Dueling biological and social contagions

Numerous models explore how a wide variety of biological and social phenomena spread in social networks. However, these models implicitly assume that the spread of one phenomenon is not affected by the spread of another. Here, we develop a model of “dueling contagions”, with a particular illustration of a situation where one is biological (influenza) and the other is social (flu vaccination). We apply the model to unique time series data collected during the 2009 H1N1 epidemic that includes information about vaccination, flu, and face-to-face social networks. The results show that well-connected individuals are more likely to get vaccinated, as are people who are exposed to friends who get vaccinated or are exposed to friends who get the flu. Our dueling contagion model suggests that other epidemiological models may be dramatically underestimating the R0 of contagions. It also suggests that the rate of vaccination contagion may be even more important than the biological contagion in determining the course of the disease. These results suggest that real world and online platforms that make it easier to see when friends have been vaccinated (personalized vaccination campaigns) and when they get the flu (personalized flu warnings) could have a large impact on reducing the severity of epidemics. They also suggest possible benefits from understanding the coevolution of many kinds of dueling contagions.

to be able to obtain current friendship information. Of these 1,300 students, 396 (30%) agreed to participate. All of these students were in turn asked to nominate up to three friends, and a total of 1,018 friends were nominated (average of 2.6 friends per nominator). This yielded 950 unique individuals to whom we sent the same invitation as the initial group. Of these, 425 (45%) agreed to participate. However, 77 of these 950 subjects were themselves members of the original, randomly selected group and hence were already participants. Thus, the sample size after the enrollment of the random group and the friend group was 744.
Nominated friends were sent the same survey as their nominators; hence, the original 425 friends also nominated 1,180 of their own friends (average of 2.8 friends per nominator), yielding 1,004 further, unique individuals. Although we did not send surveys to these friends of friends, 303 (30%) were themselves already enrolled either in the friends group or in the initial randomly selected group. survey questions asked of all students), so we include these in the data as well.
Notably, we do not expect cases of flu to meaningfully alter the social networks and friendship patterns of Harvard undergraduates, let alone over a two-month period. And, we assume that the friendship network of Harvard students in our sample did not change meaningfully over the period September to December. That is, we treat the network as static over this time interval.
Beginning on October 23, 2009, we also collected self-reported flu symptom information from participants via email twice weekly (on Mondays and Thursdays), continuing until December 31, 2009. The enrolled students were queried about whether they had had a fever or flu symptoms since our last email contact, and there was very little missing data (47% of the subjects completed all of the biweekly surveys, and 90% missed no more than two of the surveys). Students were deemed to have a case of flu (whether seasonal or the H1N1 variety) if they report having a fever of greater than 100 F (37.8 C) and at least two of the following symptoms: sore throat; cough; stuffy or runny nose; body aches; headache; chills; or fatigue.
Hence, we had two measures of flu incidence. The medical-staff standard was a formal diagnosis by a health professional and typically reflected more severe symptoms.
The self-reported standard captured cases that did not come to formal medical attention.
As expected, the cumulative incidence of the latter was approximately four times the former (32% versus 8%) by the time of cessation of follow-up on December 31, 2009.
We checked the sensitivity of our findings by using this self-reported measure of flu (see below).
As part of the foregoing biweekly follow-up, and to supplement the UHS vaccination records, we also ascertained whether the students reported having been vaccinated (with seasonal flu vaccine or H1N1 vaccine or both) at places other than (and including) UHS.
For analyses using the network data we included 750 pairs of directed friendships among the participants (there were 158 mutual ties and 592 pairs of undirected friendships). The network degree of each subject is defined as the number of undirected friendships he/she has in this social network. Data collection and analysis was approved by the Harvard IRB committee.

II. ESTIMATION METHOD
We use the method of least squares to estimate the model parameters. We choose as the starting point the day when the very first incidences of both vaccination and infection take place. Our goal is to find parameter estimates that minimize the deviation between the predicted and the real trajectories of the temporal incidence of infected and vaccinated individuals over the time course under consideration. Denote by q * = denotes the 2-norm. We constrain the search of the parameter space within the simplex We use simulated annealing to handle the presence of local minima in our parameter optimization search. We run the iteration n = 10 6 times, starting with a guess of the parameter values b 0 . The temperature T i of each iteration is chosen to be ((n−i)/n) 4 . For each iteration i, we randomly choose a new sets of parameters b 1 drawn from the close neighbourhood of b 0 (using the Gaussian deviation N(0, 0.01)). If e(b 1 ) < e(b 0 ), we set this search process, we obtain the best fitting parameters b. The residual standard error where k is the number of observations and m the number of parameters. Denote by J the Jacobian matrix, where J i j is given by The Hessian matrix can be approximated by H ≈ J T J. Accordingly the standard error of the parameter estimation b i can be approximated by

III. MODEL SELECTION CRITERION
Following previous practice [1][2][3], we use the estimation method above to fit our  where the factor k 2 / k 2 accounts for the effects of network heterogeneity on disease transmission (for homogeneous populations, it reduces to β/γ) [4,5]. Parameter estimates for other candidate models give poorer fitting results and can be found in Table 1 of the main text. Furthermore, we repeated the same fitting procedure but using all available information regarding flu infections (i.e., self-reported flu cases [6]) and obtained somewhat larger estimates of the epidemiological parameters because of the higher incidence of self-reported flu (Table S1), but these additional results confirm that social contagion is a key determinant of vaccination behaviour.

IV. NETWORK-SPECIFIC DUELING CONTAGION MODEL
Aside from the "coarse-grained" results reported in Figs and use it to represent both the biological contagion network and the social contagion network. We report the best parameter estimates in the last column of Table S1.

V. PARAMETER ESTIMATES USING SELF-REPORTED FLU DATA
We performed similar parameter estimates as in the main text but using all available information regarding flu infections (through self-report of subjects). Not surprisingly, the total number of self-reported flu illness far exceeds that of diagnosed flu, that is, 237 versus 57. As a result, self-reported flu data gives a much higher estimation of the epidemiological parameters (R 0 ∼ 2.9). It is worth noting that, albeit with much elevated flu incidence (namely, higher perceived risk of infection) in this situation, the 'rational response' model does not fit better than the 'social contagion' and the full model. In line with our conclusion in the main text, this additional result suggests that social contagion is a key determinant of vaccination. Detailed fitting results using self-reported flu data can be found in Table S1.

VI. TIME-SCALE SEPARATION ANALYSIS
We describe the dueling contagion processes of vaccination and infection based on the following aggregate fractions: We find no closed-form solutions to the ordinary differential equations above, but we can obtain analytical approximations using the time-scale separation technique for extreme values of ω.
Specifically, for ω → 0 (when individuals show very little or no responsiveness to either vaccination or infection), the spread of vaccination occurs much more slowly, compared to the spread of infection. Thus, the dueling contagion dynamics will con- following the fast dynamic of infection in which the spread of vaccination is decoupled and has almost no effects on the disease transmission: It yields: Integrating the above equation, we get: Using the initial condition, ρ S (0) ∼ 1 and ρ R (0) ∼ 0, and ρ S (∞) = 1−ρ R (∞) at the end of the epidemic spreading, we obtain the transcendental equation below, which determines the final epidemic size, ρ R : For R 0 = 1.56, ρ R is approximately 0.61.
On the other limit, for sufficiently large values of ω (when the population shows high levels of responsiveness), the spread of vaccination behavior is a fast dynamic and rapidly converges to the equilibrium level κ, while the spread of flu infection is a slow dynamic. Therefore, the epidemic spreading dynamic recovers to the classic SIR model with the pre-emptive vaccination level κ. In our study, the equilibrium level of vaccination is 0.415, which exceeds the herd immunity threshold, 1 − 1/R 0 (0.35), so the disease cannot persist in the population, ρ R ∼ 0, in this scenario.
For intermediate values of ω, the dueling contagion dynamics exhibit strong mutual interdependence (i.e., no time-scale separation): the epidemic size ρ R monotonically