lustig | Neutralizing Response against Variants after SARS-CoV-2 Infection and One Dose of BNT162b2[END_REF][START_REF] Milman | Community-level evidence for SARS-CoV-2 vaccine protection of unvaccinated individuals[END_REF][START_REF] Pritchard | Impact of vaccination on new SARS-CoV-2 infections in the United Kingdom[END_REF].

However, there is continuing concern about vaccine hesitancy in some countries and some communities within them [START_REF] Kluge | COVID-19 vaccines for the European region: an unprecedented challenge[END_REF]. Proving that a COVID-19 vaccine is safe and effective can only be a turning point if the uptake of the vaccine is high enough to stop the circulation of the virus [START_REF] Chevallier | COVID-19 Vaccine Hesitancy: Shortening the Last Mile[END_REF]. Therefore, there is an urgent need to better understand how to reach out to those who have the opportunity to be vaccinated but refuse to do so. Previous research has identified several promising levers for reducing vaccine hesitancy. These include the motivation to be altruistic [START_REF] Shim | September). The influence of altruism on influenza vaccination decisions[END_REF], emphasis on the dangers of the disease [START_REF] Horne | Countering antivaccination attitudes[END_REF], rational calculation of pros and cons, and anticipated regret at not having been vaccinated [START_REF] Betsch | Using Behavioral Insights to Increase Vaccination Policy Effectiveness[END_REF][START_REF] Brewer | Anticipated Regret and Health Behavior: A Meta-Analysis[END_REF]. However, analysing cepticism about vaccination is difficult, because of the critical issue of disentangling so many factors involved in vaccination decisions.

In this paper, we provide one of the first formalizations of a vaccination campaign in a decision-theoretic framework. We build a model that can be used to compare self-protection and behaviours before a vaccine trial is implemented, during the trial and when immunization is scaled-up. In a nutshell, we analyze a model in which an ambiguity averse individual must decide how much effort to invest in prevention in the context of a rampant disease. The individual may decide to participate in a randomized vaccination clinical trial and later in a vaccination campaign. The main objective is to study how ambiguity aversion affects the effort exerted by the individual and the estimation of the vaccine efficacy. The model shows that individuals participating in clinical trials behave differently than individuals not participating, with behavioural differences owing to risk attitude. These differences may in turn compromise the validity of the vaccine efficacy calculation. The paper also finds that vaccine hesitancy may persist even if vaccination has beneficial effects. Specifically, our benchmark model in the absence of a vaccine simply starts by showing that efforts (to invest in prevention) increase with income and decrease with exogenous improvements in the probability of not developing a severe form or dying from COVID-19.

The extension of this simple model includes vaccination and adverse effects during a clinical trial. As perceived adverse events may differ across individuals, we introduce a certain degree of pessimism related to vaccination. We show that when individuals participate in a clinical trial, the more pessimistic they are, the more they will invest in prevention, whereas optimists make fewer efforts. This is indeed crucial in clinical trials as informed participation is on a voluntary basis. Another consequence of voluntary participation is that the utility of participating in a clinical trial is greater than not participating. Clearly, this may depend on the health system. Participating in a trial may mean access to free care in the US, whereas in France, this type of motivation does not exist. This may generate selection bias and also differential efforts between optimists and pessimists in an endogenous way.

Put differently, only people with relatively low pessimism may participate in the trial. We show that the share of pessimists in the trial may in turn affect vaccine efficacy. When the probability of catching COVID-19 and the probability of developing a severe form after being infected, are two independent features, we show that it is indeed possible to identify the treatment (vaccine) effect. Otherwise, the treatment effect remains undetermined. Whether these events are independent is debatable [START_REF] Chidambaram | Factors associated with disease severity and mortality among patients with COVID-19: A systematic review and meta-analysis[END_REF][START_REF] Izcovich | Prognostic factors for severity and mortality in patients infected with COVID-19: A systematic review[END_REF][START_REF] Liu | Risk factors associated with COVID-19 infection: a retrospective cohort study based on contacts tracing[END_REF][START_REF] Pijls | Demographic risk factors for COVID-19 infection, severity, ICU admission and death: a meta-analysis of 59 studies[END_REF][START_REF] Rozenfeld | A model of disparities: risk factors associated with COVID-19 infection[END_REF]. There are probably some joint confounding factors and the availability and evolution of screening guidelines, individual factors, or clinical judgement may also affect who is tested [START_REF] Rozenfeld | A model of disparities: risk factors associated with COVID-19 infection[END_REF]. The amount of evidence remains limited on this question, though comparing age groups, children seem to be infected in large numbers without developing such severe symptoms as the elderly.

Lastly, we examine widespread inoculation in society. We show that during an immunization campaign, when the risk premium associated with wealth, expressed in terms of effort, is lower than the level of effort without any vaccine, then the individual may get vaccinated under certain conditions related to factors that are exogenous to individual behaviours. We therefore provide a discussion on vaccination hesitancy between genders.

Section 2 relates our work to the literature. Section 3 provides a benchmark model before an infectious threat is introduced to society and therefore before any clinical trial is undertaken. Section 4 extends this model during an experimental phase aiming at finding new vaccines. The optimal effort (to invest in prevention) during the vaccination clinical trial is examined. Section 5 examines optimal efforts and vaccine efficacy during the vaccination campaign. Section 6 discusses the COVID-19 vaccine's gender paradox to illustrate the model and concludes.

Background literature

This paper contributes to several strands of the literature. First, various approaches have been used in economics to model vaccination decisions. In deciding whether to get vaccinated or not (a binary choice), individuals consider the risk of morbidity from vaccination, the probability of becoming infected, and the risk of morbidity or mortality from that infection.

The decisions of individuals are indirectly influenced by externalities, i.e. the decisions of all other individuals, because the sum of these decisions yields the vaccine coverage levels in the population and hence the course of epidemics. One interesting prediction from this literature is that it is impossible to eradicate a disease through voluntary vaccination when individuals act according to their own interests [START_REF] Geoffard | Disease eradication: private versus public vaccination[END_REF][START_REF] May | Simple Rules with Complex Dynamics[END_REF].

Game theory has also explored vaccination decisions in such settings [START_REF] Bauch | Vaccination and the theory of games[END_REF][START_REF] Bauch | Group interest versus self-interest in smallpox vaccination policy[END_REF]. Interestingly, [START_REF] Auld | Choices, beliefs, and infectious disease dynamics[END_REF] indicates that a partially effective prophylactic vaccine (one which reduces probability of infection) may alter incentives and induce perverse behaviours (see [START_REF] Peltzman | The Effects of Automobile Safety Regulation[END_REF] for a similar discussion on car safety and traffic accidents). In particular, pessimistic expectations over the future of the epidemic induce more current risky behaviour. Medical researchers recognize that such reactions can occur, and this concern has contributed to some existing vaccines not being made available [START_REF] Blower | Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco[END_REF]. Direct evidence from clinical trials of prophylactic vaccines also suggests that vaccination leads to increased risk behaviours [START_REF] Chesney | Risk Behavior for HIV Infection in Participants in Preventive HIV Vaccine Trials: A Cau-tionary Note[END_REF][START_REF] Johnson | Belief of vaccine receipt in HIV vaccine trials: further cautions[END_REF].

Second, a key factor in explaining vaccination success is the perception of risks associated with infection and vaccination [START_REF] Brewer | Meta-analysis of the relationship between risk perception and health behavior: The example of vaccination[END_REF][START_REF] Streefland | Pat-terns of vaccination acceptance[END_REF][START_REF] Wheelock | Social and psychological factors underlying adult vaccination behavior: lessons from seasonal influenza vaccination in the US and the UK[END_REF]. [START_REF] Voinson | Beyond Rational Decision-Making: Modelling the Influence of Cognitive Biases on the Dynamics of Vaccination Coverage[END_REF] show for instance that sceptical individuals (negative opinion) overestimate infec-tion cost, while pro-vaccine individuals (positive opinion) overestimate vaccination cost. In decision theory under risk, vaccination is similar to self-protection. In both cases, the individual's objective is to reduce the probability of occurrence of an undesirable event. The literature on health risk attitudes has explored self-protection and prevention following the seminal paper by Ehrlich and Becker (1972). Health models in health economics have also used a similar framework [START_REF] Brianti | Optimal choice of prevention and cure under uncertainty on disease effect and cure effectiveness[END_REF][START_REF] Courbage | Decision Thresholds and Changes in Risk for Preventive Treatment[END_REF]. However, for vaccination there is a new cost that may be important, namely the risk of side effects [START_REF] Crainich | Vaccination as a trade-off between risks[END_REF].

Risk assessment is likely to be related to many factors that may or may not be related to the infection. These factors are psychological and social: media coverage, salient past experiences, beliefs about contagion and information, family and peers, trust in health care providers, health care professionals, and the state. The link between the environment and the cognitive or meta-cognitive mechanisms underlying decisions -rather than the health outcomes of decisions -has also been highlighted as a critical factor [START_REF] Oraby | Bounded rationality alters the dynamics of paediatric immunization acceptance[END_REF].

We argue rather that in the case of COVID-19, the overall picture is better described as one of ambiguity than of risk. In particular, the informational structure can be represented by a set of available scenarios that describe the effect of vaccination and the possible negative side effects due to vaccination, each associated with a distribution function. Among models that have been put forward to represent preferences under uncertainty, we have opted for the widely used α-Maxmin Expected Utility criterion (α-MEU) that generalizes the Hurwicz criterion (1951) and whose properties have been studied by [START_REF] Ghirardato | Ambiguity made precise: A comparative foundation[END_REF].1

This model has the advantage of allowing a separation between ambiguity and ambiguity attitudes, and also between risk and ambiguity attitudes.2 Moreover, the α-MEU model is of great interest for the applied decision-theoretic literature as it can model ambiguity aversion and ambiguity seeking.

Third, in the experimental context, vaccination has also been shown to be an efficient way of reducing infection, and this is naturally the first step before market authorization.

However, some authors have warned about the effects of behaviours in experimental con-texts. [START_REF] Malani | Identifying Placebo Effects with Data from Clinical Trials[END_REF] focuses on these beliefs. By manipulating an individual's beliefs while holding constant the pharmacological effects, he shows that it is possible to generate and identify placebo effects. Indeed, [START_REF] Malani | Identifying Placebo Effects with Data from Clinical Trials[END_REF] points out a neglected feature of clinical trials, namely that they manipulate patient expectations, and suggests that this feature can be used to identify the effects of expectations on outcomes, that is, placebo effects. He next uses variation in probability of treatment to identify placebo effects in several trials, and also advocates for incorporating variation in treatment probabilities into randomized trials [START_REF] Malani | Identifying Placebo Effects with Data from Clinical Trials[END_REF][START_REF] Malani | Patient enrollment in medical trials: Selection bias in a randomized experiment[END_REF]. Similarly, [START_REF] Chassang | Accounting for Behavior in Treatment Effects: New Applications for Blind Trials[END_REF] address complementarity between treatment and behaviour. They show that the gold standard double-blind randomized controlled trial (DBRCT) fails to fully account for the efficacy of treatment if there are interactions between treatment and behaviour, for example, if a treatment is more effective when patients change their exercise or diet. The decision to drop out of a trial may also be affected. [START_REF] Chan | Learning, Private Information, and the Economic Evaluation of Randomized Experiments[END_REF] exploit drop-outs to estimate the subject's utility associated with the receipt of treatment, as revealed by drop-out behaviour, and evaluate treatment effects.

Fourth, some exogenous characteristics may be particularly important in vaccination decisions. This is the case of gender for instance. Gender heterogeneity with respect to preferences, risk and ambiguity has been widely debated in the economic literature. In general, most results from abstract gamble experiments indicate that women are more risk averse than men [START_REF] Croson | Gender Differences in Preferences[END_REF]Eckel & Grossman, 2008a). With respect to ambiguity aversion, the results are inconclusive on whether women are more ambiguity averse than men [START_REF] Aggarwal | Ambiguity attitudes and myopic loss aversion: Experimental evidence using carnival games[END_REF][START_REF] Agnew | Who Chooses Annuities? An Experimental Investigation of the Role of Gender, Framing, and Defaults[END_REF][START_REF] Borghans | Gender Differences in Risk Aversion and Ambiguity Aversion[END_REF][START_REF] Kray | Preventive health behaviors influenced by self-perceptions of aging[END_REF][START_REF] Yang | Ambiguity vs risk: An experimental study of overconfidence, gender and trading activity[END_REF]. [START_REF] Borghans | Gender Differences in Risk Aversion and Ambiguity Aversion[END_REF] find that women initially respond to ambiguity much more favourably than men (i.e. their reservation price does not decline), but as ambiguity increases, men and women show similar marginal valuations of ambiguity. While the evidence from abstract gamble experiments suggests greater risk aversion among women, the evidence from experiments with a contextual environment is less conclusive (among others, see [START_REF] Eckel | Forecasting risk attitudes: An experimental study using actual and forecast gamble choices[END_REF]; [START_REF] Schubert | Gender specific attitudes towards risk and ambiguity: an experimental investigation[END_REF]). With respect to ambiguity aversion, [START_REF] Schubert | Gender specific attitudes towards risk and ambiguity: an experimental investigation[END_REF] find that women are more ambiguity averse in an investment context but not in an insurance context. [START_REF] Kocher | Ambiguity aversion is not universal[END_REF] find that ambiguity attitudes depend on the outcome domain and likelihood range, and no evidence for universal ambiguity aversion as is assumed in many applications in economics. Most of the ambiguity aversion evidence involves money, and there is little evidence on ambiguity and health. [START_REF] Curley | An Investigation of Patient's Reactions to Therapeutic Uncertainty[END_REF] found ambiguity aversion for health, which differed from ambiguity aversion for money. This domain specificity of ambiguity preferences is consistent with studies on risk and time preferences, which found that findings for money cannot be directly translated to health [START_REF] Attema | Discounting health and money: New evidence using a more robust method[END_REF][START_REF] Chapman | Temporal discounting and utility for health and money[END_REF][START_REF] Weber | A domain-specific risk-attitude scale: measuring risk perceptions and risk behaviors[END_REF]. Attema, Bleichrodt, and L'Haridon (2018) show that individuals are more pessimistic under ambiguity in the loss domain, but little is said on gender heterogeneity.

A benchmark model prior to vaccination

We start with a model where individuals decide about their optimal behaviour in the presence of an epidemic disease that may have an impact on their health.

Definitions and notations

Let us consider two health states, H i with i ∈ {0, 1}, which are such that H 1 > H 0 . In state H 0 , individuals are infected and have a severe form of the disease, while in H 1 , they have not been infected or do not have a severe form of the disease. The probability of being in good health (i.e. in H 1 state) is given by a function λ(e, µ), which increases with the preventive behaviour of individuals, denoted e, and with an exogenous biological factor, denoted µ; both e and µ are positive real numbers. We thus have λ e (e, µ) > 0 and λ µ (e, µ) > 0. Prevention may be interpreted as the effort such as physical distancing, wearing a mask, etc. Parameter µ combines all factors that are, by nature, exogenous to individual behaviour such as age, sex, immunity characteristics and also the existence of an efficient preventive treatment or vaccine. To keep the model general, we allow for substitution effects between behaviour and biological factors in the probability function. We do not at this stage impose a sign on λ eµ , even if it is reasonable to think that a more efficient biological parameter reduces the efficiency of the preventive behaviour. Taking the example of age as a proxy for the biological parameter, this means that the effect of preventive behaviors on the probablility to be in good health increases with age. This appears to be true for COVID-19 as suggested by the efficiency of vaccination. It mostly benefited to old and very old adults [START_REF] Bernal | Effectiveness of the Pfizer-BioNTech and Oxford-AstraZeneca vaccines on covid-19 related symptoms, hospital admissions, and mortality in older adults in England: test negative case-control study[END_REF][START_REF] Heaton | The Covid-19 Vaccine-Development Multiverse[END_REF][START_REF] Soiza | Efficacy and safety of COVID-19 vaccines in older people[END_REF]. However, this assumption can be discussed and is certainly not valid in all settings [START_REF] Burton | Determinants of physical activity initiation and maintenance among community-dwelling older persons[END_REF]Levy & Myers, 2004).

Individuals derive utility from consumption, c := w -e, where w > 0 is the endowment, and from their health status, H i . We suppose that the utility function, denoted U (c, H i ), is an increasing function of consumption and health, U c (c, H i ) > 0, U H (c, H i ) > 0, and is strictly concave with respect to both arguments, U cc (c, H i ) < 0, U HH (c, H i ) < 0. Hence, individuals are risk averse towards health risk. Moreover, we suppose that the second cross derivative is non-negative, U cH (c, H i ) ≥ 0, which corresponds to correlation loving, namely the complementarity between wealth and health. This relationship has been empirically supported by [START_REF] Viscusi | Utility Functions That Depend on Health Status: Estimates and Economic Implications[END_REF], [START_REF] Sloan | Alternative approaches to valuing intangible health losses: the evidence for multiple sclerosis1This article was written when Chesson was a student at Duke University.1[END_REF] and [START_REF] Finkelstein | Approaches to Estimating the Health State Dependence of the Utility Function[END_REF] among others.

Optimal effort

We assume that the epidemic has been present for a sufficiently long time and that the value of the biological parameter µ is known. In the case without vaccination, we set it equal to µ 0 . The expected utility of a non-vaccinated person is:

W (e) = λ (e, µ 0 ) U (w -e, H 1 ) + (1 -λ (e, µ 0 )) U (w -e, H 0 ) . (1)
We denote by e * the optimal effort, that is the solution of max e

W (e).

Proposition 1 There exists a unique e * if λ ee (e, µ 0

) ≥ 0. Moreover, e * ∈ (0, w) if lim λ e (0, µ 0 ) = lim U c (0, H i ) = +∞.
As the effort is costly, there is an optimal effort that is obtained provided that its marginal effect on the probability of being in the good health state is infinite at the origin and decreasing. Moreover, an infinite marginal utility when consumption tends to zero makes it possible to ensure that the effort is lower than the endowment. One could extend this framework to a pure utility cost of the effort, but the results we derive below will be unchanged.

Proposition 2 The optimal effort increases with the endowment and, provided that λ eµ ≤ 0, decreases with the biological parameter.

Proposition 2 is intuitive. We find that effort increases with income and decreases with exogenous improvements in the probability of not developing a severe form. For example, wearing a mask will certainly increase with income (the poorest households cannot afford to buy several masks per day even subsidized) and will also decline with the probability of dying from COVID-19 or developing a severe form. As a result, older individuals are more likely to wear a mask [START_REF] Haischer | Who is wearing a mask? Gender-, age-, and locationrelated differences during the COVID-19 pandemic[END_REF]. If λ eµ > 0, the effort may increase with µ, but this is less intuitive.

Randomized vaccination clinical trials

When a new epidemic emerges, the research community immediately looks for treatments and preventive solutions. Vaccine development during COVID-19 is one such example, as they reduce the likelihood of developing a severe form of infection with COVID-19. More specifically, vaccination makes it possible to act on the value of the biological parameter µ, and possibly improve it by increasing the immune defences of vaccinated people. But, we do not know the new value of µ before testing the vaccine. The gold standard in clinical trials is the double-blind placebo control clinical trials. This is a medical study involving human participants in which neither side knows who is getting treatment and who the placebo. In pragmatic trials, people who participate are not the same as those who do not participate [START_REF] Schwartz | Explanatory and pragmatic attitudes in therapeutical trials[END_REF]. The trial population also differs from the non-enrolled population, especially in randomized, blinded trials (much more so than in observational studies). This may be assimilated to the Hawthorne effect, in which the results of an experiment are due not to the experimental factors but to the fact that subjects are aware that they are participating in an experiment in which they are being tested, which usually results in greater motivation.

Below, we propose to characterize the behaviour of participants in such a test using a decision theory framework. To our knowledge, there is in the literature no model of this type under uncertainty and ambiguity.

The optimal effort of participants in a trial

We consider here that µ is likely to increase with vaccination. We consider it as a random variable, μ, of support [µ 0 , 1]. A the lower bound, µ 0 , vaccination does not modify the probability of suffering from a severe form. At the uppur bound µ = 1, the vaccination means never undergoing a serious form, which is translated into λ(e, 1) = 1. However, prior to the trial, the distribution of µ is not known.

= [0, q max ε ]
whose distribution is not known. We are aware that drop-outs induced by adverse effects may generate selection bias and imbalance in randomized average characteristics [START_REF] Chan | Learning, Private Information, and the Economic Evaluation of Randomized Experiments[END_REF], thus revealing preferences. But we do not consider this here, as we are studying a static framework.

We consider the α-MEU model of preferences representation under ambiguity. This model evaluates decisions as a weighted sum of the highest and the lowest expected utility compatible with the set of priors. The weight granted to the lowest expected utility measures the DM's ambiguity aversion. To this end, we compute the worst possible expected utility of a participant. It is characterized by a vaccine that has no effect (when the realization of μ is µ 0 ) and adverse effects that are the highest possible (when the realization of ε1 is ε) and that occur which the highest probability (when the realization of qε is q max ε ). This state of nature provides the following expected utility:

U min (e) = q max ε [λ(e, µ 0 )U (w -e, H 1 -ε) + (1 -λ(e, µ 0 ))U (w -e, H 0 -θε)] +(1 -q max ε ) [λ(e, µ 0 )U (w -e, H 1 ) + (1 -λ(e, µ 0 ))U (w -e, H 0 )] . (2)
Conversely, the best possible expected utility is given for a vaccine whose effect is maximal (when the realization of μ is 1) and has no adverse effects (when the realization of qε is 0).

This gives the following satisfaction:

U max (e) = U (w -e, H 1 ).

(3)

Each individual displays a degree of ambiguity aversion (or pessimism) about vaccination, which is denoted by α ∈ [0, 1]. Some articles of the literature document a differential vaccination hesitancy across genders, hesitancy being higher among women despite the fact that they are more likely to perceive the pandemic as a very serious health problem and to agree and comply with restraining measures [START_REF] Galasso | Gender differences in COVID-19 attitudes and behavior: Panel evidence from eight countries[END_REF]. This might partly explain, along with sex-based immunological differences, the share of women in specific sectors, or other health-related behaviours such as smoking [START_REF] Wenham | COVID-19: the gendered impacts of the outbreak[END_REF], why women die less than men of COVID-19. Still vaccine hesitancy is higher amongst women, an observation that has been called the COVID-19 vaccine's gender paradox [START_REF] Galasso | COVID-19 Vaccine's Gender Paradox[END_REF], as women are more concerned about COVID-19 and more likely to believe themselves to be infected and become seriously ill. We note that gender might not be the only source of heterogeneity in the aversion towards vaccination. In France, where vaccination hesitation is historically high (Neumann-Böhme et al., 2020), Hacquin, Altay, Araujo, [START_REF] Hacquin | Sharp rise in vaccine hesitancy in a large and representative sample of the French population: reasons for vaccine hesitancy[END_REF] show that vaccine-hesitant individuals are more likely to be women, young, less educated, vote for political extremes, be dissatisfied with the government's response to the COVID-19 crisis, and feel less at risk of COVID-19.

In our framework, those differences translate as different α.

The objective function of a person who is vaccinated is given as the convex combination of the expected utilities in the worst and best cases: (5)

V (e; α) = αU min (e) + (1 -α) U max (e). ( 4
Let e * * (α) be the solution of this problem. We first show this solution exists.

Proposition 3 There exists a unique e * * (α) ∈ (0, 1) if the conditions in Proposition 1 are satisfied and if

H 1 -H 0 ≥ ε (1 -θ).
The conditions that guarantee the existence of an optimal effort for the participants in the trial are the same as those that prevail in the benchmark model, except one that is related to the adverse effects associated with the trial. More precisely, it is required that minimal health in the good health state, H 1 -ε, should be better than that in the bad health state, H 0 -θε. We observe that it is always the case for θ ≥ 1, which means that the maximal adverse effect is greater in the bad health state than in the good one. Notice that if H 1 -H 0 < ε (1 -θ), the optimal effort might be equal to zero.

We now assume that the conditions stated in Proposition 3 are satisfied and propose some comparative statics.

Proposition 4 The optimal effort e * * decreases with µ 0 if λ eµ (e, µ 0 ) ≤ 0, increases with θ, and increases with ε and q max ε if θ ≥ 1. Moreover, the effort increases with ambiguity aversion, α.

Since U cH (w, h) ≥ 0, the marginal utility of wealth is lower in the disease state. First, the reduction in the probability of disease, by changing µ 0 , raises the expected marginal utility of wealth and thus lowers the level of effort. Second, when µ rises, it changes the efficiency of effort on probability of disease, 1 -λ e . If it increases efficiency, λ eµ < 0, this second effect reinforces the first one and the effort increases with µ 0 . Otherwise, we cannot conclude.

When θ increases, the side effects on bad health status increase, as does the gap between good and bad health status. This incites individuals to increase their effort.

The effect of side effects, ε, on optimal effort depends on the way it impacts initial health status, whether bad or good health. First, when ε rises, U c (w -e, H 1 -ε) and U c (w -e, H 0 -θε) fall since we assume that U cH (w, H) ≥ 0, lowering the expected marginal utility of wealth and thus the marginal cost of effort. This incites the individual to increase the level of effort. Second, an increase in ε raises the gap between being sick and being healthy, H 1 -H 0 + ε(θ -1) if θ ≥ 1. Thus, it raises the marginal benefit of effort. If θ < 1, an increase in ε decreases the gap between being sick and being healthy, decreases the marginal benefit of effort and goes in the opposite direction to the first effect. Therefore, we cannot conclude.

q max ε is the worst scenario in the sense that it is the highest probability of adverse side effects occurring. In our set-up, a change in the degree of ambiguity/model uncertainty corresponds to a change in the size of the set Ω q without affecting the probability distributions that already belong to the set. The arrival of a new worst scenario potentially alters the optimal level of effort. If q max ε increases, the worst scenario is even worse, and this "bad news" encourages individuals to raise their level of effort if θ ≥ 1 (else we cannot conclude).

Similarly, if q max ε decreases, the worst scenario is less severe, and this "good news" encourages individuals to reduce their level of effort (if θ ≥ 1).

Finally, ambiguity aversion encourages individuals to increase their level of effort. This result is similar to the ones obtained by [START_REF] Alary | The Effect of Ambiguity Aversion on Insurance and Self-protection[END_REF] or [START_REF] Berger | Characterizing ambiguity attitudes using model uncertainty[END_REF] in different contexts.

We now characterize the behaviour of those who participate in the trial with respect to the average behaviour e * observed prior to the trial.

Proposition 5 There exists θ ∈ (1 -(H 1 -H 0 ) /ε, 1) such that (i) e * * (α) < e * for all

α ∈ [0, 1] if θ ≤ θ; (ii) there exists a unique α ∈ (0, 1) such that e * * (α) ≥ e * ⇔ α ≥ α if θ > θ.
As a result, the most optimistic individuals (or least ambiguity-averse) make less effort during the trial. We note that they may still make an effort because they do not know if they received the vaccine dose during the trial or only a placebo. They are just doing less than what was observed before the clinical trial.

If θ is sufficiently high, that is, the adverse effect is larger in the bad health state than in the good one, the more pessimistic individuals are, the more they will invest in prevention and make efforts. If θ is low, all the participants will invest less in prevention whenever their level of pessimism.

The level of ambiguity aversion (or seeking) is a major determinant of the effort made by individuals participating in the trial. In addition, it is a determining factor in the decision whether or not to participate in these trials. Since the participants are volunteers, we introduce a participation constraint, that is the utility of participating in the trial is greater than the one of not participating:

pV (e * * (α) ; α)

+ (1 -p) W (e * * (α)) ≥ W (e * ) . (6)
Proposition 6 For sufficiently high p, there exists a unique α ∈ (0, 1) such that agents with relatively low pessimism α ∈ [0, α] participate in the trial whereas those with relatively high pessimism α ∈ [α, 1] do not participate.

We show that only the less pessimistic will participate in the trial. Since effort increases with pessimism, one may expect that those who participate will not do much prevention and particularly less than those who do not participate. Actually, this intuition is not entirely correct. Indeed, there are two cases. Either, α ≥ α: all participants involved in a clinical trial have a lower level of effort compared to the one that prevailed before the trial, i.e. we have e * * ≤ e * . Or, α < α: some participants (those with α ∈ [0, α]) have a lower level of effort and some (those with α ∈ [ α, α]) have a higher level of effort. The difficulty here is that both α and α are endogenous variables. We thus claim that the amount invested in prevention does not only depend on ambiguity aversion but also on the adverse effect size.

The change in behaviour according to whether or not individuals participate in the trial is thus linked to an external effect, ε, and preferences, α. We highlight the role of adverse effect in the following.

Proposition 7 If the adverse effect ε is sufficiently weak, then α < α. If ε is sufficiently large, it is possible to have α > α.

Importantly, the behaviour of the participants in the trial does not necessarily imply a lower effort than in the population that does not participate in the trial. Combining the results of propositions 6 and 7, we obtain that there are two thresholds to note for ambiguity aversion. First, only the most optimistic participate in the vaccine trial (α ≤ α). On the other hand, the most optimistic of them decrease their level of effort by participating (α ≤ α). vaccines have been demonstrated to be safe and effective in both clinical trials and widespread inoculation for both males and women with limited side effects, our model proposes a potential framework and explanation to fuel this debate. It might not be so surprising that vaccine-hesitant individuals are more likely to be women rather than men, at least declaratively, as men may be more optimistic or less adverse to ambiguity. In our model, optimistic individuals participate more in a trial. However, our model suggests that it will be difficult to predict efforts and vaccine efficacy, a result that has been partly explored in the literature. [START_REF] Malani | Identifying Placebo Effects with Data from Clinical Trials[END_REF] argues that "more optimistic patients may modify their behaviourlike the ulcer patient who reduces their consumption of spicy food or the cholesterol patient who exercises more often -in a manner that complements their medical treatment. If an investigator does not measure these behavioural changes (as is commonly the case), the more optimistic patient will appear to have a better outcome, that is, to have experienced placebo effects." Understanding how women differ from men in response to treatment is clearly critical for enhancing treatment efficacy. However, gender-specific efforts and behaviours are rarely measured during clinical trials, where only the medical outcome is measured. One avenue for research might be to develop protocols and measures of efforts that would help to understand this gender paradox.

We now turn to the evaluation of vaccine efficacy.

Estimating Vaccine Efficacy

Let z (α) be the share of persons with ambiguity avecrsion α within the population, with 1 0 z (α) dα = 1. In a clinical trial, the vaccine is considered effective if the proportion of uninfected is higher in the treated group than in the control group. Let us note μ, the true value of µ in case of vaccination, it is enough that μ > µ 0 for it to be the case because the probability of not having a severe form of COVID-19 in the treated group is λ (e * * , μ) while the same probability in the control group is λ (e * * , µ 0 ).

At the end of the clinical trial, the efficacy of the vaccine is determined by counting the infected among the treated and among the controls. We note that, a priori, we observe neither the e * behaviours of the participants, nor the 'true' state of nature µ.

Vaccine efficacy V E is defined as the (number of people in the control group developing a severe form -number of people in the vaccinated control group developing a severe form ) / number of people in the control group developing a severe form . If the distribution of α is the same among the controls and the vaccinated -which is justified if the samples are large enough and nobody drops out -, we obtain:

V E = α 0 z (α) (1 -λ (e * * (α) , µ 0 )) dα - α 0 z (α) (1 -λ (e * * (α) , µ)) dα α 0 z (α) (1 -λ (e * * (α)) , µ 0 ) dα = α 0 z (α) [λ (e * * (α) , µ) -λ (e * * (α) , µ 0 )] dα α 0 z (α) (1 -λ (e * * (α)) , µ 0 ) dα
Remark. Let us assume that 1 -λ(e, µ) = (1 -π(e))(1 -µ), which can be interpreted as the product of two independent probabilities: 1 -π(e) is the probability of catching COVID-19 and (1 -µ) is the probability of developing a severe form if one is infected by COVID-19, then we have:

V E = µ -µ 0 1 -µ 0 .
But if λ is not separable, we cannot deduce µ because we do not observe e * . As a result, after clinical trials and during the scale-up phase of a vaccination campaign, precise vaccine efficacy may be unknown, and we revert to the previous trial case. This may happen because vaccine efficacy depends on unobserved behaviours and efforts, but also for many other reasons, including ,for instance, fake news that will introduce noise around vaccine efficacy. In this case, our results show that preventive efforts are extremely difficult to predict, a result that is consistent with the literature [START_REF] Courbage | Prevention and precaution[END_REF]. This may call for further trials connected with real clinical practice.

Thus, provided that λ is separable, the trial enables researchers to determine the values of ε and θ and the probability of occurrence of an adverse reaction, noted q. Because this case differs from the trial case, the next section discusses this option.

Optimal efforts during the vaccination campaign

As explained above, in this section, we assume that the probabilities λ are separable, which implies that the uncertainties associated with vaccination have been removed. The biological parameter and side effect characteristics are now assumed to be known and given by the triplet ( µ, ε, q ε ). The expected utility of an agent vaccinated during the vaccination campaign is:

EU (e; µ, ε, q ε ) = q ε [λ(e, µ)U (w -e, H 1 -ε) + (1 -λ(e, µ))U (w -e, H 0 -θ ε)] +(1 -q ε ) [λ(e, µ)U (w -e, H 1 ) + (1 -λ(e, µ))U (w -e, H 0 )] . (7)
Let e * * * be the solution of: max e EU (e; µ, ε, q ε ) while e * is the effort of a non-vaccinated agent, who solves max e W (e) where W (e) is given by (1).

Let us now examine under what conditions an individual will come forward for vaccination. The condition for voluntary participation in the campaign is:

EU (e * * * ; µ, ε, q ε ) ≥ W (e * ) . (8)
Let us define the wealth risk premium in a bi-dimensional environment, expressed in terms of effort, ē, as the difference between the initial health and the global certainty equivalent of health risk. ē is therefore the solution of the following equation:

(

1 -q ε ) U (w, H 1 ) + q ε U (w, H 1 -ε) = U (w -ē, H 1 ). (9)
We can state that:

Proposition 8 If the wealth risk premium, expressed in terms of effort, is lower than the level of effort without any vaccine, i.e. if ē < e * , then individuals get vaccinated if µ is high enough.

The relationship between the demand for vaccination and the degree of risk aversion is unclear. This is consistent with the well-known results on the ambiguous impact of risk aversion on self-protection [START_REF] Dionne | Self-insurance, self-protection and increased risk aversion[END_REF]. However, in the context of vaccination, our result is close to the one obtained by [START_REF] Crainich | Vaccination as a trade-off between risks[END_REF]. In our set-up, the vaccination trial reveals the value of the effect of vaccines on the probability of having a severe form of the disease. We show that if the risk premium is relatively low, that is for weakly risk-averse individuals, then individuals decide to get vaccinated if the impact of vaccines is sufficiently high. Otherwise, they prefer not to get vaccinated and make more effort (e.g. protective measures). Our setting extends standard cost-benefit analysis, which considers only the health impacts of vaccination, by introducing risk aversion toward the possible adverse effects induced by vaccination. This may help in understanding why some individuals may still be reluctant about vaccination even if the balance between health benefits and costs is positive.

Conclusion

In this paper, we propose a novel theoretical framework to analyze vaccination trials and campaigns. It emphasizes the role of ambiguity and ambiguity attitudes by using a α-MEU model of preferences that captures the ambiguous nature of adverse effects of vaccination.

This attitude is assumed to be heterogeneous within the population. We show that individuals enrolled in a clinical trial are not likely to be representative of the total population in terms of risk exposure to the disease and, during the trial, will behave differently according to their own attitude towards ambiguity. After the trial, we characterize the condition that establishes the biological effect of the vaccine. Under that latter condition, we analyse whether the general population is likely to participate in the vaccination campaign and we

show the crucial role of the degree of risk aversion of the population.

What are the implications of these findings for how to design clinical trials and vaccination campaigns? Although we do not yet have a toolkit, this article could motivate an entire research agenda on how to anticipate the expansion of clinical trials to vaccination campaigns from a behavioral perspective. [START_REF] Chassang | Accounting for Behavior in Treatment Effects: New Applications for Blind Trials[END_REF] for example, proposes a research design when there are interactions between treatment and behaviours. Optimism bias has previously been studied in clinical research, but not in relation to ambiguity aversion [START_REF] Chalmers | What are the implications of optimism bias in clinical research[END_REF][START_REF] Sharot | The optimism bias[END_REF]. Innovative research designs accounting for ambiguity-aversion could thus be one avenue for research. Clinical trials should also aim at estimating and classifying the bias in vaccine efficacy more systematically so that the vaccination campaign could adjust this bias in information campaigns. This may be anticipated at an early stage of randomized controlled trials (RCTs) -when patients are invited to participate -by improving data collection on ambiguity measurements [START_REF] Izhakian | A theoretical foundation of ambiguity measurement[END_REF][START_REF] Jewitt | Ordering ambiguous acts[END_REF]. Finally, vaccination campaigns could naturally run out of steam if vaccine hesitancy persists despite the effectiveness of the treatment. Recently, some studies have shown the efficiency of nudges on vaccination decisions [START_REF] Dai | Behavioural nudges increase COVID-19 vaccinations[END_REF][START_REF] Patel | Text-message nudges encourage COVID vaccination[END_REF].

Experimenting different ways to nudge decision making and communicating risks in such context is certainly crucial [START_REF] Kahneman | Thinking, fast and slow[END_REF].

Last but not least, this paper complements other frameworks that have e.g. studied the political economy of vaccination [START_REF] Iverson | Optimal social distancing and the economics of uncertain vaccine arrival[END_REF][START_REF] Manski | Mandating vaccination with unknown indirect effects[END_REF][START_REF] Nganmeni | Vaccine and inclusion[END_REF]. Our model is used to illustrate the vaccination episode of the COVID-19 pandemic but do not consider other preventive behaviors such that antibody testing [START_REF] Guimarães | Antibody tests: They are more important than we thought[END_REF] or self-isolation [START_REF] Baril-Tremblay | Self-isolation[END_REF]. Moreover, our work could be usefully extended by introducing a dynamic framework in which uncertainty evolves over time. This would make it possible to characterize the waiting strategies of part of the population. It could also be extended to a more economic framework that would evaluate the cost and benefits of vaccination, and their consequences for firms and markets (see, e.g. [START_REF] Morimoto | Ambiguity in a pandemic recession, asset prices, and lockdown policy[END_REF]).

C Proof of Proposition 3

The first order condition is:

pα {-q max ε [λ(e, µ 0 )U c (w -e, H 1 -ε) + (1 -λ(e, µ 0 ))U c (w -e, H 0 -θε)] -(1 -q max ε ) [λ(e, µ 0 )U c (w -e, H 1 ) + (1 -λ(e, µ 0 ))U c (w -e, H 0 )] +λ e (e, µ 0 ) [q max ε (U (w -e, H 1 -ε) -U (w -e, H 0 -θε)) +(1 -q max ε ) (U (w -e, H 1 ) -U (w -e, H 0 ))]} -p (1 -α) U c (w -e, H 1 ) + (1 -p) W (e) = 0. (C.5)
We note that the left-hand side of the above expression is strictly positive for e = 0 if lim λ e (0, µ 0 ) = +∞ and U (w,

H 1 -ε) > U (w, H 0 -θε), which reduces to H 1 -H 0 ≥ ε (1 -θ).
It is strictly negative for e = w if lim U c (0, H i ) = +∞. Moreover, the second order condition

is pα {-2q max ε λ e (e, µ 0 ) [U c (w -e, H 1 -ε) -U c (w -e, H 0 -θε)] -2(1 -q max ε )λ e (e, µ 0 ) [U c (w -e, H 1 ) + U c (w -e, H 0 )] +q max ε [λ(e, µ 0 )U cc (w -e, H 1 -ε) + (1 -λ(e, µ 0 ))U cc (w -e, H 0 -θε)] +(1 -q max ε ) [λ(e, µ 0 )U cc (w -e, H 1 ) + (1 -λ(e, µ 0 ))U cc (w -e, H 0 )] +λ ee (e, µ 0 ) [q max ε (U (w -e, H 1 -ε) -U (w -e, H 0 -θε)) +(1 -q max ε ) (U (w -e, H 1 ) -U (w -e, H 0 ))]} +p (1 -α) U cc (w -e, H 1 ) + (1 -p) W (e) , (C.6) which is negative if λ ee (e, µ 0 ) ≤ 0 and H 1 -H 0 ≥ ε (1 -θ).

D Proof of Proposition 4

We use equations (A.1) and (C.5) as implicit functions and compute the following:

sign de * * dµ 0 = sign [pα {-q max ε λ µ (e, µ 0 ) [U c (w -e, H 1 -ε) -U c (w -e, H 0 -θε)] -(1 -q max ε )λ µ (e, µ 0 ) [U c (w -e, H 1 ) -U c (w -e, H 0 )] +λ eµ (e, µ 0 ) [q max ε (U (w -e, H 1 -ε) -U (w -e, H 0 -θε)) +(1 -q max ε ) (U (w -e, H 1 ) -U (w -e, H 0 ))]} + (1 -p) λ eµ (e, µ 0 ) [U (w -e, H 1 ) -U (w -e, H 0 )] -(1 -p) λ µ (e, µ 0 ) [U c (w -e, H 1 ) -U c (w -e, H 0 )]] , (D.7)
which is negative if λ eµ (e, µ 0 ) ≤ 0 (as in Proposition 2). We also have: which is positive if θ ≥ 1 (as we have U cH ≥ 0 and U HH < 0). Finally, we have:

sign de * * dθ = sign [(1 -λ(e, µ 0 ))U cH (w -e, H 0 -θε) + λ e (e, µ
sign de * * dq max ε = sign [λ(e, µ 0 ) [U c (w -e, H 1 ) -U c (w -e, H 1 -ε)] +(1 -λ(e, µ 0 )) [U c (w -e, H 0 ) -U c (w -e, H 0 -θε)] -λ e (e, µ 0 ) [(U (w -e, H 1 ) -U (w -e, H 1 -ε)) -(U (w -e, H 0 ) -U (w -e, H 0 -θε))]] . (D.10)
The latter expression is equal to 0 for ε = 0 and is increasing with ε if -U H (w -e, H 1 -ε) + θU H (w -e, H 0 -θε) ≥ 0, which is the case if θ ≥ 1. We therefore conclude that the sign is

positive if θ ≥ 1.
Finally, the sign of de * * dα is given by the sign of ∂Ve(e * * ;α) ∂α .Using equation (A.1), we obtain: D.11) or equivalently:

∂V e (e * * ; α) ∂α = U c (w -e * * , H 1 ) + q max ε λ e (e * * , µ 0 ) [U (w -e * * , H 1 -ε) -U (w -e * * , H 0 -θε)] -q max ε [λ(e * * , µ 0 )U c (w -e * * , H 1 -ε) +(1 -λ(e * * , µ 0 ))U c (w -e * * , H 0 -θε)] , (
∂V e (e * * ; α) ∂α = [U c (w -e * * , H 1 ) -q max ε U c (w -e * * , H 0 -θε)] +q max ε λ e (e * * , µ 0 ) [U (w -e * * , H 1 -ε) -U (w -e * * , H 0 -θε)] +q max ε λ(e * * , µ 0 ) [U c (w -e * * , H 0 -θε) -U c (w -e * * , H 1 -ε)] . (D.12)
In the previous equation, the first expression in brackets is positive since q max ε ≤ 1 and

U cH ≥ 0; due to U H > 0 the second is also positive if one assumes that H 1 -H 0 ≥ ε (1 -θ),
as required in Proposition 3; finally, due to U cH ≥ 0, the third is also positive under the same assumption. We conclude that ∂V e (e * * ; α) /∂α > 0.

E Proof of Proposition 5

To characterize the difference between e * * (α) and e * , we use the concavity of both optimization problems to state that V e (e * ; α) ≥ 0 ⇔ e * * (α) ≥ e * . It is thus sufficient to analyse the sign of V e (e * ; α), noting that e * is independent of α, ε and θ. We have:

V e (e; α) = α∆ (e) -U c (w -e, H 1 ), (E.13) where:

∆ (e) = U c (w -e, H 1 ) -q max ε [λ(e, µ 0 )U c (w -e, H 1 -ε) + (1 -λ(e, µ 0 ))U c (w -e, H 0 -θε)] -(1 -q max ε ) [λ(e, µ 0 )U c (w -e, H 1 ) + (1 -λ(e, µ 0 ))U c (w -e, H 0 )] +λ e (e, µ 0 ) [q max ε (U (w -e, H 1 -ε) -U (w -e, H 0 -θε)) +(1 -q max ε ) (U (w -e, H 1 ) -U (w -e, H 0 ))] . (E.14)
The proof proceeds in three steps. First, we note that V e (e; 0) < 0 whatever e, which implies that V e (e * ; 0) < 0 and therefore that e * * (0) < e * . Second, following the previous proposition, we know that ∂V e (e * ; α) /∂α > 0.

Third, we examine the sign of V e (e * ; 1). If V e (e * ; 1) < 0, we can conclude that e * * (α) < e * for all α ∈ [0, 1]. If V e (e * ; 1) > 0, we can conclude that there exists a unique α ∈ (0, 1] such that e * * (α) ≥ e * ⇔ α ≥ α. We have: which is equal to 0 for ε = 0 and is negative for the maximal value of ε, i.e. for ε = H 1 -H 0 . Thus, V e (e * ; 1)| θ=0 < 0 if ε is sufficiently large. Therefore, there exists θ ∈

(1 -(H 1 -H 0 ) /ε, 1) such that V e (e * ; 1) ≥ 0 ⇐⇒ θ ≥ θ.

F Proof of Proposition 6

Let the function f (α) be defined as: We have g(µ 0 ) < 0 and g (µ) > 0. Therefore, if g(1) > 0, there exists µ ∈ (µ 0 , 1) such that agents participate in the vaccination campaign µ ≥ µ and do not participate otherwise. If g(1) < 0, agents do not participate in the vaccination campaign. Since λ(e, 1) = 1, and e * * * = 0 for µ = 1, we can write:

f (α) = pV
g(1) = q ε U (w, H 1 -ε) + (1 -q ε )U (w, H 1 )

-[λ (e * , µ 0 ) U (w -e * , H 1 ) + (1 -λ (e * , µ 0 )) U (w -e * , H 0 )] . (H.26) Thus, a sufficient condition for g(1) > 0 is: q ε U (w, H 1 -ε) + (1 -q ε )U (w, H 1 ) > U (w -e * , H 1 ) . (H.27)

Or using (9), U (w -ē, H 1 ) > U (w -e * , H 1 ) , which is equivalent to ē < e * .

)

  Since we are considering double-blind placebo control clinical trials, not all participants in the trial are vaccinated. Let p be the probability of being vaccinated during the trial, which is known by the participant as it is defined by the trial organizer and usually has to be disclosed. The problem of an individual with a degree of ambiguity aversion α who participates in the trial is: max e pV (e; α) + (1 -p) W (e) .

Figure 1

 1 Figure 1 summarizes these results.

Fig. 1 :

 1 Fig. 1: Participating in trial and effort

  0 )U H (w -e, H 0 -θε)] , (D.8) which is positive. Moreover, we have: sign de * * dε = sign [λ(e, µ 0 )U cH (w -e, H 1 -ε) + (1 -λ(e, µ 0 ))θU cH (w -e, H 0 -θε)-λ e (e, µ 0 ) [U H (w -e, H 1 -ε) -θU H (w -e, H 0 -θε)]] , (D.9)

V

  e (e * ; 1) = q max ε λ e (e * , µ 0 ) [U (w -e * , H 1 -ε) -U (w -e * , H 0 -θε)] e * , µ 0 )U c (w -e * , H1 -ε) + (1 -λ(e * , µ 0 ))U c (w -e * , H 0 -θε)] .Let us compute∂V e (e * ; 1)∂θ = q max ε ε [λ e (e * , µ 0 )U H (w -e * , H 0 -θε) + (1 -λ(e * , µ 0 ))U cH (w -e * , H 0 -θε)] , (E.16)which is positive due to U H > 0 and U cH ≥ 0. Let us now evaluate:V e (e * ; 1)| θ=1 = q max ε λ e (e * , µ 0 ) [U (w -e * , H 1 -ε) -U (w -e * , H 0 -ε)] -q max ε [λ(e * , µ 0 )U c (w -e * , H 1 -ε) +(1 -λ(e * , µ 0 ))U c (w -e * , H 0 -ε)] . (E.17)Since V e (e * ; 1)| θ=1,ε=0 = q max ε W (e * ) = 0 and since ∂V e (e * ; 1)∂θ = q max ε λ e (e * , µ 0 )εU H (w -e * , H 0 -ε) + q max ε (1 -λ(e * , µ 0 ))εU cH (w -e * , H 0 -ε) > 0 and ∂V e (e * ; 1)| θ=1 ∂ε = -q max ε λ e (e * , µ 0 ) [U H (w -e * , H 1 -ε) -U H (w -e * , H 0 -ε)] +q max ε [λ(e * , µ 0 )U cH (w -e * , H 1 -ε) + (1 -λ(e * , µ 0 ))U cH (w -e * , H 0 -ε)] ,is positive due to U HH < 0, we can conclude that V e (e * ; 1) > 0 for ε > 0 and θ ≥ 1. Now, the minimal value for θ is max {1 -(H 1 -H 0 ) /ε, 0}. Let us first compute:V e (e * ; 1)| θ=1-H 1 -H 0 ε = -q max ε U c (w -e * , H 1 -ε) < 0. (E.18)Moreover, we have: V e (e * ; 1)| θ=0 = q max ε λ e (e * , µ 0 ) [U (w -e * , H 1 -ε) -U (w -e * , H 0 e * , µ 0 )U c (w -e * , H 1 -ε) + (1 -λ(e * , µ 0 ))U c (w -e * , H 0 )] ,

  (e * * (α) ; α) + (1 -p) W (e * * (α)) -W (e * ) . (F.20)The agent participates in the trial if and only if f (α) ≥ 0. Since we have:f (α) = p U min (e * * (α)) -U max (e * * (α)) < 0, (F.21)it is sufficent to evaluate f (0) and f (1) to conclude. We havef (1) = pU min (e * * (1)) + (1 -p) W (e * * (1)) -W (e * ) = pq max ε [λ(e * * (1), µ 0 )U (w -e * * (1), H 1 -ε) +(1 -λ(e * * (1), µ 0 ))U (w -e * * (1), H 0 -θε) -W (e * )] +(1 -pq max ε ) [W (e * * (1)) -W (e * )] . (F.22)would do. Then, equation (G.24) rewrites:V (e; α) -W (e) = [(1 -α) (1 -λ(e, µ 0 )) -αq max ε λ(e, µ 0 )] [U (w -e, H 1 ) -U (w -e, H 0 )] . (G.25) Thus, we have α ≤ α, for all ε ∈ [0, H 1 -H 0 ] if λ(e * , µ 0 ) ≤ 1/ (1 + αq max ε / (1 -α)) . Otherwise, there is a value, ε in ]0, H 1 -H 0 [, such that ε > ε ⇔ α ≥ α.H Proof of Proposition 8Let us define g(µ) := EU (e * * * ; µ, ε, q) -W (e * ), org(µ) = q ε [λ(e * * * , µ)U (w -e * * * , H 1 -ε) + (1 -λ(e * * * , µ))U (w -e * * * , H 0 -θ ε)] +(1 -q ε ) [λ(e * * * , µ)U (w -e * * * , H 1 ) + (1 -λ(e * * * , µ))U (w -e * * * , H 0 )] -[λ (e * , µ 0 ) U (w -e * H 1 ) + (1 -λ (e * , µ 0 )) U (w -e * , H 0 )] .

  COVID-19 vaccines validated in Europe and North America are considered safe globally but negative side effects or adverse events, such as arterial thromboembolic and haemorrhagic events, have been observed for some vaccines. Moreover, even if adverse effects are relatively low in practice, individuals may overweight them. In the case of vaccination, one may

experience an adverse reaction that reduces health; it is noted that one experiences the adverse reaction whether infected with COVID-19 or not, and that this adverse effect depends on one's health status. Adverse effects on health are measured with the random variable ε of support [0, ε] if one is in state H 1 and by θε with θ ≥ 0 if one is in state H 0 . Again, we assume that the distribution of ε is unknown. The probability of experiencing an adverse event (regardless of health status) is noted qε . It is also a random variable of support Ω q

See Etner, Jeleva, and Tallon (2012) or[START_REF] Gilboa | Ambiguity and the bayesian paradigm, advances in economics and econometrics: theory and applications[END_REF], for a survey on ambiguity literature.

Although the α-MEU model is one of the most popular models in decision making under ambiguity, there is no known axiomatic characterization in a general domain of preference actions. However, axiomatic foundations have recently been better understood(Klibanoff et al., 2021).

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

A Proof of Proposition 1

The first order condition is W (e) = 0, where W (e) can be written as:

W (e) = λ e (e, µ 0 ) [U (w -e, H 1 ) -U (w -e, H 0 )] -[λ (e, µ 0 ) U c (w -e, H 1 ) + (1 -λ (e, µ 0 )) U c (w -e, H 0 )] .

(A.1)

The second order condition is W (e) < 0, where W (e) can be written as:

a sufficient condition for W (e) to be negative is λ ee (e * , µ 0 ) ≥ 0. Moreover, e * > 0 if

B Proof of Proposition 2

We use equation (A.1) as an implicit function and derive the following relation:

Similarly, we have:

which is negative since λ eµ ≤ 0.

The first term in square brackets is negative since:

λ(e * * (1), µ 0 )U (w -e * * (1), H 1 -ε) + (1 -λ(e * * (1), µ 0 ))U (w -e * * (1), H 0 -θε) < λ(e * * (1), µ 0 )U (w -e * * (1), H 1 ) + (1 -λ(e * * (1), µ 0 ))U (w -e * * (1), H 0 ) = W (e * * (1))

and by definition, W (e * * (1)) < W (e * ). As a result, f (1) < 0. Moreover,

The first term between brackets is positive because e * * (0) < e * implies U (w -e * * (0), H 1 ) > U (w -e * , H 1 ) > W (e * ). The second term between brackets is negative by definition. We conclude that f (0) > 0 if p is sufficiently high. In that case, there exists α ∈ (0, 1) such that

G Proof of Proposition 7

We evaluate in α = α the sign of pV (e * * (α) ; α) + (1 -p) W (e * * (α)) -W (e * ); In α, we have e * * ( α) = e * , which implies that one must evaluate the sign of V (e * ; α) -W (e * ). We get:

V (e; α) -W (e) = αU min (e) + (1 -α) U max (e) -W (e)

We note that this expression is positive for ε = 0 (no adverse event) and that it is decreasing in ε. This implies that if ε is sufficiently small, then α < α. On the other hand, if ε is large, it should be possible to have α > α. This can be shown for a specific case such that θ = 0, i.e. there is no adverse reaction for those who got the disease, and, ε = H 1 -H 0 the adverse reaction leads to a deterioration in the health of a healthy person as much as the disease


Influenza Mortality in French Regions after the Hong Kong Flu Pandemic

Florian Bonnet 1 -Hippolyte d'Albis 2 -Josselin Thuilliez 3 Influenza mortality has dramatically decreased in France since the 1950s. Annual death rates peaked during two pandemics: the Asian flu and the Hong-Kong flu (1969)(1970). This study's objective is to evaluate whether the second pandemic created a structural change in the dynamics of influenza mortality in France. We employ a new database on influenza mortality since 1950 at the subnational level (90 geographic areas) to estimate statistical models to find whether a structural change happened and to explain the differences in mortality rates across geographic areas. Influenza mortality increased between 1950 and 1969, and decreased from 1970 onward. The Hong-Kong flu is identified as the event of a structural break. After the break, geographical differences are less explained by regional characteristics such as income, density or aging ratio. Hong Kong flu was found to be associated with a major change in influenza mortality in France. Change in health practices and policies induced a decline in mortality that started in 1970, just after the pandemics. The health benefits are notably important for senior citizens and for the poorest regions.

Introduction

The Covid-19 pandemic has strongly afflicted France. Yet, little attention has been given to the health outcomes following previous pandemics. Pandemics are dramatic shocks which have the ability to positively transform health systems and practices over the long run, as they generate political and economic support for medical innovation and healthcare improvement [START_REF] Dehner | WHO Knows Best? National and International Responses to Pandemic Threats and the "Lessons" of 1976[END_REF][START_REF] Morens | The Persistent Legacy of the 1918 Influenza Virus[END_REF][START_REF] Jordà | Longer-run Economic Consequences of Pandemics[END_REF][START_REF] Iskander | Pandemic Influenza Planning, United States, 1978-2008[END_REF]. Before Covid-19, and aside from the global HIV epidemic, only two pandemics severely affected France after WWII: the Asian flu and the Hong-Kong flu (1969)(1970). Death tolls of these pandemics were much lower than that of the current crisis, even adjusted for population and age distribution changes: 33,232 deaths from the Asian flu and 32,062 deaths from the Hong-Kong flu. The main difference is that in earlier pandemics, the mortality rates of infants and young adults were higher than that of today.

In the aftermath of these two pandemics, international health authorities and industrialized countries' governments carefully crafted responses, shared information, established surveillance, strengthened vaccine studies, and implemented vaccination campaigns against influenza [START_REF] Tyrrell | Vaccination against Hong Kong Influenza in Britain, 1968-9: A Report to the Medical Research Council Committee on Influenza and Other Respiratory Virus Vaccines[END_REF][START_REF] Carrat | Influenza mortality among the elderly in France, 1980-90: how many deaths may have been avoided through vaccination?[END_REF][START_REF] Sands | The Neglected Dimension of Global Security -A Framework for Countering Infectious-Disease Crises[END_REF][START_REF] Viboud | Multinational Impact of the 1968 Hong Kong Influenza Pandemic: Evidence for a Smoldering Pandemic[END_REF][START_REF] Aranzazu | Le réseau mondial de surveillance de la grippe de l'OMS. Modalités de circulation des souches virales, des savoirs et des techniques, 1947-2007[END_REF][START_REF] Vagneron | Surveiller et s'unir ? Revue d'anthropologie des connaissances[END_REF]. The aim of our study is to evaluate whether this was effective and, more precisely, whether these pandemics created a structural break in the trend line of France's flu mortality. To do so, we examine a unique historical database that tracks flu mortality by region. Our contribution is threefold.

First, we provide a refined description of influenza pandemics and seasonal influenza in 90 French geographical units from 1950 to 2015. Second, we establish that winter 1969-1970 does indeed represent a structural break in the spread and control of seasonal and pandemic influenza in France, especially for the elderly, confirming a major turning point in France's flu management. Third, we show that, after 1970, influenza mortality became regionally random, no longer dependent on income, density, and the share of elderly people.

The Hong Kong flu led to a sharp increase in the death rate of flu: in 1969, it was 30 per 100,000 in France, compared to an average of 14 per 100,000 between 1965 and 1974. Moreover, Figure 1 reveals that geographic variations were significant: death rates were lower in the North of France (with a minimum of 12.5 per 100,000 in Seine) and higher in the South-West (with a maximum of 76.4 per 100,000 in Creuse). Despite a multilevel response to the Hong Kong flu aimed at preventing future such health shocks, the current crisis reveals that France was not fully prepared. However, we show that pandemics can change health practices for the better, as the Hong-Kong flu episode is found to be associated with a reduction in both mortality and inequalities therein across regions.

Methods

Data

We gathered original death data by cause, geographical unit (département in French, hereby "department"), and sex for the period 1950-2015. The data was retrieved from the archives of the Our study focuses on mainland France, and does not include overseas territories. We use the classification of 90 departments that prevailed before 1968, but we also provide analysis by aggregating the data into 22 geographical units known as regions (hereby "regions").

For the statistical analysis, we also use population data by department, sex and age at 1 st January. These populations were calculated from age-specific populations in census years and annual vital statistics (age-specific deaths and births), using a methodological protocol close to that of the Human Mortality Database [START_REF] Bonnet | Computations of French lifetables by department, 1901--2014[END_REF][START_REF] Barbieri | Data resource profile: the human mortality database (HMD)[END_REF]. To obtain the populations on January 1 of each year of an intercensal period, the author adjusts the populations at the initial census by both observed deaths and by an estimate of the migration flow since the initial census. The overall intercensal migration flow is estimated by comparing the observed population and the theoretical population at the final census. The theoretical population is equal to the initial census population minus the observed deaths during the intercensal period. The intercensal migration flow is then distributed temporally in proportion to the time elapsed since the initial census. To compute death rates in year t, we use mean of population at 1st January of year t and t+1.

We also use income data by department (Bonnet and Sotura 2021;[START_REF] Bonnet | Les inégalités de revenu entre les départements français depuis cent ans/Income Inequality across French Departments over the Last 100 Years[END_REF]. These incomes were calculated using tax tabulations for the period 1960-1969 and 1986-2015 and aggregate income tax statistics for the periods 1922-1959 and 1970-1985. This income includes public transfers related to the pension system, but does not take into account taxes due or public subsidies provided to households.

Statistical Analysis

We aim to identify whether and when there was a structural break in the control of influenza epidemics in France. We use a Poisson model, that prevents from negative values for death rates, and regress deaths on a continuous year variable, a post-year dummy, and the interaction of post x year dummy, in addition to department-specific time trend, department fixed-effects, and specific controls (population density per km 2 , per capita income in 2015 euros, and the share of people aged 65 and more in the population).

To take into account the difference in department sizes, we use population as an offset in this regression.

We repeat the procedure, allowing the post-year dummy to vary from 1960 to 1980. The year with maximum in log-likelihood is considered to be the year of a structural break in the data. This test is similar to the Quandt likelihood ratio test that has been shown to be a reliable test for structural breaks in the case of unknown break points [START_REF] Quandt | Tests of the hypothesis that a linear regression system obeys two separate regimes[END_REF][START_REF] Andrews | Tests for parameter instability and structural change with unknown change point[END_REF][START_REF] Piehl | Testing for structural breaks in the evaluation of programs[END_REF].

We also perform a spatial analysis. We examine the differences in deaths across department using Poisson model and three specific variables: population density per km 2 (D), income per capita in 2015 euros (Inc) and the share of people aged 65 and more in the population (𝑆ℎ 𝑑 65+ ). For each year of the period 1950-2015, we estimate the coefficients of the following equation:

log (𝑚 𝑑 𝐹𝑙𝑢 ) =∝ +𝛽 1 𝐷 𝑑 + 𝛽 2 𝐼 𝑑 + 𝛽 3 𝑆ℎ 𝑑 65+ + 𝜀 𝑑 , ( 1
)
where d is the department and 𝜀 𝑑 is the difference between observed and estimated mortality due to flu.

If positive, it is defined as excess mortality. To compute confidence intervals around our estimates, we consider over-dispersion in the model. Finally, we compute pseudo R2 of our model for each year as the square of the correlation between estimated and observed departmental values. All our computations are done using R. software.

Results

The test for a trend break in mortality between 1950 and 2015 identifies 1968 as the year with the maximum in log-likelihood. Values are roughly similar for each year of the period 1968-1971, and highly decrease in 1972. There are other spikes related to influenza outbreaks (1974 and 1977), but they are smaller in magnitude. We note that sensitivity testing of our regression does not affect our main conclusions. We retain 1970 as the year of the structural break for our subsequent analyses since it is the end of the Hong-Kong flu and the central year of the period 1968-1971.

We present a range of graphs to illustrate our empirical results. effort to predominantly protect the elderly. We also notice that the structural break holds for both men and women, the death rate of the latter being slightly higher due to differences in age structure.

Discussion

The Hong Kong flu pandemic started in July 1968 and lasted until 1970. Despite the staggering death toll, the Hong Kong flu pandemic and its repercussions are surprisingly poorly documented in France [START_REF] Flahault | Influenza pandemics: past, present and future challenges[END_REF]. Beyond natural immunity which may have played a role that is difficult to assess in the medium run [START_REF] Jackson | Estimates of the Transmissibility of the 1968 (Hong Kong) Influenza Pandemic: Evidence of Increased Transmissibility Between Successive Waves[END_REF], what could be behind such a strong decline of influenza across all departments in France after 1970? It is difficult to identify any major progress in hygiene at that time [START_REF] Flahault | Influenza pandemics: past, present and future challenges[END_REF]. The discovery of single-use handkerchiefs dates back to the 1930s and massive use of hydro-alcoholic gel, for instance, dates much later. Surveillance has indeed been improved, but this progress seems to have taken place mainly during the 1980s. Finally, there is also little chance that France's influenza is strongly correlated with economic growth or crises [START_REF] Ruhm | Are Recessions Good for Your Health?[END_REF][START_REF] Brüning | Mortality and Macroeconomic Conditions: What Can We Learn From France?[END_REF].

As we examine what caused the decline, the distinction between pandemic and seasonal flu is important [START_REF] Brüning | Mortality and Macroeconomic Conditions: What Can We Learn From France?[END_REF]. France has not experienced a severe pandemic from 1969 onwards, either because the world was better prepared, or just by chance. Put differently, reduced seasonal flu mortality after 1970 may not have been influenced by pandemics, but rather part of a pattern of seasonal flu [START_REF] Carrat | Influenza mortality among the elderly in France, 1980-90: how many deaths may have been avoided through vaccination?[END_REF][START_REF] Rizzo | Trends for Influenza-related Deaths during Pandemic and Epidemic Seasons, Italy, 1969-2001[END_REF][START_REF] Guérin | Histoire de la vaccination: de l'empirisme aux vaccins recombinants[END_REF][START_REF] Gottfredsson | Lessons from the past: Familial aggregation analysis of fatal pandemic influenza (Spanish flu) in Iceland in 1918[END_REF][START_REF] Valleron | Transmissibility and geographic spread of the 1889 influenza pandemic[END_REF][START_REF] Hannoun | The evolving history of influenza viruses and influenza vaccines[END_REF][START_REF] Brès | Surveillance epidemiologique des grippes humaines dans le monde: Le programme de lOMS[END_REF]. Figure A4 and A5 in Appendix hence test for the sensitivity of our results

to the presence of pandemics and provide the change in the slope when we exclude the 1957 and 1968 pandemics, assuming a value for these particular years as equivalent to the average mortality before 1970. The difference between the slopes gives us the contribution of pandemics to the average mortality trend. When both pandemics are eliminated, the slopes before 1970 (in blue) move downward for all regions and age groups and mostly for those age groups mostly affected by pandemics, but the structural break remains apparent. This confirms the influence of pandemics in the change in flu mortality.

A change in vaccination practices could be amongst the most plausible explanations of the structural break [START_REF] Meslé | A spectacular decline in influenza mortality: the role of vaccination[END_REF]. Lacking historical data, we were unable to find reliable figures on the production of influenza vaccines in France from 1950 to 1976. However, the WHO delivered vaccination recommendations in the early 1970s and various public health organizations were also founded at that time [START_REF] Brès | Surveillance epidemiologique des grippes humaines dans le monde: Le programme de lOMS[END_REF]. Figure A6 in Appendix provides tentative estimates of influenza vaccination in France from 1950 to 2015 from unpublished data and they indicate an increase after 1970. We do not have data on vaccination at sub-national level, by age groups or SES over this period of time, to the best of our knowledge. Vaccination rate before 1977 is therefore estimated according to 4 scenarios because we do not have data for this period in France. However, influenza vaccination principles were already well known at that time and the population was aware of vaccination benefits since the Pasteurian revolution.

Figure A7

in Appendix illustrates that the epidemic was well covered by the media (MacCarthy 1969), as exemplified in an interview of Robert Boulin, the Health Minister, who got vaccinated and strongly recommend it to the population. Another article documents the high demand pharmacies faced for the vaccines. In addition, Figure A8 in Appendix shows that flu-specific mortality for senior citizens is highly correlated with vaccination in the recent period, as we observe a decline in vaccination, notably during the 2009 A(H1N1) influenza pandemic [START_REF] Vaux | Influenza vaccination coverage against seasonal and pandemic influenza and their determinants in France: a cross-sectional survey[END_REF][START_REF] Blondel | Failure of the vaccination campaign against A(H1N1) influenza in pregnant women in France: Results from a national survey[END_REF], and an increase in mortality since 2010. Other technological and medical advances, some related to co-morbidities or complications of influenza, may also have played a role. Pneumococcal vaccines, for example, help to reduce influenza mortality [START_REF] Fedson | Pneumococcal Vaccination in the United States and 20 Other Developed Countries, 1981-1996[END_REF]. Nevertheless, in France, whether the public is willing to accept vaccination when it is available remains important [START_REF] Schwarzinger | Low Acceptability of A/H1N1 Pandemic Vaccination in French Adult Population: Did Public Health Policy Fuel Public Dissonance?[END_REF][START_REF] Peretti-Watel | A future vaccination campaign against COVID-19 at risk of vaccine hesitancy and politicisation[END_REF].

Studies on this topic may also include pneumonia deaths. The counterpart of Figure 2 for pneumonia related death is provided in Figures A9a andA9b in Appendix. The patterns are quite different for that of influenza. Interestingly, we observe a hump in the 90's, certainly due to the HIV-AIDS epidemics.

The decline also coincides with the introduction of highly active antiretroviral therapies [START_REF] Mccullers | Do specific virus-bacteria pairings drive clinical outcomes of pneumonia?[END_REF][START_REF] Metersky | Epidemiology, microbiology, and treatment considerations for bacterial pneumonia complicating influenza[END_REF][START_REF] Shrestha | The role of influenza in the epidemiology of pneumonia[END_REF]. Post-influenza bacterial pneumonia is a major cause of morbidity and mortality associated with both seasonal and pandemic influenza virus illnesses.

However, understanding the role that influenza plays on the population-level epidemiology of bacterial pneumonia and, reciprocally, the role of bacterial pneumonia complicating influenza, remains a challenge, the reason why we prefer to keep both causes of deaths separated [START_REF] Feikin | Global strategies to prevent bacterial pneumonia in adults with HIV disease[END_REF][START_REF] Franquet | Pulmonary Infections in People Living with HIV[END_REF][START_REF] Roux | iPneumocystis jirovecii/iPneumonia in Patients with or without AIDS, France[END_REF]. We also replicate Figure 2 for all-cause mortality in Figure A10 in Appendix. There is no break in this period, which reinforces our argument about a specificity of the flu.

A "dry tender" effect may mitigate our story on structural changes in the short-run [START_REF] Rizzi | High excess deaths in Sweden during the first wave of COVID-19: Policy deficiencies or `dry tinder[END_REF]. Indeed, harvesting could partly explain the fall in mortality, and also the decline in the role of income after 1970. The "Dry Tinder" effect has been well documented by epidemiologists to describe a situation where a soft year creates a larger group of susceptible individuals the following year. We recognize that such a selection effect could partly explain the fall in mortality in the short-run.

However, after a few years, the most vulnerable age groups (infants and elderly) are replaced by new cohorts and the effect is less likely to explain the change we observe. Moreover, Figure A6 in Appendix shows that vaccine uptake increases substantially after 1969.

We also use our model over the entire period (rather than year by year) to predict the values at the national level. The difference between the observed values and the predicted values is excess deaths.

We reported this in Figure A11 in Appendix. We can see clearly two periods, before and after the Hong-Kong flu. Years of excess deaths are associated to the main epidemics, and from the 1970s onwards, we observe a permanent under-mortality. 2009 H1N1 pandemic had relatively low mortality effects in

France. However, it is true that the 2009 H1N1 pandemic changed public attitudes towards vaccination in France [START_REF] Lemaitre | Comparative age distribution of influenza morbidity and mortality during seasonal influenza epidemics and the 2009 H1N1 pandemic[END_REF][START_REF] Lemaitre | Mortality Burden of the 2009 A/H1N1 Influenza Pandemic in France: Comparison to Seasonal Influenza and the A/H3N2 Pandemic[END_REF][START_REF] Boëlle | Excess cases of influenza-like illnesses synchronous with coronavirus disease (COVID-19) epidemic, France[END_REF][START_REF] Fuhrman | Severe hospitalised 2009 pandemic influenza A (H1N1) cases in France, 1 July-15[END_REF]. We note that in mainland France, the 2010-2011 season was characterized in the community by an influenza epidemic of moderate intensity, which occurred between late December 2010 and mid-February 2011.

Severe cases admitted to the ICU had epidemiological characteristics and lethality comparable to those observed during the 2009-2010 pandemic and vaccine hesitancy increased in France after the H1N1 pandemic [START_REF] Schwarzinger | Low Acceptability of A/H1N1 Pandemic Vaccination in French Adult Population: Did Public Health Policy Fuel Public Dissonance?[END_REF][START_REF] Ward | Vaccine hesitancy and coercion: all eyes on France[END_REF][START_REF] Blanchon | Surveillance épidémiologique et virologique de la grippe en France, saison 2010-2011[END_REF]).

Lastly, this paper has some limitations. First, Figure 2 shows subnational heterogeneity of levels and trends. When pooling all regions, we obtain a break in 1968, and when we separate our dataset by region we obtain 1968 or 1969 as a break in 18 regions out of 22 (2 for 1970, 1 for 1967, 1 for 1974). Therefore, overall our results seem to be consistent across regions despite different magnitudes in the breaks observed at the regional level. They mostly confirm that the break is at least in 1968 and not before this date except for one region. Disaggregating even more at the department level may of course lead to more heterogeneity. Second, we use crude death rates in this paper. Crude death rates describe how many people die out of a given population over a specific time period but take no account of the population's age distribution when age-standardized measurements of mortality are not available. We use as a control in each model the share of the population aged 65 and over, ages for which influenza mortality is highest. We could control for the share of each five-year age group. Because we stratify the data and present the results without aggregating the stratum-specific information over the strata, we preferred to use crude death rates. Third, our graphical results and regression methods are not fully causal but our approach and our robustness tests concerning the date of the break are suggestive of a causal explanation.

In conclusion, we found the Hong Kong flu to be associated with a major change in influenza mortality in France. Using a unique dataset of 90 geographical units since 1950, we found a decline in mortality beginning in 1970, just after the major pandemics. The change was most notable for senior citizens and the poorest regions. These facts are useful to put the recent Covid-19 crisis in perspective. First, they reveal that health behaviors and policies are impacted by these dramatic events [START_REF] Morse | Pandemic influenza: Studying the lessons of history[END_REF]; we can thus expect profound changes to our health system in the coming years. Second, they suggest that the value of elderly life became more important in the late 1960's, a paradox for a period that is usually perceived as a golden age for youth. Finally, the flu's geographical differences were strongly reduced after the 1970's, with mortality rates that became uncorrelated to regional mean income. Geographical prevalence become a random event, and pandemics may thus hit regions differently, as we have seen in the Covid-19 crisis. The Covid-19 pandemics is thus a good example that seems to indicate that preparedness and former pandemics experience played a key role in the crisis management. However, better preparedness cannot hinder the probability that pandemics occurs, all the more if the disease was not circulating before its sudden emergence. 1) over the entire period rather than year by year and define excess deaths as the difference between estimated and observed deaths caused by flu.

Figure 1 .

 1 Figure 1. Map of flu death rates by departments in 1969 (deaths per 100,000 population).

  Statistique des Causes de Décès published by Institut National des Statistiques et des Etudes Economiques (INSEE) for the period 1950-1978, and from Institut National de la Santé et de la Recherche Médicale (INSERM) for the period 1979-2015. Figures come from the International Classification of Diseases short list. It follows the classification of the 6th revision from 1950 to 1957, the 7th revision from 1958 to 1967, the 8th revision from 1968 to 1978, the 9th revision from 1979 to 1999 and the 10th revision from 2000 to 2015. We retrieved deaths due to influenza as well as total deaths by department and sex for the entire period.

Figure 2

 2 figure provides flu specific mortality at the national level while Figures A1, A2 and A3 in Appendix

Figure 2 .

 2 Figure 2. Flu death rates by region (deaths per 100,000 population).

Figure 3

 3 Figure 3 plots mean flu death rates in France's 90 departments between 1950 and 1968 against the

Figure 3 .

 3 Figure 3. Declines in flu specific mortality across departments (deaths per 100,000 population).

Figure 4

 4 Figure 4 provides a decomposition by age group of flu specific mortality in France. These results

Figure 4 .

 4 Figure 4. Death rates of flu by age groups and sex in France (deaths per 100,000 population).

Figure 5

 5 Figure 5 plots the coefficients estimated in the linear regression characterizing geographical death rates

Figure 5 .

 5 Figure 5. Explanations of differences in department flu death rates.

Figure A2 .

 A2 Figure A2. Death rates of flu by departments Haute-Garonne to Oise (deaths per 100,000 population).

Figure A3 .

 A3 Figure A3. Death rates of flu by departments Orne to Territoire de Belfort (deaths per 100,000 population).

Figure A4 .

 A4 Figure A4. Death rates of flu by region, with and without 1957 and 1968-1969 pandemics (deaths per 100,000 population).

Figure A5 .

 A5 Figure A5. Death rates of flu by age groups, with and without 1957 and 1968-1969 pandemics.

Figure A6 :

 A6 Figure A6: Flu death rate and vaccination in France from 1950 to 2015 (deaths per 100,000 population).

Figure A7 :

 A7 Figure A7: Evidence from newspapers (12/05/1969; 12/14/1969;01/07/1970) -France Soir, Paris Presse, L'intransigeant.

Figure A8 .

 A8 Figure A8. Death rates of flu (in log) for the older age groups in France, 1980 -2015.

Figure A9a .

 A9a Figure A9a. Pneumonia death rates by region (deaths per 100,000 population).

Figure A9b .

 A9b Figure A9b. Pneumonia death rates by age (deaths per 100,000 population).

Figure A10 .

 A10 Figure A10. All cause death rates by region (deaths per 100,000 population).

Figure A11 .

 A11 Figure A11. Excess deaths in France, 1950-2015.

Table 1

 1 presents descriptive statistics for variables used in Equation (1) and specific years.

Table 1 .

 1 Descriptive statistics. We provide mean and standard deviation (in parenthesis) for 90 departmental values. Flu death rates are per 100.000, income per capita are in 2015 euros.

		1950	1960	1970	1980	1990	2000	2010
	Flu death rate	5.25	32.03	18.1	2.41	5.69	3.45	0.24
		(2.98)	(11.96)	(8.67)	(1.67)	(4.84)	(2.17)	(0.27)
	Density	183.21	205.11	233.26	234.86	239.67	245.53	263.25
		(1069.73) (1209.71) (1394.3)	(1342.6) (1334.17) (1349.34) (1452.56)
	Income per	2407.63	3855.42	6720.35	9896.5	11327	13402.45 15488.09
	adult	(615.77)	(788.22) (1153.78) (1328.44) (1415.24) (1487.54) (1465.79)
	Share of 65+	12.46	12.56	13.94	15.6	15.66	18.07	19.03
		(2.07)	(2.09)	(2.53)	(3.15)	(3.2)	(3.44)	(3.24)
	Notes:

Appendix