Date Published: November 9, 2017
Publisher: Public Library of Science
Author(s): Lucila G. Alvarez-Zuzek, Cristian E. La Rocca, José R. Iglesias, Lidia A. Braunstein, Yamir Moreno.
Through years, the use of vaccines has always been a controversial issue. People in a society may have different opinions about how beneficial the vaccines are and as a consequence some of those individuals decide to vaccinate or not themselves and their relatives. This attitude in face of vaccines has clear consequences in the spread of diseases and their transformation in epidemics. Motivated by this scenario, we study, in a simultaneous way, the changes of opinions about vaccination together with the evolution of a disease. In our model we consider a multiplex network consisting of two layers. One of the layers corresponds to a social network where people share their opinions and influence others opinions. The social model that rules the dynamic is the M-model, which takes into account two different processes that occurs in a society: persuasion and compromise. This two processes are related through a parameter r, r < 1 describes a moderate and committed society, for r > 1 the society tends to have extremist opinions, while r = 1 represents a neutral society. This social network may be of real or virtual contacts. On the other hand, the second layer corresponds to a network of physical contacts where the disease spreading is described by the SIR-Model. In this model the individuals may be in one of the following four states: Susceptible (S), Infected(I), Recovered (R) or Vaccinated (V). A Susceptible individual can: i) get vaccinated, if his opinion in the other layer is totally in favor of the vaccine, ii) get infected, with probability β if he is in contact with an infected neighbor. Those I individuals recover after a certain period tr = 6. Vaccinated individuals have an extremist positive opinion that does not change. We consider that the vaccine has a certain effectiveness ω and as a consequence vaccinated nodes can be infected with probability β(1 − ω) if they are in contact with an infected neighbor. In this case, if the infection process is successful, the new infected individual changes his opinion from extremist positive to totally against the vaccine. We find that depending on the trend in the opinion of the society, which depends on r, different behaviors in the spread of the epidemic occurs. An epidemic threshold was found, in which below β* and above ω* the diseases never becomes an epidemic, and it varies with the opinion parameter r.
In 1796 Edward Jenner invented and tested a vaccine against the smallpox, an illness that had a very high index of mortality in the 18TH century . The idea of Jenner was so successful that nowadays smallpox is practically eradicated and after this pioneering essay different vaccines were elaborated to prevent a long list of infectious diseases, from poliomyelitis to influenza. However, vaccines may present some lacks of efficiency and also some collateral effects. For example, in recent years some publications wrongly associated vaccination with autism [2, 3]. In spite of the overwhelming scientific evidence that such correlation is not actual, the belief that the results of vaccination could be worse than the illness itself spread through social networks and generated groups and movements against vaccination. Sometimes these groups are also related to some religion beliefs and/or rightists political tendencies or candidates. The debate about the efficiency of vaccination and its possible risks is then a very actual debate and a typical example of propagation of opinions, for and against vaccination. Thus, considering that opinions and contagion spreads in different ways, we will perform this study on a Network on Network. In recent years the study of complex Network of Networks (NoN) has been a subject of great interest for the scientific community, due to the large number of real word systems that can be mimic and study using these kind of topological structures [4–7]. A NoN is a system formed by single networks interacting through external connections between them. Many researches on NoN were focused in the study of cascade of failures [8–10], propagation of epidemics [11–14], and opinion dynamics [15–19] due to the ubiquitous of these processes that are present in the real scenarios. In particular, we are interested in processes that develop on NoN in which nodes belonging to different networks represents the same entities. This type of NoN are usually called multiplex networks. Epidemic spreading models have been particularly successful in understanding and predicting an epidemic outbreak and its period of extinction. Also, some models have incorporated a factor of human behavior, by considering the information and sources of information that individuals must handle, rational decisions and behavioral changes, in order to reach a more comprehensive understanding about the epidemic spreading . A commonly-used model for reproducing spreading diseases dynamics in networks is the susceptible-infected-recovered (SIR) model [21–25]. This model has been successfully used to reproduce non recurrent diseases such as the H5N5 flu or the Severe Acute Respiratory Syndrome (SARS) . Besides, it has been extensively studied under the topology of multiplex networks [27–30]. The model groups the population of individuals to be studied into three compartments according to their state: the susceptible (S), the infected (I), and the recovered (R). When a susceptible node is in contact with an infected node it becomes infected with an intrinsic probability β, which we called the virulence of the disease, and after a period of time tr it recovers and becomes immune. Usually, the type of disease that this model describes has a period of infection that lasts for six or seven days on average, flu, for example.
We are interested in studying how the propagation of diseases is influenced by the opinion formation of individuals in favor or against of getting vaccinated. The opinions will be formed and/or modified through the interaction and exchange of ideas with other individuals, which have their own opinion and co-evolves with the health condition of those individuals. In this way, the group of individuals develop a dynamic of formation of opinions in which individuals interact expressing opinions about the importance or not of being vaccinated. When an individual has a fully positive opinion about the vaccine, he acts accordingly and gets vaccinated. In our model we do not consider parental decisions on children, so the opinion on vaccination motivates just individuals, not family groups. While the process of spreading a disease generally requires face-to-face physical contact, the process of formation of opinions is more flexible because opinions can be transmitted via other media: phone calls, online social networks, video conference and instant messaging services, etc.
We study the model described in the previous section by means of extensive Monte Carlo simulations with synchronous update using a two-layer network of the same size N = 105. Nodes in each layer represent the same agent, thus we connect through an external link a pair of nodes, each from different layer, allowing only one interlink by node. We construct each layer using the Molloy-Reed algorithm  considering the Erdős-Rényi (ER)  degree distribution with 〈k〉 = 4. The propagation of the disease takes place in layer A and we fix the recovery time in tr = 6, which is in days the characteristic period of infection for a flu. Layer B is the social network, where the M-model rules the dynamic, with M = 2. As initial conditions we use for the layer B an uniform distribution for the densities of opinion, i.e., the same initial probability P+2,+1,−1,−2 = 1/4. In layer A we have initially only one agent infected, which is considered the patient zero and whose opinion is chosen at random between the four possible opinion states, a fraction 1/4 of the agents are vaccinated ones, as a consequence of their opinion state +2, and the rest are susceptible. We chose one source node of infection because this is the standard approach used by epidemiologists where most outbreaks starts with one person. At each time steps, we first let evolve the epidemic dynamic and then the opinion process. In layer A, we allow all the infected individuals to infect each one of their susceptible neighbors with probability β and the vaccinated neighbors with a probability (1 − ω)β. Then, in the opinion layer, we iterate over all the individuals and give each one of them the chance to interact with only one of its neighbors. This neighbor is chosen among those who can change the individual opinion. In case there is no neighbor that can change the opinion, nothing happen. Finally, we update all the opinions and epidemic states at the next time step. Notice that those infected individuals who had tr time step to spread the disease recover and those susceptible individuals whose opinion change to +2 change into the vaccinated state. All numerical results correspond to an average over 105 independent realizations.
In this paper, we studied the propagation of a disease in a population where all the individuals are continuously debating about getting vaccinated, considering that a susceptible individual is vaccinated if he is completely convinced about the benefits of the vaccine. For this purpose we used two-layer network where in one layer we use the SIR-model with vaccination for the propagation of a disease, and in the other layer we used the M-model (with M = 2), for the opinion formation process, where compromise and persuasion are the two processes involved and are controlled by the parameter r. We found that, in all the cases, the number of recovered agents decreases as ω increases, and this is due to the fact that as the vaccine becomes more effective, more people remain vaccinated and the propagation of the disease slows down. We found an epidemic threshold ω* above which we ensure that an epidemic will not develop. Furthermore, we found that above a certain value of β* the propagation of the disease is enhanced and it is impossible to prevent it from becoming an epidemic. Even for ω = 1 there will be a final macroscopic number of recovery individuals in the steady state. We computed this threshold as a function of r, and we found that a neutral society is the best scenario to prevent an epidemic (r ≈ 1). When compromise dominates the process of opinion formation (r << 1), the agents tend to have a moderate opinion, making difficult that they become extremist in favor of vaccination. The disease spreads through the non vaccinated agents very easily, even when the virulence is small. On the other hand, when persuasion dominates the process of opinion formation (r >> 1) the agents tend to have an extremist opinion. All the extremist agents in favor of the vaccine will be vaccinated, but those agents with a negative opinion, which are an important fraction of the population, will be easily infected. In a neutral society it is more likely to convince those agents with a negative opinion in favor on vaccination, to become extremist in favor. With compromise and persuasion in the same proportion it is easier to convince people to get vaccinated, blocking the propagation of the disease and preventing it to expand all over the population. We can conclude that the influence of the opinion on the vaccination determines, in certain cases, whether or not the disease becomes in an epidemic.