The Hard Pursuit of Optimal Vaccination Compliance in Heterogeneous Populations

2.1 Setup and Assumptions

At the beginning of date 0 (time is discrete), a vaccinable disease outbreak emerges. The Public Health Authority (PHA, hereafter) communicates the fact to the population, clarifying that vaccination can help to reduce the severity: that is, people are publicly invited to take precautions. Define s _t as the true severity of the disease (that is, the average cost of its consequences for the population of infected subjects), as can be observed at the end of date t by the PHA. We assume that s _t is a Normal random variable with mean α − β π t and variance equal to 1, where π _t is the immunization coverage. Formally:

Randomness is due to different factors: individual genetic predisposition, exposure, precautions, etc. The normality assumption seems reasonable, besides allowing for better tractability of the model. Concerning the linear relation α − βπ _t , observe that it refers to the mean of the distribution of s _t . Although linearity may appear simplistic, it allows us to keep the analysis tractable, while capturing the relevant elements of the relation between average severity and immunization coverage (see the comment in footnote 10 below). The unit-variance assumption is without loss of generality. α and β are the structural parameters of the model, and both are unknown.

At the end of each date t, the PHA, though not knowing α and β,^[4] observes the average severity s _t and the overall immunization coverage π _t . These observations are communicated to the population.

The general population is composed of two components: subpopulation P is averagely “prompt” to get vaccinated, while subpopulation H is more “hesitant”. The weight of subpopulation P is γ and the weight of subpopulation H is 1−γ. Population promptness/hesitancy is endogenous to the model and is due both to individual costs to get vaccinated and to beliefs.

In particular, individual j belonging to subpopulation i bears an individual cost to get vaccinated equal to c _j,i that includes both costs of immunization and other monetary and non-monetary costs, depending on individual risk aversion, fear of side effects of vaccination, etc. We assume that c _j,i is uniformly distributed within each subpopulation:

The clause θ _P < θ _H indicates that people in subpopulation P bear a lower maximum cost to be vaccinated: therefore, we expect that – ceteris paribus – subpopulation P displays a greater readiness to get vaccinated.

When the outbreak emerges and this is communicated by the PHA, people correctly assume that the severity of the disease is a random variable N ∼ s i , t e , 1 , with s i , t e depending on the immunization coverage. However, since the true relation between s _t and π _t is unknown, each subpopulation i shares a common subjective prior on the structural parameters α and β, such that s i , t e = α i , t − β i , t π i , t e where π i , t e is the expected immunization coverage.

Formally, the prior of subpopulation i on the structural parameters α and β is a Normal bivariate; the mean and precision^[5] of this subjective distribution at date t are, respectively, the following vector z _i,t and the symmetric precision matrix H _i,t :

As for date t = 0, we assume α _i,0 > 0, η _i,α,0 > 0, η _i,β,0 > 0, η _i,αβ,0 = 0. In general, we define the parameters α _i,t and β _i,t as the mean hyperparameters, and η i , α β , t as the precision hyperparameters of the model. At the beginning of each date, subpopulation i expects π i , t e = π t − 1 ,^[6] so that the expected severity is s i , t e = α i , t − β i , t π t − 1 .

At the end of date t, once immunization decisions have been taken, people learn s _t and π _t from the PHA and update their prior along Bayesian lines.^[7] The posterior at date t becomes the prior of date t + 1: see Figure 1.

Figure 1:

Timing of the model on date t.

2.2 The Decision to Get Vaccinated: Immunization, Severity, and Learning

Given their costs and conjectures, people must decide whether to get vaccinated on each date, because immunization is temporary. Individuals make this decision by comparing the expected consequences (severity) of the disease and their own cost of being vaccinated. Individual j belonging to population j decides to get vaccinated if and only if

The expected disease severity s i , t e is common to all individuals of each subpopulation and depends on the overall expected coverage (embedding all individuals’ decisions); the latter is computed adaptively ( π i , t e = π t − 1 ). At the start of t = 0, since no immunization decision has yet been taken, π _t−1is equal to 0 and, therefore, in order to make decisions during t = 0, the expected severity s i , t e is set equal to α _i,t .

Given the uniform distribution of individual costs, the share of each subpopulation deciding to get immunized is:

The vaccination coverage of subpopulation i is obviously inversely related to θ _i : hence, as expected, the vaccination coverage in population P is higher than in population H.

The overall immunization coverage on date t is the average of the two subpopulations:

In particular, at t = 0 we have π 0 = γ θ P α P , 0 + 1 − γ θ H α H , 0 .

Given the immunization coverage π _t , the expected value of severity on period t is:

By inspecting (1) and (2), we observe that the overall immunization coverage on a given date t depends on both structural parameters and hyper-parameters. Not surprisingly, we conclude that the immunization coverage depends (i) positively on the expected severities without any immunization coverage as conjectured by both subpopulations (α _i,t ); (ii) negatively on the effectiveness of vaccine as conjectured by both subpopulations (β _i,t ); (iii) positively on the weight γ of the prompt-to-be-vaccinated subpopulation; (iv) negatively on the maximum individual costs to get vaccinated (θ _i ).

Once vaccination decisions are taken and π _t and s _t are observed and communicated by the PHA, the subpopulations update their priors. As shown in the Appendix, by using the previous definitions of z _i,t and H _i,t , and defining the vector x t = 1 − π t ′ , the posterior at the end of date t (corresponding to the prior at the beginning of date t + 1) is:

In (3) we define e i , t = s t − x t ′ z i , t = α − β γ θ P α P , t − β P , t π t − 1 + 1 − γ θ H α H , t − β H , t π t − 1 − α i , t + β i , t π t : the term e _i,t measures the forecasting error made by each subpopulation i at date t. Therefore, the updated z _i,t+1 turns out to be equal to z _i,t , corrected by a term containing the forecasting error “deflated” by the inverse of the posterior precision matrix H i , t + 1 = H i , t + x t x t ′ . The precision matrix grows over time (in the sense of the positive-definite matrix ordering, see the Appendix), so that its inverse decreases in time: therefore, the updating process becomes increasingly slower. Expression (3) also shows that when initial precisions (H _i,0) are very high, the updating process is very sluggish right from the start. When this happens, communications from the PHA have a small impact on the updated conjectures right from the beginning.

The forecasting errors e _i,t , entering the recursive adjustment of priors, depend negatively on α _i,t (the higher α _i,t , the higher the expected severity), and positively on β _i,t (the higher β _i,t , the lower the expected severity). The impact of priors on the forecasting error clearly depends also on the structural parameters of the model.

Combining equations (1)–(3) recursively, we can simulate the subpopulations’ immunization coverages (and, of course, the average severities) at different dates, starting from given initial conjectures. By inspecting Figures 2 and 3, that illustrate the evolution of immunization coverage in time for given parameters, we observe that the model captures some relevant effects that are commonly associated with vaccination hesitancy.

Figure 2:

Vaccination coverage: t from 0 to 20 – impact of different structural parameters.

Figure 3:

Vaccination coverage: t from 0 to 20 – impact of different initial hyperparameters.

First, given that the actual average severity depends on the overall coverage, individuals who opt for vaccination suffer from a negative “externality” due to the choice of those who do not opt for immunization. Vice versa, people who prefer avoiding vaccination benefit from the positive externality of a high overall coverage (low average severity); the latter attitude can be seen as freeriding.^[8] These external effects operate both among individuals within each of the two subpopulations, and between the two subpopulations.

Second, an overshooting effect can be detected with respect to the long-run coverage (that, as clarified in Section 3, does not necessarily correspond to a socially optimal coverage). In fact, during the first dates, when adjustments are large because learning has a relevant impact, the vaccination coverage temporarily overreacts to changes in observed levels of severity/coverage. Indeed, we see from expression (3) that changes in beliefs, z _i,t+1 − z _i,t , move in the same direction as the forecasting error e _i,t , meaning that subpopulation’s mean hyperparameters – and hence subpopulation’s vaccination coverages – negatively depend on their past values.

Third, we observe a sort of herding effect due to learning: each subpopulation adjusts its behavior accounting for the overall immunization coverage and the related average severity, partially induced by the other subpopulation’s behavior.

Furthermore, in Figure 3 we observe that the initial precision hyperparameters determine the pace of convergence, because they determine the readiness of the updating process. In particular, low precisions allow faster convergence (3A vs 3B). The differences in initial subpopulations’ conjectures on α and β matter as well: very distant priors imply stronger oscillations and a longer-lasting overshooting (3C vs 3D).

2.3 Self-Fulfilling Equilibria and their Dynamic Stability

In order to analyze the impact of changes in structural parameters, also in terms of dynamic stability, we introduce a proper concept of equilibrium. Define a self-fulfilling equilibrium ^[9] (SE, hereafter) as a situation in which the subpopulations’ priors are correct at the information set that is reached in the equilibrium position, but not necessarily correct out of equilibrium. Consequently, a SE is reached when people guess (locally) the average severity given the existing vaccination coverage, and both subpopulations have no reason to modify their beliefs and the related vaccination decisions. This means that a stationary state of the dynamical learning process has been reached, and no further change will take place. Recalling that s _t is a random variable, the reasoning below is cast in expected-value terms, that is, for an SE we require s i , t e = E s t . We underline that a SE has nothing to do with social optimality (see Section 3).

The SE condition can be represented by the following system (time subscripts are omitted since variables stay constant in equilibrium):

From (4) we obtain the SE values of both the vaccination coverage π _SE and the related average severity s _SE (5), and the SE conjectures of the two subpopulations (6).

A few comments are in order. First, as can be verified by looking at (5), the SE coverage-severity couple is unique, and depends only the on structural parameters: the precise location in the stationary state of π _SE and s _SE does not depend on the subpopulations’ priors. The structural parameters monotonically affect the SE coverage and severity as expected: the SE coverage is increasing in α and γ, while being decreasing in β and θ _i ; the SE severity in increasing in α and θ _i , and is decreasing in β and γ (see the derivatives provided in the Appendix).

Second, by looking at (6), we conclude that there exist a doubly-infinite set of quadruplets of mean hyperparameters that are compatible with the SE. This infinity is easily interpreted from the geometrical point of view. Indeed, as illustrated in Figure 4, it consists of all possible intersections between the true average severity function s = α − βπ and the average severities as conjectured by the two subpopulations s _SE = α _SE,i − β _SE,i π.^[10] Given any discretional α _i , the hyperparameter β _i must satisfy (6), and vice versa. From the learning perspective, the infinity of possible SE priors implies that the subpopulations end up, in equilibrium, with (possibly very) different interpretations of the stationary state: even if guessing the true values of immunization coverage and average severity, subpopulations envisage different relations between the two variables. Therefore, if a shock hit the system, they would expect (possibly very) different consequences: a subpopulation could even believe that vaccination is noxious (β _SE,i < 0, third graph in Figure 4).

Figure 4:

Three possible SEs for given structural parameters (α = 0.9, β = 0.5, γ = 0.8, θ _P = 1, θ _H = 2).

Notably, the existence of a multiplicity of possible equilibrium priors (equilibrium hyperparameters) is proved despite the unicity of the equilibrium severity-coverage couple. The location of equilibrium, as seen above, depends only on the structural parameters of the model; however, its stability features crucially depend also on the priors of the populations, as we are going to discuss.

Different structural parameters are associated with different SEs; therefore, a shock affecting those parameters displaces the system from the pre-existing SE position. However, after a shock, people observe an unexpected change, so that a new learning process starts. As a consequence, the system may converge towards the new SE or, otherwise, it may diverge from it. In order to study the dynamic properties of the new SE, we consider the eigenvalues of the Jacobian matrix of the dynamical system evaluated at the new SE (details provided in the Appendix). The analytical study of the relation between the eigenvalues and the values of structural parameter turn out to be quite difficult (due to the highly complex dependence of the former on the latter), so that we had to resort to numerical computations in order to obtain the time of convergence (TOC, hereafter) and the period of oscillation associated with different parameter combinations.^[11]

Concerning the period of oscillation, in most cases it is equal to 2, meaning that there are alternating “up and down” movements towards the SE (as observed in Figures 2 and 3). In very few cases, the period is a number greater than 2, implying that movements around the SE are slower (they are waves of lower frequency): therefore, we do not further comment on the periods of oscillation.

On the other side, the TOC is quite sensitive to parameter changes. Consider that a positive TOC associated with a SE indicates how rapidly the variables converge to that SE if they are slightly displaced from it. On the contrary, a negative TOC means divergence (the system would converge if it moved backwards in time). A lower positive TOC means faster convergence, while a lower negative TOC indicates slower divergence.

The TOC has important consequences for the promptness of public policies in achieving the desired new level of severity/coverage (see Section 3). In fact, if a particular equilibrium is aimed at by the PHA by means of some intervention, a longer TOC associated with that equilibrium implies that reaching the objective is harder, and presumably more costly for society. This aspect must be kept logically distinct from the overshooting oscillations discussed in Section 2.2, that of course are socially costly as well since severity and coverage are not at their optimal levels.

In Figure 5 we plot the stability of SEs, measured by the TOC, against different parameter values. This is done for different scenarios (different lines in the same graph). We display a baseline scenario,^[12] plus some variations thereof: the variations consist in modifying a single parameter, keeping constant the remaining ones. Upswinging lines correspond either to a decreasing rate of convergence (if located in the positive zone), or to an increasing rate of convergence (in the negative zone). The first five graphs of Figure 5 refer to the effects of structural parameters, and the last graph refers to the effects of the precision hyperparameters of subpopulation P:^[13] recall that, while the location of SEs does not depend on the value of hyperparameters, it does depend on structural parameters.

Figure 5:

Time of convergence (TOC) against different parameters, under various scenarios.

Regarding the first five graphs in Figure 5, we see that increases either in α or in θ _i lead to higher stability or lower instability (decreasing TOC) in all possible scenarios.^[14] Vice versa, higher values of β and γ induce lower stability or higher instability in all scenarios. On the other hand, the last graph in Figure 5 shows what we know from the analysis of Section 2: higher initial precisions (matrices H _i,0) cause a more sluggish learning rate and a longer convergence rate.

Therefore, we are able to conclude that changes in structural parameters resulting in a lower SE average severity (namely, higher β or γ, and lower θ _i or α) tend to push in the direction of slower convergence or faster divergence. In fact, any change/intervention that, starting from an existing SE, reduces the true mean severity s causes a negative forecasting error of subpopulations. Hence, the learning mechanism forces the two subpopulations to reduce both α _i and β _i (see expression (3)): this may give rise to a greater difference in steepness between the conjectured and the true π–s relation. Therefore, there can result an initial lower stability or greater instability, as it happens in many Cobweb and adaptive expectations models (Ezekiel 1938; Gandolfo 1980; Nerlove 1958 chapter 4, Hommes 1994; Hommes et al. 2007).

An observation is in order. Our stability analysis and the above numerical experiments are implemented by fixing the precision hyperparameters, so that we can appreciate the characteristics of the dynamical paths only in the very first periods after a displacement from a SE. On the other side, we know from Section 2 that precisions actually increase in time: this (as shown also by the last graph in Figure 5) dampens the oscillations after a number of periods. Therefore, paths that are initially unstable will not diverge indefinitely, nor stable ones will converge rapidly: in finite time, they could approach some non-equilibrium position surrounding a SE, or could be captured by the basin of attraction of another stable SE.

Concluding, when structural parameters change, not only the system shifts towards new SEs, but, in addition, the latter are characterized by different stability properties. In particular, some changes might lead to a longer convergence time/shorter divergence time. This must be very clear to the policymakers intervening on parameters to the end of reducing the average severity of a vaccinable disease. Each intervention should be carefully evaluated not only for its capacity to achieve a new desirable SE, but also in terms of its dynamic effects.

3.1 Nudging, Subsidies, and Mandatory Vaccination

In order to favor vaccination compliance, resources might be invested in subsidizing and/or nudging individual decisions to get vaccinated. Public vaccination campaigns involve nudging, besides some forms of subsidies. For example: (a) people (or specific categories like newborns, kids, elders, fragile people, etc.) can be vaccinated for free or for a capped price; (b) individuals are invited to comply with a vaccinal agenda; (c) vaccinations are facilitated, being provided by general practitioners/pediatricians, local vaccination centers, or even the pharmacies.^[16] In general, most of the PHA actions aiming at reducing vaccination hesitancy and favoring higher vaccination coverages are based on a significant reduction of individual monetary and non-monetary costs to get vaccinated. According to our model, this policy can be seen as a shrinking of the individual cost distributions. Recall that when s _SE is too high with respect to the socially optimal severity (and, then, π _SE is suboptimal), the PHA can reduce s _SE by increasing π _SE thanks to reductions in θ _i , i = P, H (see Section 2.3). Note that green-passes (Campanozzi, Tambone, and Ciccozzi 2022) and other immunization passports (Sharun et al. 2021) can be modelled as negative nudging mechanisms that, by increasing the individual cost of non-compliance, in fact result in lower θ _i .

In many countries, some vaccinations are mandatory, and noncompliance is variously sanctioned. For instance, people who do not comply with mandatory infant vaccinations can be sanctioned with monetary sanctions and children cannot attend schools. During the COVID 19 pandemic, several countries introduced mandatory vaccinations, at least for some categories of people. In our model, we can assume that a fixed penalty S > 0 is certainly paid by those who do not comply with mandatory vaccinations. In this case, the individual j of subpopulation i decides to get vaccinated on date t if and only if α _i,t − β _i,t π _t−1 > c _j,i − S. Reconsidering the passages provided in Section 2.2, we conclude that when vaccination noncompliance is punished with a sanction, coverage increases. Repeating those passages, we obtain an equilibrium immunization coverage with sanction π SE , S = γ θ P − S + 1 − γ θ H − S α 1 + γ θ P − S + 1 − γ θ H − S β , which is indeed greater than π _SE without sanctions. The SE average severity s _SE,S = α − βπ _SE,S decreases consequently. A punishment system can allow the PHA to pursue a lower disease average severity at relatively low costs (the costs of implementing the sanction system).

The introduction of a uniform sanction results in a reduction of the relevant cost thresholds to decide to comply or not, similarly to the case of nudging. In spite of this similarity, there is a significant difference between the two. The sanction, being the same for all individuals within the two subpopulations, has a higher relative impact on subpopulation P in terms of the number of new people who are induced to get vaccinated; in addition, it has a higher relative impact for the group of individuals within each subpopulation who are characterized by a low c _i . Clearly, a moderate sanction cannot convince people who are characterized by very high individual cost to get vaccinated.

3.2 Educational Campaign/Moral Suasion

A further strategy to favor vaccination compliance is promoting information or setting up educational campaigns to encourage or persuade people to vaccinate. These interventions are typically targeted to specific subpopulations – here H, the hesitant one. A successful policy might move some individuals from subpopulation H to subpopulation P, finally increasing γ. By increasing γ, s _SE decreases (see Section 2.3), once again favoring the realignment of SE levels to the socially optimal values.

3.3 Comparison among Policies for Reasonable Scenarios

Here, we will compare the effects of policies considering different scenarios, characterized as follows.

The baseline scenario (α = 0.9, β = 0.5, θ _P = 1, θ _H = 2, γ = 0.8) is such that disease displays a relatively high zero-vaccination average severity, and vaccination is fairly effective. The largest part of the population is endowed with relatively low individual costs (subpopulation P), while those with higher individual costs (subpopulation H) are a minority.

Besides the baseline scenario, we consider a low α scenario (α = 0.5) that differs from the former because the disease is less severe, independently of the vaccination coverage; a high β scenario (β = 1) that differs from the baseline scenario because the vaccination is significantly more effective; a similar costs scenario (θ _H = 1.1) that deviates from the baseline scenario because the two subpopulations differ only slightly in terms of individual costs; and a low γ scenario (γ = 0.2) that differs from the baseline scenario because now the largest part of the population is subpopulation H.

Changes in populations’ hyperparameters are not explicitly considered here, since they affect only the dynamical features of SEs (TOC), already analyzed in Section 2, while do not affect SE coverage and severity levels.

Consider first single-parameter policies (nudging on population P, nudging on subpopulation H, and moral suasion aimed at increasing γ). By comparing the various graphs in Figure 6 (especially slopes and heights), we can observe the following facts:

1. The effects on the policy target that are associated with the nudging on H people and moral suasion are significantly weaker than those induced by the nudging on P people. In fact, by reducing θ _H from 2 to 1, the SE severity moves from approx. 0.65 to approx. 0.35. By increasing γ from 0 to 1, the SE severity moves from approx. 0.72 to approx. 0.35. Something similar happens looking at the SE coverages. Focusing instead on θ _P , we observe that by decreasing it from 1 to 0, the SE severity moves form approx. 0.7 to 0.0. Analogously, the SE coverage increases from 0.3 to 1.0.

2. Nudging on subpopulation P allows to rapidly achieve minimum SE severity (maximum coverage), because these people are particularly prompt to get vaccinated (due to their individual costs) and represent the largest part of the population. In this respect, the similar costs scenario approximately overlaps the baseline scenario, while under the low γ scenario nudging is extremely effective for very low values of θ _P while becoming less effective for higher values of θ _P (the “Low γ” line becomes flatter).

3. Nudging on subpopulation H is significantly less effective (lines are almost flat), because individuals H are less prompt to get vaccinated due to their costs and represent the minority of the population. Actually, when we consider the low γ scenario, nudging of subpopulation H starts being effective (higher slope of the “Low γ” lines) because a larger part of the population benefits from that policy.

4. Moral suasion has a moderate impact on the policy targets (weakly sloping lines), and this impact tends to vanish under the similar costs scenario. Indeed, intervening on individual characteristics (in order to move part of population from H to P) might be poorly effective when (individual costs of) the two subpopulations are not too polarized.

5. For all policies, the SE severity under the low α scenario and under the high β scenario is systematically lower than under the other scenarios (the “Low α” and “High β” lines stay in the lower part of the graphs and are flatter), because such scenarios are already more favorable to a lower SE severity, though they are associated with a systematically lower vaccination coverage (see the relevant derivatives provided in the Appendix).

Figure 6:

Nudging on Ps, nudging on Hs, and moral suasion on SE: severity and immunization coverage for various scenarios.

In order to appreciate the relative impact of the policies, including mandatory vaccination with sanction and other policy mixes, it is interesting to compare the average elasticity of the policy target (SE severity/immunization coverage) with respect to each policy parameter.

The elasticity is the ratio of the percentage variation in the policy target to the percentage variation in the policy parameter, evaluated at a particular value of the parameter. The average elasticity is computed over twenty evenly spaced values of each policy parameter, from its minimum to its maximum, as it appears in the horizontal axes of Figure 6. With regard to the policy mixes involving two policy parameters, the overall elasticity is computed as the mean of the average elasticities associated with each policy parameter. The numbers, provided in Table 1 below, describe the percentage average variation in the policy target due to an overall one per cent variation (increase or decrease, depending on the policy) in the policy parameters.

The average elasticities confirm what we already observed in the graphs: the nudging on subpopulation P is more effective than the nudging on subpopulation H, at least as long as the former is preponderant. When the latter becomes preponderant (see the low γ scenario), the nudging on subpopulation H is effective, though the impact is smaller than when the preponderant subpopulation is P and it is nudged (−0.423 vs −0.433).

When vaccines are relatively more effective (high β scenario), all the policies have a greater impact on severity reduction, though having a smaller impact on the increase of coverage (there exists a substitution effect between vaccine effectiveness and immunization coverage).

Moral suasion per se has a significantly smaller impact on the policy targets than nudging. However, it can be successfully used joint with nudging (see the comment below).

Mandatory vaccination, here, is intended as a situation where those who do not comply with the immunization obligation are sanctioned. The elasticities are computed for a 1 per cent decrease of the term θ i − S . This policy looks effective under all scenarios. The same result could be achieved through a subsidy of S to both subpopulations. Obviously, the two solutions are different in redistributive terms, because in the former case those who do not comply pay for the sanction, while in the latter case those who get vaccinated are subsidized.

In the case of mixes of moral suasion and nudging, we observe that the former weakens the nudging on subpopulation P while strengthening the nudging on subpopulation H. This confirms the greater effectiveness of nudging on the promptest subpopulation.

In summary, mandatory immunization (with sanctions) appears as an easy way to promote vaccination compliance, finally resulting in a significant reduction of the SE severity. However, other policy mixes appear to be effective as well. In particular, this happens for moral suasion joint with nudging in favor of the largest component of the population. For these reasons, countries must guarantee updated vaccinal agenda, granting both easy access to immunization (nudging) and sanctions in the case of non-compliance.

Concerning the implications of the various policies in terms of dynamics, we learnt from our model (Section 2.3 and Figure 5) that the more effective is the policy in reducing the average severity, the higher is the TOC. This means that effective policies induce weaker stability (or even stronger instability) of the associated SE, at least initially. Hence, the system might eventually converge to a position that is not the one projected by the PHA. This has important policy implications in terms of greater effort that the PHA must exert in order to reach its objectives. The model, therefore, sheds light on an uneasy interaction between policies and private beliefs/costs, learning, and the related behavior. Our arguments suggest that the PHA should be very pragmatic when intervening and should persist in the implementation of its policies.

Finally, the model seems to fit quite well two relevant aspects of immunization behavior that can be observed in reality. First, the interaction between scientific evidence (the observed severity) and people conjectures/learning/decisions translates into the existence of subsequent phases of oscillatory immunization behavior (D’Onofrio and Manfredi 2020; Duclos et al. 2009; Lanza-León, Cantarero-Prieto, and Pascual-Sáez 2024; Mantel and Cherian 2020). Second, the evidence that the same policies can result in different impacts depending on the country can be explained by the fact that people in different communities are endowed with different beliefs and costs (González-Block et al. 2020; Odone et al. 2021; Paul and Loer 2019; Privor- Dumm et al. 2020; Vakili et al. 2015).