Endogenous social distancing and its underappreciated impact on the epidemic curve

Social distancing is an effective strategy to mitigate the impact of infectious diseases. If sick or healthy, or both, predominantly socially distance, the epidemic curve flattens. Contact reductions may occur for different reasons during a pandemic including health-related mobility loss (severity of symptoms), duty of care for a member of a high-risk group, and forced quarantine. Other decisions to reduce contacts are of a more voluntary nature. In particular, sick people reduce contacts consciously to avoid infecting others, and healthy individuals reduce contacts in order to stay healthy. We use game theory to formalize the interaction of voluntary social distancing in a partially infected population. This improves the behavioral micro-foundations of epidemiological models, and predicts differential social distancing rates dependent on health status. The model’s key predictions in terms of comparative statics are derived, which concern changes and interactions between social distancing behaviors of sick and healthy. We fit the relevant parameters for endogenous social distancing to an epidemiological model with evidence from influenza waves to provide a benchmark for an epidemic curve with endogenous social distancing. Our results suggest that spreading similar in peak and case numbers to what partial immobilization of the population produces, yet quicker to pass, could occur endogenously. Going forward, eventual social distancing orders and lockdown policies should be benchmarked against more realistic epidemic models that take endogenous social distancing into account, rather than be driven by static, and therefore unrealistic, estimates for social mixing that intrinsically overestimate spreading.

groups 0-4, 5-18, 19-44, 45-64 and 65+ years respectively. Further relevant questions concern how much time was spent on public transport, in enclosed indoor spaces with more than 10 others, and what the furthest distance travelled from home was.
As well as estimating incidence trends 5 , flusurvey data have been used to identify risk factors to ILI 6 , to estimate the effectiveness of influenza vaccination 7 and to quantify health-care seeking behaviour 8 . During the four influenza seasons 2009-13, social contact data were collected in addition to the ILI-related data. Participants were asked to report conversational and physical contacts by age group in three types of setting (home, work/school and other). These data have previously been used to explain the spread of H1N1v influenza 9 .
We used the total of conversational contacts reported as measure of overall contact, and assessed whether the date at which the contacts were submitted were within the start end end date of an episode of illness with ILI symptoms (one general symptom out of fever, tiredness, weakness and headache, and one respiratory symptom out of sore throat, cough and shortness of breath). The end date of an episode was considered to be a healthy date.
We cleaned the data in the following ways: We removed bad symptom dates (end date before start date, dates after the date at which a response was submitted) in 85 out of 8800 symptom reports. We further removed all participants with fewer than three symptom reports (whether reporting healthy or ill), and removed the first submitted survey report of every participant in order to remove any potential bias from participants signing up only because they were researching influenza-related information.
Where the end date of an episode was not reported, the date of the report which stated that the illness had ended was taken as the end date of the episode.
Incidence was calculated as number of episodes of illness with ILI symptoms starting in any particular week divided by the number participants submitting a report in that week.

Monte Carlo Method
We perform Monte Carlo simulations of the SEIR model 10 , which corresponds to a random sequential update, such that during a full Monte Carlo step (MCS) each node gets a chance once on average to become infected. Each full MCS consist of repeating the following elementary step n times. Firstly, select a node i uniformly at random from the whole network. Secondly, (i) If node i is in state S, choose one neighbor j uniformly at random and visit it with probability q i . If the neighbor is visited and is in state I, node i becomes infected with probability w = 0.7. If, however, the neighbor j is in states S or R nothing happens.

Comparative Statics Derivations
We first rearrange H-FOC (Equation 2 in the main document): We are interested in the partial derivative of Equation 1 along d/d f , d/dc, d/dβ S , and d/dβ H . As the calculations for the former two and the latter two are very similar we only detail them for d/d f and d/dβ S . The other predictions also follow from similar arguments. Partial derivative of Equation 1 along d/d f : Partial derivative of Equation 1 along d/dβ S : (1 + f · (−1 + [((1 − β i S ) n−|H| − 1) · β i + 1])) · u (β i )