In the simplest infectious disease models [26], individuals are classified as occupying one of two states: ‘susceptible’, meaning they do not have the disease, and ‘infected’, meaning they do have the disease. The disease can be transmitted to a susceptible person when they come into contact with an infected person. The rate of this disease transmission from infected to susceptible is defined as , the transmission rate. Once an individual is infected, they recover from the disease at a constant rate , regardless of their contacts with susceptibles or infecteds. In one class of disease models (susceptible-infected-recovered, or SIR), recovered individuals become immune to further infection and enter a ‘recovered’ state. However, behaviors, trends, health states, etc, can occur many times over an individual's life, and therefore we assume infected individuals return to the susceptible state after recovering. This form of susceptible-infected-susceptible (SIS) model is used to model infectious diseases that do not confer immunity, like many STDs.

In the standard SIS model, infection can only be transmitted by having a contact between an infected and a susceptible individual. Social ‘infections’, however, can also arise due to spontaneous factors other than transmission. Therefore, we extend the SIS model by adding a term whereby uninfected individuals spontaneously (or ‘automatically’) become infected at a constant rate , independent of infected contacts. A diagrammatic representation of our modified SIS model, which we will call SISa, is shown in Figure 1. The corresponding differential equations for a well-mixed population are described in Eq. 1(1)where is the number of infected individuals, is the number of susceptible individuals, is the population size, is the transmission rate, is the recovery rate, and is the rate of spontaneous infection. This model assumes a constant population size and neglects birth and death. The SISa model is related to infectious disease models with ‘imports’ (migration of infecteds into the population), although here the rate of spontaneous infection is proportional to the number of susceptibles, while in import models it is a constant or proportional to the total population size.

In the infectious disease literature, a disease is said to be ‘endemic’ if a stable, non-zero fraction of the population is infected at steady state. If a single infected individual is introduced to a totally susceptible population, then the average number of secondary infections they cause before recovery is called the basic reproductive ratio, . For the regular SIS model in a well-mixed population of N individuals, . An epidemic, leading to an endemic equilibrium, only occurs for , and hence is a fundamental quantity used to describe and compare infectious diseases. For the SISa model, an epidemic occurs for all parameter values, due to the spontaneous infection term. Thus, social behaviors that can be adopted independently of neighbors mean that there is no longer a threshold for the behavior to become prevalent in a population, and even in the absence of contagion there would be a non-zero steady state prevalence. Because of this, there is not an obvious definition for in the SISa model. The steady state fraction of infected individuals in a well-mixed population is given by Eq. 2.(2)

Traditional models of infection assume that the population is well-mixed. However, this assumption is unrealistic for many diseases, and also for the social spread of trends and behaviors. To account for the population structure, the infectious process can be constrained to take place on a social network. An infected individual can only pass their infection on to the suspectibles to whom they are connected. Properties of the infectious process thus depend on both the epidemiological parameters and the network structure, and there are often no longer simple analytic formulas to describe the reproductive ratio or steady state level of infection. For example, a property of disease spread on networks are spatial correlations (in the network sense) that arise between individuals in the same state. This correlation is defined as the ratio of the observed number of connections between two types of individuals to the number of connections expected if the positioning of individuals in the network was random. Spatial correlations of like individuals can be caused by an infective process spreading within a network [29], but may also be caused by confounding environmental factors which similarly influence the behavior of connected individuals, or the formation of contacts based on similar behavior (also called homophily). For a network of N individuals with a total of E connections between them, the correlation between two states X and Y is defined by:(3)

The correlation between infected individuals, , rises above one as the epidemic proceeds, due to cluster formation as infected individuals transmit to their contacts. Similarly, the correlation between infected and susceptible individuals, , drops below one. The deviation of these correlations from 1 increases with (i) the ratio of transmissive infection () to spontaneous infection () in our model (there are no correlations without a transmissive component), and (ii) the inter-connectivity (transitivity) of the network. As a result of these spatial correlations, diseases on networks can progress more slowly than their well-mixed counterparts, leading to lower basic reproductive ratios. However, heterogeneity in the number of contacts per individual acts to increase . For two networks with the same average degree, if one has a larger variance in degree, then will be increased. Thus, it is possible for diseases on networks to have lower (or nonexistent) thresholds for endemic epidemics.

There are no analytic methods to solve SIS-type dynamics on arbitrary networks without making approximations. Thus, simulations are a more accurate tool to explore theoretical disease dynamics in structured populations without making simplifying assumptions about the network structure. For scaled, well-mixed populations, the formulas given in the previous sections for and are good approximations if is replaced with , the average contacts at a given time, while fixed networks, especially if non-uniform and highly inter-connected, can deviate from these values significantly. We can use a pair-wise approximation [29], [43], [44] to formulate the infectious process on a network structure in terms of differential equations. The fundamental variables are numbers of individuals of each type, and also the pairs of individuals, [XY] (where the edges are not directional). Because [XY] = [YX], and the total individuals and total edges is constant, the system can be reduced to three equations.(4)

Here [XYZ] represents the number of situations where and X individual is connected to a Y individual who in turn is connected to a Z individual. We can approximate all these triples in terms of pairs, using a moment closure approximation ([43], Text S1), which then reduces the number of variables to three also. Then these equations can be simplified to(5)with(6)where is the number of contacts each individual has and is the transitivity of the network (the ratio of triangles to triples). Having a simplified set of equations is very useful for understanding contagion dynamics in structured populations. Integrating equations is much faster than running simulations on large networks, and from them analytic results can be derived which allows determination of parameter dependence. These equations assume that the local neighborhood for each individual is identical, that is, everyone has the same number of contacts () and the same . They thus take into account the effects of fixed network structure but not heterogeneities between individuals. In the Supplementary Information (Text S1) we have included the extension of these equations to include heterogeneities. These equations can be used to easily simulate disease spread and get expected steady state prevalences and correlations, which are very useful approximations and give insight into parameter dependence. Later, we will compare these equations to results from full simulations on realistic networks. When (which is approximately the case for most random graphs) we can get a closed-form solution for the prevalence at steady state:(7)(8)(9)

The result of a network structure is that the number of partnerships between susceptible and infected individuals quickly becomes less than if random, and so . We can compare Eq. 7 to the well mixed result (Eq. 2), and see that the effect of the network is to lower the effective transmission rate by a factor of , and hence lower the prevalence, due to these correlations that build up locally. The larger is compared to , the more network effects are important. If infection is mostly automatic (when 0), the network no longer matters. Equation 7 actually holds generally (for any homogeneous network and any value), while Equations 8 and 9 are only applicable with  = 0.

Analyzing the n-regular pair-wise equations allows us to get analytic results and determine how and under what conditions network structure affects the spread of behaviors which are both spontaneously acquired and spread interpersonally. Although simple closed-form solutions do not exist when is non-zero, these equations can easily be integrated or numerically solved to get solutions. These equations ignore heterogeneities in the number of edges for different individuals, which can facilitate spread under some conditions (see supplement Text S1 for extension). Full stochastic simulations on large networks can be carried out to determine how and when the results differ.

The SISa model provides a formal way for assessing the social contagion of trends and behaviors that may be repeatedly caught and recovered from. Using data from the Framingham Heart Study (FHS) [45] we tested the validity of this model and estimated transmission parameters for various health related behaviors, though the focus here is on obesity as an example. To both demonstrate that obesity can display infectious-disease-like dynamics, and to estimate values for the model parameter , and , we use dynamic information about transitions between states based on our multiple time points of data. For data points separated by time intervals () smaller than the average time between transitions, the transition probabilities can be linearized. The probability of a transition from susceptible to infected after a time can be given by , and the probability of transition from infected to susceptible after time , by . It is necessary for the time between measurements to not be comparable to or greater than the average lifetime of a state to keep the probability of double transitions within a time interval low.

This epidemiological approach to social contagion has important differences from other models which look at correlations in present and past states of connected individuals. Here, similar to others [16], [19], [20], [46] we look at how contacts influence the transitions between states, which better captures the nature of contagion. Since we use pre-existing social ties, we do not see effects from selection bias in choosing friends with similar states. Additionally, time invariant confounding events that lead to concurrent changes in connected individuals will not show up as contagion effects in this model.

The dataset we use is a subset of individuals from the Framingham Heart Study [45]. This study was initiated in 1948 in Framingham, Massachusetts and has continued enrolling subjects through the present. We examined individuals in the Offspring Cohort, enrolled starting in 1971. Subjects come to a central facility at regular intervals (approximately every 4 years) for medical examination and collection of other survey data. Body mass index (BMI) was measured at each exam, and obesity was defined as BMI30 [47]. All other, lower, weights, which include underweight, normal range weight and over-weight, were classified as ‘not obese’. In additional to information on mental and physical health, subjects were asked to name at least one close friend at each exam, and were also connected to all first-order relatives, as well as coworkers and residential neighbors. For each subject, the following social connection data is available: (i) each other person to whom they were connected, (ii) the dates of initiation and termination of that relationship, (iii) the type of relationship (neighbour, coworker, first-degree relative, or friend), and (v) the geographic distance between the two subjects. The social network for each exam was constructed by creating a network matrix G, where if subject nominated subject as a connection before or during the time that subject was administered exam . All relationship types are mutual except for friendships, which are self-nominated, such that is possible for friendships.

To study the transmission of obesity, we examine changes in BMI between sequential exams. Seven exams were administered to the Offspring Cohort between 1971 to 2001, with network data collected for each. We examine transitions occurring between each exam. The average fraction of the network that was classified as obese increased between these seven exams, suggesting the transmission process is not yet at steady state (Exam 1: 14 obese; Exam 7: 29 obese). Each set of exams were closely and consistently spaced ( (exam 1), (exam 7)). In general when modeling an infectious process, the rates of infection and recovery are assumed to be constant over time, with the prevalence changing as the infectious process begins and finally reaches equilibrium or is eliminated. When examining the spread of obesity using longitudinal data on transitions between exams, we can actually test this assumption and detect changes in the rates themselves.

A given state is considered infectious if having more contacts in state makes you more likely to switch to state . That is, a positive relationship between the number of contacts in state and the probability to transition from state to state indicates that state is infectious with respect to state . Therefore, to test whether a given state is infectious with respect to another state , we perform an ordinary least squares (OLS) linear regression as follows. Each subject in state in exam N is coded as either having transitioned to state (transition = 1) or not (transition = 0) in exam N+1. We then regress this binary transition variable for each subject against the number of contacts in state that subject had during exam N. A significant positive correlation indicates that having more friends in state at the earlier exam makes you more likely to switch to state in the later exam. If state is infectious (a significant positive correlation exists), then the value of can be calculated from the slope of the regression line, and the value of can be calculated from the intercept. If state is not infectious (no significant correlation exists), then the value of can be calculated from the intercept. was taken as the average time between examinations, which varied between exams from 3 to 8 years. Using logistic regression as opposed to OLS regression gives very similar results, as the datapoint line is within the linear range of the logistic model.

The structure of the Framingham Heart Study social network varies over the course of time, ranging from 7500 individuals with an average of 5.3 connections each at the first exam, to 3500 individuals with 2.8 connections on average at the seventh exam. Summary statistics are presented in the supplement (Table S1). These changes in population size and average degree occur because individuals may die or drop out of the study but new individuals are not added. The network is approximately Poisson distributed (see Figure 2), although with some subjects having no connections. The transitivity is consistent over time at approximately 0.64. While neighbors were included as contacts in the study, like Fowler and Christakis [20] we find no significant trends when including neighbors, and so did not include these contacts. For friendships, we only consider the contacts of an individual to be those other individuals whom they nominated (other relationships are all mutual), and so the network is directional.

The results of infectiousness analysis for the spread of obesity between exams 4 and 5 are shown in Figure 3 as an example. Consistent with the SISa model formulation, we find a significant positive correlation between the probability of transitioning from ‘not obese’ to ‘obese’ and the number of ‘obese’ contacts (Figure 3A, coeff = 0.016, p = 0.0001), and no significant relationship between the transition from ‘obese’ to ‘not obese’ and the number of ‘not obese’ contacts (Figure 3D, coeff = 0.006, p = 0.15). Additionally we find no significant relationship between the probability of transitioning from ‘not obese’ to obese and the number of ‘not obese’ contacts (Figure 3B, coeff = −0.0005, p = 0.75), or the probability of transitioning from ‘obese’ to ‘not obese’ and the number of obese contacts (Figure 3C, coeff = −0.002, p = 0.85). The same analysis was repeated for each interval between sequential exams and very similar results were found. The full results from the regression analysis are presented in the supplement (Table S2). This suggests that obesity can indeed be modeled as an infectious process in the SISa framework, with ‘not obese’ susceptibles becoming ‘obese’ infecteds, and transmitting obesity to other susceptibles. The parameters for the SISa model can be calculated from the transition probabilities mentioned earlier, by dividing slope and intercept values by , the average time between exams. These values are reported for each exam in Figure 4, and the values at the latest exam interval are summarized in Table 1. For most recent exam, the transmission rate, , is found to be /year. The spontaneous transmission parameter is found to be /year. The recovery parameter is found to be /year. From these SISa model parameters, other values of interest can be calculated. The ‘average lifetime’ of a state is the average length of time and individual spends in this state before recovering, which was found to be 24 years for this time period. The ‘influence’ of a state is the cumulative probability that the infection will be passed from an infected to a susceptible connection before the infected individual recovers, and is observed here to be 13. The ‘cycle length’ is the average length of time between spontaneous infections, and is 56 years. The basic reproductive ratio is approximately 0.35, which implies that without spontaneous appearance, the obesity epidemic would not be self-sustaining based on transmission alone. However this calculation is an approximation since uses the formula for a population that is well-mixed but only effectively contacting a fraction of the total population at each time ( contacts), so does not factor in fixed network structure (there is no analytic formula for this situation). We observed a correlation in the positioning of obese and non-obese individuals of  = 1.3 and  = 0.9.

Since these rates were measured for 6 different inter-exam transitions over 30 years, we can look at how the value of these rates changes over time. Figure 4 shows the measured automatic infection (a), transmission (), and recovery rates (g) for each exam interval. Error bars are 95 confidence intervals on measurements from analyses like Figure 3. While the rate of recovery () has remained relatively constant since the 1970s, the rate of spontaneous infection () has steadily increased over time. The transmission rate, , also appears to have increased over time. These trends were tested using weighted regression (to include the different errors for each measurement) and found to be significant for and but constant for . For the rest of the study we used the time-averaged value of , . This suggests that the obesity epidemic may be driven by increasing rates of becoming obese, both spontaneously and transmissively, but not by decreasing rates of losing weight.

We also found that both happiness and depression fit the SISa model, both being contagious from a neutral emotional state [18], that smoking cessation, though not smoking itself, also fit, and that both alcohol consumption and abstinence were contagious from the opposite state (data not shown). For all of the above cases, we tested if the transition probability depended instead on the fraction of contacts in a state, instead of the number, and found no significant dependence. We also tested for dependence on other personal attributes such as age, sex and education, and found no dependence in most cases. For obesity, the transition probability from not obese to obese decreased slightly with age (coeff = −0.0012, p = 0.04). Our results show that many models of social influence make assumptions about interpersonal interactions that are not supported by this longitudinal data. One of these assumptions is the ‘majority rules’ interaction, which assumes that people will be most likely to switch to the state most of their contacts are in [40]. Here, transitions depend on the number of contacts, and only certain states (those we class as ‘infectious’) actually influence transitions (in other words, contagion is only in one direction). This has significant effects on the predictions for epidemic progression. For example, ‘majority rules’ models predict 100 infected at steady state, and that weight loss behavior spreads and so an effective intervention is to ‘pin’ certain individuals at low weights. Also, many models assume that the probability of transitioning to a state is zero if no contacts are in that state, but these results show that there is a constant probability of spontaneously becoming ‘infected’. Finally, using this framework, we can get rates for transitions, and hence have an idea for the time-course of the progression, not just the final outcome.

In this section, we will use the SISa model to make predictions and evaluate interventions for the obesity epidemic, using the parameters observed in the FHS data. For simplicity and generality, we will keep the parameters and constant at the values observed for the most recent exams, and use the time-averaged value of . Since we are mostly interested in predicting future trends, and the parameters seem to have relatively constant values over the final decade, this simplification should not affect these predictions. We also keep the network fixed at the structure observed at Exam 6, except when we compare to historic data. While the simplified pair-wise equations we present are designed for symmetrical networks, they can be approximately adapted to directional networks by letting represent the average out-degree (average number of influential contacts) instead of the total number of contacts. In the Framingham data, greater than 90 of contacts are symmetrical, and so there is little error in this approximation. For hypothetical networks were the contacts formed by out-degree and in-degree are very different sets of individuals, deviations are expected. Figure 5 shows the results of both the n-regular pair-wise equations and a full simulation on the FHS network for the spread of the obesity epidemic. The parameters used were those measured from FHS as discussed earlier. One of the important properties of the SISa model is that it always leads to a stable coexistence of both infected and susceptible individuals, with infecteds becoming 100 prevalent only in the limit as or approaches infinity. This is very different from statistical-physics-based interaction models where the population always ‘coarsens’ to everyone in a single state [40]. These results show that for the parameters measured for obesity, the pair-wise equations are not significantly different from the full simulations for predicting prevalence, and hence provide a good substitute. The reason is that the spontaneous rate () is significantly larger than the transmissive component (). For larger values of , there is a noticeable difference (shown in the next section).

This model predicts that, assuming the rates do not further change over time, the steady state proportion of obese individuals will be 42. While not great, this is a much more optimistic estimate than 100 [40]. However, all of the parameters observed in this study have an error associated with them, and so there is some uncertainty in this prediction. Figure 4 shows the ranges of the 95 confidence intervals for these values. We can estimate the uncertainty in this prediction by using first the values of these parameters, within the range of one standard deviation, that would give the highest prevalence () and then those that would give the lowest (). We used  = 0.05,  = 0.015 and  = 0.002 to get the minimum and  = 0.03,  = 0.023 and  = 0.008 to get the maximum. These simulations suggest the confidence interval for the expected prevalence can be approximated as 25 to 54. This model also allows us to estimate the time-course of the epidemic, and suggests it would take around 40 more years for the obesity prevalence to be within 1 of this maximum value. At the first time point in our data (1970), we measured the rates to be  = 0.008,  = 0.03 and  = 0.001, and the prevalence to be 14. These parameters would have led to a steady state prevalence of 24, which suggests that the rates of becoming obese must have originally been much lower than this.

We can also compare historical data on the obesity prevalence (from both national studies [47] and the FHS data) to the predicted time course shown here. To generate the model prediction, we simulated an epidemic with the pair-wise equations but allowed the rate values and network parameters to change as measured from the data (see Figure 4 and Table S1). We kept constant at the average value observed, 0.035, and varied and as observed. The value for parameter measured for the transition between exam and () was used in the simulation for times (years) between the average examination dates of exams and , and then increased to for the next time interval. The same was done for . For times before the earliest data points in FHS for which we have measured rate constants(pre 1970), we assumed the epidemic was at a steady state of 14. This could be achieved, for example, with and . Figure 6 shows that there is a good match in the time course of the model with reality after 1970, with similar rates of increase in the prevalence.

We can use the pair-wise equations to see how the steady state prevalence depends on various parameters, which is especially useful to see how interventions that aim to change a certain parameter may affect the prevalence. Figure 7 shows these results. For the parameter values for obesity, although is quite large, is still important. If changes from 0 to 0.005, the expected steady state changes from around 0.35 to 0.42. However, much larger changes can be realized by decreasing or increasing . For the obesity parameters, completely removing the contagious component is only expected to change the steady state prevalence by around 7. However, changing the spontaneous infection term can have much larger effects. While a 50 change in will result in only a 3 decrease in I, cutting in half will reduce the prevalence by 15. However, a similar absolute decrease of 0.005 would also lead to a 7 difference. The efficiency of changing one parameter versus the other can be looked directly at for various parameters, which will be shown in the next section.

In this section we will examine the more general properties of ‘infections’ following SISa model dynamics. While Figure 5 showed excellent agreement between the pair-wise equations and full simulations for the time dynamics, this is not true for all parameter regimes. When is larger and is smaller (as shown in Figure 8), and the network is strongly heterogeneous (as the Framingham network is), the pair-wise model deviates more. The reason is that heterogeneous network effects become more important for larger , and the pair-wise approximations are best for homogeneous networks. The extension of the pair-wise equations to heterogeneous networks is described in the supplement (Text S1).

We can use the pair-wise equations to see how the steady state prevalence depends on various parameters, which is especially useful to see how interventions that aim to change a certain parameter may affect the prevalence. Figure 9 shows how the steady state changes with the rate of transmission, . The blue line () shows what would happen in a classical epidemic, with no spontaneous infection. When is below a certain value (), the infection does not spread. The fraction infected increases rapidly with in this regime. As soon as we add , this thresholding behavior disappears. When the steady state is less sensitive to . The red line () shows the approximate parameter values for obesity. Here although is quite large, is still important. As with classical infectious disease models [29], disease spread on a network leads to decreased , the spatial correlation between infected and susceptible individuals, and increase and , the correlation between pairs of infected individuals and pairs of susceptible individuals, respectively. If we look at , we can see that we expect there to be some correlations of infected people at some values, but not all. So while seeing spatial correlation may hint there is a inductive process, it is definitely not necessary. You can have an infectious process without seeing correlations, just like you can see correlations without it being caused by the dynamics of influence. Spatial correlation is much higher when is small.

Figure 10 shows the dependence on the rate of spontaneous infection, . The more spontaneous infection, the more infected. When is larger (red line), increasing has less effect. The green line is for the parameters measured for obesity. We can use these graphs to compare the effects of various interventions which may reduce the rate of infection. In Figure 9 (vs ), we can see the expected decrease in the prevalence of the infection for a given decrease in . Changing has more effect when is small. The rate of recovery from an infection is , and in the obesity case, represents the rate at which obese people lose weight and transition to normal BMI values, in probability per year. Higher rates of recovery lead to lower fraction infected (Figure 11). One possible intervention is to increase the rate of recovery. For low recovery values, this has a large effect on , but for around 0.04 (the value for obesity), only small changes in result from changing .

In general, the spatial correlations () are negatively correlated with the fraction infected (I); more correlations are observed when a disease is not too infectious. If the spatial correlations were fixed to be a certain value (for example obese people cluster together due to selection bias in friendships or confounding factors), then this would actually serve to slow infection. Since we do not observe contagion of losing weight, it does not seem like it would be beneficial to have an intervention which broke up obese clusters.

The most direct way to compare various parameters for spread, and therefore interventions that reduce one of the parameters, is to look directly at for various parameters ( is the steady state fraction infected, is the parameter of interest. Figure 12 shows that for most parameter regimes, it is always best to increase the recovery rate, , as a method to reduce the fraction infected, . However, for low and low , it is best to decrease the spontaneous infection term , and for a window of intermediate , it is best to decrease the transmissive component . The third plot shows the results for the value measured for obesity, and because is low here we are in a regime where it decreasing has the most effect, so this is the best intervention.

Many analytic models of network phenomenon assume the transitivity, , is zero, meaning there are no triangles in the network. This is done to get the analytic expression presented here (Eq. 2), which is not necessary to numerically integrate the pair-wise equations, as presented in the results above. In the FHS network, we observed that is 0.64, suggesting human social networks are quite transitive. We want to examine the importance of in predicting the fraction infected. For the observed value, the effect of is negligible, as shown in Figure 13. The reason is that the dominant effect here is the spontaneous infection, which does not depend on the network structure. This justifies ignoring for infections that have only low infectivity terms. However, for large values (the equivalent of 2 is shown in Figure 14) has a more pronounced effect. While for a purely infectious process (blue line), at high , a disease can die out, even for , when , this doesn't occur, but still slightly reduces the spread. It also results in more observed spatial correlation of infected individuals. Overall, there is very little effect of in the SISa model.

We've already discussed how changes in parameters of infection affect the steady state prevalence, and we can consider this an analysis of different types of public health interventions that change rates of recovery, infection or network structure. In previous analysis of the obesity epidemic done by Bahr et al [40] they suggest a strategy of ‘pinning’ groups of people to stay in a non-obese state, similar to vaccinating against an microbial disease, as a method to remove the ‘infection’ from the population. However, in the Bahr model this intervention works (if enough people are ‘pinned’) because becoming non-obese is also contagious, which we don't find in this analysis. In the classical infectious disease setting, vaccinating can lower below the threshold for disease invasion, but in the SISa model there is no threshold, and so neither mechanism makes this an effective strategy in the SISa model. Two other possible intervention strategies come out of this model. Firstly, from Eq. 7 we can see that the fraction infected decreases with , the correlation of susceptible and infected people. If an intervention actively reduced this number, by isolating or clustering infected people, this could reduce the prevalence. Secondly, the fraction infected could be reduced if it were possible to make the ‘susceptible’ state also contagious through contacts.