Vaccination and epidemics in networked populations – An introduction

Zhen Wang,1 Yamir Moreno,2, 3, 4 Stefano Boccaletti,5, 6 and Matjaz Perc7, 8, ∗

1Center for OPTical IMagery Analysis and Learning (OPTIMAL),Northwestern Polytechnical University, Xi’an 710072, China

2Institute for Biocomputation and Physics of Complex Systems, University of Zaragoza, E-50018 Zaragoza, Spain3Department of Theoretical Physics, University of Zaragoza, E-50009, Zaragoza, Spain

4Complex Networks and Systems Lagrange Lab, Institute for Scientific Interchange, Turin 10126, Italy5CNR – Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy

6The Italian Embassy in Israel, 25 Hamered st., 68125 Tel Aviv, Israel7Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska cesta 160, SI-2000 Maribor, Slovenia

8CAMTP – Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia

This is an introduction to the special issue titled “Vaccination and epidemics in networked populations” thatis in the making at Chaos, Solitons & Fractals. While vaccination is undoubtedly one of the most importantpreventive measures of modern times, epidemics are feared as one of the most damaging phenomena in hu-man societies. Recent research has explored the pivotal implications of individual behavior and heterogeneouscontact patterns in networked populations, as well as the many feedback loops that exist between vaccinating be-havior and disease propagation [1, 2]. Interdisciplinary explorations in the realm of statistical physics, networkscience, nonlinear dynamics, and data analysis have given rise to theoretical epidemiology, as well as to the the-ory of epidemic processes in complex networks. From classical models assuming well-mixed populations andignoring human behavior, to recent models that account for behavioral feedback and population structure, wehave come a long way in understanding disease transmission and disease dynamics, and in using this knowledgeto devise effective prevention strategies. This special issue is aimed at helping the further development of thesesynergies. We hope that it contributes to enhance our understanding of vaccination and epidemics in networkedpopulations, by featuring works related to vaccination and epidemics using techniques ranging from complexand temporal networks to network of networks and show-casing the possibilities of interdisciplinarity via com-plex systems science to tackle the challenges in our quest for a healthier future. Topics of interest include butare not limited to epidemiological modeling and vaccination, behavior-vaccination dynamics, reaction-diffusionprocesses and metapopulation models, evolutionary and game theoretical models in epidemiology, as well as toinfluence maximization and digital epidemiology.

I. NETWORKS

During the past two decades network science has emergedas a central paradigm behind some of the most fascinating dis-coveries of the 21st century [3, 4]. From the mathematical for-mulation of small-world properties and their omnipresence inseemingly diverse systems such as electric power grids, foodchains, brain networks, protein networks, transcriptional net-works and social networks [5], to universal scaling proper-ties due to growth and preferential attachment that likewisepervade biological, social and technological networks [6], thefield of research today known as network science has beengoing from strength to strength, as evidenced by the manyreviews devoted to this field of research [7–14]. Network sci-ence has provided models, methods and algorithms that haverevived not just statistical physics, arguably the nurturer tothe field, but indeed many other fields of natural and socialsciences, including of course research concerning vaccinationand epidemics to which this special issue is devoted to.

In addition to static networks, network science allows us tostudy and take into account network evolution over time, forexample due to changes in external factors, the onset of dis-ease, targeted attack, or simply due to random failure. Such

∗Electronic address: matjaz.perc@uni-mb.si

changes can be studied in the realm of temporal networks[11, 15, 16], where the theoretical framework accounts forthe addition or removal of nodes, or similarly for the changesin the links between nodes, over time. A beautiful exampleof how changing links over time between different individu-als might affect disease spreading, reproduced from [11], isshown in Fig. 1.

Also of importance, networks exist between different lay-ers of each studied system, and this is particularly apparentin social systems, where one person can be simultaneouslymember of many different networks that are to various degreesinterdependent [17–21]. This can be accommodated in thetheoretical framework of multilayer networks, or more gener-ally networks of networks, which acknowledge that not onlyis the range of our interactions limited and thus inadequatelydescribed by well-mixed models, but also that the networksthat should be an integral part of such models are often in-terconnected, thus making the processes that are unfolding onthem interdependent [12, 13, 22–24]. From the world econ-omy and transportation systems to the spread of epidemics, itis clear that processes taking place in one network can signifi-cantly affect what is happening in many other networks. Fromthis point of view, the consideration of multilayer or interde-pendent networks is crucial for a comprehensive treatment ofvaccination and epidemics in networked populations [25, 26].

In addition to the theoretical coming of age of network sci-ence, technological breakthroughs in the acquisition and stor-

2

6,7,11

8,13,14

2,4,71,3

D

C

A

B

6,7,11

8,13,14

2,4,71,3A

B

D

C6,7

,11

8,13,14

2,4,71,3A

B

D

C

t = 0 t = 6.5 t = ∞

DC

B

A

1050 t

FIG. 1: The importance of temporal networks. Changes in the linksbetween individuals A, B, C and D affect disease spreading. In theupper row, the contact times between the individuals are indicated onthe edges. Let us assume that a disease starts spreading at individualA and spreads further as soon as contact occurs. This spreading pro-cess is illustrated for four different times from left to right. At t = ∞the spreading stops as individual D cannot get infected. However, ifthe spreading starts at individual D, then all other depicted individ-uals would eventually be infected. Aggregating the edges into onestatic graph cannot capture this effect that arises from the time or-dering of contacts. The lower row visualizes the same situation byshowing the temporal dimension explicitly. This figure is reproducedwith permission from [11].

age of vast amounts of digitized data have significantly aidedthe progress in our understanding of both vaccination and epi-demic spreading. The data revolution has had a particularlydeep impact on the social sciences, where social experimentsin the past typically involved one-shot self-reported data onrelationships and their outcomes in a small sample of people,while today the approach is to mine massive amounts of dig-itized data for both the structure and the content of relation-ships [27, 28], which may in turn relevantly inform vaccina-tion strategies [2] as well as the spread of epidemics [1].

In particular, the synergies between mathematical modelingand theoretical explorations and data-driven research, coupledwith taking into account feedbacks between disease, behav-ior and vaccination, are likely the future of complex systemsresearch aimed at a better understanding of vaccination andepidemics in modern human societies.

II. VACCINATION

As already stated, vaccination is undoubtedly one of themost important preventive measures of modern times, unfold-ing on the planetary scale with the aim of preventing thespread of infectious diseases that have had a crippling im-pact on many human societies in the not so distant past. The20th century, in particular, saw enormous progress in publichealth, especially in terms of preventing and treating infec-tious diseases. The use of vaccines was crucial to that effect.However, due to limited amounts of vaccines and knowledge,early studies simply assumed that vaccination should be donecompulsively and/or randomly. Today we know that vaccina-

tion efforts on networks are more effective if random vacci-nation is combined with targeted vaccination and the vaccina-tion of acquaintances [1, 29–31]. While random vaccinationof course does not require any information about the structureof the network, it needs in return a very broad coverage, andis thus very costly, in order to be effective [29, 32]. Moreprecisely, uniform vaccination is super inefficient for diseaseeradication in heterogeneous networks [29]. To offset this dis-advantage, targeted vaccination according to the centrality in-dexes of individuals, like degree, betweenness and closeness[33–37] should be considered. For example, because large-degree nodes are known to be responsible for the spreadingof disease, a degree-based targeted vaccination strategy wasproposed to immunize the most highly connected individuals[34, 38]. Compared with random vaccination, targeted vac-cination thus greatly reduces costs, but since the identifica-tion of centrality-defined individuals usually takes a long(er)time, the practicability of targeted vaccination is doubtful inpractice, especially in large populations [39]. There also ex-ists inconsistencies in the definitions of some centrality in-dexes in more complex networks, especially in multilayer net-works, which in turn restricts the universality of targeted vac-cination [12]. Acquaintance vaccination is more suitable forpractical applications in that a fraction of nodes is selectedat random, and then random neighbors are vaccinated further[33, 40]. This approach requires only partial knowledge of thenetwork structure, and its variants have been studied in sub-stantial detail [41, 42]. In addition to these different vaccina-tion strategies, network structure also plays an important rolein the transmission routes and infection levels [2, 9–12]. Forexample, in community networks, bridge nodes, connectingdifferent communities, provide pathways for disease propaga-tion, so that how to identify and immunize these nodes thusdecides the final success rate [43, 44].

With the latest advances in science and technology, today,safe and effective vaccines exist for many of the most com-mon infectious diseases, such as smallpox, measles, pertussis,influenza, hepatitis, chickenpox and diphtheria, and the useof vaccines has been estimated to save millions of children’slives per year [45]. However, due to high economic costs ofvaccination, and often also personal beliefs and the fear ofpossible side effects, people usually regard vaccination as avoluntary rather than a compulsive measure. In this sense,behavioral vaccination, borrowing the framework of behaviordynamics in game theory and psychology, becomes a usefulresearch framework to elucidate real-world disease spreadingand prevention [2, 46–48].

Despite of all the success, great challenges still remain, inpart due to the difficulties in providing and administering vac-cines in the world’s poorest and war-torn regions [49, 50], butalso because in many wealthy countries, such as the UnitedKingdom, Switzerland and Germany, there is considerableparental vaccine refusal or vaccine hesitancy on account ofunfounded concerns about significantly adverse side effectssuch as autism. Regarding the latter, it is important to ac-knowledge that vaccine refusal is not a new phenomenon, andeven the first vaccine to be invented – the smallpox vaccine– was subject to considerable resistance, as nicely illustrated

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FIG. 2: Satirical cartoon by James Gillray capturing public fears ofthe smallpox vaccine in the 18th century, the first “vaccine scare”.Edward Jenner stands in the middle, administering the smallpox vac-cine. Vaccinated persons grow cow parts out of various parts of theirbodies because the smallpox vaccine of the time was based on thevirus that causes cowpox, hence producing the fear among some thatthe vaccine would turn humans into cow-like hybrids. A painting ofthe Worshippers of the Golden Calf hangs in the background. Source:Wikimedia Commons.

by James Gillray in his 18th century portrayal of the “vaccinescare” (see Fig. 2). The take-home message is that it is veryimportant to account for behavioral feedback effects, and toaccount for cognition and deliberation when devising mathe-matical models of vaccination. Namely, as logistic, financial,technological and administrative barriers to vaccination con-tinue receding in the coming decades, enabling it to becomeeasier to improve vaccine coverage everywhere in the world,it is not too far fetched to speculate that vaccine refusal andvaccine hesitancy will become an important barrier – and per-haps even the most important barrier – to global eradication ofvaccine-preventable infections [2]. As a case in point, whenwe observe vaccinating behavior responding to changes in dis-ease prevalence, as in the Disneyland, California measles out-break of 2014 and other examples, it is impossible to overlookthat vaccinating behavior and disease dynamics are stronglylinked to one another, particularly near elimination thresholds.

Although human behavior is surely more difficult to math-ematically model than the motion of a billiard ball or manyof the somewhat more complex systems studied by physicists,it is nevertheless conceivable that at some levels of organiza-tion it is possible to develop simple models that have usefulpredictive power. For example, mathematical models incor-porating behavior-disease interactions have been fitted to timeseries data of infectious disease prevalence, and model selec-tion approaches have shown that models that include behaviorcan explain the data more effectively compared to models thatneglect behavior [51, 52].

Hence, there is a strong public health and scientific ratio-nale for studying behavior-disease-vaccination interactions.Existing research shows that behavior, disease and vaccina-

tion choices are strongly coupled to one another, and further,that this coupling is particularly important near the elimina-tion threshold [48]. Accordingly, vaccine refusal may becomean increasingly important barrier to achieving elimination anderadication, as logistic barriers recede into the distance. Fig-ure 3 illustrates the behavior-disease-vaccination interactions,which form a feedback loop. The fast diffusion of a diseasewill increase the risk perception, based on which people feelan incentive to take protective measures, i.e., to vaccinate. Inturn, the spreading of the disease is suppressed, to the pointwhere risk perception drops again and people opt not to vac-cinate.

Mathematical models hold the promise to help us better un-derstand such coupled dynamics, and comparing models withdata has often shown that behavior is important for under-standing observed epidemic patterns. Also of notice, thesemodels often exhibit interesting dynamics that is similar tothose studied in physics, including phase transitions, oscilla-tory solutions, and spatiotemporal pattern formation, thus in-dicating that similar methodologies to those used in physicsand complex systems research in general are likely to be help-ful in their study [2, 53].

For a more concrete perspective, we briefly review arguablythe first phenomenological model to capture the impact ofadaptive human behavior on disease transmission [54]. Themodel is given by the following ordinary differential equa-tions

dS

dt= −g(I)S,

dI

dt= g(I)S,−γI (1)

dR

dt= γI,

where S is the number of susceptible individuals, I is thenumber of infectious individuals, R is the number of recov-

disease

dynamics

behavioral

dynamics

+

_

risk perception

vaccination

FIG. 3: Illustration of behavior-disease-vaccination interactions. Thefeedback loop, on the one hand, promotes vaccination as the diseasespreads and risk perception rises, but on the other hand, people revertto abstaining when vaccination takes effect and the spreading of thedisease is effectively mitigated. But this then of course restarts theloop sooner or later.

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ered individuals, γ is the per capita recovery rate, and g(I)is the force of infection. The form of g(I) is chosen to reflectthe phenomenological impact of unspecified psychological ef-fects. One of the functional forms explored in [54] included

g(I) =βI

1 + βδI, (2)

which saturates at a high prevalence of infection I . Such sat-uration of the transmission rate at high levels of prevalencecould occur if individuals reacted to high prevalence by reduc-ing their contact rate through hand-washing or other means,which is a plausible assumption for a sufficiently dangerousinfectious disease. The model is phenomenological because,although it is motivated by psychological factors, the equa-tions only describe the impact of such psychological factorson the transmission rate without actually modeling specificpsychological processes such as individual cognition or sociallearning.

An apt theoretical framework to account for behavior-disease-vaccination interactions is also evolutionary gametheory [55–59], which has already been used to study the clashbetween what is optimal for the individual and what is optimalfor the group in a variety of contexts [16, 23, 60–65], includ-ing vaccination [66, 67]. At the heart of the choice whether ornot to vaccinate is of course a social dilemma, which invokesthe puzzle of human cooperation [68]. To vaccinate is clearlythe socially responsible choice, but it comes at a cost that hasto do with offsetting the fear and potential side effects. Butif individuals choose not to vaccinate, they may become in-fected, which will inflict larger costs for recovery. Those thatdo not vaccinate but still benefit from herd immunity, whichis generated by a large fraction of those who do vaccinate, isindividually the better choice, i.e., like free-riding in evolu-tionary games. Obviously, if everybody chooses not to vac-cinate the herd immunity is lost and the public good, whichin this case is the precious absence of an infectious disease,is lost with it. Accordingly, uninfected non-vaccinators arethe defectors who benefit from herd immunity generated byvaccinators without paying any real or perceived cost of vac-cination, while vaccinators are the cooperators.

In terms of the integration of cognitive abilities, a newmodel for looking at the evolution of intuition versus deliber-ation as well as at the evolution of cooperation versus defec-tion has recently been introduced [69], which lends itself toresearch in networked populations, and which could also bea viable starting ground for more realistic evolutionary gametheoretical models of vaccination.

To conclude this section, we note that traditional epidemi-ological models that do not take into account behavioral dy-namics have been very successful over the past century, bothin terms of yielding rich theoretical opportunities for researchas well as in terms of providing insights into epidemiologicalmechanisms and how to improve infection control. Neverthe-less, the impact of human behavior on disease dynamics andvaccination is ubiquitous, and vaccinating behavior itself iscertainly no exception in this important feedback loop.

III. EPIDEMIC PROCESSES

Research on epidemic outbreaks in complex heterogeneousnetworks has roots that go back to the beginning of the 21stcentury [70], when it was shown that large fluctuations inthe degrees of individuals in social networks considerablystrengthen the incidence of epidemic outbreaks. Scale-freenetworks, for example, were shown to altogether lack an epi-demic threshold and thus always contain a finite fraction ofinfected individuals. This particular weakness of networkedpopulations in comparison to well-mixed populations, whichwas observed also in models without immunity, at the timedefined a new epidemiological framework characterized by ahighly heterogeneous response of the system to the introduc-tion of infected individuals with different connectivity. Fastforward 15 years into the future, and today we have a com-prehensive theory of epidemic processes in complex networks[1], but with many challenges and open problems still remain-ing.

While epidemic models based on reaction-diffusion pro-cesses have arguably been the workhorse of the field since itsinception [71–74], one of the more exciting recent develop-ments that we here mention as example concerns agent basedmodels, and network epidemiology in particular [75–81]. Thisapproach draws on synergies between large-scale simulationsand data-based network construction, the outcome of whichis typically an agent-based model that evolves in a spatiallystructured population. A schematic illustration of the con-struction of a synthetic population is presented in Fig. 4,where based on census and demographic data as input the net-work of contacts is constructed from an intermediate bipar-tite network associating individuals to locations. The GLobalEpidemic and Mobility (GLEAM) model [82] is a particularexample that integrates census and mobility data in a fullystochastic metapopulation network model that allows for the

FIG. 4: Network epidemiology. At the macroscopic level (left), asynthetic population and the movements of its members can be con-structed from census and demographic data. A bipartite network as-sociating individuals to locations can be derived based on data fromthe synthetic population (middle). Lastly, the unipartite projection ofthe bipartite network provides a contact network (right), upon whichan agent-based model can unfold. This figure is reproduced withpermission from [1].

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detailed simulation of the spread of influenza-like illnessesaround the globe.

Importantly, while the spread of infectious diseases is mostcommonly the focal point of research on epidemic spreading,in particular because this is one of the most damaging phe-nomena in human societies in the past and present day, the re-sulting mathematical models can of course be generalized andadapted to describe diffusion of information [83], the spread-ing of rumors and scientific memes [84], the emergence ofonline virality [85], and indeed many other phenomena thatare inherent to our modern existence, both off and online. Re-cently, a null model for social spreading phenomena has beenproposed [86], which can reproduce several characteristics ofempirical micro-blogging data, such as for example the heavy-tailed distribution of meme popularity. The model clearly dis-tinguishes the roles of two distinct factors affecting popularity,namely the memory time of users and the connectivity struc-ture of the social network.

Along these lines, research on epidemics in the broadestpossible sense continues to draw together different fields forus to arrive at a better understanding of processes that are be-hind one of the greatest challenges that lie ahead for humansocieties, namely the maintenance of public health and themitigation of large-scale epidemics and the spread of infec-tious disease. As an added benefit, one finds that the samespreading phenomena that drive epidemics drive also socialcontagion processes, which surely adds to the arguments toactively engage in this research.

IV. FUTURE RESEARCH

As we hope we have succeeded to outline in this intro-duction, vaccination and epidemics in networked populationsmerit outstanding attention from the research community, notonly for their obvious importance to the wellbeing of our so-cieties, but also from the purely scientific point of view dueto the many open problems and fascinating challenges that re-main to be addressed. The triptych The Garden of Earthly De-lights shown in Fig. 5, painted by Hieronymus Bosch aroundthe turn of the 16th century, is a lucid portrayal of events thatmay transpire if we fail to keep the public health in the bestpossible order, and if we fail to cooperate in vaccinating ourchildren and working together to minimize the risk of epi-demics.

Particularly promising areas of research that warrant atten-tion include the consideration of mathematical and statisti-cal laws that govern the coevolution of the network and theevolution of vaccination or of the epidemic process that un-folds. Here the formalism of temporal networks is a neces-sary prerequisite [11], while further ideas how to proceed canbe sought in [15, 16], where research in closely related fieldshas already been reviewed.

Further in terms of the interaction network, recent researchhas revealed that social networks are almost inherently inter-dependent and layered, in the sense that a particular networkof contacts seldom exists in isolation [12, 13]. Consequently,processes unfolding in one network can have significant and

also to a large degree unexpected consequences in another net-work [17–21, 25, 26]. It remains subject to future researchto reveal to what degree this influences our ability to predictand control the epidemics, and what are the consequences ofvaccination abstinence in one social group for others that arecodependent.

Regarding mathematical modeling, it remains of interest tofurther integrate the many feedbacks between disease and be-havior, in particular vaccinations behavior, and to introducecognition and deliberation as processes that are inseparable tohumans. Here research in psychology and anthropology canprovide vital guidance and relevantly inform the next gener-ation of models for them to realistically represent these pro-cesses that thus far remain under explored.

Lastly, one has to mention the ever-increasing availabilityof digital data and the many exciting prospects this creates,not just for testing theoretical predictions, but also in termsof using the data during the modeling process itself, as hasalready been demonstrated, for example in [82].

To conclude, we note that this special issue is also to featurefuture research. In order to avoid delays that are sometimesassociated with waiting for a special issue to become com-plete before it is published, we have adopted an alternativeapproach at Chaos, Solitons & Fractals. The special issue willbe a virtual one, updated continuously from the publicationof this introduction onwards, meaning that new papers willbe published immediately after acceptance. The down sideof this approach is that we cannot feature the traditional briefsummaries of each individual work that will be published, butwe hope that this is more than made up for by the immediateavailability of the latest research. Please stay tuned, and con-sider contributing to “Vaccination and epidemics in networkedpopulations”.

FIG. 5: The Garden of Earthly Delights serves as a didactic warningon the perils of life’s temptations, which applies as much to shirkingon vaccination, as it does to abandoning vigilance in times of epi-demics. From left to right, we first have the meeting of Adam andEve, followed by a socially balanced state of wellbeing in the center,and ending with a hellscape that portrays the torments of damnationthat may result due to excesses in modern human societies. The pic-ture was taken by Alonso de Mendoza, and reproduced here from thepublic domain. Source: Wikimedia Commons.

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Acknowledgments

We would like to thank the publishing team at Chaos, Soli-tons & Fractals for all their support and assistance in creatingthis special issue. We would also like to thank all the au-thors and referees in advance for their valuable contributionsand help. This work was supported by the National Key Re-

search and Development Program (Grant 2016YFB0800602),the National Natural Science Foundation of China (Grant61471299), by the Government of Aragon, Spain through agrant to the group FENOL, by MINECO and FEDER funds(Grant FIS2014-55867-P), and by the Slovenian ResearchAgency (Grants J1-7009 and P5-0027).

[1] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespig-nani, Epidemic processes in complex networks, Rev. Mod.Phys. 87 (2015) 925.

[2] Z. Wang, C. T. Bauch, S. Bhattacharyya, A. d’Onofrio, P. Man-fredi, M. Perc, N. Perra, M. Salathe, D. Zhao, Statistical physicsof vaccination, Phys. Rep. 664 (2016) 1–113.

[3] M. E. J. Newman, Networks: An Introduction, Oxford Univer-sity Press, Oxford, 2010.

[4] A.-L. Barabasi, Network Science, Cambridge University Press,Cambridge, 2015.

[5] D. J. Watts, S. H. Strogatz, Collective dynamics of ’smallworld’ networks, Nature 393 (1998) 440–442.

[6] A.-L. Barabasi, R. Albert, Emergence of scaling in random net-works, Science 286 (1999) 509–512.

[7] R. Albert, A.-L. Barabasi, Statistical mechanics of complex net-works, Rev. Mod. Phys. 74 (2002) 47–97.

[8] M. E. J. Newman, The structure and function of complex net-works, SIAM Review 45 (2003) 167–256.

[9] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. Hwang,Complex networks: Structure and dynamics, Phys. Rep. 424(2006) 175–308.

[10] S. Fortunato, Community detection in graphs, Phys. Rep. 486(2010) 75–174.

[11] P. Holme, J. Saramaki, Temporal networks, Phys. Rep. 519(2012) 97–125.

[12] S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gomez-Gardenes, M. Romance, I. Sendina-Nadal, Z. Wang, M. Zanin,The structure and dynamics of multilayer networks, Phys. Rep.544 (2014) 1–122.

[13] M. Kivela, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno,M. A. Porter, Multilayer networks, J. Complex Netw. 2 (2014)203–271.

[14] L. Lu, D. Chen, X.-L. Ren, Q.-M. Zhang, Y.-C. Zhang, T. Zhou,Vital nodes identification in complex networks, Phys. Rep. 650(2016) 1–63.

[15] T. Gross, B. Blasius, Adaptive coevolutionary networks: a re-view, J. R. Soc. Interface 5 (2008) 259–271.

[16] M. Perc, A. Szolnoki, Coevolutionary games – A mini review,BioSystems 99 (2010) 109–125.

[17] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin,Catastrophic cascade of failures in interdependent networks,Nature 464 (2010) 1025–1028.

[18] R. Parshani, S. V. Buldyrev, S. Havlin, Interdependent net-works: Reducing the coupling strength leads to a change from afirst to second order percolation transition, Phys. Rev. Lett. 105(2010) 048701.

[19] R. Parshani, S. V. Buldyrev, S. Havlin, Critical effect of depen-dency groups on the function of networks, Proc. Natl. Acad.Sci. USA 108 (2011) 1007–1010.

[20] J. Gao, S. V. Buldyrev, H. E. Stanley, S. Havlin, Networksformed from interdependent networks, Nature Physics 8 (2012)40–48.

[21] S. Havlin, D. Y. Kenett, E. Ben-Jacob, A. Bunde, H. Hermann,J. Kurths, S. Kirkpatrick, S. Solomon, J. Portugali, Challengesof network science: Applications to infrastructures, climate, so-cial systems and economics, Eur. J. Phys. Special Topics 214(2012) 273–293.

[22] M. De Domenico, A. Sole-Ribalta, E. Cozzo, M. Kivela,Y. Moreno, M. A. Porter, S. Gomez, A. Arenas, Mathemati-cal formulation of multilayer networks, Phys. Rev. X 3 (2013)041022.

[23] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Evolutionary gameson multilayer networks: a colloquium, Eur. Phys. J. B 88 (2015)124.

[24] D. Y. Kenett, M. Perc, S. Boccaletti, Networks of networks – anintroduction, Chaos, Solitons & Fractals 80 (2015) 1–6.

[25] J. Sanz, C.-Y. Xia, S. Meloni, Y. Moreno, Dynamics of inter-acting diseases, Phys. Rev. X 4 (2014) 041005.

[26] G. F. de Arruda, E. Cozzo, T. P. Peixoto, F. A. Rodrigues,Y. Moreno, Disease localization in multilayer networks, Phys.Rev. X 7 (2017) 011014.

[27] D. Lazer, A. Pentland, L. A. Adamic, S. Aral, A. L. Barabasi,D. Brewer, N. Christakis, N. Contractor, J. Fowler, M. Gut-mann, T. Jebara, G. King, M. Macy, D. Roy, M. Van Alstyne,Life in the network: the coming age of computational socialscience, Science 323 (2009) 721–723.

[28] N. A. Christakis, J. H. Fowler, Connected: The SurprisingPower of Our Social Networks and How They Shape Our Lives,Little Brown, New York, 2009.

[29] R. Pastor-Satorras, A. Vespignani, Immunization of complexnetworks, Phys. Rev. E 65 (2002) 036104.

[30] Y. Liu, Y. Deng, M. Jusup, Z. Wang, A biologically inspired im-munization strategy for network epidemiology, J. Theor. Biol.400 (2016) 92–102.

[31] J. Gomez-Gardenes, P. Echenique, Y. Moreno, Immunizationof real complex communication networks, Eur. Phys. J. B 49(2006) 259–264.

[32] R. M. Anderson, R. M. May, B. Anderson, Infectious diseasesof humans: dynamics and control, Wiley, New York, 1992.

[33] R. Cohen, S. Havlin, D. Ben-Avraham, Efficient immunizationstrategies for computer networks and populations, Phys. Rev.Lett. 91 (2003) 247901.

[34] C. M. Schneider, T. Mihaljev, S. Havlin, H. J. Herrmann, Sup-pressing epidemics with a limited amount of immunizationunits, Phys. Rev. E 84 (2011) 061911.

[35] R. M. Christley, G. Pinchbeck, R. Bowers, D. Clancy,N. French, R. Bennett, J. Turner, Infection in social networks:using network analysis to identify high-risk individuals, Am. J.Epidemiology 162 (2005) 1024–1031.

[36] B. Shams, Using network properties to evaluate targeted immu-nization algorithms, Network Biology 4 (2014) 74.

[37] S. P. Borgatti, Centrality and network flow, Social Networks 27(2005) 55–71.

[38] L. K. Gallos, F. Liljeros, P. Argyrakis, A. Bunde, S. Havlin,

7

Improving immunization strategies, Phys. Rev. E 75 (2007)045104.

[39] J. C. Miller, J. M. Hyman, Effective vaccination strategies forrealistic social networks, Physica A 386 (2007) 780–785.

[40] N. Madar, T. Kalisky, R. Cohen, D. Ben-avraham, S. Havlin,Immunization and epidemic dynamics in complex networks,Eur. Phys. J. B 38 (2004) 269–276.

[41] P. Liu, H. Miao, Q. Li, A common acquaintance immuniza-tion strategy for complex network, in: Computer and Informa-tion Science, 2009. ICIS 2009. Eighth IEEE/ACIS InternationalConference on, IEEE, 2009, pp. 713–717.

[42] Z. Wang, D.-W. Zhao, L. Wang, G.-Q. Sun, Z. Jin, Immunityof multiplex networks via acquaintance vaccination, EPL 112(2015) 48002.

[43] M. Salathe, J. H. Jones, Dynamics and control of diseases innetworks with community structure, PLoS Comput. Biol. 6(2010) e1000736.

[44] K. Gong, M. Tang, P. M. Hui, H. F. Zhang, D. Younghae, Y.-C. Lai, An efficient immunization strategy for community net-works, PLoS ONE 8 (2013) e83489.

[45] P. Bonanni, Demographic impact of vaccination: A review,Vaccine 17 (1999) S120–S125.

[46] S. Funk, M. Salathe, V. A. Jansen, Modelling the influence ofhuman behaviour on the spread of infectious diseases: a review,J. R. Soc. Interface 7 (2010) 20100142.

[47] P. Manfredi, A. D’Onofrio, Modeling the interplay between hu-man behavior and the spread of infectious diseases, Springer,New York, 2013.

[48] Z. Wang, M. A. Andrews, Z.-X. Wu, L. Wang, C. T. Bauch,Coupled disease–behavior dynamics on complex networks: Areview, Phys. Life Rev. 15 (2015) 1–29.

[49] S. Cousins, Syrian crisis: health experts say more can be done,The Lancet 385 (2015) 931–934.

[50] P. V. Ganapathiraju, C. B. Morssink, J. Plumb, Endgame forpolio eradication? Options for overcoming social and politi-cal factors in the progress to eradicating polio, Global PublicHealth 10 (2015) 463–473.

[51] C. T. Bauch, S. Bhattacharyya, Evolutionary game theory andsocial learning can determine how vaccine scares unfold, PLoSComput. Biol. 8 (2012) e1002452.

[52] T. Oraby, V. Thampi, C. T. Bauch, The influence of social normson the dynamics of vaccinating behaviour for paediatric infec-tious diseases, Proc. R. Soc. B 281 (2014) 20133172.

[53] G.-Q. Sun, M. Jusup, Z. Jin, Y. Wang, Z. Wang, Pattern tran-sitions in spatial epidemics: Mechanisms and emergent proper-ties, Phys. Life Rev. 19 (2016) 43–73.

[54] V. Capasso, G. Serio, A generalization of the kermack-mckendrick deterministic epidemic model, Mathematical Bio-sciences 42 (1978) 43–61.

[55] K. Sigmund, Games of Life: Exploration in Ecology, Evolutionand Behavior, Oxford University Press, Oxford, UK, 1993.

[56] J. W. Weibull, Evolutionary Game Theory, MIT Press, Cam-bridge, MA, 1995.

[57] J. Hofbauer, K. Sigmund, Evolutionary Games and Popula-tion Dynamics, Cambridge University Press, Cambridge, U.K.,1998.

[58] M. A. Nowak, Evolutionary Dynamics, Harvard UniversityPress, Cambridge, MA, 2006.

[59] K. Sigmund, The Calculus of Selfishness, Princeton UniversityPress, Princeton, NJ, 2010.

[60] G. Szabo, G. Fath, Evolutionary games on graphs, Phys. Rep.446 (2007) 97–216.

[61] J. M. Pacheco, V. V. Vasconcelos, F. C. Santos, Climate changegovernance, cooperation and self-organization, Phys. Life Rev.

11 (2014) 573–586.[62] M. R. D’Orsogna, M. Perc, Statistical physics of crime: A re-

view, Phys. Life Rev. 12 (2015) 1–21.[63] D. Helbing, D. Brockmann, T. Chadefaux, K. Donnay,

U. Blanke, O. Woolley-Meza, M. Moussaid, A. Johansson,J. Krause, M. Perc, Saving human lives: What complexity sci-ence and information systems can contribute, J. Stat. Phys. 158(2015) 735–781.

[64] M. Perc, Phase transitions in models of human cooperation,Phys. Lett. A 380 (2016) 2803–2808.

[65] M. Perc, J. J. Jordan, D. G. Rand, Z. Wang, S. Boccaletti,A. Szolnoki, Statistical physics of human cooperation, Phys.Rep. (in press).

[66] C. T. Bauch, D. J. D. Earn, Vaccination and the theory of games,Proc. Natl. Acad. Sci. USA 101 (2004) 13391–13394.

[67] F. Fu, D. I. Rosenbloom, L. Wang, M. A. Nowak, Imitationdynamics of vaccination behaviour on social networks, Proc.R. Soc. B 278 (2011) 42–49.

[68] D. A. Rand, M. A. Nowak, Human cooperation, Trends in Cog-nitive Sciences 17 (2013) 413–425.

[69] A. Bear, D. G. Rand, Intuition, deliberation, and the evolutionof cooperation, Proc. Natl. Acad. Sci. USA 113 (2016) 936–941.

[70] Y. Moreno, R. Pastor-Satorras, A. Vespignani, Epidemic out-breaks in complex heterogeneous networks, Eur. Phys. J. B 26(2002) 521–529.

[71] W. O. Kermack, A. G. McKendrick, A contribution to the math-ematical theory of epidemics, Proc. R. Soc. A 115 (1927) 700–721.

[72] R. M. Anderson, R. M. May, Infectious Diseases in Humans,Oxford University Press, Oxford, 1992.

[73] O. Diekmann, J. Heesterbeek, Mathematical Epidemiology ofInfectious Diseases: Model Building, Analysis and Interpreta-tion, John Wiley & Sons, New York, 2000.

[74] M. Keeling, P. Rohani, Modeling Infectious Diseases in Hu-mans and Animals, Princeton University Press, Princeton,2007.

[75] S. Eubank, H. Guclu, V. A. Kumar, M. V. Marathe, A. Srini-vasan, Z. Toroczkai, N. Wang, Modelling disease outbreaks inrealistic urban social networks, Nature 429 (2004) 180–184.

[76] L. Hufnagel, D. Brockmann, T. Geisel, Forecast and control ofepidemics in a globalized world, Proc. Natl. Acad. Sci. USA101 (2004) 15124–15129.

[77] N. M. Ferguson, D. A. Cummings, S. Cauchemez, C. Fraser,S. Riley, A. Meeyai, S. Iamsirithaworn, D. S. Burke, Strate-gies for containing an emerging influenza pandemic in South-east Asia, Nature 437 (2005) 209–214.

[78] I. M. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Han-shaoworakul, D. A. Cummings, M. E. Halloran, Containingpandemic influenza at the source, Science 309 (2005) 1083–1087.

[79] V. Colizza, A. Barrat, M. Barthelemy, A.-J. Valleron,A. Vespignani, Modeling the worldwide spread of pandemicinfluenza: baseline case and containment interventions, PLoSMed. 4 (2007) e13.

[80] D. L. Chao, M. E. Halloran, V. J. Obenchain, I. M. Longini Jr,FluTE, a publicly available stochastic influenza epidemic sim-ulation model, PLoS Comput. Biol. 6 (2010) e1000656.

[81] D. Brockmann, D. Helbing, The hidden geometry of complex,network-driven contagion phenomena, Science 342 (2013)1337–1342.

[82] W. Van den Broeck, C. Gioannini, B. Goncalves, M. Quag-giotto, V. Colizza, A. Vespignani, The GLEaMviz computa-tional tool, a publicly available software to explore realistic epi-

8

demic spreading scenarios at the global scale, BMC InfectiousDiseases 11 (2011) 37.

[83] S. Bikhchandani, D. Hirshleifer, I. Welch, A theory of fads,fashion, custom, and cultural change as informational cascades,Journal of Political Economy 100 (1992) 992–1026.

[84] T. Kuhn, M. Perc, D. Helbing, Inheritance patterns in cita-tion networks reveal scientific memes, Phys. Rev. X 4 (2014)041036.

[85] S. Goel, A. Anderson, J. Hofman, D. J. Watts, The structural vi-rality of online diffusion, Management Science 62 (2015) 180–196.

[86] J. P. Gleeson, K. P. OSullivan, R. A. Banos, Y. Moreno, Effectsof network structure, competition and memory time on socialspreading phenomena, Phys. Rev. X 6 (2016) 021019.