Exploiting Cellular Data for Disease Containment and ...pejovicv/docs/Lima2013.pdf · Exploiting Cellular Data for Disease Containment and Information Campaigns Strategies in Country-Wide

Exploiting Cellular Data for Disease Containment andInformation Campaigns Strategies in Country-Wide Epidemics

A. Lima, M. De Domenico, V. Pejovic, and M. MusolesiSchool of Computer Science, University of Birmingham, United Kingdom

Human mobility is one of the key factors at the basis of the spreading of diseases in a popula-tion. Containment strategies are usually devised on movement scenarios based on coarse-grainedassumptions. Mobility phone data provide a unique opportunity for building models and definingstrategies based on very precise information about the movement of people in a region or in a coun-try. Another very important aspect is the underlying social structure of a population, which mightplay a fundamental role in devising information campaigns to promote vaccination and preventivemeasures, especially in countries with a strong family (or tribal) structure.

In this paper we analyze a large-scale dataset describing the mobility and the call patterns ofa large number of individuals in Ivory Coast. We present a model that describes how diseasesspread across the country by exploiting mobility patterns of people extracted from the availabledata. Then, we simulate several epidemics scenarios and we evaluate mechanisms to contain theepidemic spreading of diseases, based on the information about people mobility and social ties, alsogathered from the phone call data. More specifically, we find that restricting mobility does not delaythe occurrence of an endemic state and that an information campaign based on one-to-one phoneconversations among members of social groups might be an effective countermeasure.

I. INTRODUCTION

Health and well-being of populations are heavily in-fluenced by their behaviour. The impact of the habitsand local customs, including patterns of interactions andmobility at urban and regional scales, on health issuesis remarkable [1]. The diffusion of mobile technologywe are experiencing nowadays gives scholars an unprece-dented opportunity to study massive data that describehuman behavior [2]. An increasing number of peoplecarries smart mobile phones, equipped with many sen-sors and connected to the Internet, for the whole day [3].Data coming from a large number of people can describetrends in the macroscopic behavior of populations [4–6].The results of the analysis of these trends can be directlyapplied to a number of real-world scenarios, and, morein general, to several applications where cultural and lo-cal differences play a central role. Analyzing this kind ofdata can provide invaluable help to support the decision-making process, especially in critical situations. For thisreason, many public and private organizations are nowa-days increasingly adopting a data-centric approach intheir decisional process [7]. We believe that this strat-egy can be particularly useful in developing countries,which might have a lacking infrastructure1.

Among the issues that developing countries are facingtoday, healthcare is probably the most urgent [9]. In

1 We use the term “developing” to indicate countries that are as-signed a low Human Development Index (HDI) by United Na-tions Statistics Division. We are aware of the limitations of thisclassification. As reported by UN, the designations “developed”and “developing” are intended for statistical convenience and donot necessarily express a judgment about the stage reached by aparticular country or area in the development process [8].

these countries the effectiveness of campaigns is often re-duced due to low availability of data, inherent limits inthe infrastructure and difficult communication with thecitizens, who might live in vast and remote rural areas.As a result, action plans are difficult to deliver. How-ever, we believe that a data-centric approach can be aninnovative and effective way to address these issues.

In this paper, we focus on containment of epidemics.We use movement data extracted from the registrationpatterns in a cellular network to evaluate the influenceof human mobility on the spreading of diseases in a ge-ographic area. In particular, we utilize this model toinvestigate how infectious agents might spread to dis-tant locations because of human movement in order toidentify optimal strategies that can be adopted to con-trast the epidemics. We also evaluate how the collabo-rative effort of the population can be crucial in criticalscenarios. For the reasons we mentioned before, in coun-tries that are facing development challenges, vaccinationcampaigns are often hard to advertise to the population.Lack of communication and information is believed to beamong the main causes of failure for immunization cam-paigns. The same applies to awareness campaigns thattry to promote prophylaxis procedures that reduce theoccurrence of contagion. However, in these cases, we ar-gue that a collaborative effort leveraging individual socialties can be effective in propagating effective information(i.e., a sort of “immunizing information”) to a widespreadaudience. Moreover, information received by people whoare socially close can have a higher chance of leading toan actual action.

A large body of research has been conducted on modelsthat describe the diffusion of diseases, with a particularrecent interest on the role that human movement playsin spreading infections in large geographic areas [10–12],and also on the impact of human behavior on the spread-ing itself [12, 13]. With respect to the state of the art,

2

the main contributions of this paper can be summarizedas follows:

• We propose an epidemic model based on a net-work of geographic metapopulations, which de-scribes how people move between different geo-graphic regions and spread the disease.

• We evaluate containment techniques based on therestriction of mobility of the most central areas.The centrality of the areas is extracted by buildinga movement network between all the geographic ar-eas based on the mobility patterns of the individu-als.

• We extend the model with a competing informa-tion spreading where distance contagion might takeplace. In other words, we study the dynamics of thesystem considering three characterizing aspects ofthe problem: the disease epidemics, human mobil-ity and information spreading. This epidemics rep-resents the diffusion of information related to mea-sures to prevent or to combat the diseases, suchas information about the ongoing vaccination andprevention campaigns in a certain area or actionsthat will help to limit spread of the infection, suchas boiling water or avoiding contacts with peoplethat are already ill.

• We evaluate the models by using the data providedby the Orange “Data for Development” [14]. Wediscuss the effectiveness of the containment strate-gies and, in particular, for the information dissem-ination strategy, we identify the degree of partici-pation that is required to make it successful.

• We observe that restricting mobility by disallowingany movement from and to a limited set of subpre-fectures does not delay the occurrence of the en-demic state in the rest of the country. We also findthat a collaborative effort of prevention informationspreading can be an effective countermeasure.

This paper is organized as follows. In Sec. II we brieflydescribe the four different datasets provided by Orangeand we specify how they are used in the present study.In Sec. III we introduce our two models for epidemicsand information spreading by taking into account hu-man mobility and call patterns observed in Ivory Coast.In Sec. IV we present the results obtained by simulatingseveral epidemics scenarios and evaluating mechanismsto contain the epidemic spreading of diseases. Finally, inSec. V we summarize our main findings and we proposehow the present study can be improved if more detaileddata about mobility and calls will be available.

II. OVERVIEW OF THE DATASET

The data provided for the D4D challenge [14] consistof four datasets (identified by the labels SET1, SET2,

0 50 100 150 200 250

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Calls matrix C

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(a)

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250

Mobility matrix M

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10−5

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10−3

10−2

10−1

100

(b)

Figure 1: Logarithmic representation of the calls matrix (a)and the mobility matrix (b). Null values are indicated usingthe white color. For both matrices highest values are mostlyconcentrated along the diagonal, showing that communicationand movement between sub-prefectures is highly uncommon.However, the calls matrix is visibly denser than the mobilitymatrix, confirming that phone contacts between different sub-prefectures are more usual than movement.

SET3, SET4), containing information about user mobil-ity and call patterns at various levels of granularity andtime duration. We will now discuss how these datasetscan be used to build a model which accounts for usermobility and information spreading.

Two datasets contain information about mobility andcommunication patterns at macroscopic level. More pre-cisely:

• The SET1 dataset contains the number and theduration of calls between pairs of cell phone tow-ers, aggregated by hour. This dataset providesmacroscopic information about communication inthe country. We associate cell phone towers withthe sub-prefecture they are located in, by using thesupplied geographic position. Then, we evaluatethe probability of a call being established betweensub-prefectures i and j with:

cij =Cij∑k Cik

, (1)

where Cij is the number of phone calls initiatedfrom the sub-prefecture i and directed to the sub-prefecture j, during the entire period of observa-tion. The term at denominator indicates the to-tal communication flux between every pair of sub-prefectures and it is used to normalize the prob-ability. Using these values we build a calls ma-trix C, shown in Fig. 1(a). This matrix also showshigh values along the diagonal, but it is distinctlydenser, showing that calls between sub-prefecturesare more common than movement. The verticalline at x = 60 identifies calls directed to the sub-prefecture that contains the capital.

• The SET3 dataset contains the trajectories of50,000 randomly-selected individuals, at a sub-

3

(a) (b)

Figure 2: Geographic network obtained from call logs (a) and mobility traces (b). Color is used to indicate the communitystructure: nodes within the same community are represented with the same color.

prefecture level resolution, for five months.2 Thisdataset can be used to estimate the probability thatan individual moves from the sub-prefecture i to thesub-prefecture j:

mij =

∑uMu

ij∑k

∑uMu

ik

, (2)

where Muij is the number of times user u moves

from the sub-prefecture i to j. The numeratorcounts how many times users who are in i moveto j; the denominator normalizes this number bythe total number of transitions from i to any sub-prefecture k. Using these values we build a mobilitymatrix M , shown in Fig. 1(b). By using this ma-trix, we model human mobility in the country as aMarkov process [15]. We observe that the matrixis quite sparse and the highest values are concen-trated along the diagonal. As the representationis in logarithmic scale, this demonstrates that themovement between sub-prefectures is present, butrather uncommon.

In Fig. 2(a) and Fig. 2(b) we show the geographic net-works of calls and mobility, respectively. Nodes are

2 17 sub-prefectures do not have any cell phone towers and for thisreason do not appear in SET3. We discard these sub-prefecturesfrom our analysis, since their users will be considered as belong-ing to nearby sub-prefectures.

positioned using the geographic locations of the sub-prefecture they represent, and their color indicates thecommunity structure of the network based on [16].

The other two datasets provide microscopic informa-tion about mobility and communication patterns betweenindividuals. Although we do not use them for the anal-ysis in this paper, we now briefly outline how they couldbe used:

• The SET2 dataset contains fine-grained individualtrajectories of 50,000 randomly sampled individu-als over two-week periods. This dataset could beused to estimate the number of potential connec-tions that an individual might have in a certainarea, served by a cell phone tower.

• The SET4 dataset contains time-varying ego-networks of 5,000 users, describing the network ofcommunication in time-slots of 2 weeks. If twousers are connected by a link in a time-slot, itmeans that at least one call occurred during thetwo weeks under consideration3. The ego-networkaggregated over the whole observation time, builtconsidering every link that is present at least once,

3 We have found that 1.31% of the total number of edges in ego-networks connect pairs of users who are neither egos nor first-level neighbors: therefore, we do not consider such edges in ouranalysis.

4

Figure 3: Distribution of friends for the ego-networks ag-gregated over time. Error bars indicate statistical uncer-tainty, while the shaded area represent 99% confidence in-tervals around the observed value.

describes the number of people contacted by anindividual during the entire period. This datasetcould be used to estimate the number of potentialsocial connections that an individual might get intouch with. The degree distribution of the aggre-gate ego-network is shown in Fig. 3.

III. SPREADING MODELS

In this section we discuss two models: a model of dis-ease spreading as a function of the mobility patterns ofindividuals between different geographic areas inferredfrom the cellular registration records and a model for in-formation spreading among the same population, consid-ering the social structure inferred from the call records.In the following section, we will evaluate the models usingthe data provided for the Orange Data for Developmentchallenge.

A. Epidemic Spreading and Mobility

We will now present a model that represents the evolu-tion of an epidemic taking place on a network of metapop-ulations. The aim of the model is to describe how the sys-tem evolves under the action of two processes, contagionand mobility. For this dataset, each metapopulation iscomposed by the individuals located in a particular sub-prefecture. Hence, the population is distributed in n dif-ferent metapopulations, each having Ni[t] individuals attime t. We make the simplifying assumption that thereare no deaths and births in the considered time window,i.e., at each time t = 1, 2, . . . T the total population isconstant

∑ni=1Ni[t] = N .

We assume that contagion happens inside eachmetapopulation following a standard SIS model [17]. Weindicate the number of infected and susceptible individ-uals at time t in a sub-prefecture i with Ii[t] and Si[t],respectively. At each time t a person is either infected orsusceptible, therefore Ni[t] = Ii[t] + Si[t].

Simultaneously, individuals move through themetapopulation network according to the mobilitymatrix M of dimension n×n extracted from the cellulartraces. The generic element mij of the matrix representsthe probability that a person moves from the metapop-ulation i to j, as described by Eq. 24. This matrixdescribes how the state variables Ni[t] evolve over time:Ni[t + 1] =

∑nj=1mjiNi[t]. Under the assumption that

individuals inside the classes I and S move consistentlywe can write the last relation also for the state variablesIi[t] and Si[t]

5. The contagion-mobility combined systemcan then be described by the following set of equations:

Ii[t+ 1] =

n∑j=1

mji

[Ij [t] + λ

Sj [t]

Nj [t]Ij [t]− γIj [t]

]

Si[t+ 1] =

n∑j=1

mji

[Sj [t]− λ

Sj [t]

Nj [t]Ij [t] + γIj [t]

],

for each sub-prefecture i = 1, 2, . . . , n, with λ beingthe product of contact rate and contagion probabilityand γ being the recovery rate. The formulae inside thesquare brackets describe the evolution of n SIS models,one for each metapopulation. They are multiplied forthe elements of the mobility matrix, which accounts forindividuals moving between metapopulations.

This analytical model describes the expected outcomeof a stochastic model where the following actions occurat each time step:

1. Each infected person in the sub-prefecture j causes

the infection of new λSj

Njindividuals inside j. This

step is repeated for each sub-prefecture.

2. A new position i is assigned to each individual inthe sub-prefecture j according to the probabilitydensity function [mj1,mj2, . . . ,mjn]. This step isrepeated for each sub-prefecture.

B. Information Spreading

The model we presented in the last section tries to re-produce the spreading of a disease in a population whereindividuals change locations over time. The aim of this

4 In general, this matrix can be time-varying, and it can be ad-justed according to seasonal trends or real-time data at each step,for example following estimates based on historical data. In par-ticular, this matrix can be used to study the impact of policiesin real-time. However, in order to simplify the presentation, weuse a matrix not changing over time. The treatment can be gen-eralized, also applying the recent theoretical results related totime-varying networks [18, 19].

5 This assumption can also be relaxed when data about the dif-ferent classes of individuals is available, i.e., when a matrix foreach class can be defined.

5

R S

start

I

ω γ

λξ

Ustart A

ψ

Figure 4: State machines describing the state transitions ofa person with respect to the disease contagion (R=Resistant,S=Susceptible and I=Infected) and with respect to the in-formation spreading (U=unaware, A=aware), respectively. Aperson starts in the susceptible and unaware states. We as-sume that aware individuals spread the information and can-not go back to the unaware state.

work is to analyze some scenarios and study the effec-tiveness of some containment techniques. In particular,as anticipated, we would like to investigate if a collabora-tive effort of the population is able, in theory, to reduceconsiderably the spread of the disease and what propor-tions should it have to be effective. More precisely, thepopulation can disseminate information through personalsocial ties immunizing, such as information about pre-vention techniques, hygiene practises, advertisement ofnearby vaccination campaigns and in general any infor-mation that can lead to a reduction of the number ofcontagion events.

In order to take into consideration these aspects, wenow use a SIR model for each metapopulation, so thateach person either belongs to the susceptible (S), in-fected (I) or resistant (R) category. At the same time,another simultaneous epidemic happens on the networkof metapopulations, disseminating information that canmake individuals resistant to the disease. In fact, aperson also belongs to the category of unaware (U) oraware (A) individuals, with respect to the immuniz-ing information. More formally, we have that Ni[t] =Ii[t] + Si[t] +Ri[t] = Ai[t] + Ui[t].

It is worth noting that this “immunizing epidemic”goes beyond the boundaries of metapopulations (sub-prefectures): in other words, it is a distance contagion.It is also important to remark that the states “aware”and “resistant” are substantially different. An unawareperson that receives the information (i.e. has an “infor-mation contact”) becomes aware with rate ψ; since theperson is aware, he or she will start spreading the in-formation as well. An infected person that receives theinformation becomes immune with rate ω. Additionally,individuals who have acquired immunity through infor-mation can lose it with rate ξ. The transition rates be-tween states are summarized in Fig. 4. The model canbe described by the following set of equations, specifyinghow state vectors evolve over time:

Ii[t+ 1] =

n∑j=1

mji

[Ij [t] + λ

Sj [t]

Nj [t]Ij [t]− γIj [t]

]

Si[t+ 1] =

n∑j=1

mji

[Sj [t] − λ

Sj [t]

Nj [t]Ij [t] + γIj [t] + ξRj [t]+

− ωSj [t]∑nk=1 ckjAk[t]∑nk=1 ckjNk[t]

]Ri[t+ 1] =

n∑j=1

mji

[Rj [t]− ξRj [t] + ωSj [t]

∑nk=1 ckjAk[t]∑nk=1 ckjNk[t]

]

Ai[t+ 1] =

n∑j=1

mji

[Aj [t] + ψUj [t]


]

Ui[t+ 1] =

n∑j=1

mji

[Uj [t] − ψUj [t]


](3)

for every i = 1, 2, . . . , n. The fraction∑n

k=1 ckjAk[t]∑nk=1 ckjNk[t]

rep-

resents the probability that a call from an aware personoccurs in the metapopulation j. It models the distance-contagion, and it is possible to verify that if the matrixis identical (absence of contacts between populations) itreduces to Ak[t]/Nk[t], falling back to a model where con-tagion occurs only inside metapopulations.

This analytical model describes the expected value ofa stochastic model where the following actions occur ateach time step t:

1. Each infected person in the sub-prefecture j causes

λSj

Njnew individuals to get infected, inside j. This


2. Each unaware person in the sub-prefecture j be-

comes aware with probability ψ∑n


. This


3. Each person in the sub-prefecture j who issusceptible, becomes resistant with probability

ω∑n


. This step is reapeated for each sub-

prefecture.

4. A new position i is assigned to each person in thesub-prefecture j according to the probability den-sity function [mj1,mj2, . . . ,mjn]. This step is re-peated for each sub-prefecture.

IV. ANALYSIS

We initialize each scenario by allocating 22 mil-lion individuals (the estimated population size of IvoryCoast for July 2012 is 21,952,093 [20]) to different sub-prefectures across the country, according to the data inSET3. In each scenario we bootstrap the spreading pro-cess by infecting a fraction of the population (0.1%) dis-tributed across metapopulations according to differentcriteria:

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uniform

top 10

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uniform

top 10

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random

Figure 5: Fraction of infected population at the stationarystate (left panel) and time required to reach the stationarystate (right panel), for different values of r0 and for differentinitial conditions. Missing values in the curves mean that,for the corresponding values, no stationary state is reachedduring the period of observation.

• Uniform distribution: every sub-prefecture gets anumber of infected proportional to their popula-tion, i.e., every sub-prefecture has the same fractionof infected population.

• Random: a single sub-prefecture, chosen randomly,is the origin of the infection.

• Centrality based: the sub-prefectures are orderedby decreasing centrality values, then the first 1, 5or 10 highest ranked sub-prefectures are chosen, asshown in Table I.

We study the evolution of the epidemics for a periodof 6 months. We investigate multiple scenarios using theanalytical model considering a large set of ranges for thekey parameters. We conducted a series of Monte-Carlosimulations for multiple sets of parameters, confirmingthe validity of the analytical models presented in the pre-vious section. In the following, we present results basedon these models.

A. No Countermeasures

We will firstly explore the evolution of the epidemicsin the case where no countermeasures are taken. In order

Betweeness Closeness Degree Eigenvalue60 60 60 6039 58 58 5889 39 39 3958 69 69 6975 138 138 250144 250 64 138138 64 144 64165 144 250 144212 182 122 122168 122 182 182

Table I: Highest ranked sub-prefectures, according to differentdefinitions of centrality. We observe that the sets of the top10 sub-prefectures ordered by centrality are very similar.

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i∞

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curbed top10

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0

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curbed top1

Figure 6: Fraction of infected population at the stationarystate (left panel) and time required to reach the stationarystate (right panel), for different values of r0 when the epi-demic starts from a random sub-prefecture, and different lev-els of geographic quarantine are applied. Missing values in thecurves mean that, for the corresponding values, no stationarystate is reached during the period of observation.

to analyze the evolution of the system more clearly, weinvestigate two measures: the fraction of infected popu-lation i∞ at the stationary state and the time required toreach the stationary state τ . In Fig. 5 we plot their val-ues versus r0 = λ

γ , which is the basic reproductive ratio

of a classic SIS model [17]. As a future work, we plan toderive the analytical form of the basic reproductive ra-tio of our models, which take into account mobility andinformation spreading. We observe that for r0 = λ

γ < 1

there is no endemic state (i.e., the final fraction of in-fected population is zero), whereas for r0 > 1 a non-nullfraction of population is infected. Values for r0 = 1 aremissing since no stationary state is reached within ourobservation window. In other words, for this particularscenario, experimental results show that the basic repro-ductive ratio of our model is very close to r0; we expectthis to be a consequence of the low inter-subprefecturesmobility. We can also notice that the initial conditionsdo not affect i∞ at all. Before the critical point (i.e.,r0 = 1) the choice of the initial conditions has also noimpact on the delay time, whereas for r0 > 1 it slightlyaffects the delay: epidemics that initially involves moresub-prefectures are slightly faster than the others.

B. Geographic Quarantine

We now analyze the effects of curbing on the mobilitybetween sub-prefectures, i.e., forbidding all the incomingand outgoing movement of a group of sub-prefectures.In order to do so, we calculate the centrality values ofeach sub-prefecture in the mobility matrix. We presentthe results for eigenvalues centrality. As it is possible toobserve in Tab. I, the ranking based on other centralitiesis very similar. Then, for the quarantine operations, weselect those with the highest centrality values. From apractical point of view, this is achieved by simply chang-ing the i−th row and column in the mobility matrix, sothat all the elements mij and mji are null, except forthe elements mii = 1. For these scenarios, we randomly

7

0.0 0.2 0.4 0.6 0.8 1.0ξ

0.0

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ω=ψ

fraction of infected population (λ = 0.5, γ = 0.5)

0.0

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time to reach the stationary state (λ = 0.5, γ = 0.5)

0

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> 300

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=ψ


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0

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Figure 7: Fraction of infected population at the stationary state (top row) and time required to reach the stationary state(bottom row), for different values of r0 = λ

γ(from left to right 1, 2, 4, 10, respectively). White spaces show that no stationary

state is reached during the period of observation.

choose a single sub-prefecture where the initial individu-als are infected, and then we average i∞ and τ over allruns. As shown in Fig. 6, the fraction of the infectedpopulation is sensibly affected by this measure, as thepopulation inside the quarantined areas is protected fromcontagion. However, contrary to the intuition, the delayis not affected by the quarantine, even when the coun-termeasures involves 10 sub-prefectures, which accountfor almost half population. This suggests that such aninvasive, expensive and hard to enforce measure reducesconsiderably the endemic size, but does not slow downthe disease spreading in the rest of the country. For thisreason, we now investigate a radically different approachto protect the population.

C. Information Campaign (Social Immunization)

We now show how a collaborative information cam-paign could help in contrasting the spread of the disease,following the model we presented in the last section. Weinitialize the scenario by distributing the immunizing in-formation to 1% of the population, randomly chosen re-gardless of their location. These people will be informedand will be instructed to spread the information. In otherwords, we assume that they will contact their social con-nections, according to the call matrix.

In Fig. 7 we show the density plots describing i∞ and τfor various values of r0 (, for a subset of scenarios whereω = ψ, i.e., when the information that spreads amongthe population has the same chance to immunize a per-son and to involve the person in the spreading process.This is consistent with a scenario where the same set of

people who become aware also become immunized by theinformation they have received. Blank squares show thata stationary state was not reached for the correspond-ing set of parameters. The figure shows how contagious(ω=ψ) the immunizing information has to be with re-spect to how often people “forget” (ξ) in order to slowdown the disease considerably and to reduce the endemiccases. When ω = ψ = 0 we fall back to the model with-out information spreading, and the value of ξ does notaffect i∞ and τ . For ω = ψ > 0 and ξ = 0 the fractionof infected population goes to zero in all cases, becausethe number of people aware of the information does notdecrease, thus increasing the number of new immunizedindividuals at each step. We can notice that even for lowvalues of participation ω and for information that givestemporary immunization (ψ > 0), the final fraction of in-fected individuals is considerably lower than in the casewhere no countermeasures are taken.

In Figs. 8 and 9 we show the density plots for ω and ψwhen ξ is constant. In particular, we analyze the scenariofor ξ = 0 (Fig. 8), which represents for example a scenariowhere the immunizing information is about vaccinationcampaigns (individuals who have been administered vac-cination do not lose immunity). For every combination ofparameters we have absence of endemic state even withthe highest considered value of r0. The two parametersthat represent how individuals are likely to get involvedboth in the immunization and in the information spread-ing (ω and ψ) seem to have the same impact on the delayof the infection.

The value ξ = 0.5 (Fig. 9) describes the scenario whenthe information is about a good practice (e.g., boilingwater, using mosquito nets, etc.), which loses its effec-

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Figure 8: Fraction of infected population at the stationary state (top row) and time required to reach the stationary state(bottom row), for different combinations of λ

γ(from left to right 1, 2, 4, 10, respectively) and ξ = 0. White spaces show that

no stationary state is reached during the period of observation.

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Figure 9: Fraction of infected population at the stationary state (top row) and time required to reach the stationary state(bottom row), for different combinations of λ

γ(from left to right 1, 2, 4, 10, respectively) and ξ = 0.5. White spaces show that

no stationary state is reached during the period of observation.

tiveness or it is stopped being used by a person with rateξ. For this case we can notice that the fraction of in-fected population is independent from ψ, as rows in thedensity plot are of the same color. This suggests that,for this scenario, the rate at which people lose immunitydoes not affect the size of the endemic state.

V. CONCLUSIONS

In this paper we have presented a model that describesthe spreading of disease in a population where individualsmove between geographic areas, extracted from cellularnetwork records. We have showed the evolution of thedisease and we have evaluated two types of countermea-sures, namely the quarantine of central geographic areasand a collaborative “viral” information campaign among

9

the population, by inferring the underlying social struc-ture from the call records.

Our future research agenda includes the investigationof analytical aspects of the model, such as the derivationof the critical reproductive ratio R0, i.e., the value thatcorresponds to the transition between an endemic and anendemic-free infection. Currently, the model is based onthe assumption of a static mobility matrix: our goal isto refine the model by introducing time-dependent ma-trices, also exploring the application of the recent the-oretical results related to temporal networks. We alsoplan to refine the model introducing specific contact ratesfor each metapopulation, potentially based on more fine-grained information about the number of encounters and

the number of calls of each individual, if available. Fi-nally, we plan to explore hybrid countermeasures, such asconcurrent partial restrictions of mobility and targetedinformation campaigns.

Acknowledgments

The authors thank Charlotte Sophie Mayer for use-ful and fruitful discussions. This work was supportedthrough the EPSRC Grant “The Uncertainty of Iden-tity: Linking Spatiotemporal Information Between Vir-tual and Real Worlds” (EP/J005266/1).

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[14] V.D. Blondel, M. Esch, C. Chan, F. Clerot, P. Deville,E. Huens, F. Morlot, Z. Smoreda, and C. Ziemlicki, Datafor Development: the D4D Challenge on Mobile PhoneData arXiv preprint arXiv:1210.0137 (2012).

[15] J. R. Norris, Markov Chains (Cambridge UniversityPress, Cambridge, United Kingdom, 1998).

[16] V. D. Blondel, J.-L. Guillaume, R. Lambiotte, andE. Lefebvre, Fast unfolding of communities in large net-works Journal of Statistical Mechanics: Theory and Ex-periment 2008, P10008 (2008).

[17] M. J. Keeling and P. Rohani, Modeling infectious diseasesin humans and animals (Princeton University Press,Princeton, NJ, 2011).

[18] J. Tang, S. Scellato, M. Musolesi, C. Mascolo, andV. Latora, Small-world Behavior in Time-varying GraphsPhysical Review E 81, (2010), 055101(R).

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