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  1. Author Response:

    Reviewer #1 (Public Review):

    [...] In the model, the mitigation function is fitted; no actual data on deliberate versus randomly-varying behavior change is used. Given clear empirical signals of synchronous and delibate response to epidemiology, modulated by social factors (Weill et al., 2020), a persuasive demonstration that consideration of random behavioral variation is necessary and/or sufficient to explain observed US COVID-19 dynamics would need to start from mobility data itself, and then find some principled way of partitioning changes in mobility into those attributable to random variation versus deliberate (whether top-down or bottom-up) action.

    As suggested by our referees and the editor, we undertook a principled analysis of the US COVID-19 data that took into account Google mobility patterns. The average mobility reflects systematic changes in social activity due to both government-imposed mitigations and knowledge-based adaptation of the population. We identified a range of dates (July 2020- February 2021) during which there has been only modest and slow changes in the average mobility. This time range allows for a direct test of our model, accounting for stochastic changes in social activity uncorrelated across the population (see Figure 6 and Appendix 5. Figure 1A).

    In the new version, we also present a direct comparison of the predictive power of our SSA model vs the traditional SIR model within this time range (see Figure 8 and Appendix 5. Figure 2).

    My other main concern is that the central result of transient epidemiological dynamics due to transient concordance of abnormally high versus low social activity-stems from the choice to model social behavior as stochastic but also mean-seeking. While I find this idealization plausible, I think it would be good to motivate it more.

    In other words, the central, compelling message of the paper is that if collective activity levels sometimes spike and crash, but ultimately regress to the mean, so will transmission. The more that behavioral model can be motivated, the more compelling the paper will be.

    We included an additional justification of our form of stochastic social dynamics and expanded the discussion of relevant prior studies. Especially revealing are the studies of burstiness in virtual communication such as e-mail (Vazquez et al. (2007); Karsai et al. (2012)). Studies of digital communications can be easily studied over a substantial time interval, which is more problematic for field studies of face-to-face contact networks. These studies unequivocally show the regression of individual activity levels towards its long-term mean value. This regression happens over a well-defined relaxation time ranging from days to months depending on the context. Note that the value towards which the activity regresses may not be identical for different individuals. In the context of our model, such persistent heterogeneity is captured by the distribution of \alpha_i with the dispersion parameter \kappa.

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  2. Evaluation Summary:

    This manuscript will be of interest to epidemiologists and population biologists interested in outbreak dynamics in populations with complex social structures, such as emergent viral infections of humans. The study offers a functional, differential-equation (DE)-based framework for capturing the transition from emergence to endemicity without the huge over-compensation cycles typically predicted by DE models but rarely seen in natural populations. The model framework currently offers insights into the drivers of epidemic dynamics and, after further testing and calibration, may be useful for assessing control strategies for emerging infectious diseases.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 agreed to share their name with the authors.)

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  3. Reviewer #1 (Public Review):

    Summary of the paper

    The authors' model is an extension of standard disease models (Kermack & McKendrick, 1927; Yang & Brauer, 2008) that track the spread of an infectious disease within a host population. The authors consider the possibility that individuals' level of activity (and thus their probability of contacting others and potentially transmitting or contracting the infectious disease) may vary in time. Importantly, individual activity levels vary according to a stochastic processes that is not in any way affected by the current disease dynamics in the population or by the individuals' own disease states.

    The authors' key result is that if individual social activity levels can spike or crash but then tends to return to their mean value, then synchronous spikes and/or crashes among many individuals' activity levels can lead to corresponding transient changes in the epidemiological dynamics. Waves take off when many individuals are active, but may peak well before herd immunity is reached, because individual activity levels regress to the mean.

    Nowhere in the authors' model does individual behavior depend upon individual disease state or population-level disease dynamics. There are many epidemiological models featuring adaptive host behavior; in these, individuals respond behaviorally to the disease. Those adaptive behavior models show disease dynamics that would not be seen in the standard (i.e. constant contact rate) Susceptible-Infectious-Recovered (SIR) model (see for instance Epstein et al., 2008; Fenichel et al., 2011; see Bauch et al., 2013 for an extensive review).

    This, then, is the authors' key result: behavioral change that is not responsive to the disease itself can still produce transient plateaus, sub-herd immunity peaks, etc. The authors thus offer a valuable null model that should be considered when responsive behavioral change models are proposed to explain observed epidemiological dynamics.

    I believe that this is an important result, especially in light of the explosion of adaptive behavior epidemiology that has accompanied the COVID-19 pandemic thanks to an unprecedented wealth of both epidemiological (e.g. case / hospitalization / death) and behavioral (e.g. Google Mobility) data (Nouvellet et al., 2021). Claims that responsive behavior explains observed epidemiology will need to improve upon this null model in some way in order to be persuasive. My principal reservation about the paper is that the model is presented less as such a null model and more as a mechanistic explanation of observed COVID-19 dynamics. I did not find the case for this interpretation sufficiently convincing, for reasons I will explain below.

    The authors find a number of other interesting results, including that stochastically time-varying behavior can reduce the likely "overshoot" of the disease attack rate beyond the herd immunity threshold, and can produce states of "Transient Collective Immunity". These results are a property of a previously-presented model developed by the same authors, in which individual activity levels may vary in time but not necessarily according to a defined stochastic process (Tkachenko et al., 2021). In general, given that this paper builds on that work, I would encourage the authors to be clearer about distinguishing their current results from their prior findings.

    The authors characterize the potential endemic state for a pathogen under their model (in the case that previously-exposed individuals can become once again susceptible on some timescale), and show that time-varying heterogeneous contact behavior again alters the dynamics of the approach to endemicity. Notably, they find that behavioral variation can reduce the amplitude of peaks and troughs on the way to endemicity, potentially avoiding stochastic extinction of the disease during troughs.

    The authors compare their analytical results to stochastic simulations based on the underlying stochastic process, and find good agreement. Finally, the authors fit their model to COVID-19 death data from United States geographical regions and compare predicted model trajectories to observed deaths.

    Key contribution of the paper

    In my view, the greatest strength of the paper is in providing a plausible null model for how adaptive behavior can modulate epidemiology even when it does not respond directly to disease, and in developing analytical results that give further insight into the origin and magnitude of these effects given the underlying model parameters.

    Concerns regarding the paper

    My principal concern about the paper is the implicit claim that the model explains the epidemiological patterns of COVID-19 in the United States during summer and fall 2020.

    The authors fit their model to US death data by estimating parameters related to the degree of mitigation as a function of time M(t), as well as some seasonality parameters affecting R0 as a function of time. It is not clear whether baseline R0 was also estimated, since it is not listed as a fixed.

    As the authors point out,monotonically increasing R0M(t) in a standard well-mixed SIR far from herd immunity would result in a single peak that overshoots the (ever-increasing) HIT. In the authors' fitted model, deaths in fact initially decline in the northeast and midwest before rising again, and the epidemic in the south displays two peaks separated by a trough.

    But I am not sure this is a particularly convincing demonstration of the correctness of a model as an explanation for the observed dynamics. Official distancing policies may have monotonically become more lax over the period June 1 through to, e.g., the fall. But restrictions were tightened in winter in response to surges, and there was clear signal of behavioral response to incresasing transmission that seems unlikely to have been mere regression to the mean.

    In the model, the mitigation function is fitted; no actual data on deliberate versus randomly-varying behavior change is used. Given clear empirical signals of synchronous and delibate response to epidemiology, modulated by social factors (Weill et al., 2020), a persuasive demonstration that consideration of random behavioral variation is necessary and/or sufficient to explain observed US COVID-19 dynamics would need to start from mobility data itself, and then find some principled way of partitioning changes in mobility into those attributable to random variation versus deliberate (whether top-down or bottom-up) action.

    My other main concern is that the central result of transient epidemiological dynamics due to transient concordance of abnormally high versus low social activity-stems from the choice to model social behavior as stochastic but also mean-seeking. While I find this idealization plausible, I think it would be good to motivate it more.

    In other words, the central, compelling message of the paper is that if collective activity levels sometimes spike and crash, but ultimately regress to the mean, so will transmission. The more that behavioral model can be motivated, the more compelling the paper will be.

    References

    Bauch, C., d'Onofrio, A., & Manfredi, P. (2013). Behavioral epidemiology of infectious diseases: An overview. Modeling the interplay between human behavior and the spread of infec- tious diseases, 1-19.

    Epstein, J. M., Parker, J., Cummings, D., & Hammond, R. A. (2008). Coupled contagion dy- namics of fear and disease: Mathematical and computational explorations. PLoS One, 3(12), e3955.

    Fenichel, E. P., Castillo-Chavez, C., Ceddia, M. G., Chowell, G., Parra, P. A. G., Hickling, G. J., Holloway, G., Horan, R., Morin, B., Perrings, C., et al. (2011). Adaptive human behav- ior in epidemiological models. Proceedings of the National Academy of Sciences, 108(15), 6306-6311.

    Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London, Series A, 115(772), 700-721.

    Nouvellet, P., Bhatia, S., Cori, A., Ainslie, K. E., Baguelin, M., Bhatt, S., Boonyasiri, A., Brazeau, N. F., Cattarino, L., Cooper, L. V., et al. (2021). Reduction in mobility and covid-19 transmission. Nature communications, 12(1), 1-9.

    Tkachenko, A. V., Maslov, S., Elbanna, A., Wong, G. N., Weiner, Z. J., & Goldenfeld, N. (2021). Time-dependent heterogeneity leads to transient suppression of the covid-19 epidemic, not herd immunity. Proceedings of the National Academy of Sciences, 118(17).

    Weill, J. A., Stigler, M., Deschenes, O., & Springborn, M. R. (2020). Social distancing responses to covid-19 emergency declarations strongly differentiated by income. Proceedings of the National Academy of Sciences, 117(33), 19658-19660.

    Yang, C. K., & Brauer, F. (2008). Calculation of R0 for age-of-infection models. Mathematical Biosciences & Engineering, 5(3), 585.

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  4. Reviewer #2 (Public Review):

    Current mechanistic modeling approaches lack a unified framework that agrees with data for connecting local well-mixed outbreaks with long-term steady state dynamics of epidemics in highly heterogeneous (e.g., national or global) populations. The model presented here combines overdispersion in how many contacts people have on average (e.g., node degree distribution in a static network) with temporally correlated variability in each individual's degree of sociality. They show the framework can recapitulate agent-based simulations of these processes and provide a qualitative comparison with data from SARS-CoV-2. They suggest that their model describes the time series of COVID-19 deaths observed in different regions of the U.S.

    Key features of their model include:

    1. progression toward the 'herd immunity threshold' without an overshoot

    2. lack of pathogen extinction following an initial outbreak even for relatively small population sizes

    3. a long plateau in infection rates after the first (not so huge) wave, which can easily be excited to subsequent waves in the face of changes in transmission rates (e.g., seasonal forcing, lockdown, NPIs, etc)

    4. Interactions between multiple time-scales (i.e., infectious period, 'burstiness' of individual social behavior through time, waning of immunity / antigenic variation)

    5. An emergent time scale of epidemic relaxation toward the steady state

    This model formulation is a step forward for outbreak biologists and infectious disease epidemiologists that can help integrate multiple time scales and achieve a more nuanced understanding of their interplay by quantitatively incorporating them into models. The model formulation may also be useful in design and assessment of control measures, though I would like to see more rigorous comparisons with varied data sets and sensitivity analysis of some of the model parameters before I would use it for prediction. In particular, the fits to SARS-CoV-2 death data are not entirely compelling, and the final sentence of the paper is overstated, claiming quantitative agreement with data on secondary waves of COVID-19 in the US.

    While the match with particular dynamic data is not very compelling, their general argument is strong: epidemics rarely burn through populations as deterministic, non-agent-based models predict, even models that include persistent heterogeneity in transmission rates. This modeling framework provides a functional approximation of assumptions regarding individual heterogeneities, and show that it can recapitulate many broad features of transitions from emergence to endemicity.

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