key: cord-0824262-exauiuql authors: Pires, Marcelo A.; Oestereich, Andre L.; Crokidakis, Nuno; Queir'os, S'ilvio M. Duarte title: Antivax movement and epidemic spreading in the era of social networks: Nonmonotonic effects, bistability and network segregation date: 2021-01-19 journal: Physical review. E DOI: 10.1103/physreve.104.034302 sha: 1120c6fd490c8503d05676c150b04a266ded2b4d doc_id: 824262 cord_uid: exauiuql In this work, we address a multicoupled dynamics on complex networks with tunable structural segregation. Specifically, we work on a networked epidemic spreading under a vaccination campaign with agents in favor and against the vaccine. Our results show that such coupled dynamics exhibits a myriad of phenomena such as nonequilibrium transitions accompanied by bistability. Besides we observe the emergence of an intermediate optimal segregation level where the community structure enhances negative opinions over vaccination but counterintuitively hinders - rather than favoring - the global disease spreading. Thus, our results hint vaccination campaigns should avoid policies that end up segregating excessively anti-vaccine groups so that they effectively work as echo chambers in which individuals look to confirmation without jeopardising the safety of the whole population. In such a complex contemporary society where elements -people and events -influence one another and feedback at different scales [1] , the application of tools set forth in Statistical Physics in order to cope with collective phenomena has gained prominence in other areas such as Biology and Medicine, Social Sciences, and Humanities, which have put quantitative tools in the methodologies they apply [2] [3] [4] [5] [6] [7] . Accordingly, phenomena in which there is a change in the collective behavior displayed by a social system have turn into an appropriate field for the application of such techniques [8] [9] [10] ; among the several different instances we can find important contributions within the spreading of epidemics as well as opinion dynamics (see eg [11] and Sec. II). In spite of the fact that the two subjectmatters are not related at first, the dissemination of a causal relationship between neurological disorders and vaccinations [12] has prompted an urban myth that ultimately has jeopardized the elimination of the disease in countries with a very high Human Development Index as the USA [13] . Next the manuscript is organized as follows: in Sec. II, we establish the state-of-the-art of the problem; in Sec. III, we introduce our model for the combined dynamics of opinion and contagion; in Sec. IV, we discuss the results for the model; and in Sec. V, we present our final observations on the work and future perspectives about it. Modular networks [14] are generated by an algorithm that leads to networks with an architecture of communities. A given node in each community can be connected to nodes of the same community (intracommunity links) and/or to nodes of the other community (intercommunity links). The impact of the network modularity in spreading processes has been investigated in recent years. Since the results introduced in Ref. [14] , a series of works were published regarding the subject of optimal network modularity; therein, the authors showed that modular structure may have counterintuitive effects on information diffusion. Indeed, it was discussed that the presence of strong communities in modular networks can facilitate global diffusion by improving local intracommunity spreading. Still in relation to modular networks, it was recently found that an optimal community structure that maximizes spreading dynamics which can pave the way to rich phase diagrams with exhibiting first-order phase transitions [15] . Within the same context, the authors in Ref. [16] discussed the impact of social reinforcement in information diffusion. They also found optimal multicommunity network modularity for information diffusion, i.e., depending on the range of the parameters the multicommunity structure can facilitate information diffusion instead of hindering it. Regarding biological systems, it was recently found there is a nonlinear relation between modularity and global efficiency in animal networks, with the latter peaking at intermediate values of the former [17] . In addition, in neural networks there exists an optimal modularity for memory performance, where a balance between local cohesion and global connectivity is established, allowing optimally modular networks to remember longer [18] . The authors in Ref. [19] studied the importance of close and ordinary social contacts in promoting large-scale con-tagion and found an optimal fraction of ordinary contacts for outbreaks at a global scale. With respect to correlations in complex networks, it was found that constraining the mean degree and the fraction of initially informed nodes, the optimal structure can be assortative (modular), core-periphery or even disassortative [20] . Other recent works leading with optimal modularity in networks can be found in [21, 22] . In a recent work [23] , it was proposed a model of disease spreading in a structural modular complex network and studied how the number of bridge nodes n that connect communities affects disease spreading. It was verified that near the critical point as n increases, the disease reaches most of the communities, but each community has only a small fraction of recovered nodes. Moreover, a combination of social networks with game theory was studied in Refs. [24, 25] . Disease information can spark strong emotions like fear -or even panic -that would affect behavior during an epidemic. The authors in [26] considered an agentbased model that assumes that agents can obtain a complete picture of the epidemic via information from local daily contacts or global news coverage. Those results helped conclude that such model can be used to mimic real-world epidemic situations and explain disease transmission, behavior changes, and distribution of prevalence panic. Game theory was also considered to reproduce the decision-making process of individuals during the evolution of a disease. In [27] a spatial evolutionary game was coupled to a SIR model, and the results showed that protective behaviors decrease the numbers of infected individuals and delay the peak time of infection. The study also concluded that increased numbers of risk-averse individuals and preemptive actions can more effectively mitigate disease transmission; however, changes in human behavior require a high social cost (such as avoidance of crowded places leading to absences in schools, workplaces, or other public places). A recent work considered a coupled behavior-change and infection in a structured population characterized by homophily and outgroup aversion [28] . It was found that homophily can either increase or decrease the final size of the epidemic depending on its relative strength in the two groups. In addition, homophily and outgroup aversion can also produce a 'second wave' in the first group that follows the peak of the epidemic in the second group. Models of opinion dynamics were applied in the context of opinions about vaccination (pro versus anti-vaccine) without coupling an epidemic process [29] . Later, kinetic opinion dynamics were coupled to classical epidemic models in order to study the feedback among risk perception, opinions about vaccination, and the disease spreading. In [30] it was found that the engagement of the pro-vaccine individuals can be crucial for stopping the epidemic spreading. On the other hand, the work [31] found counterintuitive outcomes like the fact that an increment in the initial fraction of the population that is pro-vaccine can lead to smaller epidemic outbreaks in the short term, but it also contributes to the survival of the chain of infections in the long term. Recently, the anti-vaccine sentiment was treated as a cultural pathogen. The authors in [32] modeled it as a 'infection' dynamics. The authors showed that interventions to increase vaccination can potentially target any of three types of transitions -decreasing sentiment transmission to undecided individuals, increasing pro-vaccine decisions among undecided individuals, or increasing sentiment switching among anti-vaccine individuals. We previously cited anti-vaccine opinions, thus it is important to mention some recent discussion about the global anti-vaccine movement. Since the online discussions dominate the social interactions in our modern world, the propagation of such anti-vaccine opinions is growing fast. A recent report noted that 31 million people follow anti-vaccine groups on Facebook, with 17 million people subscribing to similar accounts on YouTube [33] . The authors in [34] recently pointed that if the current trends continue, anti-vaccine views will dominate online discussion in 10 years. The importance of antivaccine movement is fundamental for the evolution of COVID-19 outbreak. Indeed, the authors in [35] called attention to the fact that it is a key point to qualitatively assess how the administration of a vaccine could affect the COVID-19 outbreak, taking into account of the behavioral changes of individuals in response to the information available on the status of the disease in the community. According to a study published in August 2020, nearly one in four adults would not get a vaccine for COVID-19 [36] and in some countries, more than half of the population would not get it, including Poland and France [37] . In September 2020, it was verified that only 42 percent of Americans said yes to receiving a future COVID-19 vaccine, across all political sides. It means that even in a best-case scenario where a future high performing vaccine is 95% effective in an individual, it would only impact 42x95≈ 40% of the population, which is way below predicted thresholds for herd immunity [38] . Even though payoff-based models have been employed to address the problem of vaccination dynamics (for instance see [24, 25, 39] and the references therein), there is an alternative approach that is based on the coupling of epidemic and psychosocial factors that have been provided a successful modelling of phenomena related to vaccination dynamics [28, 32, [40] [41] [42] [43] [44] . In this work, we follow such second methodology. Specifically, based on Refs. [31] , [45] we consider an agent-based dynamics in which the opinion about vaccination, o i ∈ [−1, 1], of each agent, i, evolves with 1(−1), then it adopts the extreme value 1(−1) [45] . The opinion dynamics regarding the vaccination campaign is coupled with the epidemic dynamics, due to the factor I n(i) (t) in Eq. (1). Based on [30, 31] (and references therein), we define the transitions among the epidemic compartments as follows: • I α → S: an Infected agent i recovers with probability α; • R φ → S: a immune agent i becomes Susceptible again with the resusceptibility probability φ. We assume that Vaccinated and Recovered agents are in the same compartment [46] [47] [48] [49] [50] . The vaccination probability g i of an agent i is proportional to his opinion about vaccination −1 ≤ o i ≤ 1: Despite the differences, the modeling of the coupling between disease and opinion evolution is still a open subject. In this work, we consider the two dynamics have the same time scale. An overview of our model is shown in Fig.2 . An element in this problem which is still focus of debate concerns the timescale of each dynamics, epidemic and opinion. On the one hand, it is often assumed in the epidemiological literature [41] [42] [43] 51 ] that the two timescales are equivalent. At first, this can be understood as a simplification it captures the mass vaccination campaigns governments swiftly implement in order to avoid disease outbreaks. On the other hand, it is possible to assume different timescales of evolution of the diseases and opinions about the disease [52, 53] . In this work we consider the first approach of equality between the two timescales. Based on Ref. [54] and related literature, we start by picking the first N 1 = N/2 of the N nodes and attaching them to the community 1, and assigning the other N 2 = N − N 1 nodes to community 2. We then proceed by randomly assigning (1 − µ)M connections among pairs of nodes from the same community and µM connections are randomly distributed among pairs of nodes that belong to distinct communities, where M = N k/2 and k is the network average degree [54] . The parameter µ regulates the community strength: large values of µ means more ties between the two communities consequently a weaker community organization. Another way to control the network structure -especially in the formation of the echo chambers -is by considering rewiring [55] . We consider that community 1 holds a positive stance on vaccination, whereas the community 2 holds a negative opinion about that. We also assume the chain of infections starts in community 2, because o i < 0 leads to a low propensity for the agents to get vaccinated, which is naturally more relevant. If the epidemic started in community 1, pro-vaccine opinions, o i > 0, would induce a higher probability for an agent to get vaccinated that ultimately would end up disrupting the chain of contagions. Let U (a, b) be a single random value from a uniform distribution in the range [a, b]. At t = 0, we set: In this section, we present our results come from Monte Carlo simulations of networks with N = 10 4 nodes and k=20. In all simulations, we set α = 0.1 and φ = 0.01, without loss of generality. In Figs. 3-6 , we show the steady-state density of infected agents in the community u, I u . We also depict the behavior of the stationary opinion in the community u, m u . In turn, I tot and m tot refer to the global proportion of infected individuals and global mean opinion, respectively. The results in Fig. 3 show that in the community 2 -the seed community -there is a transition from the absorbing phase (extinction of the epidemic) to the epidemic survival phase. In the community 1, there is no survival of the chain of infections in the long term. In this setting with µ = 0.1 -which can be understood as yielding a weak modular structure because of the small value of the parameter -the seed community remains with the negative opinion about vaccination, which weakens the vaccination campaign and thus facilitates the local permanence of the disease. Similarly, there is a persistence of the initial opinion in the community 1, which in this case is pro-vaccine and therefore favors the vaccine uptake that makes the epidemic spreading unsustainable. This means that a low number of intercommunity ties hinders the change in the community stance over vaccination; that creates a strong distinction in the epidemic spread between both communities with community 1 being unfavorable to epidemic spreading since m 1 > 0, and community 2 being favorable since m 2 < 0. In Fig. 4 dominance of the negative opinion about vaccination in the community 2. The epidemic contagion spreading is halted in the community 2, even though the agents have a negative opinion about the vaccination, due to the intermediate number of bridges, µ = 0.2, to the other community. These bridges are just strong enough to drain the infected agents of the community 2, but not strong enough to change its average opinion. In Fig. 5 , with µ = 0.3 there is a high number of intercommunity links. This additional connectivity between communities weakens the initial epidemic spreading in the community 2, but it is sufficient to introduce the possibility of a wide opinion change in the community 1. The opinion change in the community 1 facilitates the epidemic spreading in that community. This effect is limited because we can see for high infection probabilities λ > 0.8 the epidemic spread vanishes. So, we have a counterintuitive effect, because for higher transmissibility the epidemic spread vanishes. The reason behind this is the risk perception, wI in Eq. (1), which promotes vaccination, so higher transmissibility leads to a bigger outbreak that in turn results in better opinions about vaccination which ends up stopping the epidemic outbreak. The emergence of an intermediate range of µ that blocks the local and global epidemic spreading is visible in Fig. 6 . Regarding the opinion dynamics, an initial increase in µ leads to a decrease in m 1 and an increase in m 2 , that is the collective opinions tend to be less extremist for an initial rise in the amount of inter-communities routes. Then a further increase in µ promotes a sudden rise in m 1 and m 2 which means a speed up in the switch of opinions in the community 2. A further rise in µ leads to a bistable behavior in both communities. This intermediary range of inter-community connectivity that promotes a minimal epidemic spreading seems to also come from a perceived increment in the probability of an infected individual having a vaccinated neighbor. The increment of the bridges between communities the initially infected agents have a bigger probability of having a neighbor that was vaccinated because initially most of the infected people are in the community 2 and most agents with a positive opinion about vaccination are in the community 1. This effect does not persist for higher values of µ because then both communities tend to adopt the same average opinion about vaccination and this opinion can, in some cases, be negative. A negative global opinion about vaccines does not guarantee that the epidemic spread will persist, as can be seen in some cases for µ ≈ 0.23 where all samples had no infected individual but some of them had negative opinions about the vaccination. This can occur due to the fact that the number of infected agents can become zero before the negative global consensus about vaccines is reached. While in Figs. 3-4 there is a single stable steadystate (either extinction or persistence), Fig. 5 displays bistable solutions depending on the randomness 'embedded' in the dynamics. Moreover, the results in Fig. 3 suggest the absorbing-active epidemic transition is continuous for strong communities (such as µ = 0.1) whereas the results shown in Fig. 5 signalize this extinctionpersistence epidemic transition is discontinuous for weak communities (such as µ = 0.3). Therefore, the structural factors present in the modular networks can induce the emergence of bistability in the epidemic-vaccinationopinion dynamics as well as a change in the nature of the absorbing-active transitions. An overall look into Figs. 3-6 reveals a sudden transition can emerge from structural factors (increasing µ) or epidemiological factors (increasing λ). The transitions from the Disease-Free phase to the active phase and viceversa (epidemic resurgence) highlight the nonmonotonic behavior of the full dynamics with the transmissibility λ. Comparing with other works, we see that while in [14] there is an optimal modularity for enhancing information spreading, here there is an optimal modularity for hindering epidemic spreading. In figs. 7 and 8 we can see a wide range of results for two different settings of risk perception, i.e. w = 0.1 and w = 0.2. These results are similar but they show how increasing the risk perception reduces the range of parameters that present an endemic state. Other than that, we can also see that in community 1 the endemic state is more prevalent for higher values of modularity µ, this is to be expected since initially only community 2 has infected agents. In community 2 the increment in modularity initially reduces the fraction of infected agents, but at a certain point when the endemic state appears in community 1 it surges back in community 2. This further reinforces that optimal modularity reduces the epidemic spreading. In previous work, namely Ref. [40] , it was shown with a binary opinion dynamics that the spread of opinions against vaccination is one of the potential responsible for the large outbreaks of vaccine-preventable diseases in many high-income countries. In this work, we have gone farther afield to show the emergence of a networked SIRSV model that the spectrum of scenarios arising from the competition of pro-vs anti-vaccine views during an epidemic spreading is highly complex. The several outcomes shown in Figs.3-8 point out that our model produces a diverse phenomenology where the social and biological scenarios exhibit a nonmonotonic dependence with spreading rate λ. From the perspective of the dynamical systems, our results provide a new mechanism for bistability in a biological-social setting. From a practical point of view, our work offers new perspectives for the development of novel strategies for halting epidemic spreading based on tuning the modularity to an optimal degree. Some pro-vaccine strategies can have as side effect the segregation between individuals with conflicting views about the vaccines and clustering of similars. In [56] the authors found that in scenarios with effective vaccines, the impact of clustering and correlation of belief systems become stronger. Alternatively, the authors in Ref. [57] shown that segregation of anti-vaxxers can potentially extend the duration of an epidemic spreading, whereas in Ref. [58] it was found that an increase in the contact between vaccine refusers and the rest of the society can lead to a scenario where vaccination alone may not be able to prevent an outbreak. Here we show that too much or too low segregation of anti-vaxxers favors the chain of contagion, but an intermediate level of segregation disfavor the epidemic spreading. Therefore, our results indicate that vaccination campaigns should avoid strategies that have as a side effect too much informa-tional segregation of anti-vaccine groups so that reliable pro-vaxx information can reach those groups whilst enforcing a minimum degree of physical distancing as it occurs in countries where childhood vaccination is required at some degree, namely school entry [59] . Our work produces a thought-provoking analogy. In a small-world architecture, there is an intermediate number of long-range bridges that lead the full network to have unusual properties such as high clustering and low path lengths. Here, a structure with an intermediate number of inter-community ties leads the dynamics in the full network to produce an interesting outcome, namely the suppression of the epidemics. Thus, it would be interesting to consider further sophisticated network architectures, like multiplex networks. Despite the rich phenomenology we observed in our model, some limitations can be discussed which can be targeted in future work. The structure social contacts' structure of modular networks, presenting communities, is relevant to study several dynamical processes [14] - [23] . However, it could be more realistic to consider two distinct layers, one for the spreading of each dynamics (epidemic and opinion ones), but with each dynamics influencing the other. Such multiplex network structure can model better the coupled opinion-epidemic dynamics. Other rules for the opinion dynamics, distinct of the kinetic exchanges, could also be considered. Besides, it will be worthwhile to consider the interplay between several sources of heterogeneity in agent's bias, namely plurality and polarization [60] . The authors acknowledge financial support from the Brazilian funding agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação Carlos Chagas Filho de Amparò a Pesquisa do Estado do Rio de Janeiro (FAPERJ). Complex systems: a physicist's viewpoint Introduction to the Theory of Complex Systems sociology, geology by computational physicists Statistical physics of social dynamics Non-Traditional Applications of Computational Statistical Physics Sociophysics: A review of galam models Sociophysics: an introduction Epidemic processes in complex networks Fundamentals of spreading processes in single and multilayer complex networks Unification of theoretical approaches for epidemic spreading on complex networks Coevolution spreading in complex networks The "urban myth" of the association between neurological disorders and vaccinations Resurgence of measles in the united states: how did we get here? Optimal network modularity for information diffusion Optimal community structure for social contagions Optimal multicommunity network modularity for information diffusion Social transmission in networks: global efficiency peaks with intermediate levels of modularity Optimal modularity and memory capacity of neural reservoirs Close and ordinary social contacts: How important are they in promoting large-scale contagion? Optimal information diffusion in stochastic block models Optimal modularity in complex contagion, in Complex spreading phenomena in social systems Network modularity controls the speed of information diffusion Epidemic spreading on modular networks: The fear to declare a pandemic Imitation dynamics of vaccination behaviour on social networks Imperfect vaccine and hysteresis Modeling learning and forgetting processes with the corresponding impacts on human behaviors in infectious disease epidemics Risk perception and human behaviors in epidemics Coupled dynamics of behavior and disease contagion among antagonistic groups Public debates driven by incomplete scientific data: the cases of evolution theory, global warming and h1n1 pandemic influenza Dynamics of epidemic spreading with vaccination: impact of social pressure and engagement Sudden transitions in coupled opinion and epidemic dynamics with vaccination Modelling anti-vaccine sentiment as a cultural pathogen The online anti-vaccine movement in the age of covid-19 The online competition between pro-and anti-vaccination views Effects of information-dependent vaccination behavior on coronavirus outbreak: insights from a siri model Three in four adults globally say they would get a vaccine for covid-19 Vaccination strategies against covid-19 and the diffusion of anti-vaccination views Not sure? handling hesitancy of covid-19 vaccines Statistical physics of vaccination The effect of opinion clustering on disease outbreaks Dynamics of public opinion under the influence of epidemic spreading Dynamic modeling of vaccinating behavior as a function of individual beliefs Beyond rational decision-making: modelling the influence of cognitive biases on the dynamics of vaccination coverage Dueling biological and social contagions Opinion formation in kinetic exchange models: Spontaneous symmetry-breaking transition Complexity and asymptotical behavior of a sirs epidemic model with proportional impulsive vaccination Complicated endemics of an sirs model with a generalized incidence under preventive vaccination and treatment controls Threshold and stability results for an sirs epidemic model with a general periodic vaccination strategy Complete global stability for an sirs epidemic model with generalized non-linear incidence and vaccination Optimal vaccination strategies and rational behaviour in seasonal epidemics Interacting opinion and disease dynamics in multiplex networks: discontinuous phase transition and nonmonotonic consensus times Role of time scale in the spreading of asymmetrically interacting diseases Epidemic spreading with awareness and different timescales in multiplex networks Threestate opinion dynamics in modular networks Public discourse and social network echo chambers driven by socio-cognitive biases Effect of clustering and correlation of belief systems on infectious disease outbreaks, medRxiv Epidemic dynamics with homophily, vaccination choices, and pseudoscience attitudes Immune life history, vaccination, and the dynamics of sars-cov-2 over the next 5 years Which countries have mandatory childhood vaccination policies? Hysteresis and disorder-induced order in continuous kinetic-like opinion dynamics in complex networks