Modeling the interplay between seasonal flu outcomes and individual vaccination decisions
MModeling the interplay between seasonal fluoutcomes and individual vaccination decisions
Irena Papst* , Kevin P. O’Keeffe , and Steven H. Strogatz Center for Applied Mathematics, Cornell University, Ithaca, NY 14853 Senseable City Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 Department of Mathematics, Cornell University, Ithaca, NY 14853
January 21, 2021 *Corresponding author: [email protected]
Abstract
Seasonal influenza presents an ongoing challenge to public health. The rapidevolution of the flu virus necessitates annual vaccination campaigns, but the decisionto get vaccinated or not in a given year is largely voluntary, at least in the United States,and many people decide against it. In early attempts to model these yearly flu vaccinedecisions, it was often assumed that individuals behave rationally, and do so withperfect information—assumptions that allowed the techniques of classical economicsand game theory to be applied. However, the usual assumptions are contradicted by theemerging empirical evidence about human decision-making behavior in this context.We develop a simple model of coupled disease spread and vaccination dynamicsthat instead incorporates experimental observations from social psychology to modelannual vaccine decision-making more realistically. We investigate population-leveleffects of these new decision-making assumptions, with the goal of understandingwhether the population can self-organize into a state of herd immunity, and if so,under what conditions. Our model agrees with established results while also revealingmore subtle population-level behavior, including biennial oscillations about the herdimmunity threshold.
Keywords: seasonal influenza, vaccination, decision-making, socialpsychology, SIR model 1 a r X i v : . [ q - b i o . P E ] J a n Introduction
Annual influenza epidemics are a significant public health challenge, with up to 650,000individuals dying from respiratory diseases associated with the flu each year (World HealthOrganization 2017). In the United States alone, the total economic burden of seasonalinfluenza, including direct medical costs and lost earnings due to illness or death, has beenestimated as $26.8 billion annually (Molinari et al. 2007).One of the main challenges of controlling seasonal influenza spread is that the virusesevolve quickly (on the same time scale as the annual epidemics), with multiple strainscirculating concurrently. A key adaptation mechanism, antigenic drift, gives rise to newinfluenza strains by randomly changing segments of viral surface proteins. Given that ahost’s immune system uses these surface proteins to identify the virus so that it may beneutralized (Taubenberger and Kash 2010), antigenic drift thus acts as an evolutionarycountermeasure. It helps the flu evade the immune system and thereby promotes its spreadthrough the host population.The rapid evolution of the flu results in the constant threat of a pandemic, and it alsomakes it challenging to develop effective, long-lasting vaccines. The seasonal flu vaccine isupdated every year to protect against the strains that seem to pose the largest upcoming threat.Seasonal influenza vaccination is largely voluntary in the United States, so individuals mustdecide whether or not to vaccinate each year.For many people, this decision is not easy. It involves many quantities that are effec-tively impossible for an individual to estimate accurately, such as their likelihood of beingvaccinated successfully, the probability of an adverse reaction to the vaccine, as well astheir increased risk of catching the flu by foregoing the vaccine. Yet although the decisionmay be difficult at an individual level, at a societal level the benefits of vaccinating can beimmense; if a critical mass of individuals choose to immunize themselves, “herd immunity”can be achieved. In this desirable state, the density of susceptible individuals is so low thatan infection chain cannot be sustained and so an epidemic cannot occur (Fine 1993).Somewhat paradoxically, the possibility of achieving herd immunity makes an indi-vidual’s decision to vaccinate or not even more complex, at least when viewed throughthe lens of classical game theory (wherein agents are assumed to be purely self-interested,and to behave rationally with perfect information about the situation at hand). The issueis that as vaccination coverage increases, individuals are increasingly incentivized not tovaccinate. Each person would do better relying on others to bear any burden associated withvaccination, while everyone reaps the benefits of widespread immunity. Such “free-riding”logic makes it impossible to ever actually achieve herd immunity with rational hosts.The free-riding problem is common in models where agents rationally weigh a delayed2ollective group benefit against immediate individual costs. Individuals are assumed tomake decisions by selecting the strategy that maximizes an objective, individual payofffunction, as prescribed by classical economics (Hardin 2013). Some early models ofvoluntary vaccination decisions involve such assumptions and inevitably yield agents thatutilize free-riding logic, which precludes herd immunity (Geoffard and Philipson 1997;Bauch and Earn 2004).However, a more recent empirical study suggests that free-riding logic is uncommonwhen individuals specifically consider whether or not to get the seasonal influenza vaccine(Parker et al. 2013). In fact, this study finds that the majority of individuals surveyeddo not account for the vaccination decisions of others when making their own decision.Nevertheless, despite increasing evidence that assumptions from classical economics maynot appropriately capture human behavior in the context of infectious disease spread, somebehavior-disease models continue to be built on such foundations (Verelst et al. 2016).Other studies have challenged these assumptions by replacing them with those frombehavioral economics, which leverages social psychology in its models of human decision-making. Voinson et al. (2015) develop a behavior-disease model that incorporates cognitivebiases and differing vaccine opinions among individuals to study vaccination coverage overtime. Oraby and Bauch (2015) study pediatric acceptance of vaccines by incorporatingprospect theory into their disease model.In this paper, we consider a simplified model for the interplay between annual vaccinationdecisions and seasonal influenza spread, in which individual voluntary vaccination decisionsare informed by observed social psychology in this context. Unlike previous work, we model repeated vaccination decisions to reflect the annual vaccination decision necessitated by therapid evolution of influenza viruses. We investigate population-level effects of these newdecision-making assumptions, with the goal of understanding whether the population canself-organize into a state of herd immunity, and if so, under what conditions. Despite ourmodel’s idealized nature, we find that its results align with those utilizing assumptions basedon classical economics, although our model also predicts more nuanced population-levelbehaviors, such as oscillations in and out of herd immunity on a biennial basis.
Decision theory and social psychology suggest that, in general, individuals tend to useheuristics, or rules of thumb, rather than a “rational” cost-benefit analysis in complexdecision-making (Tversky and Kahneman 1974). Moreover, decision-making tends to obeythe law of inertia: choices generally remain unchanged but are sensitive to both small nudgesand unfavorable resulting outcomes (Thaler and Sunstein 2009). Our model is based on3oth of these ideas.For simplicity, assume that each year, an individual chooses whether or not to receive theseasonal influenza vaccine based solely on evaluating their most recent outcome with boththe vaccine and the disease. The vaccine carries a cost, or risk, of adverse reaction, whichcan be interpreted as a cost to an individual’s health due to vaccine side effects (morbidity),a direct economic cost from paying for the vaccine, and/or an indirect economic cost suchas taking unpaid leave from work to get vaccinated. In what follows, we will interpretvaccine cost as morbidity, but the modeling framework is flexible enough to accommodateother interpretations. There is also some probability that vaccination successfully confersimmunity upon the recipient.Before we present the model in detail in Section 3, let us first describe it intuitively. Inany given year, individuals make the decision of whether or not to vaccinate, then followthrough with their choice, and then the flu season occurs. The epidemic resolves itself, eachindividual assesses their personal outcome from the past year, and then decides whether ornot to get vaccinated prior to next year’s flu season. The decision rule is a simple heuristic:if a person “won” last year (did not get sick and did not have an adverse reaction to thevaccine), they stick to their vaccination choice and make the same decision the followingyear. If they “lost” (got sick or had a bad reaction to the vaccine), they are nudged to changetheir behavior (switch from vaccinating to not, or vice versa) in hopes of eliciting a betterduring the next flu season.Figure 1 shows a schematic of the model. As an example of how to flow through thechart, let us first consider an individual who has decided to vaccinate in a given year (adecision made by a proportion 𝑝 𝑛 of the population in year 𝑛 ). This decision correspondsto the top left fork of the tree in Figure 1. Next, suppose the vaccine succeeds in conferringimmunity (which occurs with probability 𝑠 ), and the vaccine does not elicit an adversereaction (which occurs with probability 1 − 𝑟 ). This favorable outcome does not pushthe individual away from their default (winning) strategy of vaccinating, so they decide tovaccinate again the following year; the vaccine seems to have succeeded in protecting themfrom the flu. (We are assuming here that only the failure of the vaccine can be observedby the individual, and only if they happen to get ill that year. Otherwise, the success of thevaccine is presumed, since there is no evidence to the contrary.)On the other hand, an individual may choose not to vaccinate in a given year (a decisionmade by a proportion 1 − 𝑝 𝑛 of the population, shown by following the top right forkin Figure 1). If such an individual then happens to contract the flu (which occurs with aprobability that we will calculate below), then since this non-vaccinating individual’s choicewas a “losing” strategy, they decide to vaccinate the following year. The various other pathsthrough the tree can be understood similarly.4 otal populationVaccinate Do not vaccinateSick Not sickBad reactionVaccine succeedsNo reactionVaccine failsNo reactionBad reaction Not sickSick Year n Do not vaccinate
Year n+1
Vaccinate
Figure 1: All possible decisions and outcomesfor an individual in year n , leading to differentdecisions for year n + . Boxes with solid bor-ders denote decisions and possible intermediaterepercussions in year 𝑛 . Boxes with dashed anddotted borders denote final repercussions that de-termine the decision in year 𝑛 +
1: vaccinate (boxwith dashed border) or do not vaccinate (box withdotted border). In the model, 𝑝 𝑛 is the propor-tion of vaccinators in year 𝑛 , 𝑟 is the probabilitythat the vaccine induces a cost, 𝑠 is the probabil-ity that the vaccine succeeds, 𝜙 is the final sizeof the epidemic (normalized as a fraction of thewhole population), and 𝑓 ( 𝜙 ) = 𝜙 ( 𝑠𝑝 𝑛 )/( − 𝑠𝑝 𝑛 ) is the fraction of the susceptible population thatwas infected in the epidemic occurring in year 𝑛 . A “bad reaction” results when one incurs acost from the vaccine ( e.g., vaccine side effectson health, or economic burden). See Section 3for model details. In the next section, we write down the governing equations for our model. We beginby recalling some standard results for a simple epidemiological model and then couple thatmodel to a social psychological model for individual vaccination decisions.
We choose the susceptible-infected-removed (SIR) model of infectious disease dynamics(Kermack and McKendrick 1927), both because it is a well-established epidemiologicalmodel and because there exists an analytical expression for the final size of epidemicspredicted by the model. The SIR model is adequate for modeling each annual influenzaoutbreak individually, though we note that a more realistic flu model could be substitutedinto our vaccination coverage model, provided that the final size could at least be calculatednumerically.Suppose 𝑆 ( 𝑡 ) and 𝐼 ( 𝑡 ) represent proportions of a population that are susceptible andinfected, respectively, at a time 𝑡 . Infected individuals are assumed to be immediately5nfectious; there is no latency period in this model. There may also be individuals that areremoved from the infection process as they have already recovered from the illness (denotedby proportion 𝑅 ( 𝑡 ) ), but since we assume this disease propagates in a closed population, wehave 𝑅 ( 𝑡 ) = − 𝑆 ( 𝑡 ) − 𝐼 ( 𝑡 ) , which means we do not need to track the removed individualsexplicitly.The SIR model is defined by a set of two coupled, nonlinear, ordinary differentialequations: 𝑑𝑆𝑑𝑡 = − 𝛽𝑆𝐼, (3.1a) 𝑑𝐼𝑑𝑡 = 𝛽𝑆𝐼 − 𝛾 𝐼, (3.1b)where 𝛽 is the disease transmission rate and 𝛾 is the disease recovery rate. The derivedquantity R = 𝛽 / 𝛾 is the basic reproduction number of the disease; it gives the averagenumber of secondary cases generated by an infectious individual in a fully susceptiblepopulation over the course of their illness.In our model, we assume the initial conditions 𝑆 ( ) = − 𝑠 𝑝, < 𝐼 ( ) << , 𝑅 ( ) = 𝑠 𝑝 (3.2)to incorporate a vaccine uptake level at proportion 𝑝 with success probability 𝑠 . For this model, one can derive an implicit equation for the final size of the epidemic 𝜙 ( 𝑥 ) ,where 𝑥 is the proportion initially immune to the disease (Ma and Earn 2006): 𝜙 ( 𝑥 ) = ( − 𝑥 ) ( − 𝑒 −R 𝜙 ( 𝑥 ) ) . (3.3)The solution to Equation 3.3 can be written in terms of the principal branch of theproduct log function ( i.e., the Lambert W-function), denoted by 𝑊 [·] : 𝜙 ( 𝑥 ) = − 𝑥 + R 𝑊 (cid:104) −R ( − 𝑥 ) 𝑒 −R ( − 𝑥 ) (cid:105) (3.4)This expression for the final size of the yearly influenza epidemic is used later inEquation 3.7 to complete the model. 6 .3 Critical vaccination threshold for herd immunity An epidemic cannot be sustained if the average number of secondary cases provoked by aninfected individual in the population is below one (since this infected individual cannot evenreplace themselves in the infection chain, let alone generate further infections). In otherwords, for the population to achieve herd immunity, the effective reproduction number (thebasic reproduction number times the proportion currently susceptible), R eff = R ( − 𝑠 𝑝 ) ,must be driven below 1. Thus the critical vaccination threshold, 𝑝 crit , satisfies the equation R ( − 𝑠 𝑝 crit ) =
1, and so 𝑝 crit = 𝑠 (cid:18) − R (cid:19) . (3.5) R for seasonal influenza Estimates of R for vary depending on year, location, and influenza subtype since the basicreproduction number depends not only on the immunological properties of the virus, butalso on the social behavior of the host population. A systematic review by Biggerstaff et al.(2014) catalogues many estimates of both the basic and effective reproduction numbers ofpandemic, zoonotic, and seasonal influenza.The most relevant estimates for our study are those for the 1976-1981 outbreak ofH1N1/H3N2/B in the USA. Two studies were performed to estimate the basic reproductionnumber in this outbreak, and they both use serologically confirmed infections for their data,which make these estimates particularly reliable . One study found R = .
70 (Fergusonet al. 2006), while another found R = .
16 (Britton and Becker 2000) for this outbreak.We average these two values and take R = . Let 𝑝 𝑛 be the proportion of the population that vaccinates in year 𝑛 . Our goal in this sectionis to derive a discrete map for the vaccine coverage 𝑝 𝑛 + in year 𝑛 + It is difficult to distinguish seasonal influenza from other upper respiratory tract infections by symptomsalone, so studies based purely on reported symptoms may not yield a good estimate for the R of seasonalinfluenza. Instead, studies based on serologically confirmed infections are more reliable. ≤ 𝑟 ≤ ≤ 𝑠 ≤ 𝑓 ( 𝜙 ) to denote the proportion of all susceptible individuals who get sick during an epidemicof size 𝜙 . To calculate 𝑓 in terms of 𝜙 , note that the fraction of the total population that issusceptible in year 𝑛 is 1 − 𝑠 𝑝 𝑛 . Of these individuals, a fraction 𝑓 · ( − 𝑠 𝑝 𝑛 ) will get infected,by definition of 𝑓 . But since this fraction also equals the number of infected individualsdivided by the total population, it simply equals 𝜙 , the fractional size of the epidemic, asgiven by Equation 3.4. Therefore, 𝜙 = 𝑓 · ( − 𝑠 𝑝 𝑛 ) , from which we conclude that 𝑓 ( 𝜙 ( 𝑠 𝑝 𝑛 )) = 𝜙 ( 𝑠 𝑝 𝑛 ) − 𝑠 𝑝 𝑛 . (3.6)In other words, the proportion of susceptible individuals who end up infected is simply thefinal size of the epidemic renormalized to the susceptible population.With these preliminaries out of the way, we can deduce the vaccine coverage rate 𝑝 𝑛 + in year 𝑛 + 𝑛 , and by counting the proportion of the populationflowing down each of the branches in Figure 1 into vaccinating in year 𝑛 +
1. We assumethat every individual is susceptible to that year’s flu strain at the start of each flu season, soeveryone must make the choice of whether or not to vaccinate each year.First consider the group of non-vaccinating individuals, which make up a proportion1 − 𝑝 𝑛 of the population. These individuals will only vaccinate in year 𝑛 + 𝑛 , an event that occurs to a fraction 𝑓 = 𝑓 ( 𝜙 ( 𝑠 𝑝 𝑛 )) of them. Thus, the equation for 𝑝 𝑛 + will include a term 𝑓 · ( − 𝑝 𝑛 ) , which accounts for those that did not vaccinate andgot sick.For vaccinating individuals, either the vaccine succeeds, with probability 𝑠 , or it doesnot, with probability 1 − 𝑠 . If the vaccine succeeds, there is still an independent chance thatthe individual will have side effects that discourage them from vaccinating the followingyear, which occurs at the vaccine morbidity rate, 𝑟 . However, those for whom the vaccinesuccessfully conferred immunity and provoked no side effects will once again vaccinate thefollowing year since they have no reason to change strategy, which adds the term ( − 𝑟 ) · 𝑠 · 𝑝 𝑛 to the equation for 𝑝 𝑛 + .If the vaccine fails for an individual (with probability 1 − 𝑠 ), but did not cause anydiscouraging side effects (with probability 1 − 𝑟 ), the only reason they would continueto vaccinate would be if they thought the vaccine succeeded; that is, they happened not8o get sick, even though they were not successfully immunized. A proportion 1 − 𝑓 of susceptible individuals avoid infection, so the final term of the equation for 𝑝 𝑛 + is ( − 𝑓 ) · ( − 𝑟 ) · ( − 𝑠 ) · 𝑝 𝑛 .Putting all of these contributions together, we find that the discrete map for 𝑝 𝑛 + , theproportion of the population vaccinating in year 𝑛 +
1, is given by 𝑝 𝑛 + = 𝜙 ( 𝑠 𝑝 𝑛 ) − 𝑠 𝑝 𝑛 ( − 𝑝 𝑛 ) + ( − 𝑟 ) 𝑠 𝑝 𝑛 + (cid:20) − 𝜙 ( 𝑠 𝑝 𝑛 ) − 𝑠 𝑝 𝑛 (cid:21) ( − 𝑟 ) ( − 𝑠 ) 𝑝 𝑛 , (3.7)where the function 𝜙 is given by Equation 3.4. This map is biologically sensible; if0 ≤ 𝑝 𝑛 ≤
1, one can check that 0 ≤ 𝑝 𝑛 + ≤
1. Hence, as long as the initial condition issensible (0 ≤ 𝑝 ≤ [ , ] . The predictions of the model depend on the relative magnitudes of its parameters: thevaccine parameters (morbidity or cost, 𝑟 , and success, 𝑠 ), and the disease parameter, R .The basic reproduction number R gives a sense of the “infectiousness” of the disease;in our analysis, we estimate the basic reproduction number of seasonal influenza in amodern US population to be R = . 𝑝 crit . Whenvaccine coverage meets or exceeds this threshold, no epidemic occurs. One might expectthe model to self-organize into herd immunity if the population can collectively make useof the memory of the previous flu seasons in a lasting way.To our disappointment, we find that if there is any cost to the vaccine ( 𝑟 > .0 0.25 0.5 0.75 1.0Vaccine morbidity, r V a cc i n e s u cce ss , s (I) No lastingherd immunity (II) Herd immunityevery other year Figure 2: Long-term model behavior for R = . and p = , as a function of vaccine morbidity( < r ≤ ) and vaccine success ( ≤ s ≤ ). Behavior in these regions of parameter space was deducedby iterating the vaccine coverage map (Equation 3.7) numerically until it converged to a fixed point. Themajority of parameter space is dominated by convergence to vaccine levels below the herd immunity threshold,which results in no lasting herd immunity (region I: the system converges to a period 1 fixed point, 𝑝 ∗ , thatsatisfies 𝑝 ∗ < 𝑝 crit ). For higher vaccine success, there is a possibility of achieving herd immunity every otheryear, provided that vaccine morbidity 𝑟 is sufficiently large (region II: the system converges to a period-2 fixedpoint, ( 𝑝 ∗ , 𝑝 ∗ ) ). In this regime, the system oscillates between sub-optimal vaccine coverage ( 𝑝 ∗ < 𝑝 crit ) andherd immunity with overvaccination ( 𝑝 ∗ > 𝑝 crit ): see Figure 3. igure 3: Vaccine cover-age level over time in theregime where herd im-munity eventually occursevery other year ( R = . , r = . , s = . ). The system converges to astate where the vaccine cov-erage level oscillates asym-metrically about the crit-ical vaccination threshold, 𝑝 = 𝑝 crit , denoted by thedashed line. n . . . . . . P r o p o r t i o n o f p o pu l a t i o n v a cc i n a t e d , p n p crit munity impossible to achieve in the long term, our model predicts that it is possible for thepopulation to achieve herd immunity every other year, even if there is a cost and moderatefailure rate to the vaccine. This oscillatory behavior (Figure 3) is the result of the systemconverging to a state where it alternates between the population bearing a significant diseaseburden (a large epidemic in the previous year encouraging vaccination in the following year)and the population bearing a significant vaccine cost (which incentivizes non-vaccination en masse in the following year). When a vaccine has a moderate-to-high success rate, anda sufficiently high cost, the system is constantly balancing an illness-vaccine cost tradeoff.Even in the case where there is no lasting herd immunity, the system may neverthelessspend a significant length of time in the herd immunity interval before eventually droppingout (Figure 4). This effect is especially pronounced when vaccine morbidity is low, andeven when the vaccine is only moderately successful, both of which are properties of thereal seasonal influenza vaccines. The transient herd immunity period increases as vaccinesuccess increases and/or vaccine cost decreases.When there is no cost to the vaccine ( 𝑟 =
0; Figure 5), the system can self-organizeinto herd immunity in three ways (regions whose label contains “lasting herd immunity”),in addition to yielding no lasting herd immunity as before (region I). The system may startin the herd immunity region and therefore stay in it indefinitely (region II), since there isno vaccine cost to drive the coverage level down. Alternatively, the system may convergeto lasting herd immunity which is either inefficient as it involves overvaccination (regionIII), or it may converge to optimal, lasting herd immunity precisely at the herd immunitythreshold (region IV).The mechanisms that drive the population to either inefficient or optimal self-organized11 .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . r Y e a r s Vaccine success, s Figure 4: Number ofyears spent in the herdimmunity interval ( p >p crit ) during the transientperiod in the regime ofno lasting herd immunity( R = . , p = ). As the vaccine improves inquality (either vaccine mor-bidity decreases, or vaccinesuccess increases), the timeperiod spent in the herd im-munity interval lengthens. herd immunity are markedly different (Figure 6). In the case of sustained overvaccination,the population starts at a relatively low level of vaccination initially (lighter curve). Asubstantial epidemic occurs in the first year, encouraging a large proportion of individualsto vaccinate in the following year: too many, in fact. The population springs itself into theherd immunity interval after that first year, and since there is no opposing force pushingvaccination coverage down, overvaccination continues indefinitely. In the case of optimalvaccination, the population may also start at a relatively low level of initial vaccination, butthe first epidemic sustained is not as devastating as in the previous case (darker curve). Amoderate proportion of the population is affected by the disease and switches to vaccinatingin the following year. The epidemic sustained in this next year is not as large as the onebefore it (thanks to the increase in vaccination), and encourages another (smaller) group ofindividuals to switch to vaccinating next year. This process gradually guides the populationto the herd immunity threshold, eventually achieving the optimal level of vaccination.
While the simple model studied here cannot self-organize into sustained herd immunitywhen there is any cost to the vaccine, it may still achieve herd immunity every other year.When the vaccine success rate is sufficiently large for a given cost, there is an ongoingbattle between the disease and the vaccine. If the population undervaccinates in one year,it undergoes an epidemic which drives the system to overvaccinate in the following year. Anon-trivial proportion of the population then bears some cost associated with the vaccine,which discourages those individuals from getting vaccinated the following year, driving the12 .0 0.25 0.5 0.75 1.0Initial vaccination proportion, p V a cc i n e s u cce ss , s (I) No lastingherd immunity(II) Lastingherd immunityfrom start(III) Converge toine cient, lastingherd immunity(IV) Converge tooptimal, lastingherd immunity Figure 5: Long-term model behavior with no vaccine cost ( r = ) for R = . . Behavior in theseregions of parameter space can be deduced by iterating the vaccine coverage map (Equation 3.7) numericallyuntil it converges to a fixed point, but the regions correspond to the analytical criteria detailed in the Appendix.There is still a region with no lasting herd immunity (region I: the system converges to a fixed point, 𝑝 ∗ ,that satisfies 𝑝 ∗ < 𝑝 crit ); in this regime, the system never achieves herd immunity ( 𝑝 𝑛 < 𝑝 crit for all 𝑛 ≥ 𝑝 ≥ 𝑝 crit ), it may converge to “inefficient” lasting herd immunity (region III: sustained overvaccination), orit may converge to “optimal” lasting herd immunity (region IV: vaccination approaching the herd immunitythreshold 𝑝 crit ). n . . . . . . P r o p o r t i o n o f p o pu l a t i o n v a cc i n a t e d , p n p crit Initial vaccination level, p Figure 6: Vaccine cover-age level over time withno vaccine cost ( r = )in the regime of self-organized herd immunity R = . , s = . ) . Ifthe initial population levelis too low (lighter curve),the population springs intothe interior of the herd im-munity interval, [ 𝑝 crit , ] ,which results in sustainedovervaccination. If thepopulation initially vacci-nates at a more moderatelevel (darker curve), vac-cine coverage converges tothe optimal herd immu-nity threshold, 𝑝 crit , in anasymptotic way. system back down to an undervaccinated state, and the cycle repeats. Provided the amplitudeof this (asymmetric) oscillation about the herd immunity threshold is sufficiently small, thisregime effectively achieves herd immunity as any epidemic that occurs is relatively small.While the goal of disease eradication has not strictly been achieved, the resulting epidemicsare so small in the model that a bit of stochasticity may be enough to push the circulatingflu strain into extinction (in a closed population).Although this promising biannual behavior is possible, the region of no lasting herd im-munity dominates vaccine parameter space, particularly for vaccine morbidity and successlevels that are realistic for the seasonal influenza vaccine (cost near zero, success rate around50%) (Figure 2). The system may not drive itself to herd immunity asymptotically for thesetypes of vaccines, but a significant length of time is spent in the herd immunity intervalduring the transient period (Figure 4). This effect opens the door for other public healthinterventions ( e.g., vaccination- and disease-awareness campaigns) which have not been in-cluded in the model but may help push the population into lasting herd immunity. Increasesto the length of time spent in the herd immunity interval can be achieved by improving thevaccine, by increasing vaccine success, and/or by decreasing vaccine morbidity.In the case where there is no cost to the vaccine, the system can achieve self-organizedherd immunity that is either inefficient (due to overvaccination) or optimal (at the criticalvaccination threshold). If the population initially vaccinates at a very low level, it undergoesa large epidemic, and the following year, the population overreacts, propelling itself intothe herd immunity interval much like a diver on a springboard. Overvaccination continuessince there is no cost to the vaccine, and thus no force pushing population vaccine coverage14own. On the other hand, moderate initial vaccination leads the population to converge tothe optimal vaccination level at the herd immunity threshold. In this case, the populationgradually learns from year to year through successively smaller epidemics. Each suchepidemic recruits smaller and smaller proportions of the population to vaccinate until herdimmunity is achieved. Notably, this result occurs even with a moderately effective vaccine,like that of real seasonal influenza.While it may not be realistic to assume that a vaccine can be considered costless to anindividual, this extreme case illustrates that if a vaccine can be perceived as costless, thepopulation can self-organize into sustained herd immunity. Such a result is still possibleeven if the vaccine is only moderately successful and even if the population does notimmediately take to getting vaccinated. We have presented an intentionally simple model for seasonal influenza vaccination thatchallenges the usual assumption that individuals make use of perfect, global informationcompletely rationally when making annual flu vaccination decisions. The usual assumptionwrongly predicts widespread use of free-rider logic, which is not typically observed in thiscontext. We make use of established results in social psychology to inform our model,which gives rise to both interesting and interpretable dynamics.In the case where there is some cost to the vaccine, our model still predicts regimes wherevaccination coverage is below the herd immunity threshold in the long term, which agreeswith previous models. However, our model also predicts new regions where herd immunityis achieved every other year: a result of the population oscillating between vaccine-basedand disease-based morbidity. When we further assume that the vaccine has no cost, it is stillpossible for the model to predict no lasting herd immunity. We also observe convergence toenduring herd immunity, either at the optimal level, where the population vaccinates exactlyenough to reach this protected state, or inefficiently, where the population overvaccinates.Our disease-behavior model is deliberately simple as a first step, to focus on the effect ofincorporating a more realistic decision model on top of a well-established model for diseasespread. Future work should focus on making this model more realistic and validating itwith appropriate data.Currently, both the decision-making and disease processes are deterministic; a stochasticversion of this model in either respect would be closer to reality. Since agents only relyon the current state of the system to inform their next decision, our model could easily becast in a Markov chain framework. Modeling the disease spread on a socio-spatial network15ould also provide greater realism, by mimicking the way hosts interact and thus spreadinfectious diseases like the flu (Chao et al. 2010).Seasonal influenza is an immensely complex phenomenon, and we have not accountedfor issues such as that of multiple concurrently circulating strains (Prosper et al. 2011),cross-reactivity of vaccines between strains (Iorio et al. 2012; Moa et al. 2016), or waningvaccine immunity over the course of the flu season (Rambhia and Rambhia 2018). Suchadditions to the model would also serve to make it more realistic.
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0, the model map (Equation 3.7) reduces to 𝑝 𝑛 + = 𝜙 ( 𝑠 𝑝 𝑛 ) − 𝑠 𝑝 𝑛 ( − 𝑝 𝑛 ) + 𝑠 𝑝 𝑛 + (cid:20) − 𝜙 ( 𝑠 𝑝 𝑛 ) − 𝑠 𝑝 𝑛 (cid:21) ( − 𝑠 ) 𝑝 𝑛 . (A1)Fixed points, 𝑝 , of this map must satisfy 𝑝 = 𝜙 ( 𝑠 𝑝 ) − 𝑠 𝑝 ( − 𝑝 ) + 𝑠 𝑝 + (cid:20) − 𝜙 ( 𝑠 𝑝 ) − 𝑠 𝑝 (cid:21) ( − 𝑠 ) 𝑝, (A2)which simplifies to 𝜙 ( 𝑠 𝑝 ) − 𝑠 𝑝 ( 𝑝 ( − 𝑠 ) − ) = . (A3)Equation A3 is satisfied when either (i) 𝜙 ( 𝑠 𝑝 ) = 𝑝 ( − 𝑠 ) − = 𝑝 ∈ [ 𝑝 crit , ] since 𝜙 ( 𝑠 𝑝 ) = 𝑝 ≥ 𝑝 crit . Provided 𝑝 crit <
1, the "herd immunity" interval [ 𝑝 crit , ] is an invariant set of neutrally stable fixedpoints that exists in the map’s domain of [ , ] . In other words, if 𝑝 𝑛 ∈ [ 𝑝 crit , ] for any 𝑛 ,the trajectory is then trapped in the herd immunity interval for all remaining time (preciselyat the value 𝑝 𝑛 ).Region II in Figure 5 is given by all ( 𝑝 , 𝑠 ) that satisfy 𝑝 ≥ 𝑝 crit = 𝑠 (cid:16) − R (cid:17) . In otherwords, the population starts at herd immunity and remains at herd immunity indefinitely.Region III is given by all ( 𝑝 , 𝑠 ) such that 𝑝 < 𝑝 crit but 𝑝 ≥ 𝑝 crit . This region representspopulations whose first epidemic was so large that it propels the population into herdimmunity immediately after the first year.Case (ii) is satisfied by 𝑝 ∗ = /( − 𝑠 ) . This fixed point is disjoint from the herd immunityinterval when 𝑝 ∗ < 𝑝 crit , (A4)12 − 𝑠 < 𝑠 (cid:18) − R (cid:19) , (A5) 𝑠 < − R − . (A6)19et us define 𝑠 crit = − /( R − ) ; when 𝑠 ≥ 𝑠 crit , the fixed point 𝑝 ∗ = /( − 𝑠 ) disappearsinto the herd immunity interval.Numerical simulations suggest that 𝑝 ∗ is stable when it exists disjoint from the herd immu-nity interval ( i.e., when 𝑠 < 𝑠 crit ), with the basin of attraction being [ , 𝑝 crit ) ; such trajecto-ries will converge to 𝑝 ∗ as 𝑛 → ∞ . Region I is given by all ( 𝑝 , 𝑠 ) that satisfy 𝑝 < 𝑝 crit and 𝑠 < 𝑠 crit . In other words, vaccine coverage starts below the herd immunity thresholdand never surpasses it. Instead, vaccine coverage converges to 𝑝 ∗ = /( − 𝑠 ) < 𝑝 crit .Lastly, region IV is given by all ( 𝑝 , 𝑠 ) with both 𝑝 < 𝑝 crit and 𝑝 < 𝑝 crit , but also 𝑠 ≥ 𝑠 crit .In this case, the fixed point 𝑝 = /( − 𝑠 ) does not exist distinct from the herd immunityinterval, but also the first epidemic is not strong enough propel vaccine coverage overthe herd immunity threshold ( 𝑝 < 𝑝 crit ). Under these conditions, numerical simulationssuggest that { 𝑝 , 𝑝 , ... } is a monotonically increasing sequence where each 𝑝 𝑛 < 𝑝 crit , butlim 𝑛 →∞ 𝑝 𝑛 = 𝑝 critcrit