Vaccination and Epidemics in Networked Populations – an Introduction

Vaccination and epidemics in networked populations – An introduction

Zhen Wang,1 Yamir Moreno,2, 3, 4 Stefano Boccaletti,5, 6 and Matjazˇ 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, Spain 3Department of Theoretical Physics, University of Zaragoza, E-50009, Zaragoza, Spain 4Complex Networks and Systems Lagrange Lab, Institute for Scientiﬁc Interchange, Turin 10126, Italy 5CNR – 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, Israel 7Faculty of Natural Sciences and Mathematics, University of Maribor, Koroskaˇ 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” that is in the making at Chaos, Solitons & Fractals. While vaccination is undoubtedly one of the most important preventive measures of modern times, epidemics are feared as one of the most damaging phenomena in human societies. Recent research has explored the pivotal implications of individual behavior and heterogeneous contact patterns in networked populations, as well as the many feedback loops that exist between vaccinating behavior and disease propagation [1, 2]. Interdisciplinary explorations in the realm of statistical physics, network science, nonlinear dynamics, and data analysis have given rise to theoretical epidemiology, as well as to the theory of epidemic processes in complex networks. From classical models assuming well-mixed populations and ignoring human behavior, to recent models that account for behavioral feedback and population structure, we have come a long way in understanding disease transmission and disease dynamics, and in using this knowledge to devise effective prevention strategies. This special issue is aimed at helping the further development of these synergies. We hope that it contributes to enhance our understanding of vaccination and epidemics in networked populations, by featuring works related to vaccination and epidemics using techniques ranging from complex and temporal networks to network of networks and show-casing the possibilities of interdisciplinarity via complex systems science to tackle the challenges in our quest for a healthier future. Topics of interest include but are not limited to epidemiological modeling and vaccination, behavior-vaccination dynamics, reaction-diffusion processes and metapopulation models, evolutionary and game theoretical models in epidemiology, as well as to inﬂuence maximization and digital epidemiology.

I. NETWORKS changes can be studied in the realm of temporal networks [11, 15, 16], where the theoretical framework accounts for During the past two decades network science has emerged the addition or removal of nodes, or similarly for the changes as a central paradigm behind some of the most fascinating dis- in the links between nodes, over time. A beautiful example coveries of the 21st century [3, 4]. From the mathematical for- of how changing links over time between different individu- mulation of small-world properties and their omnipresence in als might affect disease spreading, reproduced from [11], is seemingly diverse systems such as electric power grids, food shown in Fig. 1. chains, brain networks, protein networks, transcriptional net- Also of importance, networks exist between different lay- works and social networks [5], to universal scaling proper- ers of each studied system, and this is particularly apparent ties due to growth and preferential attachment that likewise in social systems, where one person can be simultaneously pervade biological, social and technological networks [6], the member of many different networks that are to various degrees field of research today known as network science has been interdependent [17–21]. This can be accommodated in the going from strength to strength, as evidenced by the many theoretical framework of multilayer networks, or more gener- reviews devoted to this field of research [7–14]. Network sci- ally networks of networks, which acknowledge that not only ence has provided models, methods and algorithms that have is the range of our interactions limited and thus inadequately revived not just statistical physics, arguably the nurturer to described by well-mixed models, but also that the networks the field, but indeed many other fields of natural and social that should be an integral part of such models are often in- sciences, including of course research concerning vaccination terconnected, thus making the processes that are unfolding on and epidemics to which this special issue is devoted to. them interdependent [12, 13, 22–24]. From the world econ- In addition to static networks, network science allows us to omy and transportation systems to the spread of epidemics, it study and take into account network evolution over time, for is clear that processes taking place in one network can signifi- example due to changes in external factors, the onset of dis- cantly affect what is happening in many other networks. From ease, targeted attack, or simply due to random failure. Such this point of view, the consideration of multilayer or interdependent networks is crucial for a comprehensive treatment of vaccination and epidemics in networked populations [25, 26]. In addition to the theoretical coming of age of network sci- ∗Electronic address: [email protected] ence, technological breakthroughs in the acquisition and stor- 2

8,13,14 8,13,14 8,13,14 B C B C B C tion efforts on networks are more effective if random vacci- 1 1 1 1 1 1 7, nation is combined with targeted vaccination and the vaccina- 6,7, 6, 6,7, A 1,3 A 1,3 7 A 1,3 tion of acquaintances [1, 29–31]. While random vaccination 2,4,7 2,4, 2,4,7 of course does not require any information about the structure D D D of the network, it needs in return a very broad coverage, and is thus very costly, in order to be effective [29, 32]. More t = 0 t = 6.5 t = ∞ A precisely, uniform vaccination is super inefficient for disease B eradication in heterogeneous networks [29]. To offset this dis- C advantage, targeted vaccination according to the centrality in- D dexes of individuals, like degree, betweenness and closeness 1050 [33–37] should be considered. For example, because large- t degree nodes are known to be responsible for the spreading of disease, a degree-based targeted vaccination strategy was FIG. 1: The importance of temporal networks. Changes in the links proposed to immunize the most highly connected individuals between individuals A, B, C and D affect disease spreading. In the [34, 38]. Compared with random vaccination, targeted vac- upper row, the contact times between the individuals are indicated on cination thus greatly reduces costs, but since the identifica- the edges. Let us assume that a disease starts spreading at individual tion of centrality-defined individuals usually takes a long(er) A and spreads further as soon as contact occurs. This spreading pro- time, the practicability of targeted vaccination is doubtful in cess is illustrated for four different times from left to right. At t = ∞ practice, especially in large populations [39]. There also ex- the spreading stops as individual D cannot get infected. However, if ists inconsistencies in the definitions of some centrality in- the spreading starts at individual D, then all other depicted individ- dexes in more complex networks, especially in multilayer net- uals would eventually be infected. Aggregating the edges into one works, which in turn restricts the universality of targeted vac- static graph cannot capture this effect that arises from the time or- cination [12]. Acquaintance vaccination is more suitable for dering of contacts. The lower row visualizes the same situation by showing the temporal dimension explicitly. This figure is reproduced practical applications in that a fraction of nodes is selected with permission from [11]. at random, and then random neighbors are vaccinated further [33, 40]. This approach requires only partial knowledge of the network structure, and its variants have been studied in sub- age of vast amounts of digitized data have significantly aided stantial detail [41, 42]. In addition to these different vaccina- the progress in our understanding of both vaccination and epi- tion strategies, network structure also plays an important role demic spreading. The data revolution has had a particularly in the transmission routes and infection levels [2, 9–12]. For deep impact on the social sciences, where social experiments example, in community networks, bridge nodes, connecting in the past typically involved one-shot self-reported data on different communities, provide pathways for disease propaga- relationships and their outcomes in a small sample of people, tion, so that how to identify and immunize these nodes thus while today the approach is to mine massive amounts of dig- decides the final success rate [43, 44]. itized data for both the structure and the content of relation- With the latest advances in science and technology, today, ships [27, 28], which may in turn relevantly inform vaccina- safe and effective vaccines exist for many of the most com- tion strategies [2] as well as the spread of epidemics [1]. mon infectious diseases, such as smallpox, measles, pertussis, In particular, the synergies between mathematical modeling influenza, hepatitis, chickenpox and diphtheria, and the use and theoretical explorations and data-driven research, coupled of vaccines has been estimated to save millions of children’s with taking into account feedbacks between disease, behav- lives per year [45]. However, due to high economic costs of ior and vaccination, are likely the future of complex systems vaccination, and often also personal beliefs and the fear of research aimed at a better understanding of vaccination and possible side effects, people usually regard vaccination as a epidemics in modern human societies. voluntary rather than a compulsive measure. In this sense, behavioral vaccination, borrowing the framework of behavior dynamics in game theory and psychology, becomes a useful II. VACCINATION research framework to elucidate real-world disease spreading and prevention [2, 46–48]. As already stated, vaccination is undoubtedly one of the Despite of all the success, great challenges still remain, in most important preventive measures of modern times, unfold- part due to the difficulties in providing and administering vac- ing on the planetary scale with the aim of preventing the cines in the world’s poorest and war-torn regions [49, 50], but spread of infectious diseases that have had a crippling im- also because in many wealthy countries, such as the United pact on many human societies in the not so distant past. The Kingdom, Switzerland and Germany, there is considerable 20th century, in particular, saw enormous progress in public parental vaccine refusal or vaccine hesitancy on account of health, especially in terms of preventing and treating infec- unfounded concerns about significantly adverse side effects tious diseases. The use of vaccines was crucial to that effect. such as autism. Regarding the latter, it is important to ac- However, due to limited amounts of vaccines and knowledge, knowledge that vaccine refusal is not a new phenomenon, and early studies simply assumed that vaccination should be done even the first vaccine to be invented – the smallpox vaccine compulsively and/or randomly. Today we know that vaccina- – was subject to considerable resistance, as nicely illustrated 3

tion choices are strongly coupled to one another, and further, that this coupling is particularly important near the elimination threshold [48]. Accordingly, vaccine refusal may become an increasingly important barrier to achieving elimination and eradication, 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 disease will increase the risk perception, based on which people feel an incentive to take protective measures, i.e., to vaccinate. In turn, the spreading of the disease is suppressed, to the point where risk perception drops again and people opt not to vaccinate. Mathematical models hold the promise to help us better un- derstand such coupled dynamics, and comparing models with data has often shown that behavior is important for understanding observed epidemic patterns. Also of notice, these models often exhibit interesting dynamics that is similar to FIG. 2: Satirical cartoon by James Gillray capturing public fears of those studied in physics, including phase transitions, oscilla- the smallpox vaccine in the 18th century, the first “vaccine scare”. tory solutions, and spatiotemporal pattern formation, thus in- Edward Jenner stands in the middle, administering the smallpox vac- dicating that similar methodologies to those used in physics cine. Vaccinated persons grow cow parts out of various parts of their bodies because the smallpox vaccine of the time was based on the and complex systems research in general are likely to be help- virus that causes cowpox, hence producing the fear among some that ful in their study [2, 53]. the vaccine would turn humans into cow-like hybrids. A painting of For a more concrete perspective, we briefly review arguably the Worshippers of the Golden Calf hangs in the background. Source: the first phenomenological model to capture the impact of Wikimedia Commons. adaptive human behavior on disease transmission [54]. The model is given by the following ordinary differential equa- tions by James Gillray in his 18th century portrayal of the “vaccine dS = −g(I)S, scare” (see Fig. 2). The take-home message is that it is very dt important to account for behavioral feedback effects, and to dI account for cognition and deliberation when devising mathe- = g(I)S, −γI (1) dt matical models of vaccination. Namely, as logistic, financial, dR technological and administrative barriers to vaccination con- = γI, tinue receding in the coming decades, enabling it to become dt easier to improve vaccine coverage everywhere in the world, where S is the number of susceptible individuals, I is the it is not too far fetched to speculate that vaccine refusal and number of infectious individuals, R is the number of recov- vaccine hesitancy will become an important barrier – and per- haps even the most important barrier – to global eradication of vaccine-preventable infections [2]. As a case in point, when we observe vaccinating behavior responding to changes in dis- risk perception ease prevalence, as in the Disneyland, California measles out- + break of 2014 and other examples, it is impossible to overlook that vaccinating behavior and disease dynamics are strongly linked to one another, particularly near elimination thresholds. disease behavioral dynamics dynamics Although human behavior is surely more difficult to math- ematically model than the motion of a billiard ball or many of the somewhat more complex systems studied by physicists, _ it is nevertheless conceivable that at some levels of organiza- vaccination tion it is possible to develop simple models that have useful predictive power. For example, mathematical models incor- porating behavior-disease interactions have been fitted to time series data of infectious disease prevalence, and model selec- tion approaches have shown that models that include behavior FIG. 3: Illustration of behavior-disease-vaccination interactions. The can explain the data more effectively compared to models that feedback loop, on the one hand, promotes vaccination as the disease neglect behavior [51, 52]. spreads and risk perception rises, but on the other hand, people revert Hence, there is a strong public health and scientific ratio- to abstaining when vaccination takes effect and the spreading of the nale for studying behavior-disease-vaccination interactions. disease is effectively mitigated. But this then of course restarts the Existing research shows that behavior, disease and vaccina- loop sooner or later. 4 ered individuals, γ is the per capita recovery rate, and g(I) III. EPIDEMIC PROCESSES is the force of infection. The form of g(I) is chosen to reflect the phenomenological impact of unspecified psychological ef- Research on epidemic outbreaks in complex heterogeneous fects. One of the functional forms explored in [54] included networks has roots that go back to the beginning of the 21st century [70], when it was shown that large fluctuations in βI the degrees of individuals in social networks considerably g(I) = , (2) 1 + βδI strengthen the incidence of epidemic outbreaks. Scale-free networks, for example, were shown to altogether lack an epi- which saturates at a high prevalence of infection I. Such sat- demic threshold and thus always contain a finite fraction of uration of the transmission rate at high levels of prevalence infected individuals. This particular weakness of networked could occur if individuals reacted to high prevalence by reduc- populations in comparison to well-mixed populations, which ing their contact rate through hand-washing or other means, was observed also in models without immunity, at the time which is a plausible assumption for a sufficiently dangerous defined a new epidemiological framework characterized by a infectious disease. The model is phenomenological because, highly heterogeneous response of the system to the introduc- although it is motivated by psychological factors, the equa- tion of infected individuals with different connectivity. Fast tions only describe the impact of such psychological factors forward 15 years into the future, and today we have a com- on the transmission rate without actually modeling specific prehensive theory of epidemic processes in complex networks psychological processes such as individual cognition or social [1], but with many challenges and open problems still remain- learning. ing. While epidemic models based on reaction-diffusion pro- An apt theoretical framework to account for behavior- cesses have arguably been the workhorse of the field since its disease-vaccination interactions is also evolutionary game inception [71–74], one of the more exciting recent develop- theory [55–59], which has already been used to study the clash ments that we here mention as example concerns agent based between what is optimal for the individual and what is optimal models, and network epidemiology in particular [75–81]. This for the group in a variety of contexts [16, 23, 60–65], includ- approach draws on synergies between large-scale simulations ing vaccination [66, 67]. At the heart of the choice whether or and data-based network construction, the outcome of which not to vaccinate is of course a social dilemma, which invokes is typically an agent-based model that evolves in a spatially the puzzle of human cooperation [68]. To vaccinate is clearly structured population. A schematic illustration of the con- the socially responsible choice, but it comes at a cost that has struction of a synthetic population is presented in Fig. 4, to do with offsetting the fear and potential side effects. But where based on census and demographic data as input the net- if individuals choose not to vaccinate, they may become in- work of contacts is constructed from an intermediate bipar- fected, which will inflict larger costs for recovery. Those that tite network associating individuals to locations. The GLobal do not vaccinate but still benefit from herd immunity, which Epidemic and Mobility (GLEAM) model [82] is a particular is generated by a large fraction of those who do vaccinate, is example that integrates census and mobility data in a fully individually the better choice, i.e., like free-riding in evolu- stochastic metapopulation network model that allows for the tionary games. Obviously, if everybody chooses not to vaccinate the herd immunity is lost and the public good, which in this case is the precious absence of an infectious disease, is lost with it. Accordingly, uninfected non-vaccinators are the defectors who benefit from herd immunity generated by vaccinators without paying any real or perceived cost of vaccination, while vaccinators are the cooperators. In terms of the integration of cognitive abilities, a new model for looking at the evolution of intuition versus deliberation as well as at the evolution of cooperation versus defec- tion has recently been introduced [69], which lends itself to research in networked populations, and which could also be a viable starting ground for more realistic evolutionary game theoretical models of vaccination. To conclude this section, we note that traditional epidemiological models that do not take into account behavioral dy- FIG. 4: Network epidemiology. At the macroscopic level (left), a namics have been very successful over the past century, both synthetic population and the movements of its members can be con- in terms of yielding rich theoretical opportunities for research structed from census and demographic data. A bipartite network as- as well as in terms of providing insights into epidemiological sociating individuals to locations can be derived based on data from mechanisms and how to improve infection control. Neverthe- the synthetic population (middle). Lastly, the unipartite projection of less, the impact of human behavior on disease dynamics and the bipartite network provides a contact network (right), upon which vaccination is ubiquitous, and vaccinating behavior itself is an agent-based model can unfold. This figure is reproduced with certainly no exception in this important feedback loop. permission from [1]. 5 detailed simulation of the spread of influenza-like illnesses also to a large degree unexpected consequences in another net- around the globe. work [17–21, 25, 26]. It remains subject to future research Importantly, while the spread of infectious diseases is most to reveal to what degree this influences our ability to predict commonly the focal point of research on epidemic spreading, and control the epidemics, and what are the consequences of in particular because this is one of the most damaging phe- vaccination abstinence in one social group for others that are nomena in human societies in the past and present day, the re- codependent. sulting mathematical models can of course be generalized and Regarding mathematical modeling, it remains of interest to adapted to describe diffusion of information [83], the spread- further integrate the many feedbacks between disease and be- ing of rumors and scientific memes [84], the emergence of havior, in particular vaccinations behavior, and to introduce online virality [85], and indeed many other phenomena that cognition and deliberation as processes that are inseparable to are inherent to our modern existence, both off and online. Re- humans. Here research in psychology and anthropology can cently, a null model for social spreading phenomena has been provide vital guidance and relevantly inform the next gener- proposed [86], which can reproduce several characteristics of ation of models for them to realistically represent these pro- empirical micro-blogging data, such as for example the heavy- cesses that thus far remain under explored. tailed distribution of meme popularity. The model clearly dis- tinguishes the roles of two distinct factors affecting popularity, Lastly, one has to mention the ever-increasing availability namely the memory time of users and the connectivity struc- of digital data and the many exciting prospects this creates, ture of the social network. not just for testing theoretical predictions, but also in terms Along these lines, research on epidemics in the broadest of using the data during the modeling process itself, as has possible sense continues to draw together different fields for already been demonstrated, for example in [82]. us to arrive at a better understanding of processes that are be- To conclude, we note that this special issue is also to feature hind one of the greatest challenges that lie ahead for human future research. In order to avoid delays that are sometimes societies, namely the maintenance of public health and the associated with waiting for a special issue to become com- mitigation of large-scale epidemics and the spread of infec- plete before it is published, we have adopted an alternative tious disease. As an added benefit, one finds that the same approach at Chaos, Solitons & Fractals. The special issue will spreading phenomena that drive epidemics drive also social be a virtual one, updated continuously from the publication contagion processes, which surely adds to the arguments to of this introduction onwards, meaning that new papers will actively engage in this research. be published immediately after acceptance. The down side of this approach is that we cannot feature the traditional brief summaries of each individual work that will be published, but IV. FUTURE RESEARCH we hope that this is more than made up for by the immediate availability of the latest research. Please stay tuned, and con- As we hope we have succeeded to outline in this intro- sider contributing to “Vaccination and epidemics in networked duction, vaccination and epidemics in networked populations populations”. merit outstanding attention from the research community, not only for their obvious importance to the wellbeing of our societies, but also from the purely scientific point of view due to the many open problems and fascinating challenges that remain to be addressed. The triptych The Garden of Earthly De- lights shown in Fig. 5, painted by Hieronymus Bosch around the turn of the 16th century, is a lucid portrayal of events that may transpire if we fail to keep the public health in the best possible order, and if we fail to cooperate in vaccinating our children and working together to minimize the risk of epidemics. Particularly promising areas of research that warrant attention include the consideration of mathematical and statistical laws that govern the coevolution of the network and the evolution of vaccination or of the epidemic process that un- folds. Here the formalism of temporal networks is a neces- FIG. 5: The Garden of Earthly Delights serves as a didactic warning sary prerequisite [11], while further ideas how to proceed can on the perils of life’s temptations, which applies as much to shirking be sought in [15, 16], where research in closely related fields on vaccination, as it does to abandoning vigilance in times of epi- has already been reviewed. demics. From left to right, we first have the meeting of Adam and Further in terms of the interaction network, recent research Eve, followed by a socially balanced state of wellbeing in the center, has revealed that social networks are almost inherently inter- and ending with a hellscape that portrays the torments of damnation dependent and layered, in the sense that a particular network that may result due to excesses in modern human societies. The pic- of contacts seldom exists in isolation [12, 13]. Consequently, ture was taken by Alonso de Mendoza, and reproduced here from the processes unfolding in one network can have significant and public domain. Source: Wikimedia Commons. 6

Acknowledgments search and Development Program (Grant 2016YFB0800602), the National Natural Science Foundation of China (Grant We would like to thank the publishing team at Chaos, Soli- 61471299), by the Government of Aragon,´ Spain through a tons & Fractals for all their support and assistance in creating grant to the group FENOL, by MINECO and FEDER funds this special issue. We would also like to thank all the au- (Grant FIS2014-55867-P), and by the Slovenian Research thors and referees in advance for their valuable contributions Agency (Grants J1-7009 and P5-0027). and help. This work was supported by the National Key Re-

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