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.