Effects of confinement and vaccination on an epidemic outburst: a statistical mechanics approach
?scar Toledano, Begoña Mula, Silvia N. Santalla, Javier Rodríguez-Laguna, ?scar Gálvez
EEffects of confinement and vaccination on an epidemic outburst: a statisticalmechanics approach ´Oscar Toledano, Bego˜na Mula, Silvia N. Santalla, Javier Rodr´ıguez-Laguna, and ´Oscar G´alvez Dto. F´ısica Interdisciplinar, UNED, Madrid (Spain) Dto. F´ısica Fundamental, UNED, Madrid (Spain) Dto. F´ısica & Grupo Interdisciplinar de Sistemas Complejos,Universidad Carlos III de Madrid, Legan´es (Spain) (Dated: February 20, 2021)This work describes a simple agent model for the spread of an epidemic outburst, with specialemphasis on mobility and geographical considerations, which we characterize via statistical mechan-ics and numerical simulations. As the mobility is decreased, a percolation phase transition is foundseparating a free-propagation phase in which the outburst spreads without finding spatial barriersand a localized phase in which the outburst dies off. Interestingly, the number of infected agents issubject to maximal fluctuations at the transition point, building upon the unpredictability of theevolution of an epidemic outburst. Our model also lends itself to test with vaccination schedules.Indeed, it has been suggested that if a vaccine is available but scarce it is convenient to selectcarefully the vaccination program to maximize the chances of halting the outburst. We discussand evaluate several schemes, with special interest on how the percolation transition point can beshifted, allowing for higher mobility without epidemiological impact.
I. INTRODUCTION
Epidemic containment has been a crucial problem forhumankind throughout our history, which has becomeof paramount importance since the COVID-19 outbreaklate in 2019. The efforts to understand the spread ofinfectious diseases have attracted a large variety of pro-fessionals from all scientific fields, ranging from biology tosociology (see recent studies from different fields: [1–3]).Mathematical modeling has also provided very relevanttools to analyze the stream of data concerning the in-fected population, and has substantially contributed topolicy design (see e.g. [5, 6, 15]). Among different ap-proximations employed, agent-based models have beenextensively used to provide policy recommendations evenin the COVID-19 case [7], for which simulations based onstratified population dynamics were recently carried out[8]. Multiscale approaches are known to improve our abil-ity to explain the geographical expansion of the disease,and they have been also used, along data-driven simula-tions, to analyze the epidemic of COVID-19 in Brazil [9].Epidemic waves have also been considered [9, 10] employ-ing stratified population dynamics and non-autonomousdynamics, where mitigation effects are subsequently im-posed and relaxed.During the COVID-19 epidemic, and due to the scarceamount of vaccination doses in the first instants of thevaccination campaign, immunization schedules have alsoattracted attention of the modeling community with theaim of stifling the expansion of an epidemic burst by re-moving a number of nodes substantially below the perco-lation threshold [11] through the search of certain typesof motifs in the contact network.However, predictions about the evolution of an epi-demic burst are known to be difficult and unreliable,because the uncertainty in the initial data propagates exponentially [12]. Yet, on occasions the inherently un-predictability of the models can be turned in our favor.Here we claim that the fluctuations in the number of in-fected people during the evolution of an epidemic burstcan provide useful information regarding our ability tostifle an ongoing epidemic.In this work, we propose a very simple agent model [13]of the susceptible-infected-recovered (SIR) type [14],[15].Agents follow a random walk in the vicinity of theirhomes, which are randomly distributed on a square lat-tice, wandering up to a maximum distance R , whichmay vary from agent to agent. When R is very short,agents are effectively confined at home, and an infec-tious outbreak will very likely die off. As this distanceis increased, outbreaks have larger chances of spreadingthroughout the system, until a percolation phase transi-tion is reached, [16] in which the presence of an infinitecluster becomes certain. Further increases of mobilitywill have a very limited impact on the spread of the in-fectious disease. Interestingly, predictions about the fu-ture evolution of the epidemic are more difficult at thetransition point. Indeed, above the percolation thresholdthe mean-field theory associated with the SIR equationsprovide a very accurate prediction of the evolution of themodel, while below the transition point typical outburstswill not propagate beyond a certain correlation length.But when the agent mobility approaches to the percola-tion value, the precise geographical origin of the outbreakbecomes crucial to predict the outcome. Even though aninfinite connected cluster exists, the probability that theinitial infected agent will be part of it is minimal at thatpoint, leading to maximal uncertainty. Thus, we showthat the fluctuations in the number of infected peoplebecome a very useful observable in order to pinpoint thephase transition.Of course, our model is far too simple to be taken deci-sively in order to provide policy recommendations, which a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b should be always undertaken with the assessment of ex-perts from several fields. Yet, our results suggest thatlock-down measures are either useless, when the systemremains much above the percolation threshold, or effec-tive, once it has been crossed. Finding that phase tran-sition is therefore of paramount importance, and a verydifficult task. We can not provide a complete recipe toestablish that threshold in practice, but we provide aninteresting proxy: fluctuations in the infection reach.If an effective vaccine is available but scarce, a vacci-nation schedule becomes unavoidable, i.e.: the determi-nation of the agents which must be immunized in orderto minimize the spread of the infection. Of course, manyindividuals should receive the vaccine because of otherconsiderations, such as age, health condition or profes-sion and we will not discuss these very relevant aspects.Our model lends itself very easily to the evaluation of theefficiency of a vaccination program. In it, individuals areonly distinguished through their relations: some homesare relatively isolated, so their inhabitants are not likelyto spread the disease. Yet, a naive approach would con-sider that individuals with many connections should bethe first candidates to be immunized. We will prove thatthis criterion is not optimal. Indeed, some agents withfew connections act as natural bridges between differentclusters. Thus, if they receive the vaccine the clusterswill become isolated and the outburst will be halted.This article is organized as follows. Sec. II discusses indetail our agent model, and how our numerical simula-tions are performed. In Sec. III and IV we describe ourresults regarding the percolation phase transition as theagent mobility or recuperation probability are increased.The efficiency of several vaccination schemes is discussedin Sec. V, specially in connection with the shift of thepercolation threshold. Our conclusions and ideas for fu-ture work are discussed in Sec. VI. II. THE CONFINED-SIR MODEL
We propose a very simple model to characterize the ef-fects of partial confinement in epidemic expansion, whichwe call the confined-SIR model.
A. Description of the model
Let us consider N agents moving on an L × L squarelattice with periodic boundary conditions. Agent i pos-sesses a home , determined by a fixed lattice point, (cid:126)H i .Agents can move freely within their wandering circles ,centered at (cid:126)H i and with radius R i . Time advances indiscrete steps ∆ t = 1, in arbitrary units. At each time,each agent takes a random step within their wanderingcircles with equal probability, and if the step leads to aforbidden position, a new step is attempted. We will as-sume that homes are distributed randomly over the wholelattice. Figure 1. Illustration of the confined SIR model. The redagent at the center is infected, and can move freely within itswandering circle, shown in magenta. The blue agents are sus-ceptible, and can move within their wandering circles, show-ing in cyan. Notice that the magenta circle intersects all cyancircles, thus providing a finite probability of infection for allthose agents.
Agents can be classified into three groups: susceptible( S ), infected ( I ) and recovered ( R ). Susceptible agentsget infected with probability β when they share cell withan infected agent. Infected agents may get recovered ateach time-step, with probability γ . An illustration canbe seen in Fig. 1.Our aim is to determine the probability distributionfor the number of infected agents, with special interestin the asymptotic values, i.e. the total number of agentsthat have suffered the infection in the long run. In somecases, an effective SIR model can be written down andsolved analytically.The dynamics of the model can be approximated by aMarkov chain. Let us consider that, at time t , we knowthe epidemiological status of each agent, s i ∈ { S, I, R } ,the location of their home H i and their wandering radii, R i . From these data we can estimate the transition prob-abilities for each agent.Let A i stand for the number of lattice points in thewandering circle of agent i , while A ij will stand for thenumber of lattice points in the intersection between thewandering circles of agents i and j . The probability thatagent i and agent j will collide at time t can be estimatedas C ij = A ij A i A j . (1)Thus, if agent s i = I and s j = S , the probability perunit time that agent j will get infected becomes P Iij = βC ij . Thus, we can find the probability per unit timethat an agent i ∈ S will get infected just summing overall possible infection sources P Ii = β (cid:88) j ∈ I C ij = β (cid:88) j ∈ I A ij A i A j . (2)The evolution of the expected number of infectedagents can now be found, (cid:104) I ( t + 1) (cid:105) = (1 − γ ) (cid:104) I ( t ) (cid:105) + β (cid:42)(cid:88) i ∈ Sj ∈ I C ij (cid:43) , (3)Notice that the C ij do not change , but the right handside changes due to the evolution in the identity of theagents in sets I and S .In the homogeneous limit, R i ∼ L and all agents wan-der around the whole region. Therefore, all C ij ∼ /L .Thus, in a mean-field approximation Eq. (3) becomes (cid:104) I ( t + 1) (cid:105) ≈ (1 − γ ) (cid:104) I ( t ) (cid:105) + βL (cid:104) S ( t ) (cid:105)(cid:104) I ( t ) (cid:105) , (4)and we obtain the usual SIR differential equations. In-troducing fractional variables, s = (cid:104) S (cid:105) /N , i = (cid:104) I (cid:105) /N and r = (cid:104) R (cid:105) /N we reach: s ( t + 1) ≈ s ( t )(1 − βρi ( t )) (5) i ( t + 1) ≈ (1 − γ ) i ( t ) + βρs ( t ) i ( t ) (6) r ( t + 1) ≈ r ( t ) + γi ( t ) , (7)where ρ is the population density defined as: ρ = N/L . (8)For early times, s ( t ) ≈ R = βργ . (9)We will always assume R >
1, i.e.: the infection willpropagate in the total mixing approximation.
B. Effective network
Within the confined-SIR model, the infection can prop-agate between agents i and j only if C ij >
0. This condi-tion determines an effective network, G , such as the onesshown in Fig. 2, in which the homes are represented as Figure 2. Effective networks obtained from a confined-SIRsystem using N = 100 homes on a 50 ×
50 lattice. Nodesare associated with the homes of the agents, while links aredrawn if their respective dwellers are able to meet during theirwandering. Top: using R = 2; bottom: using R = 5. nodes of a graph, and the links denote the pairs of agentsthat are able to meet. Both networks are obtained for N = 100 agents on 50 ×
50 lattices, using R = 2 (top)and R = 5 (bottom). It can be readily seen that thenetwork for R = 2 contains many disconnected clusters,while for R = 5 it contains a large cluster spanning mostof the nodes. C. Numerical simulations
We have run numerical simulations of our confined-SIR model on a square lattice, placing N random homesand using the same value of the wandering radius R forall agents. Unless otherwise specified, we always averageover N S = 5000 samples, with fixed values for β = 1 and γ = 0, N = 1000 agents and ρ = 5 · − . Thus, theaverage distance between homes, ¯ r (eq. 10), and the sizeof the lattice are determined by density values and thenumber of agents considered.In the case of large values of the wandering radius R ,we expect the behavior to correspond to the mean-fieldapproximation provided by the SIR equations for perfectmixing. We have checked that conjecture numerically,and the results are shown in Fig. 3 (top panel). The cor-respondence between our parameters and the parametersof the SIR model is immediate. However, for low values ofthe mobility parameter ε , we see in bottom panel of Fig.3 that the average curve significantly deviates from thesolutions of the SIR equations because the assumptionsof perfect mixing do not hold. III. PERCOLATION PHASE TRANSITION
The theory of bond percolation has been one of theforemost paradigms of statistical mechanics for morethan fourty years [17–20], with applications to magnetism[21], wireless communications [22], ecological competition[23] or sequence alignment in molecular biology [24]. Re-cently, a very relevant connection was described betweenthe geodesics in strongly disordered networks and bondpercolation [29]. Application of percolation theory to epi-demics has been carried out by previous authors, such as[25, 26] or more recently [27, 28].Bond percolation on a fixed lattice is characterized bya single parameter, p , the probability that a given bondwill be present. Above a certain threshold value, p > p c ,the probability that the system will contain an infiniteconnected cluster reaches one, with p c = 1 / p . Let us consider N agents on an L × L lattice. The average distance betweenhomes, ¯ r , can be estimated as¯ r = L √ N . (10)Now, let us notice that the wandering radius R is onlymeaningful when compared to this average distance be-tween homes. Thus, we introduce a mobility parameter , ε = R ¯ r = R √ NL . (11)
Figure 3. Top: Average number of susceptible, infected andrecovered agents as a function of time, depicted with contin-uous black, red and blue lines respectively, as obtained using β = 0 . γ = 5 · − , L = 141, N = 1000 and ε = 9 . ε = 2 .
0, arerepresented. Probability distributions of the values obtainedin the simulations are also represented with the logarithmiccolor scale.
We will readily show that the mobility parameter ε is theonly relevant variable to determine the geometry of oursystem.Critical points, such as the percolation transition, typi-cally lead to large fluctuations. Thus, we have considereda new observable: the standard deviation of the fractionof infected agents, σ I , which is shown in Fig. 4 (toppanel) as a function of the mobility parameter, for dif-ferent lattice sizes and densities. Indeed, we can observethat the fluctuations in the number of infected agentspresent a maximum for a certain value of ε which onlydepends on the ratio between the recovery and infectionprobabilities, β/γ .Let us provide evidence that this critical value of themobility parameter, ε c , corresponds to the position of thepercolation phase transition. In the vicinity of the per-colation phase-transition many observables show criticalbehavior in the form of power laws. The most salient of . . . . . . . .
35 0 . .
45 0 . .
55 0 . .
65 0 . σ I εN = 500 N = 1000 N = 1500 N = 2000 N = 4000 N = 8000 σ I ερ = 5 · − ρ = 1 · − ρ = 5 · − ρ = 1 · − ρ = 5 · − ρ = 1 · − Figure 4. Top: Standard deviation of the fraction of infectedagents as a function of the mobility parameter ε , for differentnumbers of agents and a fixed density ρ = N/L = 5 · − .Bottom: Standard deviation of the number of infected agentsas a function of the mobility parameter using N = 1000 agentsfor several values of the agent density ρ . Notice the collapseof all curves, using β = 1 and γ = 0. those is given by the average size of a cluster, s . Belowthe transition, we have (cid:104) s (cid:105) ∼ | p − p c | − η , (12)with η = 43 / ≈ .
39 for a square lattice [16]. Fig. 5shows the average cluster size of our effective networksas a function of the mobility parameter ε , for differentsystem sizes and populations. We can observe that, forall system sizes considered, the average size of the clus-ter diverges as we approach a critical value ε c , with anexponent which slightly differs from the value obtainedin the square lattice, η ≈ .
35. This new value for thecritical exponent η seems to be robust under changes inthe lattice size and the density. .
01 0 . h s i | ε − ε c | N=500N=1000N=1500N=2000N=4000N=8000 η = 2 . , ε c = 0 . Figure 5. Average cluster size as a function of ε − ε c , using ρ = 5 · − , β = 1 and γ = 0. Notice the power-law behavior,following Eq. (12), with an exponent η ≈ . IV. NON-ZERO RECUPERATIONPROBABILITY
It is relevant to discuss how does the situation changewhen there is a non-zero recuperation probability, γ > τ ij ≈ C − ij is the expected time be-fore the infection may propagate from agent i to agent j (or viceversa), while τ R ≈ γ − corresponds to the ex-pected time before recovery. Thus, if τ ij (cid:29) τ R we may as-sume that the infection will not be able to propagate fromagent i to agent j , while in the opposite case, τ R (cid:29) τ ij we may neglect the possibility of recovery. We may claimthat a finite recuperation probability provides an effectivecutoff for the local infection probabilities, thus removingweak links from the graph.Fig. 6 (top panel) shows the standard deviation or fluc-tuations in the number of infected nodes as a function ofthe mobility parameter as we increase the recovery prob-ability per unit time, γ . We notice that the fluctuationlevel at the maximum does not depend strongly on γ . Inaddition, the value of mobility parameter at which thismaximum takes place grows quickly with γ . Thus, weare led to claim that the recovery probability per unittime affects the value of the percolation threshold. Alsowe can see in Fig. 6 that for low γ values the fluctuationspeak in a maximum as it was shown in Fig. 4, while it isreach a plateau for higher values of γ . When the numberof final infected agent along increasing γ values is calcu-lated (bottom panel of Fig. 6), we observed a continuousdecrease of (cid:104) s (cid:105) for higher recuperation probability, as ex-pected. . . . . . . . . .
45 0 . . . σ I ε γ = 0 γ = 5 · − γ = 1 · − γ = 2 . · − γ = 5 · − γ = 1 · − . . . h s i ε γ = 0 γ = 5 · − γ = 1 · − γ = 2 . · − γ = 5 · − γ = 1 · − Figure 6. Top: Standard deviation of the number of infectednodes as function of the mobility parameter for different val-ues of the γ parameter. Bottom: number of final infectedagents as function of the mobility parameter for different val-ues of the γ parameter. Simulations were performed with N = 1000, N S = 5000 samples, β = 1 and ρ = 5 · − . V. VACCINATION SCHEDULES
Let us consider the possibility of providing immunitythrough vaccination to a certain (small) fraction of thepopulation, f v . In this section we will attempt to answerthe following questions: how to select the agents thatwill receive the vaccine, if our only aim is to minimizethe spread of a future epidemic outburst. Notice that,in practice, many other issues must be considered in thissituation, such as the health conditions or the age of thepatients.The simplest vaccination schedule is merely to selectrandomly the individuals. Of course, we do not expectthis schedule to be very efficient. We have consideredseveral observables which can be employed to determinehow useful it will be to provide the vaccine to a certainagent. The simplest one is the total degree , defined as thenumber of first neighbors in the effective graph. Naively,we might sort the agents by their degrees, and immunizethe first f v N . Yet, it is more efficient to recompute thedegree of each agent after each selection, and we will doso unless otherwise specified. We can also define a fragmentation degree associated toan agent, intuitively defined as the relative increment ofthe local distance when he/she is removed. For any agent i , we can define its local distance S i ( A i ) as sum of all thedistances between the agents A i = { j | C ij > , j (cid:54) = i } ,i.e. the neighbors of i . Fragmentation of agent i is thusdefined as F i = 1 − S ( A i ∪ i ) S ( A i ) (13)The most promising observable is, nonetheless, the betweenness-centrality [30] associated to each agent, BC i ,defined as the fraction of the total number of geodesicswhich go through agent i . This measure, indeed, isglobal, and takes O ( N ) steps to compute using Dijk-stra’s algorithm to evaluate the geodesics, while we areconsidering the case where the BC i is recalculated af-ter each vaccination [31]. In previous studies vaccinationschemes based on immunizing the highest-betweennesslinks have proved to bet very efficient (see e.g. [32]).Thus, we have considered the following five vaccinationschedules.0. No vaccination, considered the base case.1. Select randomly.2. Select the agents with highest degree (HD).3. Select the agents with highest fragmentation (HF).4. Select the agents with highest betweenness-centrality (BC).The vaccination programs effectively change the topo-logical properties of the network whenever the removedagents are not selected at random. Thus, in Fig. 7 wecan observe an specific example of a network where dif-ferent vaccination schemes have been performed over thesame amount of agents. In order to get insight, let usobserve the cluster structure. For random vaccination,Fig. 7 (A), the large clusters remain untouched. For adegree-based vaccination scheme, Fig. 7 (B), we can seethat links have been removed from the core of the clus-ters, but the clusters themselves remain connected. In-deed, immunizing the individuals with a large number ofconnections seems to have a low impact on the networkstructure in our case. The reason is that high-degreeagents tend to be neighbors of other high-degree agents.Panel (C) of Fig. 7, on the other hand, shows that remov-ing agents with a large fragmentation fraction can leadto much smaller clusters, but still some very large clus-ters remain active, leading to a likely propagation of anepidemic outburst to a large fraction of the population.Fig. 7 (D) shows the resulting network when the agentswith a largest betweenness-centrality have been removed(with recalculation), and we can readily see that all largeclusters have indeed disappeared. (A ) (B ) (C ) (D ) Figure 7. Effect of different vaccination schedules on a fixed network: (A) Random vaccination, (B) Highest degree, (C)Highest fragmentation, (D) Highest betweenness-centrality. Black dots denote susceptible agents, and they are joined byblack links. Immunized agents are colored green, and the dead links are green. The parameters for all cases are N = 1000, ε = 0 .
89 and a 15% vaccination rate. Videos of numerical simulations for these vaccinations schemes can been seen in https://sites.google.com/view/epidemic-modelling-uned/home [33]
In order to describe the quantitative effect of the differ-ent vaccination schedules, the top panel of Fig. 8 depictsthe expected value of the total maximal number of in-fected agents as a function of the mobility parameter.The no vaccination case, marked with the black curve,reaches the total number of agents, N = 1000, with thetransition at around ε c ≈ .
6. The effect of the ran-dom vaccination scheme is analogous to that obtained byperforming a reduction of the population density with a factor 1 − f v , and thus the mobility parameter at the tran-sition point will follow the relation ε c = ε c (1 − f v ) − / ,where ε c stands for the critical value of the mobility pa-rameter when no agents are vaccinated. Interestingly,random vaccination and highest degree vaccination pro-vide similar outcomes, with a slightly higher critical valueof the mobility parameter for the random case. Indeed,this shows that highest degree vaccination is not effectiveat all. Highest fragmentation vaccination, on the other . . . . . . . h s i ε BCRandomHDHFNo Vaccination0 . .
81 0 5 10 15 20 ε c % vaccinatedBCRandomHDHF Figure 8. Top: Number of final infected agents for each vac-cination schedule as a function of the mobility parameter ε with 5% of vaccinated agents. Bottom: Mobility parametermarking the percolation phase transition, ε c , for each vac-cination schedule along on the fraction of vaccinated agentsconsidered. hand, results in a substantial shift of the critical mobil-ity parameter, being necessary to vaccinate only about1 / /
10 to obtain the same ε c than in the case of random vaccination. This resultis in agreement with previous studies [32]. Indeed, allschemes seem to reach a similar value of the total numberof infected agents for large enough mobility parameters.Thus, we conclude that an effective vaccination schemewill increase substantially the mobility threshold underwhich an epidemic outburst will die off. Yet, the effectsof vaccination can be substantially reduced above thatmobility threshold.How effective are the vaccination schedules shifting thepercolation transition point? In order to answer thatquestion we have traced the percolation threshold as afunction of the ratio of immunized agents in the bottompanel of Fig. 8. The base value, for no vaccination, is ε c ≈ . ε c ≈ . ≈
10% vaccinationrate. Highest-fragmentation vaccination performs worse(0 .
74 for ≈
10% vaccination), but still provides a sig-nificant improvement of the epidemiological situation forscarce vaccines. Random and high degree vaccinationsperform similarly (0 .
62 for ≈
10% vaccination), and noneof them are quite effective.
VI. CONCLUSIONS AND FURTHER WORK
In this work we have presented a very simple agentmodel for epidemic expansion in which the effects of par-tial confinements and vaccinations can be tested, and wehave characterized its properties using tools from statisti-cal mechanics, specially from percolation theory. Pleasenotice that our model is far too simple to lead to pol-icy recommendations without further insight from ex-perts from different fields, ranging from virology to soci-ology. The main shortcoming of our model when appliedto human epidemics is that human mobility is not geo-graphically restricted in the same way as in our model.Indeed, even under lock-down essential workers must at-tend their workplaces, which might be very far away fromtheir homes. Yet, we expect that some of our conclusionscan be interesting for researchers in epidemic expansionand lead, in combination with insights from other spe-cialists, to sensible policy recommendations which willhelp alleviate the effects of present and future epidemicoutbursts.Our first conclusion, in the confined-SIR model, is thatconfinement is basically useless much above the percola-tion threshold, and basically effective reasonably belowit. Thus, it is of paramount importance to design lock-down measures so that mobility is restricted sufficientlybelow the percolation threshold, but it is useless to re-duce mobility much below that point. The determinationof the percolation threshold is not easy in practice, buta good hint is provided by the fluctuations in the sizesof the outbursts. Above the transition point, most out-bursts reach a huge fraction of the population, and be-low it, it will only reach a few individuals. Yet, near thetransition, the number of infected agents may vary enor-mously, depending on the location of the initial infectedagent. In addition, the effect of increasing the recoveryprobability in our model causes a decrease of the numberof final infected agents since it provides a efficient cut-off for the infection probability between pairs of agentswhich present low overlapping in their wandering areas.Our second conclusion is that in order to determine anefficient vaccination schedule, disregarding other health-care considerations, bridge individuals should be tar-geted for vaccination, i.e. individuals which move be-tween different clusters. Their immunization will leadto an effective confinement of an epidemic outburst toits initial cluster, thus creating effective firewalls be-tween them. Bridge individuals can be detected via betweenness-centrality , which is a global and computa-tionally expensive measure, or through easier proxies,such as the individual fragmentation . Fragmentation an-swers the question: are your friends friends among them?
Individuals whose friends form a clique are not good can-didates for vaccination, but individuals whose friends donot know each other are.Confined SIR models seem an appropriate tool in or-der to improve our intuition regarding the effectivenessof different strategies to stifle an epidemic outburst andto find relevant observables in order to provide relevantobservables which characterize the current situation andallow us to make meaningful predictions. For example,in this work we have emphasized the very interesting roleprovided by the fluctuations in the maximal number of infected agents. Indeed, a very promising line of researchis the statistical analysis of these fluctuations during realepidemic outbursts, such as COVID-19. These analysispresent a very interesting challenge: fluctuations shouldbe compared ceteris paribus , i.e. removing major differ-ences between the different geographical areas and times.
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