Flexible imitation suppresses epidemics through better vaccination
FFlexible imitation suppresses epidemics through better vaccination
Soya Miyoshi, Marko Jusup, and Petter Holme ∗ Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Yokohama, Japan
ABSTRACT
The decision of whether or not to vaccinate is a complex one. It involves the contribution both to a socialgood—herd immunity—and to one’s own well being. It is informed by social influence, personal experience,education, and mass media. In our work, we investigate a situation in which individuals make their choicebased on how social neighbourhood responded to previous epidemics. We do this by proposing a minimal-istic model using components from game theory, network theory and the modelling of epidemic spreading,and opinion dynamics. Individuals can use the information about the neighbourhood in two ways—eitherthey follow the majority or the best-performing neighbour. Furthermore, we let individuals learn which ofthese two decision-making strategies to follow from their experience. Our results show that the flexibility ofindividuals to choose how to integrate information from the neighbourhood increases the vaccine uptakeand decreases the epidemic severity if the following conditions are fulfilled. First, the initial fraction of in-dividuals who imitate the neighbourhood majority should be limited, and second, the memory of previousoutbreaks should be sufficiently long. These results have implications for the acceptance of novel vaccinesand raising awareness about vaccination, while also pointing to promising future research directions.
I. INTRODUCTION
The development of vaccines is one of the greatestachievements of modern medicine. They save millionsof lives yearly, not only by giving immunity to people ex-posed to an infection but also by stopping disease out-breaks. Most famously, perhaps, vaccine drove the eradi-cation of smallpox. [1]
At the time of writing, vaccines are themain hope for a pharmaceutical solution to the COVID-19crisis. [2–4]
To be able to evaluate interventions involving vaccina-tion, we need to model the selection of who gets vac-cinated and how that affects epidemics. [5]
Creating suchmodels is a very challenging task to which this paper seeksto contribute. [6, 7]
The main difficulty lies in the complexfeedback mechanisms between the epidemics itself and thedecision to get vaccinated. [8–10]
Not only are there irra-tional anti-vaccination sentiments that themselves spreadthrough social contagion, [11, 12] but sometimes not gettingvaccinated is a perfectly rational choice.When the population level of immunity is high enough,an outbreak will die out by itself. If this is the case, the pop-ulation is said to have herd immunity . Even if the vaccine iseffective, the marginal benefit of getting vaccinated in a so-ciety with herd immunity is small. [13]
On top of this, a vac-cine could be costly, inefficient, laden with side-effects, orinconvenient to administer. [14]
For an individual, the realityis often between these extremes and not a choice between acheap lifesaver or a costly unnecessity. Thus, this choice isknown in the literature as the vaccination dilemma . [15, 16] Assume the vaccine is effective but has some side-effects(although much milder than the disease itself). Then onecan model the rational-choice aspect of vaccination within ∗ [email protected] the framework of game theory. [7, 17] As often is the case,behavioural and economic game theorists both take aninterest in this problem, with somewhat different start-ing points. The behavioural game-theory line of researchtypically focuses on herd immunity as a public good—something valuable and accessible without competition toanyone in society. [9, 10]
Like other public goods, herd immu-nity is prone to free-riding people who undermine the goodby avoiding vaccination. [18]
Economic game theorists in-stead think of vaccination as a decision based on an individ-ual’s costs and benefits, and regard herd immunity as a pos-itive externality. The truth lies in between these pictures. [8]
Benefiting a public good is not the most common drivingforce behind an individual’s vaccination decisions. [19, 20]
Itwould also make little sense to get vaccinated to contributeto herd immunity if very few others were vaccinated, whichis precisely when the private benefits are the largest. [13]
Onthe other hand, herd immunity and, ultimately, eradicationof a disease are the primary goals at a national level, [3, 4] andthus more than a mere externality.There is a growing research interest in game-theoreticstudies of vaccination. [6, 17]
For articles in the economicgame-theory literature, see e.g. Geoffard and Philipson [8] orFrancis [13] and further references therein. In the evolution-ary game-theory literature, early works coupled game the-ory and epidemic dynamics by differential-equation basedmodels. [21]
Later, authors recognised that social interactionstructures are better modelled by networks. For example,in Fu et al. [15] individuals compare their fitness to randomlyselected network neighbours to determine whether or notto imitate the neighbour. The phrases “fitness” and “payoff”(that in this paper are synonymous) come from the game-theory literature and capture the ability to avoid infectionminus the cost associated with the vaccination. Other au-thors have extended the use of imitation dynamics. Zhanget al. [22] , for example, considered the possibility that deci- a r X i v : . [ q - b i o . P E ] O c t (a) (b)(c) (d) TimeTime TimeTime
FIG. 1.
Time evolution of a simulation run.
We show the outbreak size ((a) and (b)) and the fraction of conformists ((c) and (d)) for a short m = m = c = c = β =
2. The networks are constructed by the Erd˝os–Rényi model with N =
64 nodes and the average degree k =
2, which ensures theexistence of a giant component (that essentially all nodes belong to). Time is measured in the number of vaccination cycles. sions are neighbour-dependent by defining an individual’sfitness depending on their entire neighbourhood. Zhang etal. [23] and Han et al. [24] studied individuals with memory,calculating an individual’s fitness as the weighted averageof the past payoffs. Moreover, models by Xia et al., [25]
Ichi-nose and Kurisaku, [26] and Iwamura and Tanimoto [27] dis-regarded payoffs altogether in certain situations by allow-ing individuals to follow the neighbourhood majority with anon-zero probability, or, otherwise, use fitness comparisonto decide their action.Our study extends the above models by unifying twoideas. First, when making a vaccination decision basedon the performance of their neighbours, individuals canfollow different decision rules. Second, an individual canlearn by experience and thus change its decision rules astime goes on. We use two empirically observed decisionrules. First, individuals can follow the best performer inthe extended neighbourhood (the neighbourhood of a fo-cal node and the node itself). [28, 29]
We call such individuals performists . Second, they can also follow the majority in theextended neighbourhood. [28]
We will refer to individuals fol-lowing this strategy as conformists . Furthermore, we will letthe individuals chose between these two rationales basedon experience. The duration of the individuals’ memory isone of the model-parameters we explore. In the remain-der of this article, we will go over the technical details of our model, present our simulation results, and discuss thebroader implications of our findings.
II. MODEL DESCRIPTION
We construct a network-based epidemiological modelwith vaccination. The model comprises individuals embed-ded in a social network through which epidemics spread.Probably the closest scenario is that of seasonal influenza.Occasionally, the population has the opportunity of gettingvaccinated. Then individuals evaluate the performance oftheir social surroundings, and, based on this information,decide about their vaccination. They have two ways (imita-tion mechanisms) of performing this evaluation—the men-tioned conformist and performist strategies. The individu-als then update these imitation mechanisms based on ex-perience.Each cycle of epidemic outbreaks and vaccinations playsout in three stages. First, individuals choose whether or notto vaccinate following their current strategy and availableinformation. Second, we calculate the expectation value ofthe outbreak size for the particular configuration of vacci-nated nodes. This is done by averaging over 640 runs ofthe Susceptible-Infectious-Recovered (SIR) simulation (dis- (a) (b)(c) (d)
TimeTimeTimeTime
FIG. 2.
Time evolution of typicalruns.
We show the outbreak size((a) and (b)) and the fraction of con-formists ((c) and (d)) for a short m = m = c = c = cussed further below). Finally, individuals choose betweenthe conformist or performist imitation strategy. We simu-late these vaccination cycles over 150 times to reach a steadystate.At the beginning of every simulation, we randomly vac-cinate 10 % of the population and let a fraction c be con- FIG. 3.
Low initial conformism is good, moderate initial con-formism is best, and high initial conformism is bad.
We showthe outbreak size, averaged over the last 50 vaccination cycles, asa function of the infection rate. The three curves correspond tothree different levels of initial conformism. Low initial conformismyields good results in terms of the outbreak size, especially at largeinfection rates ( c = c = c = m = formists (otherwise performists). From the second vaccina-tion, all individuals make decisions following their strate-gies. We allow the choice of strategy to depend on the expe-rience gathered through vaccination. We assume that indi-viduals choose the imitation mechanism that gives them alarger average payoff over the period of m preceding vacci-nation cycles.For the epidemic simulation, we used the stan-dard Markovian Susceptible-Infectious-Recovered (SIR)algorithm. [30] This is the canonical model of diseases thatmake people immune upon recovery. It has nominallytwo parameters—the infection rate β and the recov-ery rate ν . However, ν only sets the time scale of theoutbreak, and since we are only interested in the finaloutbreak size, we can set ν = [30] and implemented at github.com/pholme/sir .When the individuals evaluate their performance, theyassume a cost c V per vaccination and a cost c I for getting in-fected. For an approved vaccine, we can assume c V < c I . Ac-cordingly, a vaccinated individual receives the payoff Π i =− c V per vaccination cycle, whereas someone who is not vac-cinated gets the payoff Π i = − c I I ( i ), where I ( i ) is the frac-tion of outbreak simulations when i got infected. We set c V = c I = N =
64 nodes and equallymany edges. We try other, larger, sizes as well, but these givequalitatively similar results. Since this paper will not con-cern the large-size limit, to cut computation times, we stickwith the smaller systems.We will primarily explore three parameters in oursimulations— c , m , and β and average the simulations de- (a) (b)(c) (d) FIG. 4.
Updating of imitation mechanisms often leads to disease-curbing, moderate levels of conformism, but only if memory is longenough.
We show heatmaps of the outbreak size ((a) and (b)) and the fraction of conformists ((c) and (d)) as a function of the infectionrate and initial conformism. When the infection rate is low, the outbreak size is independent of the initial conformism, which is evidentfrom the horizontal contour curves for β (cid:47) scribed above over 10 simulation runs. We use the OACISframework [31] to manage our simulations. FIG. 5.
Long memory is better than short.
The figure shows theaverage outbreak size as a function of the infection rate. The threecurves correspond to three different memory lengths. The out-break size for infection rates β (cid:39) c = III. RESULTS
Time evolution.
In Fig. 1, we show a representative sim-ulation run of the model. We find that the outbreak sizecannot be reduced by having a large initial fraction of con-formists, irrespective of the memory length (Fig. 1, yellowcurves in (a) and (b)). Conversely, if the initial presenceof conformists is moderate, the outbreak size is greatly re-duced (Fig. 1, blue curves in (a) and (b)). Short memoryturns out to have almost no effect on the choice of imita-tion mechanisms, causing the population to end with a sim-ilar fraction of conformists as in the initial state (Fig. 1(c)).Increasing memory changes the situation. In this case, theupdating of imitation mechanisms increases, allowing thepopulation to converge to a fraction of conformists notice-ably different than in the initial state (Fig. 1(d)). The popu-lation is even able to eliminate outbreaks in the case whenthe initial fraction of conformists is moderate (Fig. 1, bluecurve in panel (b)). We confirm that the selected simula-tion run is indeed representative by averaging across a largenumber of runs (Fig. 2), which exhibit the same qualitativecharacteristics as the described single run.
Initial conformism.
To further clarify the role of initialconformism, we inspect how the outbreak size depends onthe infection rate for various values of the parameter c whenmemory is relatively long (Fig. 3). At low infection rates, β (cid:47) (a) (b)(c) (d) FIG. 6.
The level of conformism evolves only if memory is long enough and initial conformism is not moderate.
In panels (a) and (b), weshow heatmaps of the outbreak size. In panels (c) and (d), we show the fraction of conformists as the functions of the infection rate and thememory length. As before, when the infection rate is low, the outbreak size is independent of the memory length, which can be seen by thehorizontal contour curves for β (cid:47) m < m >
4, drives conformism to moderate levels between 40–60 % (for β (cid:39) β and m . formism. After that, for β (cid:39) < c < c .A detailed examination of the phase space reveals thata population of individuals who dynamically interchangeimitation mechanisms, as envisioned in our model, oftenevolves to moderate levels of conformism, but only if mem-ory is long enough (Fig. 4). The outbreak size is thus reducedin populations with longer memory relative to those withshorter memory, for all infection rates β (cid:39) β (cid:39) (cid:47) c (cid:47) Memory length.
Our results so far have suggested thatmemory length plays a vital role in shaping the fate of dis-ease outbreaks. We now inspect this role in further detail.In particular, looking at how the population performs as thememory length increases, we find that long memory is bet-ter than short, even in situations when initial conformism is in the disease-curbing, moderate range (Fig. 5). The effect iseven more pronounced when initial conformism is outsideof the moderate range (Fig. 6(a)), the reason being that longmemory, characterised by m >
4, entices a substantial dy-namical interchange of imitation rules when the infectionrate is sufficiently large ( β (cid:39) m <
3, cannotachieve the same as initial conformism is largely preservedthroughout simulation runs (Fig. 6(c)). That moderate con-formism indeed curbs the disease effectively is revealed bysimulation runs for c = Network structure.
To confirm that the described re-sults are robust to the choice of network structure, we haverun additional simulations on two-dimensional square lat-tices and scale-free networks. Interestingly, only the formerstructure exhibits any noteworthy differences compared tothe Erd˝os-Rényi networks. Our previous observation thata longer memory length reduces the outbreak size as theinfection rate increases is valid for all networks structures(Fig. 7(a) and (b)). The square lattice quantitatively dif-fers from the Erd˝os-Rényi network in that the former has amuch larger diameter which, for a given infection rate, im-pedes disease outbreaks. This, in turn, somewhat weakens (c)(a) (d)(b)
FIG. 7.
Network structure quantita-tively impacts the results, but gen-eral conclusions stand.
In panels(a) and (b), we display the outbreaksize and in panels (c) and (d) thevaccination coverage as functions ofthe rate of infection for two differ-ent network structures. Panels (a)and (c) pertain to the square latticewith a periodic boundary, whereaspanels (b) and (d) pertain to theErd˝os–Rényi network of the averagedegree k =
2. Our observation that alonger memory length decreases theoutbreak size as the rate of infectionincreases is valid irrespective of thenetwork structure. Quantitative dif-ferences between networks arise be-cause the lattice has a large diam-eter that impedes outbreaks giventhe same infection rate. Initial con-formism is c = N = the effect of long memory on suppressing diseases. Indeed,a longer memory always leads to more vaccination cover-age, but such coverage is much smaller in the lattice than inthe Erd˝os-Rényi network when infection rates are relativelylarge (Fig. 7(c) and (d)). Individuals thus have an easier timelearning the best course of action to combat epidemics intighter, more compact networks. IV. DISCUSSION
We have constructed a model of vaccination—based inequal proportions on game theory, network epidemiology,and models of social influence—in which individuals in ad-dition to deciding whether to vaccinate or not, also havethe option to choose a preferred imitation mechanism. In-dividuals called conformists thus rely on a simple heuristicby which they imitate the behaviour of the neighbourhoodmajority. Performists, by contrast, imitate neighbours thatperform the best in terms of payoff. The results show thatthe dynamic interchange of these imitation mechanismssuppresses disease outbreaks through better vaccination.Two notable phenomena are that (i) too much initial con-formism in the population and (ii) short memory of indi-viduals are obstructive for the vaccination coverage.The reason why too much conformism leads to a lowvaccination coverage is that too many individuals imitatewhat they see in their neighbourhoods, and they mostlysee defection. This, in turn, is because our simulations start with the initial vaccination coverage of 10 %, whichis a valid starting point from the perspective of novel vac-cine acceptance. Such an angle is relevant at the timeof writing. On the one hand, we are amidst a pandemic(COVID-19) for which vaccines seem like the best hope fora solution. [2–4]
On the other hand, recent years have seena trend of increasing vaccine hesitancy. [10, 20, 32]
This trendcould partly be explained by the kind of mix of rationalthinking and predisposition to following the crowd as ismanifested by the conformists. [11, 33]
However, probably itwould be more appropriate to extend our model to includezealots—individuals who do not let the dynamics of thegame to affect their choices. [14, 18]
The other notable phenomenon revealed by our model,specifically, that more extended memory helps boost thevaccination coverage, is suggestive in the sense that edu-cational measures could be used to bolster the collectiveawareness of the role of vaccines in controlling infectiousdiseases. [19]
Classrooms are an ideal setting to raise aware-ness about the burden of infectious diseases as well as thesuccess of vaccines in limiting these. [1, 34]
Finally, we note that in our simulations, the individualslearn strategies to make the vaccination choice. This meansthey are manifesting “wisdom of crowds” [35] —i.e., that pop-ulations can perform distributed computation tasks, inte-grating information without central control.
ARTICLE INFORMATION
Acknowledgements
Authors appreciate the supportfrom the Japan Society for the Promotion of Science (KAK-ENHI grants 20H04288 and 18H01655) and the SumitomoFoundation (grant for basic science research projects).
Author contributions
M. J. and P. H. devised research.S. M. performed research. All authors discussed the resultsand wrote the manuscript.
Conflict of interest
Authors declare no conflict of inter-est. [1] D. A. Henderson,
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