Epidemic dynamics with homophily, vaccination choices, and pseudoscience attitudes
IIs segregating anti–vaxxers a good idea? ∗ Matteo Bizzarri a , Fabrizio Panebianco b , and Paolo Pin c,da PhD in Economics, Universit`a Bocconi, Milan, Italy b Department of Economics and Finance, Universit`a Cattolica, Milan,Italy c Department of Economics and Statistics, Universit`a di Siena, Italy d IGIER & BIDSA, Universit`a Bocconi, Milan, ItalySeptember 2020
Abstract
Some pro-vaccine policies (e.g. compulsory vaccination in publicschools) may have the effect of separating those that are in favor fromthose that are against vaccines, inducing a segregating effect. We studyan SI-type model, with the possibility of vaccination, where the pop-ulation is partitioned between pro-vaxxers and anti-vaxxers . We showthat, during the outbreak of a disease, segregating people that areagainst vaccination from the rest of the population decreases the speedof recovery, may increase the number of cases, and can make the dis-ease become endemic. Then, we include endogenous choices based onthe tradeoff between the cost of vaccinating and the risk of gettinginfected. We show that the results remain valid under endogenousopinion choices, unless people are very flexible in determining their pro–vaxxers or anti–vaxxers identity. ∗ We thank seminar participants at the University of Antwerp, Universit`a di Bologna,Oxford University, at the 2nd EAYE Conference at Paris School of Economics, BocconiUniversity and Politecnico di Torino. We also would like to thank Daniele Cassese, An-drea Galeotti, Melika Liporace, Matthew Jackson, Alessia Melegaro, Luca Merlino, DuniaLopez–Pintado, Davide Taglialatela and Fernando Vega–Redondo for useful commentsand suggestions. Fabrizio Panebianco and Paolo Pin gratefully acknowledges fundingfrom the Spanish Ministry of Economia y competitividad project ECO2107-87245-R, andItalian Ministry of Education Progetti di Rilevante Interesse Nazionale (PRIN) grants2015592CTH and 2017ELHNNJ, respectively. a r X i v : . [ q - b i o . P E ] S e p EL classification codes: C61
Optimization Techniques, Program-ming Models, Dynamic Analysis –
D62
Externalities –
D85
Network For-mation and Analysis: Theory –
I12
Health Behavior –
I18
GovernmentPolicy, Regulation, Public Health
Keywords:
Seasonal diseases; vaccination; anti–vaccination movements;SI–type model; segregation; endogenous choices.
We model an economy that is facing the possible outbreak of a disease,for which a vaccine with temporary efficacy is available. This mimics whathappens every year for seasonal flu, but it could also be the case in the nearfuture for Covid19. Even before Covid19, vaccination has been almost unanimously consid-ered the most effective public health intervention by the scientific community(see e.g. Larson et al., 2016 or Trentini et al., 2017). However, in recent yearsthere have been many people who either refuse drastically any vaccinationscheme, or reduce or delay the prescribed vaccination. This phenomenon hasbecome more pronounced in the last decades, especially in Western Europeand in the US, and many public health organizations have issued publiccalls to researchers to enhance the understanding of the phenomenon andits remedies. Even in the present times of Covid19 epidemics, the oppositionto vaccination policies is alive. The focus of this paper is on the effects of containment measures thataim at reducing contacts between vaccinated and unvaccinated people, andtheir interaction with vaccination choices of agents and with the dynamics At present, we know that the virus of Covid19 mutates very rapidly (Korber et al.,2020; Pachetti et al., 2020) and that it seems to be seasonal (Carleton and Meng, 2020).Scientists and politicians are considering the possibility that for the next year it couldbecome like a seasonal flu that deserves a new vaccine every year: for example, see thisreport from April 2020. There is also another reason for which vaccination against Covid19may not be permanent: more recent studies like Seow et al. (2020) have shown thatCovid19 antibodies fall rapidly in our body, so that it could be the case that people willneed to vaccinate regularly (e.g. every 2 years) against the virus. See Larson et al. (2016) for a general and recent cross country comparison. Moststudies are based on the US population: Robison et al. (2012), Smith et al. (2011), Nadeauet al. (2015) and Phadke et al. (2016) are some of the more recent ones. Funk (2017)focuses on measles in various European countries. Rey et al. (2018) analyzes the case ofFrance. On this, see the recent reports of Johnson et al. (2020), Ball (2020) and Malik et al.(2020).
2f anti–vaxxer movements. During the Covid19 outbreak, governments haveimplemented very strong and drastic temporary containment and quarantinepolicies. However, such stringent policies cannot be permanent measures,and in normal times the policy makers are able to implement only milderpolicies that may segregate people in certain loci of activity. Typical mildermeasures of this kind, often implemented in the recent years, are the limita-tions for attending schools. In order to protect the public, in many countriesrecent laws forbid enrollment of non vaccinated kids into public schools, andthis is believed to have brought to an increase in enrollment in more tolerantprivate schools. , On an abstract level, this corresponds to a change in the homophily of interactions: the policy has a segregating effect, incentivizingpeople with anti-vaccination beliefs to interact more together.First, we consider a mechanical model in which vaccination choices areexogenous. The policy parameter that the policy maker can tune is the segre-gation between two groups of people: those that are against vaccination andall the others, which we call for simplicity anti-vaxxers and pro-vaxxers (orjust vaxxers), respectively. The two groups differ in the judgement about thereal cost of vaccination, which is deemed higher by anti-vaxxers. This canbe thought of as a psychological cost, a sheer mistake, or any phenomenonthat may lead to a difference in perceived cost: we remain agnostic on thecause of it, our aim is to study its consequences. We think of this differencein perceived cost as a basic cultural trait, which, as the literature on culturaltransmission, affects the preferences (or beliefs) of agents, who are still freeto choose to vaccinate or not, on the basis of a heterogeneous component ofthe cost. This means that an anti-vaxxer in our model may still vaccinate,if the heterogeneous component of the cost is small enough. Through thiswe mean to capture not the extremists that would never take a vaccine,but the much more general phenomeon of vaccine hesitancy , which is muchmore widespread and, so, potentially much more dangerous (Trentini et al.,2017).We model the segregation policy as a parameter h ∈ [0 , This phenomenon is documented for California by Silverman and Yang (2019). Recentnews show that similar trends happened in Italy and have been considered a cause of themeasles outbreak in Manhattan in April 2019. Another application, in light of the policies enforced during the Covid19 crisis, are the rotation schemes that companies and other public and private organizations have appliedand can apply to limit physical interaction between people (on this, see the recent work byEly et al., 2020): it is admissible that a policy maker may want to include unvaccinatedpeople all in the same group. h can be thought as a policy parameter that regardsthose people that are unvaccinated without valid medical reasons, and onwhom the policy maker decides some form of social distancing or segregation.We think at h as a number that is far from one (which would be the case oftotal segregation). This partial segregation policy is implemented against adisease that is seasonal, before the epidemics actually takes place. We showthat the policy may backfire: more segregation may cause the disease to dieout more slowly, and cause more infection in the whole population, or evenmore infection among vaxxers . In particular, our results suggest care bothto a social planner concerned with total infection in the population and to asocial planner concerned only with infection among the vaxxers. The choicebetween the two approaches depends on the attitude toward society we wantto model, and in particular on the specific interpretation of the difference inperceived cost, e.g. as a pure bias that the social planner should consideras such, or as a form of real psychological cost that we may want to factorin the welfare computation. As a consequence of these considerations,we remain agnostic on a general welfare criterion and explore instead thephysical outcome of the amount of infection, which in such an environmentis likely to be a prominent, if not the only, element of any welfare analysis.The reason why an increase in h may generate more infection is thathomophily protects the group with less infected, because it decreases thecontacts and so the diffusion of the disease across groups. Which group hasmore infected will in turn depend on initial conditions and on the differencein vaccinations rates between the two groups. If the total number of agentsinitially infected is the same across the two groups then homophily hasno effect on total infection. Hence, a planner that cares only about theinfection among vaxxers has no clear choice: she will desire an increase of h (e.g., in case of an outbreak among anti-vaxxers, but would have oppositepreferences in case of an outbreak among vaxxers).If, instead, the two groups differ in the number of infected agents, theeffect on total infection depends on the interplay of initial conditions andvaccinations. If the less vaccinated group happens to have more infections(because suffered a larger share of the initial outbreak), we know homophilyfurther increases infections in such group. The crucial observation is that These are complex issues at the forefront of research in behavioral economics, seeBernheim (2009). This can happen, e.g., if the initial seeds are unequally distributed, and initially morevaxxers are infected, see Section 3.
4t increases infections at a disproportionally larger rate than in the case inwhich it is the more vaccinated group that has more infections. As a re-sult, if the outbreak is among anti-vaxxers, total infection in the populationincreases with h , while if the outbreak is among vaxxers it decreases.In the second part of the paper we endogenize the vaccination choicesof people. Vaccination choices are taken before the disease spreads out. Weview this as a classical trade–off between the perceived cost of vaccinatingand the expected cost of getting sick. In the model, the difference betweenanti–vaxxers and pro–vaxxers is only in the perceived costs of vaccination.We show that even if we endogenize these choices, the qualitative predictionsof the mechanical model are still valid: a policy that increases segregationis counterproductive.Finally, we endogenize the choice of people on whether to be anti–vaxxeror pro–vaxxer. This choice is modelled as the result of social pressure, withthe transmission of a cultural trait. There is a well-documented fact aboutvaccine hesitancy that seems hard to reconcile with strategic models: thegeographical and social clustering of vaccine hesitancy. Various studies,reviewed e.g. by Dub´e and MacDonald (2016), find that people are morelikely to have positive attitudes toward vaccination if their family or peershave. This is particularly evident in the case of specific religious confessionsthat hold anti-vaccination prescriptions, and tend to be very correlated withsocial contacts and geographical clustering. These studies, though observa-tional and making no attempt to assess causal mechanisms, present evidenceat odds with the strategic model: if the main reason not to vaccinate is freeriding, people should be less likely to vaccinate if close to many vaccinatedpeople, and not vice versa. In addition, Lieu et al. (2015) show that vaccinehesitant people are more likely to communicate together than with otherpeople. Edge et al. (2019) document that vaccination patterns in a networkof social contacts of physicians in Manchester hospitals are correlated withbeing close in the network. It has been also shown that, in many cases,providing more information does not make vaccine hesitant people changetheir mind (on this, see Nyhan et al., 2013, 2014 and Nyhan and Reifler,2015). However, people do change their mind about vaccination schemes,but they do so under psychological rules that look irrational, as documentedrecently by Brewer et al. (2017), for example. In a review of the literature,Yaqub et al. (2014) finds that lack of knowledge is cited less than distrustin public authorities as a reason to be vaccine-hesitant. This is true bothamong the general public and professionals: in a study of French physicians,Verger et al. (2015) finds only 50% of the interviewed trusted public healthauthorities. They both find a correlation between vaccine hesitancy and the5se or practice of alternative medicine.When we fully endogenize the choices of people (both membership togroups and vaccination choices) we find that the predictions of the simplemechanical model remain valid only if the groups of the society are rigidenough, and people do not change easily their mind about vaccines. Ifinstead people are more prone to move between the anti–vaxxers and pro–vaxxers groups, then segregation policies can have positive effects. Thesimple intuition for this is that when anti–vaxxers are forced to interactmore together, they internalize the higher risk of getting infected, and as aresult they are more prone to become pro–vaxxers.We contribute to three lines of literature, related to three steps of ouranalysis highlighted above: the analysis of the effects of segregation in epi-demiological models, the economics literature on vaccination and its equilib-rium effects, and the literature on diffusion of social norms and transmissionof cultural traits.The medical and biological literature using SI-type models is wide anda review of it is beyond our scopes. We limit ourselves to note that recentlysome papers have considered dynamic processes with superficial similarity toours. Jackson and L´opez-Pintado (2013) and Izquierdo et al. (2018) are thefirst, to our knowledge, to study how homophily affects diffusion. Pananoset al. (2017) analyze critical transitions in the dynamics of a three equationmodel including epidemics and infection.The literature on strategic immunization has analyzed models wheregroups are given, and the immunization choice is taken, as in Galeotti andRogers (2013), or both the immunization and the level of interaction areendogenous, as in Goyal and Vigier (2015). At an abstract level, the differ-ence with respect to our setting is that we endogenize the group partition,through a dynamics of diffusion of social norms. The economics of social norms and transmission of cultural traits is alively field, surveyed by Bisin and Verdier (2011). Common to this litera-ture is the use of simple, often non-strategic, dynamic models of evolutionof preferences. We adopt this framework, finding it useful despite the dif-ferences we discuss later. A paper close to ours is Panebianco and Verdier(2017), that considers how social networks affect cultural transmission in aSI-type model, with a more concrete network specification through degree There is also a recent literature in applied physics that studies models where thediffusion is simultaneous for the disease and for the vaccination choices. On this, see thereview of Wang et al. (2015), and the more recent analysis of Alvarez-Zuzek et al. (2017)and Vel´asquez-Rojas and Vazquez (2017).
We consider a simple SIS model with vaccination and with two groups ofagents, analogous to the setup in Galeotti and Rogers (2013). To understandthe main forces at play, we start by taking all the decisions of agents asexogenous, and we focus on the infection dynamics. In the following sectionswe endogenize the choices of the players.Our society is composed by a continuum of agents of mass 1, which ispartitioned in two groups of agents. To begin with, in this section thispartition is exogenous. Agents in each group are characterized by theirattitude towards vaccination. In details, following a popular terminology,we label the two groups with a , for anti-vaxxers , and with v , for vaxxers .Thus, the set of the two groups is G := { a, v } , with g ∈ G being the genericgroup. Let q a ∈ [0 ,
1] denote the fraction of anti-vaxxers in the society, and q v = 1 − q a the fraction of vaxxers . To ease the notation, when this doesnot create ambiguity, we write q for q a .People in the two groups meet each others with an homophilous bias.We model this by assuming that an agent of any of the two groups has aprobability h to meet someone from her own group, and a probability 1 − h to meet someone else randomly drawn from the whole society. This impliesthat anti-vaxxers meet each others at a rate of ˜ q a := h + (1 − h ) q a , whilevaxxers meet each others at a rate of ˜ q v := h +(1 − h ) q v = h +(1 − h )(1 − q a ).Note that h is the same for both groups, but if q a (cid:54) = q v and h >
0, then˜ q a (cid:54) = ˜ q v .For each g ∈ G , let x g ∈ [0 ,
1] denote the fraction of agents in group g that are vaccinated against our generic disease. It is natural to assume, h is the imbreeding homophily index, as defined in Coleman (1958), Marsden (1987),McPherson et al. (2001) and Currarini et al. (2009). It can be interpreted in several ways,as an outcome of choices or of opportunities. As we assume that h can be affected bypolicies, we can interpret it as the amount of time in which agents are kept segregated bygroup, while in the remaining time that they have they meet uniformly at random. x a < x v , and by now this is actually the onlydifference characterizing the two groups. Let µ be the recovery rate of thedisease, while its infectiveness is normalized to 1. Setting the evolution of the epidemics in continuous time, we study thefraction of infected people in each group. When this does not generateambiguity, we drop time indexes from the variables. For each i ∈ G , let ρ i be the share of infected agents in group i . Since vaccinated agents cannotget infected, we have ρ a ∈ [0 , − x a ] and ρ v ∈ [0 , − x v ], respectively.The differential equations of the system are given by:˙ ρ a = (cid:0) − ρ a − x a (cid:1)(cid:16) ˜ q a ρ a + (1 − ˜ q a ) ρ v (cid:17) − ρ a µ ;˙ ρ v = (cid:0) − ρ v − x v (cid:1)(cid:16) ˜ q v ρ v + (1 − ˜ q v ) ρ a (cid:17) − ρ v µ. (1)For each g ∈ G , (cid:0) − ρ g − x g (cid:1) ∈ [0 ,
1] represents the set of agents who areneither vaccinated, nor infected, and thus susceptible of being infected byother infected agents. Moreover, the share of infected agents met by vaxxersand anti-vaxxers is given by (cid:16) ˜ q a ρ a + (1 − ˜ q a ) ρ v (cid:17) and by (cid:16) ˜ q v ρ v + (1 − ˜ q v ) ρ a (cid:17) ,respectively. Finally, ρ a µ and ρ v µ are the recovered agents in each group. Result 1 (Homophily and endemic disease) . The system (1) always admitsa trivial steady state: ( ρ a , ρ v ) := (0 , h , there exists a ˆ µ ( h ) > µ < ˆ µ ( h ), (0 ,
0) is unstable, whereas (ii) if µ > ˆ µ ( h ), (0 ,
0) isstable. This result is obtained in the standard way, by setting to zero the tworight–hand side parts of the system in (1) and solving for ρ a and ρ v . Theformal passages are in Appendix D, as those of the other results that follow.In the remaining of the paper, we focus on the case in which µ > ˆ µ ( h ),because it is consistent with diseases that are not endemic but show them-selves in episodic or seasonal waves. For those diseases, the society laysfor most of its time in a steady state where no one is infected. However,exogenous shocks increase temporarily the number of infected people untileventually the disease dies out, as it happens, for example, for the seasonaloutbreaks of flu. Note that ˆ µ ( h ) := ( T + ∆) ∈ [0 , T := ˜ q a (1 − x a ) + ˜ q v (1 − x v ) and∆ := (cid:112) T − h (1 − x a )(1 − x v ). ∆ is always positive and it is increasing in q . Moreoverˆ µ ( h ) ∈ [0 ,
1] and its value is 1 − x v + q ( x v − x a ) for h = 0 and 1 − x a for h → µ ( h ) is increasing in h , so we can highlight a first importantrole for h in the comparative statics. If h increases, it is possible that adisease that was not endemic, because µ > ˆ µ ( h ), becomes so because ˆ µ ( h )increases with h , and the sign of the inequality is reversed. Indeed, a higherhomophily counterbalances the negative effect that the recovery rate µ hason the epidemic outbreak. The main focus of our interest is to see what is the welfare loss due to theepidemics, and how this depends on the policy parameter h . In our simplesetting, the welfare loss is measured by the total number of infected peopleover time, that is cumulative infection . For analytical tractability, wewill approximate the dynamics of outbreaks with the linearized version ofthe dynamics ˆ ρ , that satisfies:˙ˆ ρ t = J (cid:18) ˆ ρ at ˆ ρ vt (cid:19) , ˆ ρ = (cid:18) ρ a ρ v (cid:19) (2)where J is the Jacobian matrix of (1) calculated in the (0 ,
0) steady state,and ( ρ a , ρ v ) (cid:48) is the initial magnitude of the outbreak. We can think of it asthe number of infected agents at the beginning of a particular outbreak, oras the expected value of the initial number of infected agents according tosome probability distribution.The cumulative infection in the two groups and in the overall populationis: CI a := (cid:90) ∞ ˆ ρ a ( t ) dtCI v := (cid:90) ∞ ˆ ρ v ( t ) dtCI := q a CI a + (1 − q a ) CI v (3)Note that, since q a is fixed, CI takes into account both the number ofinfected agents of each group at each period, and also the length of theoutbreak. In the range of parameters for which (0 ,
0) is stable, all theintegrals are finite, so here we do not add discounting, for simplicity. We willexplore the implications of the introduction of time preferences in Section2.3 below. The expressions can be found in Lemma 1 in the Appendix D.To understand the mechanics that regulates the share of agents that getinfected during the outbreak, let us consider three different types of initialconditions: The epidemics starts (i) among vaxxers ( ρ v > ρ a = 0),9ii) among anti-vaxxers ( ρ v = 0 and ρ a > ρ v = ρ a > Result 2 (Who is better off?) . The cumulative number of infected agentsis such that CI a ≥ CI v if and only if: ρ v (1 − x a ) − ρ a (1 − x v ) + µ ( ρ a − ρ v ) ≥ CI a and CI v (we derive it in Lemma 1 in the Appendix D). Inequality(4) underlines the roles of the parameters in determining the welfare of thegroups. The left–hand side is increasing in x v and decreasing in x a : the gapin vaccinations tends to penalize the less vaccinated group. Since the cu-mulative infection is an intertemporal measure, the initial conditions concuralso in determining which group is better off: the difference is increasing in ρ a and decreasing in ρ v . µ regulates the importance of this effect in thediscrepancy of initial conditions: the larger µ , the shorter the epidemics, thelarger the importance of the initial conditions. In particular we have:i) if the outbreak starts among vaxxers, vaxxers have larger cumulativeinfection;ii) if the outbreak starts among anti–vaxxers, anti–vaxxers have larger cu-mulative infection;iii) if the outbreak starts symmetrically in both groups, the group withless vaccinated (anti–vaxxers, under our assumptions) has the largestcumulative infection.In particular, the evaluation of what group is better off in terms ofinfections is independent of homophily. However, the levels of contagion dodepend on homophily, as the following result shows. It is obtained applyingdefinitions from (3) and taking derivatives. Result 3 (Effect of h and q a ) . a) CI and CI a are increasing (decreasing) in h if and only if CI a > CI v ( CI a < CI v ); CI v is decreasing (increasing) in h if and only if CI a >CI v ; Because the stability assumptions imply − x v + µ > − x a + µ > CI , CI a and CI v are increasing (decreasing) in q if and only if CI a > CI v ( CI a < CI v ).In particular, the marginal effects of h and q a for different outbreak typesare those reported in Table 1. If the outbreak is among · · · vaxxers anti–vaxxers both, symmetricallythe effect ∂CI a ∂h < ∂CI v ∂h > ∂CI a ∂h > ∂CI v ∂h < ∂CI a ∂h > ∂CI v ∂h < h is: ∂CI∂h < ∂CI∂h > ∂CI∂h > ∂CI a ∂q a < ∂CI v ∂q a < ∂CI a ∂q a > ∂CI v ∂q a > ∂CI a ∂q a > ∂CI v ∂q a > q a is: ∂CI∂q a < ∂CI∂q a > ∂CI∂q a > h and q a on CI a , CI v , and CI , when there isan outbreak among vaxxers, anti–vaxxers, or symmetrically in both groups.The previous results show how initial conditions and parameters con-tribute to determine what is the effect of an increase in h . As anticipatedin the introduction, if the initial parameters are such that CI a = CI v , thenboth the total infection, CI , and the group level ones, CI a and CI v , donot depend on homophily. If instead the initial parameters are such that CI a (cid:54) = CI v , then homophily hurts the group with more infected, because itcauses the infection to spread to more members of the group and less out-side. Table 1 helps us understand the behavior in prototypical cases, andanalyze whether a policy that increases h has the desired effect.To better understand the mechanics, let us first focus on the effects ofhomophily (first row of Table 1). First note that, if the outbreak happensjust in one of the two groups, homophily protects the group that is notinfected ex ante. So, intuitively, the outbreak has the strongest effect interms of infected agents in the group in which the outbreak has taken place.The effect of homophily on the overall CI is however ambiguous and dependson the initial condition.Consider first the case in which the outbreak takes place among vaxxers.Then, at the beginning, the infection takes over among the group with thehighest vaccination rate, since x v > x a . The higher the homophily h , themore vaxxers interact among each others, and thus the more the infectionremains within the group that is more protected against it. For this reason,the higher h , the less the CI . For the opposite reason, if the outbreaks takesplace in the anti-vaxxers group, homophily makes infection stay more in theless protected group, and CI increases.11o the crucial message is that a policy having the effect of increasing h cannot be considered unanimously beneficial neither from a planner con-cerned with total infection, nor from a planner concerned with just infectionamong vaxxers.To understand the role of q a on the CI (second row of Table 1), recallthat a higher q a means a higher share of agents less protected against thedisease. Consider first the case in which the outbreak takes place in thevaxxers group. Then, a higher q a means that the number of infected agents,which are in the v group, is lower. Thus, all CI measures are decreasing in q a . For the opposite reasoning, all CI measures are increasing in q a if theoutbreak takes place in the anti-vaxxers group. If the outbreak is symmetric,then the two forces mix. However, if q a increases, the share of agents who arenot protected against the disease increases, and thus CI measures increase. In this section we explore the implications of the degree of impatience of theplanner on the evaluation of the impact of homophily. Time preferences canbe crucial for the planner. As we have seen, for example, in the Covid19epidemics, the planner, given a CI, may prefer not to have all infected agentssoon because of some capacity constraints of the health system.For example, Figure 1 shows the time evolution of the infection of bothgroups and overall society’s in case of an outbreak among the vaxxers. Inthis case, since the outbreak starts among the vaxxers, it is among this groupthat infection is higher initially, while eventually infection becomes largeramong the anti–vaxxers, due to the lower vaccination levels. The effectson cumulative infection depends on how the planner trades off today andtomorrow infections: the more the planner is patient, the more the infectionamong anti–vaxxers becomes prominent.Moreover, since in our setting the impact of segregation polices dependson the relative amount of infected agents in the two groups, as specified inResult 3, in our context the time preference is also crucial for the evaluationof the impact of homophily on total cumulative infection.Thus, we first define the discounted cumulative infection: CI a := (cid:90) ∞ e − βt ρ a ( t ) dtCI v := (cid:90) ∞ e − βt ρ v ( t ) dtCI := q a CI a + (1 − q a ) CI v (5)12 ntivaxvaxtot ρ Figure 1: CI as a function of time in case the outbreak starts among vaxxers( ρ a = 0). Here ρ v = 0 . x a = 0 . x v = 0 . q = 0 . h = 0 . µ = 1.where β > Result 4.
Discounted cumulative infections are equivalent to cumulativeinfections in a model with recovery rate µ (cid:48) = µ + β .This is not too surprising: µ is a measure of how fast the epidemics diesout, and β is a measure of how fast the welfare loss dies out. The previousresult carries on even when, as we do in the following sections, choices onvaccination and on types are made endogenous.The impact can be made more precise if we stick to exogenous choices,as it is done below. Result 5.
In the model with discounting: CI a ≥ CI v if and only if − ρ a (1 − x v ) + ρ v (1 − x a ) + ( µ + β )( ρ a − ρ v ) ≥ β makes initial conditionsmore important in the welfare evaluation. For example, without timepreferences, we may have that ρ a < ρ v but CI a > CI v , because thedifference in vaccinated agents dominates the difference in initial out-break. However, if time preferences are introduced, or β gets larger,a planner may evaluate that CI a < CI v because she is putting moreweight on the earlier moments of the epidemics.13. An increase in the degree of impatience β can change the impact ofhomophily, as illustratd in Figure 2. To understand this point, givena population share q , there exists a β such that homophily does notimpact the CI (with time preferences). In this CI, groups get infectedat different rates along time. As we change β , the planner gives moreweight on the group getting infected earlier. As we have seen above,homophily plays a role in this process, keeping the infection more intoeach group. In Figure 2, we consider the case in which q = .
3, so thatthere are more vaxxers than anti-vaxxers, and vaxxers are also morevaccinated. Thus, the more the planner is impatient, the more sheis satisfied by the fact that the majority of agents (vaxxers) are lessinfected when homophily increases.Figure 2: Cumulative infection as a function of homophily for different valuesof time preference. Here µ = 0 . x a = 0 . x v = 0 . q = 0 . In this section we show that homophily slows down the diffusion dynamics, ashas already been studied in a different context by Golub and Jackson (2012).We consider as a measure of convergence time the magnitude of the leadingeigenvalue, which in this case is the one with the smallest absolute value.This is because the solution of our linear system is a linear combination ofexponential terms whose coefficients are the eigenvalues (which are negativeby stability). Hence, when t is large, the dominant term is the one containing14he eigenvalue which has smallest absolute value. Result 6.
Consider a perturbation around the stable steady state (0 , ,
0) is increasing in h .This result shows that homophily, by making the society more segre-gated, makes the convergence to the zero infection benchmark slower once anoutbreak occurs. This is obtained analyzing the eigenvalues of the Jacobianmatrix, computed in the steady state. All results are obtained analytically(see Appendix D), and the resulting eigenvalues are decreasing in absolutevalue in h .If we look at the effects of other parameters, we have that the eigenvaluesare increasing in absolute value in both x a and x v . This is because a largernumber of vaccinated agents means a smaller space for infection to diffuse.Finally, since x a < x v , then the smallest eigenvalue is decreasing (in absolutevalue) in q a , while the largest eigenvalue is increasing. Since the long rundynamics (i.e. asymptotic convergence) depends on the smallest eigenvalue,this means that the dynamics is asymptotically slower the larger the fractionof the population with less vaccinated agents.Results 1, 3 and 6 in this section provide clear implications that shouldbe taken into account when considering policies that affect the level of ho-mophily h in the society. Any increase in segregation between vaxxers andanti–vaxxers may induce the disease to become endemic. Additionally, alarger h , if there is a temporary outbreak, will slow down the recovery timeand in some cases (i.e. when the outbreak does not start only among vaxxers)it may increase the cumulative infection caused by the disease.When applied to the real world, the results of this section can be seenas first order effects because, in general, a policy that changes h may haveeffects also on x a , x v and q a . Indeed, in the following sections, we endogenizethe shares of vaccinated agents, and the shares of vaxxers and anti–vaxxers We should be careful, though, because this is true non–generically outside of the eigendirection of the second eigenvector. Indeed, in our case the eigenvectors are: e = (cid:18) − (1 − x v ) ˜ q a + ( x a −
1) ˜ q a + ∆2 (1 − x v ) (1 − ˜ q a ) , (cid:19) and e = (cid:18) − (1 − x v ) ˜ q a − ( x a −
1) ˜ q a + ∆2 (1 − x v ) (1 − ˜ q a ) , (cid:19) . So, we can see that the first eigendirection does not intersect the first quadrant, while thesecond does. Hence, we should remember that the first eigenvalue is a measure of speedof convergence only generically, outside of the eigendirection identified above.
15n the society. The outcomes then depend also on the second order effects:the interaction of the impact of homophily on the endogenous variables.
In this section we start introducing elements of endogeneity. First of all weconsider vaccination choices, to make the shares x a and x v endogenous. Tobegin with, we still consider the partition of our society in anti-vaxxers andvaxxers, q a and q v , as exogenously fixed (we will relax this assumption inthe next section). In our approach, people take vaccination decision ex–ante, before an epidemics actually takes place, and cannot update theirdecision during the diffusion of the epidemics. This mimics well diseases,like seasonal flu, for which the vaccine takes a few days before it is effective,and the disease spreads rapidly among the population.We model the behavior of agents who consider the trade-off between pay-ing some fixed cost for vaccinating, or incurring the risk of getting infected,and thus paying with some probability a cost associated to health. We needto set some assumptions about vaccination costs and agents’ perception ofthe risk of being infected. Now that x a and x v are endogenous, it is the dif-ference in the perception of costs that characterizes the difference betweenthe two groups. Vaccination costs
Vaxxers and anti-vaxxers have different perceptionsabout vaccination costs. For vaxxers we assume that vaccination costs are c v ∼ U [0 , c a ∼ U [ d, d ], with d >
0. This is to saythat anti-vaxxers perceive a higher cost of getting a vaccine than vaxxers do.
Risk of infection
We assume that agents think about the risk of infec-tion as proportional to the fraction of unvaccinated people that they meet.This is reasonable, because they form a belief before the actual outbreakoccurs (e.g., agents decide to vaccinate against flu a few months before win-ter). Agents multiply this fraction of unvaccinated people by a factor k > σ v be the share of unvaccinated people met by vaxxers, See, for example, Bricker and Justice (2019) and Greenberg et al. (2019) for a recentanalysis of the anti–vaxxers arguments: Those are mostly based on conspiracy theoriesthat attribute hidden costs to the vaccination practice, and not so much on minimizingthe effects of getting infected. Our model would not change dramatically if we attributethe difference in perception on the costs of becoming sick (see equations (8) and (9)), butwe stick to the first interpretation because it makes the computations cleaner. σ v = ˜ q v (1 − x v ) + (1 − ˜ q v )(1 − x a ) , (6)so that vaxxers perceive the risk of infection to be kσ v . Similarly σ a = ˜ q a (1 − x a ) + (1 − ˜ q a )(1 − x v ) , (7)so that anti–vaxxers perceive the risk of infection to be kσ a . Now, onlythose for which costs are lower than perceived risk decide to vaccinate. So: x ∗ v = min { kσ v , } , (8)whereas, for an anti–vaxxer, we have x ∗ a = max { , min { kσ a − d, }} . (9)The two solutions are both interior, for any value of the other parameters,whenever d < min (cid:110) k , kk +1 (cid:111) , and we call this the interiority condition ,which is analyzed in depth in Appendix A. We use interiority conditionas a maintaned assumption for the remainder of the paper. In this case,equations (6)–(9) imply a system that provides x a = 1 − dq a k − d (1 − q a )1 + hkx v = 1 − dq a k + dq a hk . (10)In what follows we focus on the interior solutions. In Appendix B weanalyze numerically the case in which x v = 0, and we see that the qualitativeresults are analogous to those that we present here.First of all, we note that (i) x v > x a - since vaxxers perceive a lowervaccination costs than anti-vaxxers; (ii) x a is increasing in h whereas x v isdecreasing in h - since a higher homophily makes vaxxers more in contactwith agents who are less susceptible than anti-vaxxers and, as a consequence,( x v − x a ) is decreasing in h ; (iii) x a and x v are increasing in q a - since thehigher the share of anti-vaxxers, the more agents are in touch with othersubjects at risk of infection; (iv) the total number of vaccinated people is q a x a + (1 − q a ) x v = k − dq a k , it is independent of h , but decreasing in q a -this is due to a Simpson paradoxical effect: both groups vaccinate more, butsince anti-vaxxers increase, in the aggregate vaccination decreases.We can examine also cumulative infections, in a neighborhood of the sta-ble steady state (0 , a = ρ v , that can be compared with Result 3, looking at the third column ofTable 1. Also, analytical tractability is obtained only for values of h that are small , as would be the effect of a policy that limits contacts between vaxxersand anti-vaxxers only in few of the daily activities (e.g. only in schools). Proposition 1.
Consider a perturbation around (0 , , such that ρ a = ρ v > . Then, there exists ¯ h > such that, if h < ¯ h :(i) ∂x a /∂h = − ∂x v /∂h ;(ii) CI is increasing in h ;(iii) CI is increasing in q a (but the marginal effect is lower than in theexogenous case of Result 3). This proposition analyzes what happens for an outbreak that is sym-metric in the two groups, when homophily is low enough. We have alreadyseen above that the effect of homophily is opposite for x a and x v . Here wefind that the magnitude of the effects is the same for both groups, and thatthe higher the homophily, the higher CI . Thus, homophily policy does notseem to be a good policy to be implemented in these cases. At the sametime, the more the anti-vaxxers, the more the number of infected agents.To complete our analysis of endogenous choices, in the next section weendogenize also the partition between the two groups. In the previous section we have illustrated the tradeoff faced by agents be-tween two different costs: the act of vaccinating, and the risk of being in-fected, which is in turn based on the fraction of unvaccinated agents theyexpect to meet, σ v and σ a . We now consider how the shares of vaxxers andanti-vaxxers change with time, that is how q is determined. In the real worldthis decision does not seem to be updated frequently, and can be consideredas fixed during a single flu season. So, in the model, we assume that thisdecision is taken before actual vaccination choices, which are in turn takenbefore the epidemics eventually starts. Our aim here is to offer a simple andflexible theory of the diffusion of opinions to be integrated in our main epi-demics model. As explained in the Introduction, the empirical observationsthat important drivers of vaccination opinions are peer effects and culturaldimensions in general leads us to discard purely rational models, where thedecision of not vaccinating descends from strategic considerations. Given18he complex pattern of psychological effects at play we opt for a simple re-duced form model capturing the main trade-offs. In particular, we are goingto assume the diffusion of traits in the population to be driven by expectedadvantages : the payoff advantage that individuals in each group estimate tohave with respect to individuals in the other group. This is made precise bythe next definition. Definition 1 (Expected advantage) . Consider an individual in group a .Define ∆ U a as the Expected advantage individual a estimates to have withrespect to individuals in group v . Specifically:∆ U a = U a → a − U a → v U a → a = − E c [( c + d ) kσ a
1] interval theydo not vaccinate and incur risk of infection. Note, however, that this is theevaluation from the perspective of agents in group a , and thus the cost ofvaccination is c + d instead of c . Hence U a → v is the area below the red19urve. The difference ∆ U a is given by the red area minus the blue area. U v → v and U v → v are computed accordingly. The details of the calculationand the corresponding figure are in Appendices C.1 and C.2.cost c disutility dkσ a kσ a − d kσ v kσ v Disutility of v as perceived by a Disutility of a as perceived by a Figure 3: Composition of ∆ U a . The graph represents the disutility incurredby an individual as a function of its cost c . ∆ U a is the red area minus theblue area.To ease notation, let q = q a . Then, we make the following assumptionover the population dynamics Assumption 1.
Given an α ∈ R , the level of q increases when q α ∆ U a > (1 − q ) α ∆ U v and it decreases when q α ∆ U a < (1 − q ) α ∆ U v .Clearly, the implication of the previous assumption is that the restingpoints of the dynamics are such that q α ∆ U a = (1 − q ) α ∆ U v , but stability hasto be addressed. The simplest example of dynamics satisfying Assumption1 is: ˙ q = q (1 − q )[ q α ∆ U a − (1 − q ) α ∆ U v ] , but we allow also for any non linear generalization.The dynamics obtained from Assumption 1 generalizes the standardworkhorse model in cultural transmission, the one by Bisin and Verdier(2001), in two ways: ( i ) endogenizing the socialization payoffs, as from20efinition 1, and ( ii ) introducing a parameter α regulating the stickiness agents have in changing their identity via social learning. Indeed, at thelimit α → ∞ , ˙ q = 0 and types are fixed. Note also that α regulates thestrength of cultural substitution , a phenomenon often observed in culturaltransmission settings: the tendency of members of minorities to preservetheir culture by exerting larger effort to spread their trait. Thus, we areable to encompass different types of social dynamics. ( i ) If α = 0 this isa standard replicator dynamics (see e.g. Weibull, 1997). ( ii ) If α <
0, themodel displays cultural substitution, as most standard cultural transmissionmodels. Moreover, the more α is negative, the more there is substitution. Inparticular, if α = − ( iii ) If α >
0, themodel displays cultural complementarity, so that the smaller the minority,the less the effort exerted, and the less the minority survives. Note thatcultural complementarity is increasing in α .The environment of social influence is not only shaped by physical con-tacts and it is not the same of the epidemic diffusion of the actual disease(because in the real world many contacts are online and are channeled bysocial media). Hence, any policy on h can have a limited effect on it, be-cause for us h is a restriction on the physical meeting opportunities. As aconsequence, h does not appear explicitly in Assumption 1.If ∆ U a = 0, naturally there will be no anti–vaxxers. This will happen iffor example the bias d is very high, or homophily is very high, so that theincreased infection risk from being an anti-vaxxer (the blue area in Figure3) is so large that no one wants to be an anti-vaxxer. This is of course anuninteresting case, so from now on we are going to assume the following: Assumption 2 ( Interiority conditions ) . x a , x v , and ∆ U a are interior(details in terms of exogenous parameters are in the Appendix A).The following result shows that only the case α < q ∗ = 0 or q ∗ = 1. Proposition 2.
Under the interiority conditions: ( i ) If α ≥ there are nointerior stable steady states of the dynamics for q . ( ii ) If α < , there existsa q ∗ ∈ (0 , such that q ∗ α ∆ U a = (1 − q ∗ ) α ∆ U v . Moreover, there is always See Bisin and Verdier (2011). To be precise, the model by Bisin and Verdier (2001) refers to intergenerational trans-mission. In the Appendix we show how a similar equation can be recovered in a contextof intragenerational cultural transmission =- α =- α =- Figure 4: q as a function of h . d = 0 . k = 1, µ = 1. The range of h isrestricted as prescribed by the interiority conditions A an interior stable steady state and there exists a threshold h such that, for h < h , the steady state is unique and stable. Again, the proof of this result is obtained with standard methods, ap-plying the implicit function theorem to the condition from Assumption 1,looking at results for h →
0, and using the continuity of the system to provethat results hold in an interval [0 , ¯ h ) for h . The equilibrium level of q can becomputed analytically only in the case α = − h = 0, sincethe differences in payoffs across groups become null, and q ∗ = .We are now interested in the effect of an increase in homophily on q ∗ .Figure 4 shows, on the basis of numerical examples with α = − , α = − α = −
3, that homophily has a negative effect on q ∗ and that this resultseems to extend to any α < small values of h . Proposition 3.
Under the interiority conditions, and if α < there existsa threshold h such that, for h < h , q ∗ is decreasing in h in the unique andstable steady state. The intuition is that a larger h magnifies the negative effects of beinganti-vaxxers in terms of infection, relatively to vaxxers. This is internalizedin the cultural dynamics, via the ∆ U s. This long run effect of h on anti-vaxxers share is one of the few positive effects of segregating policies.As we have done in the preliminary model with exogenous choices, we cananalyze the effects of homophily on the cumulative infection, when the initial22erturbation is symmetric across both groups (see Result 3, summarized inthe third column of Table 1). We find that the effects depend on the mag-nitude of α , the parameter regulating how agents are rigid/prone towardssocial influence. Proposition 4.
Consider the model with endogenous q , α < and interi-ority conditions. Consider an outbreak affecting both groups symmetrically,starting from the unique stable steady state and h = 0 . Then, there exists athreshold α such that: • if α < α , CI is increasing in h (cid:0) d CI d h (cid:12)(cid:12) h =0 > (cid:1) ; • if α > α , CI is decreasing in h (cid:0) d CI d h (cid:12)(cid:12) h =0 < (cid:1) . With this proposition we consider the effects of the introduction of someform of segregation policy, taking also into account the cultural dynamics.This means that if α is large in magnitude (the first bullet point, since α is negative) then the society is rigid in its opinions, and the effects arequalitatively the same that we would have if types and vaccination choiceswere fixed (Result 3). If instead α is small in magnitude (the second bulletpoint), then the reaction of q ∗ to a policy change of h is large, and this revertsthe effect: cumulative infection is decreasing in homophily. In this last case,the effects of a policy based on partial segregation will be the desired ones.In this respect, how agents are subjected to social influence can revert theeffects of a policy based on homophily. Figure 5 shows this effect for twovalues of α <
0. These are also compared with what would happen, withthe same parameters, under the assumptions of Result 3 (all choices areexogenous) and Proposition 1 (only vaccination choices are endogenous, butgroups are fixed). The figure shows that, only when α is negative and smallin absolute value, the cumulative infections decreases in homophily. In allthe other cases a policy based on homophily can increase the cumulativeinfection at various degrees.The intuition for the different marginal effects of h on cumulative in-fection seems to lie on the marginal effects on the speed of the dynamics,via the first eigenvalue (see Result 6), as Figure 6 illustrates: the cases inwhich cumulative infection increases with h are those in which the leadingeigenvalue is decreasing in magnitude, and vice versa.Note that Result 4 about the discount rates of a policy maker are stillvalid with endogenous choices, as we discussed in Section 2.3.23igure 5: Cumulative infection in the three models. Whenever exogenous, q , x a and x v are set using the median value of h = 0 .
1. The other parametersare set at k = 2, d = 0 . µ = 1, ρ a = ρ v = 0 . The problem of vaccine skepticism is a complex one, that requires analy-sis from multiple angles: psychological, medical, social. In this paper wepropose an analysis of the trade-offs faced by a policy maker interested inminimizing infection in a world with vaxxers and anti–vaxxers, having avail-able a policy inducing some degree of segregation, or homophily, h . The keyobservation is that reducing contact with anti–vaxxers may be counterpro-ductive for vaxxers and for the society as a whole, because it slows downthe dynamics of the disease to its steady state, if there is an outbreak. Ho-mophily may actually increase the duration of the outbreaks, and dependingon the impatience of the planner this might crucially change the impact ofthe policy. Further, if cultural types are endogenous, the intensity of cul-tural substitution is key in determining the impact of the policy. Our resultssuggest that the study of policy responses to the spread of vaccine hesitantsentiment would benefit from trying to pin down more precisely the intensityof these mechanisms. 24 =- α =- α =- α =- α =- α =- - - - - - Figure 6:
Left panel : Cumulative infection as function of homophily inthe interior equilibrium. The other parameters are set at k = 2, d = 0 . µ = 1, ρ a = ρ v = 0 . Right panel : corresponding leading eigenvalue ofdynamical system as a function of h .25 ppendicesA Interiority conditions In Section 3 we have included endogenous choices about vaccinations, usingtwo parameters, d and k , for the beliefs about the expected costs of vaccina-tion and of becoming infected. We assume interiority conditions , which areessentially the conditions for which ∆ U a > V a is positive is hk ( hk +1) k +1 < d . For this to be compatiblewith x a and x v being interior, we need d < min { k , k (1+ k ) } , hence we needalso 2 h (1 + hk ) < k . So in addition to k high enough we also need h smallenough. Figure 7 shows the regions in the h – d plane for which the interiorityconditions are satisfied, depending on the value of k . k = = = Figure 7: Region of parameters where all endogenous variables are interior. µ is fixed to 1. B Analysis of corner solution x a = 0 In this section we explore the case in which the interiority conditions arenot satisfied, and d is large enough so that the unique equilibrium in the26accination game is: x a = 0 ,x v = k ( h − kq + k + 1 , (14)(15)provided d < /k so that x v (cid:54) = 0. Indeed, if both x a = 0 and x v = 0 , theneverything is exogenous and we are back to the mechanical model.Note that x v maintains the properties we expect: it is decreasing in h and increasing in q , as can be directly seen from the expression above. Cumulative infection
Consider a symmetric initial conditions. Since x v is decreasing in h , it meansthat now when h increases we have the direct effect on CI which is increasing,plus a decrease in vaccination, which further increases the effect on CI. Theeffect of x a , which is of the opposite sign, disappears in the computations, sowe have exactly the same behavior as in the interior solution, and, moreover,in this case the negative effect of h on CI is even stronger. The derivativesare: ∂CI∂h = − k ( q − qρ v µ ( k ( h ( µ − q + µ − µq ) + µ − ∂CI∂q a = kρ v (( h − kq + k + 1)2( k ( h ( µ − q + µ − µq ) + µ − ∂CI∂x a = − qρ v (( h − kq + k + 1) k ( h ( µ − q + µ − µq ) + µ − ∂CI∂x v = ( q − ρ v (( h − kq + k + 1) k ( h ( µ − q + µ − µq ) + µ − (16)(17)(18)(19) This is possible if hk + khkq − kq + k +1 < d < k and either k < (cid:16) < k < (cid:0) √ (cid:1) ∧ < q < − k + k +1 k ∧ < h < − k − kq + k +1 k − kq (cid:17) ndogenous cultural types The socialization payoffs are:∆ U a = − kσ a + dkσ v + 12 k σ v + kσ v (1 − kσ v )= k (2( d − hk )(( h − kq + k + 1) − k )2(( h − kq + k + 1) ∆ U v = kσ a − k σ v − kσ v (1 − kσ v )= k (2 h (( h − kq + k + 1) + 1)2(( h − kq + k + 1) (20)(21)(22)(23)We can apply the intermediate value theorem provided ∆ U a does not be-come negative. ∆ U a ( q = 0) = − k (2( k +1)( hk − d )+ k )2( k +1) . If h < , this is positiveunder the condition hk + khkq − kq + k +1 < d . So for h small enough we get aninterior solution for α < q endogenous.Nevertheless, numerical simulations reveal a picture very similar to the onedescribed in the case of interior solution, in the main text. Specifically, themagnitude of α is crucial to determine the effect of an increase in homophily,as illustrated in Figure 8. Again, the main mechanism through which ho-mophily acts is via the increased length of the outbreak, as measured by theleading eigenvalue, as shown in Figure 9. α =- α =- α =- Figure 8: Cumulative infection as function of homophily in the equilibriumwhere x a = 0. The other parameters are set at k = 1, d = 0 . µ = 0 . ρ a = ρ v = 0 .
1. 28 =- α =- α =- - - - - - Figure 9: Eigenvalues in the equilibrium where x a = 0. The other parame-ters are set at k = 1, d = 0 . µ = 0 . ρ a = ρ v = 0 . Endogenous groups
C.1 A simple model of intragenerational cultural transmis-sion
In this section we illustrate how equation (1) with α = − influencer and the target .The incentive for the influencer is based only on other–regarding preferences,for two reasons: it is consistent with some survey evidence (K¨umpel et al.2015, Walsh et al. 2004), and in this economy every agent has negligibleimpact on the spread of the disease, so socialization effort cannot be drivenby the desire to minimize the probability of infection, or similar motivations.The timing of the model is as follows. • Before the matching, agents choose a proselitism effort level τ at , τ vt ; • When 2 agents meet, if they share the same cultural trait nothinghappens. Otherwise, one is selected at random with probability toexert the effort and try to have the other change cultural trait.The fraction of cultural types evolves according to: q at +1 = q at P aat + (1 − q at ) P vat , (24)where the transition rate P aat is the probabilities that an agent a is matchedwith another agent who, next period, results to be of type a and P vat isthe probabilities that an agent v is matched with another agent who, nextperiod, results to be of type a . These probabilities are determined by effortsaccording to the following rules: P aat = ˜ q at + (1 − ˜ q at ) 12 + (1 − ˜ q at ) 12 (1 − τ vt ) ,P vat = 12 (1 − ˜ q at ) τ vt , (25)(26)( P vvt and P avt are defined similarly) which yield the following discretetime dynamics: ∆ q at = q at (1 − q at )(1 − h )∆ τ t , (27)where ∆ τ t := τ at − τ vt . 30ffort has a psychological cost, which, as in Bisin and Verdier (2001),we assume quadratic. Hence, agents at the beginning of each period (beforethe matching happens) solve the following problem:max τ at − ( τ at ) (cid:124) (cid:123)(cid:122) (cid:125) cost of effort + q at U a → at + (1 − q at ) 12 ( τ at U a → at + (1 − τ at ) U a → vt ) (cid:124) (cid:123)(cid:122) (cid:125) expected social payoff , (28)which yields as a solution: τ at = (1 − q at ) ( U a → at − U a → vt ) (cid:124) (cid:123)(cid:122) (cid:125) ”cultural intolerance” ,τ vt = (1 − q vt )( U v → vt − U v → at ) . (29)(30)Hence, the dynamics implied by our assumptions is:∆ q at = q at (1 − q at )((1 − q at )∆ U a − q vt ∆ U v ) . (31)The steady state of this dynamics is determined by the equation:(1 − q at )∆ U a = q vt ∆ U v (32)which is precisely the steady state implied by (1) when α = − C.2 Socialization payoffs
We have: U v → v = − (cid:90) k · σ v c dc − (cid:90) k · σ v ( k · σ v ) dc ; U a → a = − (cid:90) k · σ a − d ( c + d ) dc − (cid:90) k · σ a − d ( k · σ a ) dc ; U v → a = − (cid:90) k · σ a − d c dc − (cid:90) k · σ a − d ( k · σ a ) dc ; U a → v = − (cid:90) k · σ v ( c + d ) dc − (cid:90) k · σ v ( k · σ v ) dc . (33)(34)(35)(36)31ost c disutility kσ v Disutility of v as perceived by v kσ a − d kσ v kσ a Disutility of a as perceived by v Figure 10: Composition of ∆ U v . The area below the black line is U v → v , theare below the red line is U v → a , ∆ U v is the red area.Integration and the use of 10 yields:∆ U a = 12 ( x v − x a ) − ( d − ( x v − x a )) (1 − x v )= d (cid:0) d (cid:0) − h − hk q + k + 1 (cid:1) − hk ( hk + 1) (cid:1) k + 1)( hk + 1) ∆ U v = 12 ( x v − x a ) + ( d − ( x v − x a )) (1 − x a )= d ( d (2 hk (( h − kq + k + 1) + k + 1) + 2 hk ( hk + 1))2( k + 1)( hk + 1) (37)(38)(39)(40)Figure 10 shows the composition of ∆ U v = U v → v − U v → a . It is theanalogous of Figure 1 for ∆ U a . 32 Proofs
Proof or Result 1
Proof.
To analyze stability, we need to identify the values of parameters forwhich the Jacobian matrix of the system is negative definite when calculatedin (0 , J = (cid:18) (1 − x a ) ˜ q a − µ ( x a −
1) (˜ q a − x v −
1) (˜ q v −
1) (1 − x v ) ˜ q v − µ (cid:19) We can directly compute the eigenvalues, which are: e = ˆ µ − µe = ˆ µ − µ − ∆ . where ˆ µ := ( T + ∆) ∈ [0 , T := ˜ q a (1 − x a ) + ˜ q v (1 − x v ), and ∆ := (cid:112) T − h (1 − x a )(1 − x v ).The eigenvalues are real and distinct because, given ( x + y ) > xy whenever x (cid:54) = y , we get∆ = T − h (1 − x a )(1 − x v ) ≥ q a (1 − x a )˜ q v (1 − x v ) − h (1 − x a )(1 − x v )Now ˜ q a ˜ q v = h + h (1 − h ) + (1 − h ) q (1 − q ) ≥ h , so we conclude ∆ > µ > ˆ µ . Cumulative infection
Lemma 1.
Let ( ρ a , ρ v ) be the infected share for each group at the outbreak.Then in the linearized approximation around the (0,0) steady state: CI a = 2 [ ρ a ( µ − (1 − x v )˜ q v ) + ρ v (1 − x a ) (1 − ˜ q a )]( T − µ − ∆)( T − µ + ∆) ; CI v = 2 [ ρ a (1 − x v ) (1 − ˜ q v ) + ρ v ( µ − (1 − x a )˜ q a )]( T − µ − ∆)( T − µ + ∆) ; CI = 2 [ ρ a ( µ + (1 − x v )(1 − q v )) + ρ v ( µ + (1 − x a )(1 − q a ))]( T − µ − ∆)( T − µ + ∆) . (41)(42)(43)33 roof. The linearized dynamics is:˙ dρ ( t ) = M dρ ( t ) dρ (0) = ρ where ρ = ( ρ a , ρ v ), and: M = 1∆ e t ( T − µ ) (cid:18) sinh (cid:18) ∆ t (cid:19) (cid:18) − x a ˜ q a + ˜ q a − µ + 12 (2 µ − T ) (cid:19) + 12 ∆ cosh (cid:18) ∆ t (cid:19)(cid:19) M = 1∆ (1 − x a ) (1 − ˜ q a ) sinh (cid:18) ∆ t (cid:19) e t ( T − µ ) M = 1∆ (1 − x v ) (1 − ˜ q v ) sinh (cid:18) ∆ t (cid:19) e t ( T − µ ) M = 1∆ e t ( T − µ ) (cid:18) sinh (cid:18) ∆ t (cid:19) (cid:18) − x v ˜ q v + ˜ q v − µ + 12 (2 µ − T ) (cid:19) + 12 ∆ cosh (cid:18) ∆ t (cid:19)(cid:19) The cumulative infection in time in the two groups can be calculatedanalytically by integration, since it is just a sum of exponential terms. In-tegration yield, for CI v : CI v = (cid:90) ∞ dρ v ( t ) dt = 2 ( ρ a (1 − x v ) (1 − ˜ q v ) + ρ v ( µ − (1 − x a )˜ q a ))( − ∆ − µ + T )(∆ − µ + T ) +lim t − > ∞ e t ( T − µ ) (cid:18)
2∆ cosh (cid:18) ∆ t (cid:19) ( ρ a ( x v −
1) (˜ q v −
1) + ρ v ((1 − x v )˜ q v + µ − T )) +sinh (cid:18) ∆ t (cid:19) (cid:0) ρ v (cid:0) ( T − µ ) (2 ( x v −
1) ˜ q v + T ) + ∆ (cid:1) − ρ a ( T − µ ) ( x v −
1) (˜ q v − (cid:1)(cid:19) and the limit is zero if µ > ˆ µ because the leading term is Exp (cid:0) t ( T − µ ) + ∆2 (cid:1) =ˆ µ − µ . An analogous reasoning for CI a yields: CI a = (cid:90) ∞ dρ a ( t ) dt = 2 ( ρ a ( µ − (1 − x v )˜ q v ) + ρ v (1 − x a ) (1 − ˜ q a ))( − ∆ − µ + T )(∆ − µ + T ) CI v = (cid:90) ∞ dρ v ( t ) dt = 2 ( ρ a (1 − x v ) (1 − ˜ q v ) + ρ v ( µ − (1 − x a )˜ q a ))( − ∆ − µ + T )(∆ − µ + T ) (44)(45)The total CI in the population is CI = q a CI a + (1 − q a ) CI v CI = 2( − ∆ − µ + T )(∆ − µ + T ) ( q a ( ρ a ( µ − (1 − x v )˜ q v ) + ρ v (1 − x a ) (1 − ˜ q a )) +341 − q a ) ( ρ a (1 − x v ) (1 − ˜ q v ) + ρ v ( µ − (1 − x a )˜ q a )))= ρ a q a ( µ − (1 − x v )˜ q v ) + (1 − q a ) (1 − x v ) (1 − ˜ q v ))( − ∆ − µ + T )(∆ − µ + T ) + ρ v q a (1 − x a ) (1 − ˜ q a ) + (1 − q a ) ( µ − (1 − x a )˜ q a ))( − ∆ − µ + T )(∆ − µ + T ) Proofs for Result 3
First, note that µ > ˆ µ implies: µ > − x a > h (1 − x a ) µ > − x v > h (1 − x v ) µ > h (1 − x a )1 − (1 − h ) qµ > h (1 − x v )1 − hq The expressions of the derivatives are: ∂CI a ∂h = ( q −
1) ( x a −
1) ( µ + x v −
1) ( ρ a ( µ + x v − − ρ v ( x a + µ − hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI a ∂q a = ( h −
1) ( x a −
1) ( h ( x v −
1) + µ ) ( ρ a ( µ + x v − − ρ v ( x a + µ − hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI a ∂x a = (( h − q ( x v −
1) + µ + x v −
1) ( µ ( h ( q − − q ) ( ρ a − ρ v ) − hρ a ( x v − − µρ v )2 ( hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI a ∂x v = ( h − q −
1) ( x a −
1) ( µ (( h − q ( ρ v − ρ a ) + ρ v ) + h ( x a − ρ v )2 ( hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI v ∂h = q ( x v −
1) ( x a + µ −
1) ( ρ a ( µ + x v − − ρ v ( x a + µ − hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI v ∂q a = ( h −
1) ( x v −
1) ( h ( x a −
1) + µ ) ( ρ a ( µ + x v − − ρ v ( x a + µ − hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI v ∂x a = − ( h − q ( x v −
1) ( hρ a ( µ + µ ( − q ) + x v −
1) + µqρ a + ( h − µ ( q − ρ v )2 ( hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI v ∂x v = ( h ( q −
1) ( x a −
1) + q ( − x a ) − µ + q ) ( µ (( h − q ( ρ v − ρ a ) + ρ v ) + h ( x a − ρ v )2 ( hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI∂h = µ ( q − q ( x a − x v ) ( ρ a ( µ + x v − − ρ v ( x a + µ − hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI∂q a = ( h −
1) ( ρ a ( µ + x v − − ρ v ( x a + µ − h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + x v − hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI∂x a = − q ( h ( x v −
1) + µ ) ( hρ a ( µ + µ ( − q ) + x v −
1) + µqρ a + ( h − µ ( q − ρ v )2 ( hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − ∂CI∂x v = ( q −
1) ( h ( x a −
1) + µ ) ( µ (( h − q ( ρ v − ρ a ) + ρ v ) + h ( x a − ρ v )2 ( hµ ( − qx a + x a + qx v −
1) + h ( x a −
1) ( x v −
1) + µ ( q ( x a − x v ) + µ + x v − Note that all the denominators are positive, so to control the sign fromnow on we focus on the numerators. In particular, recognising the numera-tors of the first two as precisely the terms arising in 4 we can conclude that CI is increasing in h if and only if CI a > CI v and CI is increasing in q ifand only if CI a > CI v .If initial conditions are symmetric: ∂CI a ∂h > ⇐⇒ − ( q − ρ a ( x a −
1) ( x a − x v ) ( µ + x v − > ∂CI a ∂q a > ⇐⇒ − ( h − ρ a ( x a −
1) ( x a − x v ) ( h ( x v −
1) + µ ) > ∂CI a ∂x a > ⇐⇒ − ρ a ( h ( x v −
1) + µ ) ( µ − (1 − h )(1 − q ) (1 − x v )) > ∂CI a ∂x v > ⇐⇒ ( h − q − ρ a ( x a −
1) ( h ( x a −
1) + µ ) > ∂CI a ∂h > ∂CI a ∂q a > ∂CI a ∂x a < ∂CI a ∂x v <
0. Similarly, if ρ a = 0: ∂CI a ∂h > ⇐⇒ − ( q −
1) ( x a − ρ v ( x a + µ −
1) ( µ + x v − > ∂CI a ∂q a > ⇐⇒ − ( h −
1) ( x a − ρ v ( x a + µ −
1) ( h ( x v −
1) + µ ) > ∂CI a ∂x a > ⇐⇒ − (1 − h )(1 − q ) ρ v ( µ − (1 − q )(1 − h )(1 − x v )) > ∂CI a ∂x v > ⇐⇒ ( h − q −
1) ( x a − ρ v ( h ( x a −
1) + µ (( h − q + 1)) > ∂CI a ∂h < ∂CI a ∂q a < ∂CI a ∂x a < ∂CI a ∂x v < ρ v = 0: ∂CI a ∂h > ⇐⇒ ( q − ρ a ( x a −
1) ( µ + x v − > ∂CI a ∂q a > ⇐⇒ ( h − ρ a ( x a −
1) ( µ + x v −
1) ( h ( x v −
1) + µ ) > ∂CI a ∂x a > ⇐⇒ − ρ a ( µ − (1 − q )(1 − h )(1 − x v )) ( h ( µ − (1 − x v )) + µ (1 − h ) q ) > ∂CI a ∂x v > ⇐⇒ − ( h − µ ( q − qρ a ( x a − > ∂CI a ∂h > ∂CI a ∂q a > ∂CI a ∂x a < ∂CI a ∂x v < Proof of Result 4
The linearized dynamics is (from Lemma 1):˙ ρ a = 1∆ e t ( T − µ ) (cid:18) sinh (cid:18) ∆ t (cid:19) (cid:18) − x a ˜ q a + ˜ q a − T (cid:19) + 12 ∆ cosh (cid:18) ∆ t (cid:19)(cid:19) ρ a + 1∆ (1 − x a ) (1 − ˜ q a ) sinh (cid:18) ∆ t (cid:19) e t ( T − µ ) ρ v ˙ ρ v = 1∆ (1 − x v ) (1 − ˜ q v ) sinh (cid:18) ∆ t (cid:19) e t ( T − µ ) ρ a + 1∆ e t ( T − µ ) (cid:18) sinh (cid:18) ∆ t (cid:19) (cid:18) − x v ˜ q v + ˜ q v − T (cid:19) + 12 ∆ cosh (cid:18) ∆ t (cid:19)(cid:19) ρ v In particular, it depends on µ just through the exponential term e t ( T − µ ) .So we can rewrite it as: ˙ ρ a = e t ( T − µ ) A ( t )˙ ρ v = e t ( T − µ ) V ( t )where A ( t ) and V ( t ) do not depend on µ . Now the discounted cumulativeinfection for anti–vaxxers is equal to: CI a = (cid:90) ∞ e − βt e t ( T − µ ) A ( t ) dt = (cid:90) ∞ e t ( T − µ + β )) A ( t ) dt which is precisely the expression for the non discounted cumulative infectionin a model where the recovery rate is µ (cid:48) = µ + β .37 roof of Result 6 From the proof of Result 1, the eigenvalues are: e = ˆ µ − µe = ˆ µ − µ − ∆ . Moreover, they are both decreasing in absolute value as h increases (this iseasy to see for e , given that ˆ µ is positive and increases in h , but it holdsalso for e ). Proof of Proposition 1
We have: (cid:18) dCIdq (cid:19)(cid:12)(cid:12) h =0 = (cid:18) ∂CI∂q (cid:19)(cid:12)(cid:12) h =0 + (cid:18) ∂CI∂x a ∂x a ∂q + ∂CI∂x v ∂x v ∂q (cid:19)(cid:12)(cid:12) h =0 The first term can be obtained setting h = 0 in the expressions from theproof of Result 3: (cid:18) ∂CI∂q (cid:19)(cid:12)(cid:12) h =0 = ( k + 1) ( ρ a ( µ + x v − − ρ v ( x a + µ − k + 1) ( q ( x a − x v ) + µ + x v − The correction term instead is: (cid:18) ∂CI∂x a ∂x a ∂q + ∂CI∂x v ∂x v ∂q (cid:19)(cid:12)(cid:12) h =0 = dk (( q − ρ v − qρ a )2( k + 1) ( q ( x a − x v ) + µ + x v − which is negative: so endogenizing the vaccination choices always yieldsa smaller effect of a change in q . Moreover, if the initial conditions aresymmetric the numerator of the derivative becomes:( k + 1)( x v − x a ) − dk = ( k + 1) d − dk = d > CI is still increasing in q , but at a lower rate.Similarly, the derivative with respect to h is: (cid:18) dCIdh (cid:19)(cid:12)(cid:12) h =0 = (cid:18) ∂CI∂h (cid:19)(cid:12)(cid:12) h =0 + (cid:18) ∂CI∂x a ∂x a ∂h + ∂CI∂x v ∂x v ∂h (cid:19)(cid:12)(cid:12) h =0 and the correction term is null: (cid:18) ∂CI∂x a ∂x a ∂h + ∂CI∂x v ∂x v ∂h (cid:19)(cid:12)(cid:12) h =0 = − dkq ( q −
1) ( ρ v − q ( ρ v − ρ a ))2 ( q ( x a − x v ) + µ + x v − − dk ( q − q ( µqρ a − µ ( q − ρ v )2 µ ( q ( x a − x v ) + µ + x v − = 0so the derivative is exactly the same as in Result 3.38 roof of Proposition 2 Consider the function F ( q ) = q α ∆ U a − (1 − q ) α ∆ U v . Both ∆ U a and ∆ U v arebounded from above and bounded away from 0, so when q → q α → ∞ (because α < q →
1. By the intermediate value theorem, there exist a solution q ∗ ∈ (0 , F :d F d q = d k + 1)( hk + 1) × (cid:0) aq a − (cid:0) d (cid:0) − h − hk q + k + 1 (cid:1) − hk ( hk + 1) (cid:1) − d ( h − hk ( q a + (1 − q ) a )+ a (1 − q ) a − (cid:0) d ( h − hk q + d ( k + 1)(2 hk + 1) + 2 hk ( hk + 1) (cid:1)(cid:1) If q → d F d q → −∞ , whereas if q → d F d q → + ∞ , so that, by continuity,there must be a stable steady state. If if h → d F d q → αd − α <
0, so for h in a neighborhood of 0 the steady state is unique and stable. Proof of Proposition 3
For h = 0 we have that q = . We can compute the derivative using theimplicit function theorem. The first derivative is above. The second isd F d h = d k + 1)( hk + 1) × dk ((1 − q ) a ( hk ( d ( − ( k + 2) q + k + 1) −
1) + dkq − − q a ( d ( kq ( h ( k + 2) −
1) + k + 1) + hk + 1))so that: d q d h (cid:12)(cid:12)(cid:12)(cid:12) h =0 = − d F d h d F d q = 2 k + dkα (2 d + 2 dk )and we can see that q is always decreasing with homophily, but with adifferent level of intensity according to the magnitude of α . Proof of Proposition 4
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