Infectious disease modeling of social contagion in networks

Abstract

Many behavioral phenomena have been found to spread interpersonally through social networks, in a manner similar to infectious diseases. An important difference between social contagion and traditional infectious diseases, however, is that behavioral phenomena can be acquired by non-social mechanisms as well as through social transmission. We introduce a novel theoretical framework for studying these phenomena (the SISa model) by adapting a classic disease model to include the possibility for 'automatic' (or 'spontaneous') non-social infection. We provide an example of the use of this framework by examining the spread of obesity in the Framingham Heart Study Network. The interaction assumptions of the model are validated using longitudinal network transmission data. We find that the current rate of becoming obese is 2 per year and increases by 0.5 percentage points for each obese social contact. The rate of recovering from obesity is 4 per year, and does not depend on the number of non-obese contacts. The model predicts a long-term obesity prevalence of approximately 42, and can be used to evaluate the effect of different interventions on steady-state obesity. Model predictions quantitatively reproduce the actual historical time course for the prevalence of obesity. We find that since the 1970s, the rate of recovery from obesity has remained relatively constant, while the rates of both spontaneous infection and transmission have steadily increased over time. This suggests that the obesity epidemic may be driven by increasing rates of becoming obese, both spontaneously and transmissively, rather than by decreasing rates of losing weight. A key feature of the SISa model is its ability to characterize the relative importance of social transmission by quantitatively comparing rates of spontaneous versus contagious infection. It provides a theoretical framework for studying the interpersonal spread of any state that may also arise spontaneously, such as emotions, behaviors, health states, ideas or diseases with reservoirs.

Figure 1. The SISa model of infection.

There are three processes by which an individual's state can change. (i) An infected individual transmits infection to a susceptible contact with rate . (ii) A susceptible individual spontaneously becomes infected at rate , regardless of the state of their contacts. (iii) An infected individual returns to being susceptible at rate , independent of the state of their contacts.

Figure 2. The degree distribution of the Framingham Heart Study Network.

The degree distribution of the Framingham Heart Study social network at the most recent exam (7) considered in this study. Connections include friends, family and coworkers. The average degree is around k = 3 and the transitivity is  = 0.64 (the ratio of triangles to triples).

Figure 3. Evidence for disease-like spread of obesity.

Obesity behaves like a disease agent, infecting those in a susceptible ‘not obese’ state. The probability of transitioning from ‘not obese’ to ‘obese’ increases in the number of ‘obese’ contacts (A), and doesn't depend on the number of ‘not obese’ contacts (B). Conversely, the probability of recovering to the ‘not obese’ state does not depend on the number of ‘not obese’ contacts (D) or the ‘obese’ contacts (C)). Labels above points on plot are the number of observations averaged into that data point, and error bars are the standard error of the proportion.

Figure 4. Change in observed parameters over time.

Parameter measurements for obesity from each set of consecutive exams. Data point at exam N represents the value for the transition from exam N to N+1. Error bars are 95 confidence intervals on measurements from regression of transition probability versus number of contacts of a certain type. (A) Contact-independent rates. The rate of recovery () appears to be constant within the margins of error throughout the study while the rate of automatic infection () appears to increase between exams 1 and 3, then stay constant. (B) The contact-dependent transmission rate () appears to increase over time.

Figure 5. Simulations of obesity epidemic using SISa model.

Time series of an epidemic on the Framingham Heart Study network, using full simulations (light blue) or the n-regular pair-wise equations (dark blue). Parameters used are those measured for the obesity epidemic: . In the SISa model there is a co-existence of susceptible and infected individuals at steady state. For these parameters there is a good agreement with simulations and the pair-wise equations for the fraction infected (A), but the equations predict less correlations (B), due to the neglect of heterogeneities in the number of contacts.

Figure 6. Comparing SISa model timecourse to historical data.

A comparison of historical data on the prevalence of obesity in the Framingham Heart Study (blue dots) and the National Health and Nutrition Examination Survey (red dots) with the timeseries predicted from the SISa model with time-varying parameters. For the simulation, we allowed the parameters and to vary as observed in Figure 4, but kept constant at its average value. Before 1970 (when our measurements started), the prevalence of obesity was assumed to be stable at 14. The model and the data both show very similar rates of increase, with a slow post-1970 increase, followed by a rapid increase, and then increasing more slowly. The SISa model predicts the prevalence of obesity will increase slowly to a peak at 42.

Figure 7. Fraction infected versus SISa model parameters.

Dependence of the equilibrium fraction infected on obesity interventions which act to change the rates of infection (transmission (A) and ‘automatic’ infection (B)) or recovery (C). When not varying, parameters are .

Figure 8. Pairwise equations diverge from simulations when transmission is higher.

Time series of an epidemic on the Framingham Heart Study network, using full simulations (light blue) or the n-regular pair-wise equations (dark blue). When the ratio of is larger than that observed for the spread of obesity, the pair-wise equations diverge more from the full simulations, both for the fraction infected (A) and the correlations (B). .

Figure 9. Dependence of the equilibrium fraction infected and correlations on the rate of transmission, .

Dependence of the equilibrium fraction infected (A) and correlations (:(B), :(C), :(D)) on the rate of transmission, . When , expected in most social infections, there is no longer a threshold () needed for the infection to invade the population. The network causes infected individuals to cluster away from susceptible individuals , and this is more pronounced for larger and lower fraction infected. Parameters are .

Figure 10. Dependence of the equilibrium fraction infected and correlations on the rate of automatic infection, .

Dependence of the equilibrium fraction infected (A) and correlations (:(B), :(C), :(D)) on the rate of automatic infection, . Parameters are .

Figure 11. Dependence of the equilibrium fraction infected and correlations on the rate of recovery from infection, .

Dependence of the equilibrium fraction infected(A) and correlations (:(B), :(C), :(D)) on the rate of recovery from infection, . Parameters are .

Figure 12. Determining the best parameter to target in an intervention.

This graph compares interventions which act to change different parameters of infection (transmission (A), ‘automatic’ infection (B), recovery (C)). Shown is the rate of change of the fraction infected at equilibrium with respect to a change in various parameters of infection. The y axis labels represent the absolute change in the percent infected for a change of 0.01 in one of the parameters. Changing is better for small and changing is best for larger . For intermediate , changing is best. Parameters are .

Figure 13. Dependence of the equilibrium fraction infected and correlations on the network transitivity, .

The dependence of the equilibrium fraction infected(A) and correlations (:(B), :(C), :(D)) measured from the pair-wise equations on the network transitivity, . For the parameters measured for the transmission of obesity, shown here, there is no strong dependence on . Hence for studying the obesity epidemic it is justified to ignore to simplify calculations. Parameters are .

Figure 14. Dependence on network transitivity, , for larger transmission rates.

The dependence of the equilibrium fraction infected (A) and correlations (:(B), :(C), :(D)) measured from the pair-wise equations on the network transitivity, . For larger , slightly decreases the fraction infected by leading to more spatial correlation of infected individuals. Parameters are .