Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus, and an endemic state.

It is well recognized that population heterogeneity plays an important role in the spread of epidemics. While individual variations in social activity are often assumed to be persistent, that is, constant in time, here we discuss the consequences of dynamic heterogeneity. By integrating the stochastic dynamics of social activity into traditional epidemiological models, we demonstrate the emergence of a new long timescale governing the epidemic, in broad agreement with empirical data. Our stochastic social activity model captures multiple features of real-life epidemics such as COVID-19, including prolonged plateaus and multiple waves, which are transiently suppressed due to the dynamic nature of social activity. The existence of a long timescale due to the interplay between epidemic and social dynamics provides a unifying picture of how a fast-paced epidemic typically will transition to an endemic state.

Introduction

The COVID-19 pandemic has underscored the prominent role played by population heterogeneity in epidemics. It has been well documented that at short timescales the transmission of the infection is highly heterogeneous. That is to say, it is characterized by the phenomenon of superspreading, in which a small fraction of individuals is responsible for a disproportionately large number of secondary infections (Lloyd-Smith et al., 2005; Galvani and May, 2005; Endo et al., 2020; Sun et al., 2021). At the same time, according to multiple models, persistent population heterogeneity is expected to suppress the herd immunity threshold (HIT) and reduce the final size of an epidemic (Pastor-Satorras et al., 2015; Bansal et al., 2007; Gomes et al., 2020; Tkachenko et al., 2021; Neipel et al., 2020; Britton et al., 2020). In the context of COVID-19, this observation led to a controversial suggestion that a strategy relying exclusively on quickly reaching herd immunity might be a viable alternative to government-imposed mitigation. However, even locations that have been hardest hit by the first wave of the epidemic have not gained a lasting protection against future waves (Faria et al., 2021; Sabino et al., 2021). Another puzzling aspect of the COVID-19 pandemic is the frequent occurrence of plateau-like dynamics, characterized by approximately constant incidence rate over a prolonged time (Thurner et al., 2020; Weitz et al., 2020).

These departures from predictions of both classical epidemiological models and their heterogeneous extensions have led to a greater appreciation of the role played by human behavior in epidemic dynamics. In particular, one plausible mechanism that might be responsible for both suppression of the early waves and plateau-like dynamics is that individuals modify their behavior based on information about the current epidemiological situation (Epstein et al., 2008; Funk et al., 2009; Fenichel et al., 2011; Bauch, 2013; Rizzo et al., 2014; Weitz et al., 2020; Arthur et al., 2021). Another possibility is that long plateaus might arise because of the underlying structure of social networks (Thurner et al., 2020).

Here, we study epidemic dynamics, accounting for random changes in levels of individual social activity. We demonstrate that this type of dynamic heterogeneity, even without knowledge-based adaptation of human behavior (e.g., in response to epidemic-related news) (Epstein et al., 2008; Funk et al., 2009; Fenichel et al., 2011; Bauch, 2013; Rizzo et al., 2014; Weitz et al., 2020; Arthur et al., 2021), leads to a substantial revision of the epidemic progression, consistent with empirical data for the COVID-19 pandemic. In a recent study (Tkachenko et al., 2021), we have pointed out that population heterogeneity is a dynamic property that roams across multiple timescales. A strong short-term overdispersion of the individual infectivity manifests itself in the statistics of superspreading events. At the other end of the spectrum is a much weaker persistent heterogeneity operating on very long timescales. In particular, it is this long-term heterogeneity that leads to a reduction of the HIT compared to that predicted by classical homogeneous models (Gomes et al., 2020; Tkachenko et al., 2021; Neipel et al., 2020; Rose et al., 2021; Britton et al., 2020). In particular, in our previous work (Tkachenko et al., 2021), it was demonstrated that the entire effect of persistent heterogeneity can be well characterized by a single parameter, which we call the immunity factor . This quantity is related to the statistical properties of heterogeneous susceptibility across the population and to its correlation with individual infectivity. For the important case of gamma-distributed individual susceptibilities, we show that the classical proportionality between the fraction of susceptible population and the effective reproduction number, , transforms into a power-law scaling relationship . This leads to a modified version of the result for the HIT, . However, that result assumes persistent or time-independent heterogeneity. In reality, the epidemic dynamics is likely to be sensitive to what happens at intermediate timescales, where the social activity of each individual crosses over from its bursty short-term behavior to a smooth long-term average. Due to this type of dynamic heterogeneity, the suppression of early waves of the COVID-19 epidemic, even without active mitigation, does not signal achievement of long-term herd immunity. Instead, as argued in Tkachenko et al., 2021, this suppression is associated with transient collective immunity (TCI), a fragile state that degrades over time as individuals change their social activity patterns. In this work, we present a stochastic social activity (SSA) model explicitly incorporating time-dependent heterogeneity and demonstrate that the first wave is generally followed either by secondary waves or by long plateaus characterized by a nearly constant incidence rate. In the context of COVID-19, both long plateaus and multiwave epidemic dynamics have been commonly observed. According to our analysis, the number of daily infections during the plateau regime, as well as the individual wave trajectories, are robust properties of the epidemic and depend on the current level of mitigation, degree of heterogeneity, and temporal correlations of individual social activity.

Our work implies that, once plateau-like dynamics is established, the epidemic gradually evolves towards the long-term HIT determined by persistent population heterogeneity. However, reaching that state may stretch over a surprisingly long time, from months to years. On these long timescales, both waning of individual biological immunity and mutations of the pathogen become valid concerns and would ultimately result in a permanent endemic state of the infection. Such endemic behavior is a well-known property of most classical epidemiological models (Keeling and Rohani, 2011). However, the emergence of the endemic state for a newly introduced pathogen is far from being completely understood (Wolfe et al., 2007; Engering et al., 2019; Pastor-Satorras and Vespignani, 2001a). Indeed, most epidemiological models would typically predict complete extinction of a pathogen following the first wave of the epidemic, well before the pool of susceptible population would be replenished. A commonly accepted, though mostly qualitative, explanation for the onset of endemic behavior of such diseases as measles, seasonal cold, etc., involves geographic heterogeneity: the pathogen may survive in other geographic locations until returning to a hard-hit area with a depleted susceptible pool (Wolfe et al., 2007; Engering et al., 2019). In contrast, our theory provides a simple and general mechanism that prevents an overshoot of the epidemic dynamics and thus naturally and generically leads to the endemic fixed point.

The importance of temporal effects has long been recognized in the context of network-based epidemiological models (Starnini et al., 2017; Volz and Meyers, 2007; Bansal et al., 2010; Read et al., 2008). On the one hand, available high-resolution data on real-world temporal contact networks allow direct modeling of epidemic spread on those networks. On the other hand, building upon successes of epidemic models on static unweighted networks (Lloyd and May, 2001; May and Lloyd, 2001; Moreno et al., 2002; Pastor-Satorras et al., 2015), a variety of temporal generalizations have been proposed. These typically involve particular rules for discrete or continuous network rewiring (Volz and Meyers, 2007; Bansal et al., 2010; Read et al., 2008) such as in activity-based network models (Perra et al., 2012; Vazquez et al., 2007; Rizzo et al., 2014). While important theoretical results have been obtained for some of these problems, especially regarding the epidemic threshold, many open questions and challenges remain in the field. In this paper, we start with a more traditional heterogeneous well-mixed model, which is essentially equivalent to the mean-field description of an epidemic on a network (Moreno et al., 2002; Pastor-Satorras and Vespignani, 2001b; Bansal et al., 2007), and include effects of time-variable social activity that modulates levels of individual susceptibilities and infectivities.

Results

SSA model

The basic idea behind our model is represented in Figure 1. Each individual is characterized by time-dependent social activity proportional to his/her current frequency and intensity of close social contacts. This quantity determines both the individual susceptibility to infection as well as the ability to infect others. The time evolution of contact frequency, and hence , is in principle measurable by means of proximity devices, such as RFID, Bluetooth, Wi-Fi, etc. (Salathe et al., 2010; Starnini et al., 2017; Isella et al., 2011; Pastor-Satorras et al., 2015). In fact, multiple studies of that kind have been conducted over the years, alongside more traditional approaches based on, for example, personal logs (Danon et al., 2013). In addition, virtual interactions by means of e-mail, social media, and mobile communications are commonly used as proxies for studies of interpersonal contacts (Rybski et al., 2009; Barabási, 2005; Saramäki and Moro, 2015; Nielsen et al., 2021). Digital communications can be studied over a substantial time interval for a large number of individuals, thus presenting a significant challenge for field studies of face-to-face contact networks. It is generally accepted that the presence of an underlying dynamic contact network may drastically affect epidemic dynamics. However, the sheer complexity of that network makes it hard to integrate the social dynamics into common epidemic models. The simple stochastic model of social activity proposed in this work is based on several observations that appear to be rather generic both for real and virtual interpersonal communications. Individual social activity tends to be ‘bursty’ and overdispersed when observed over short enough timescales (e.g., several days). While individuals demonstrate bursts of activity across multiple timescales, the analysis of various communication networks reveals a cutoff time, beyond which the level of activity reverts to its long-term average (Vazquez et al., 2007; Karsai et al., 2012). Note that this average may still exhibit person-to-person variations corresponding to persistent heterogeneity of the population. The mean-reversion time constant may range from days to months, depending on the context of the study (Vazquez et al., 2007; Karsai et al., 2012). In this work, we make a model assumption that a similar mean-reversion time exists for in-person social activity, that is, for .

Figure 1.

Schematic illustration of the stochastic social activity model in which each individual is characterized by a time-dependent social activity.

(a) People with low social activity (depicted as socially isolated figures at home) occasionally increase their level of activity (depicted as a party). The average activity in the population remains the same, but individuals constantly change their activity levels from low to high (arrows pointing up) and back (arrows pointing down). Individuals are colored according to their state in the susceptible-infected-removed (SIR) epidemiological model: susceptible, green; infected, red; and removed, - blue. The epidemic is fueled by constant replenishment of susceptible population with high activity due to transitions from the low-activity state. (b) Examples of individual time-dependent activity (solid lines), with different persistent levels (dot-dashed lines). S,I,R states of an individual have the same color code as in (a). Note that pathogen transmission occurs predominantly between individuals with high current activity levels.

In our SSA model, we combine a simple mathematical description of social dynamics with the standard susceptible-infected-removed (SIR) epidemiological model. Qualitatively it leads to long-term epidemic dynamics fueled by replenishment of the susceptible population due to changes in the level of individual social activity from low to high. Figure 1a illustrates this process by showing people with low social activity (depicted as socially isolated at home) occasionally increasing their level of activity (depicted as a party). Figure 1 represents the same dynamics in terms of individual functions . Note that each person is characterized by his/her own long-term average activity level (dot-dashed lines), but the transmission occurs predominantly between individuals with high levels of current social activity. This is because determines both the current susceptibility and the individual infectivity of a person. However, secondary transmission is delayed with respect to the moment of infection, by a time of the order of a single generation interval (around 5 days for COVID-19).

For any individual , the value of has a tendency to gradually drift towards its persistent average level , which itself varies within the population. In our model, we assign a single timescale to this mean reversion process. This is of course a simplification of the multiscale relaxation observed in real social dynamics. While can be treated as a fitting parameter of our model, here we simply set it to be days, several times longer than the mean generation interval of COVID-19, days. Note that from the point of view of the epidemic dynamics, variations in activity on timescales shorter than the mean generation interval may be safely ignored. For example, attending a single party would increase an individual’s risk of infection but would not change his/her likelihood of transmission to others 5 days later.

Individual social activity is assumed to be governed by the following stochastic equation:

Here, is a zero mean Gaussian noise giving rise to time-dependent variations in . We set the correlation function of the noise as , which results in diffusion in the space of individual social activity with a diffusion coefficient proportional to a and the correlation time . This stochastic process is well known in mathematical finance as the Cox–Ingersoll–Ross (CIR) model (Cox et al., 1985) and has been studied in probability theory since the 1950s (Feller, 1951). The major properties of this model are (i) reversion to the mean and (ii) non-negativity of a at all times, both of which are natural for social activity. Furthermore, the steady-state solution of this model is characterized by gamma-distributed a. This is consistent with the empirical statistics of short-term overdispersion of disease transmission manifesting itself in superspreading events (Lloyd-Smith et al., 2005; Endo et al., 2020; Sun et al., 2021). More specifically, for a given level of persistent activity , this model generates a steady-state distribution of ‘instantaneous’ values of social activity following a gamma distribution with mean and variance . Additional discussion of this model is presented in Appendix 1.

The statistics of superspreader events is usually represented as a negative binomial distribution, derived from a gamma-distributed individual reproduction number (Lloyd-Smith et al., 2005). The observed overdispersion parameter (Endo et al., 2020; Sun et al., 2021) can be used for partial calibration of our model. This short-term overdispersion has both stochastic and persistent contributions. In our model, the former is characterized by dispersion k0. In addition, we assume persistent levels of social activity to also follow a gamma distribution with another dispersion parameter, . In several recent studies of epidemic dynamics in populations with persistent heterogeneity (Tkachenko et al., 2021; Aguas et al., 2020; Neipel et al., 2020), it has been demonstrated that determines the HIT. Multiple studies of real-world contact networks (summarized, e.g., in Bansal et al., 2007) report an approximately exponential distribution of , which corresponds to . Throughout this paper, we assume a more conservative value, , that is, coefficient of variation , half way between the fully homogeneous case and that with exponentially distributed . For consistency with the reported value of the short-term overdispersion parameter (Sun et al., 2021), , we set .

Epidemic dynamics with stochastic social activity

According to Equation 1, individuals, each with their own persistent level of social activity effectively diffuse in the space of their current social activity . This leads to major modifications of the epidemic dynamics (see Appendix 1 for the detailed technical discussion). For instance, the equation for the susceptible fraction in classical epidemic models (Keeling and Rohani, 2011) acquires the following form:

Here, is the fraction of susceptible individuals within a subpopulation with a given value of persistent social activity and with current social activity , at the moment of infection, and is the current strength of infection. Its time evolution can be described by any traditional epidemiological model, such as age-of-infection, SIR/SEIR, etc. (Keeling and Rohani, 2011).

Equation 2 is dramatically simplified by writing it as . The new variables and measure persistent and, respectively, transient heterogeneity of the attack rate. As the epidemic progresses, new infections selectively remove people with high current levels of social activity . The variable measures the degree of such selective depletion of susceptibles. Conversely, the variable quantifies the extent of depletion of susceptibles among subpopulations with different levels of persistent social activity . In the long run, transient heterogeneity disappears due to stochastic changes in the levels of current social activity . Thus, asymptotically approaches 0 as . We combine this ansatz with a general methodology (Tkachenko et al., 2021) that provides a quasi-homogeneous description for a wide variety of heterogeneous epidemiological models. For a specific case of SIR dynamics, we assign each person a state variable set to 1 when the individual is infectious and 0 otherwise. Now, the activity-weighted fraction of the infected population is defined as , and the current infection strength is proportional to it:

Here, is a time-dependent mitigation factor, which combines the effects of government interventions, societal response to the epidemic, as well as other sources of time modulation, such as seasonal forcing.

Using the above ansatz, the epidemic in a population with both persistent and dynamic heterogeneity of individual social activity can be compactly described as a dynamical system with only three variables: the susceptible population fraction , the infected population fraction (activity-weighted) that, according to Equation 3, is proportional to the strength of infection , and the transient heterogeneity variable . As shown in Appendix 2, the dynamics in the -space are given by the following set of differential equations:

As discussed above, the scaling exponent in Equation 4 is the immunity factor that we introduced in Tkachenko et al., 2021 to describe the reduction of the HIT due to persistent heterogeneity. In the context of the present study, depends both on short-term and persistent dispersion parameters as described in Appendix 2. For parameters , , and days used throughout our study, one gets , consistent with our earlier estimate in Tkachenko et al., 2021.

In Figure 2, we schematically represent three feedback mechanisms that lead to self-limited epidemic dynamics. The most conventional of them relies on depletion of the susceptible population (red). Another mechanism is due to government mitigation as well as personal behavioral response to perceived epidemic risk (purple). Finally, according to our theory, there is yet another generic mechanism related to accumulated heterogeneity of the attack rate, quantified by the variable . Due to the long-term relaxation of , this feedback loop limits the scale of a single epidemic wave, but does not provide long-term protection against new ones.

Figure 2.

Schematic representation of feedback mechanisms that lead to self-limited epidemic dynamics.

In traditional epidemic models, the major factor is the depletion of the susceptible population (red). Government-imposed mitigation and/or behavioral knowledge-based adaptation to the perceived risk creates a second feedback loop (purple). Yet another feedback mechanism is due to dynamic heterogeneity of the attack rate parameterized by (black). Note that this mechanism is due to the selective removal of susceptibles with high current levels of social activity in the course of the epidemic. Therefore, it does not involve any knowledge-based adaptation, defined as modulation of average social activity in response to the perceived danger of the current level of infection. The attack rate heterogeneity is generated by the current infection and suppresses itself on the timescale of due to reversion of individual social activity towards the mean.

Origin of waves and plateaus

As demonstrated below, the theory described by Equation 4, Equation 5, Equation 6 is in excellent agreement with simulations of an agent-based model (ABM) in which social activities of 1 million agents undergo stochastic evolution described by Equation 1 (compare solid lines with shaded areas in Figure 3 and Figure 4).

Figure 3.

Comparison of the epidemic dynamics in three models.

The mitigation profile (a), the daily incidence (b), and the cumulative attack rate (c). The mitigation profile (a), the daily incidence (b), and the cumulative attack rate (c) in the SIR model with homogeneous population (black curves), the model with persistent population heterogeneity (brown curves), and our SSA model with dynamic heterogeneity (green curves). While parameters in all three models correspond to the same herd immunity threshold (HIT), the behavior is drastically different. In the persistent model, the epidemic quickly overshoots above HIT level. In the case of dynamic heterogeneity, the initial wave is followed by a plateau-like behavior with slow relaxation towards the HIT. Note an excellent agreement between the quasi-homogeneous theory described by Equation 4; Equation 5; Equation 6 (solid lines) and an agent-based model with 1 million agents whose stochastic activity is given by Equation 1 (shaded area = the range of three independent simulations).

Figure 4.

The time course of an epidemic with enhanced mitigation during the first wave.

(a) shows the progression for two different strategies. In both cases, the enhanced mitigation leads to a 50% reduction of from 2 to 1. In the first scenario (early mitigation, blue curves), the reduction lasted for only 15 days starting from day 27. In the second scenario (delayed mitigation, red curves), the mitigation was applied on day 37 and lasted for 45 days. (b and c) show daily incidence and cumulative attack rates for both strategies. As predicted, differences in the initial mitigation had no significant effect on the epidemic in the long run: the two trajectories eventually converge towards the universal attractor. However, early mitigation allows the peak of the infection to be suppressed, potentially reducing stress on the healthcare system. A delayed mitigation gives rise to a sizable second wave.

Figure 3 illustrates the dramatic effect that time-dependent heterogeneity has on epidemic dynamics. It compares three cases: the classical homogeneous SIR model (black), the same model with persistent heterogeneity (brown), and the dynamic heterogeneity case considered in this study (green). The latter two models share the same HIT (green dashed line) that is reduced compared to the homogeneous case (black dashed line). In the absence of dynamic heterogeneity (black and brown), the initial exponential growth halts once the respective HIT is reached, but the overall attack rate ‘overshoots’ beyond that point, eventually reaching a significantly larger level, known as the final size of the epidemic (FSE). Importantly, in both these cases the epidemic has only a single wave of duration set by the mean generation interval multiplied by a certain R0-dependent factor. In the case of dynamic heterogeneity (green), described by Equation 4; Equation 5; Equation 6, the epidemic is transiently suppressed at a level that is below even the heterogeneous HIT. As we argued in Tkachenko et al., 2021, this temporary suppression is due to the population reaching a state we termed transient collective immunity (TCI). That state originates due to the short-term population heterogeneity being enhanced compared to its persistent level. Stochastic contributions to social activity responsible for this enhancement eventually average out, leading to a slow degradation of the TCI state. Figure 3b illustrates that as the TCI state degrades, the daily incidence rate develops an extended plateau on the green curve. The cumulative attack rate shown in Figure 3c relaxes towards the HIT. As shown in Appendix 3, in this regime . By substituting this relationship into Equation 6, one observes that the relaxation is characterized by an emergent long timescale. This timescale of the order of governs the relaxation towards either herd immunity or the endemic state of the pathogen. Note that it may be considerably longer than the timescale for individuals to revert to their mean level of activity provided that the short-term overdispersion is strong (i.e., ).

According to (Equation 4; Equation 5; Equation 6) for a fixed mitigation level , any epidemic trajectory would eventually converge to the same curve, that is, the universal attractor. The existence of the universal attractor is apparent in Figure 4, where we compare two scenarios with different mitigation strategies applied at early stages of the epidemic. In both cases, an enhanced mitigation was imposed, leading to a reduction of by 50% from 2 to 1. In the first scenario (blue curves), the enhanced mitigation was imposed on day 27 and lasted for 15 days. In the second scenario (red curves), the mitigation was applied on day 37 and lasted for 45 days. As predicted, this difference in mitigation has not had any significant effect on the epidemic in the long run: these two trajectories eventually converged towards the universal attractor. However, short- and medium-term effects were substantial. The early mitigation scenario (blue curve) resulted in a substantial suppression of the maximum incidence during the first wave. Immediately following the release of the mitigation the second wave started and reached approximately the same peak value as the first one. If the objective of the intervention is to avoid overflow of the healthcare system, this strategy would indeed help to achieve it. In contrast, the delayed mitigation scenario (red curve) turned out to be largely counterproductive. It did not suppress the peak of the first wave, but brought the infection to a very low level after it. Eventually, that suppression backfired as the TCI state deteriorated and the epidemic resumed as a second wave, which is not as strong as the first one.

Since the late-stage evolution in our model is characterized by a long relaxation time , the possibility of waning of individual biological immunity or escape mutations of the pathogen accumulated over certain (presumably, also long) time becomes a relevant effect. It can be incorporated as an additional relaxation term in Equation 5. The analysis of our equations, modified in this way, shows that the universal attractor leads to a fixed point corresponding to the endemic state. That point is located somewhat below the heterogeneity-modified HIT and characterized by a finite residual incidence rate and, respectively, by finite values of and . Here, is the susceptible population fraction in the endemic state, which is close to but somewhat higher than that at the onset of the herd immunity. A similar endemic steady state exists in most classical epidemic models (see Keeling and Rohani, 2011 and references therein). However, in those cases, epidemic dynamics would not normally lead to that point due to overshoot. Instead, these models typically predict a complete extinction of the disease when the prevalence drops below one infected individual. This may happen before herd immunity is lost due to waning biological immunity and/or replenishment of the susceptible population (e.g., due to births of immunologically naive individuals). That is not the case when time-dependent heterogeneity is included. Furthermore, in contrast to classical models, even in closed and reasonably small populations our mechanism would lead to an endemic state rather than pathogen extinction.

Note that for most pathogens the endemic point is not fixed, but instead is subjected to periodic seasonal forcing in . This leads to annual peaks and troughs in the incidence rate. Our model is able to describe this seasonal dynamics as well as the transition towards it for a new pathogen (see Figure 5). It captures the important qualitative features of seasonal waves of real pathogens, for example, the three endemic coronavirus families studied in Neher et al., 2020. They are (i) sharp peaks followed by a prolonged relaxation towards the annual minimum and (ii) a possibility of multi-annual cycles due to parametric resonance.

Figure 5.

Multiyear dynamics of a hypothetical new pathogen.

Effects of waning biological immunity with characteristic time years, and seasonal forcing are included (see Appendix 4 for details). In the case of persistent heterogeneity without temporal variations of social activity (brown solid line), the infection becomes extinct following the initial wave of the epidemic. In contrast, dynamic heterogeneity leads to an endemic state with strong seasonal oscillations (green line). Inset: the epidemic dynamics in the phase space. The black dotted line corresponds to the universal attractor trajectory, manifested, for example, as a plateau in green line in Figure 3b. The attractor leads to the endemic state (red point).

To understand the nature of the overall epidemic dynamics, we focus on the behavior of variables and . Their evolution is described by Equation 4, Equation 6 with playing the role of a driving force. As a result of depletion of the susceptible population, the driving force is gradually reduced, and the dynamics converges towards a slow evolution along the universal attractor shown as a black dotted trajectory in coordinates at the inset to Figure 5. For initial conditions away from that trajectory (say, , ), linear stability analysis indicates that the epidemic dynamics has a damped oscillatory behavior manifesting itself as a spiral-like relaxation towards the universal attractor. A combination of this spiral dynamics with a slow drift towards the endemic state gives rise to the overall trajectory shown as the solid green line in the inset to Figure 5. The periodic seasonal forcing generates a limit cycle about the endemic point (small green ellipse around the red point).

More generally, any abrupt increase of the effective reproduction number, for example, due to a relaxed mitigation, seasonal changes, etc., would shift the endemic fixed point up along the universal attractor. According to Equation 4; Equation 5; Equation 6 this will once again trigger a spiral-like relaxation. It will manifest itself as a new wave of the epidemic, such as the secondary waves in Figure 4b.

Application to COVID-19 in the USA

In addition to stochastic changes in social activity, multiple other factors are known to affect the epidemic dynamics: government-imposed mitigation, knowledge-based adaptation of social behavior, seasonal forces, vaccinations, emergence of new variants, etc. Constructing and calibrating a model taking into account all of these factors is well beyond the scope of this study. A principled way of integrating the effects of mitigation and knowledge-based adaptation is to use average mobility data. By their nature, these data capture population-wide trends in social activity, while averaging out individual-level stochasticity. In Figure 6, we show historic Google Mobility Data for retail and recreation in four major regions of the USA: Northeast, Midwest, South, and West (China Data Lab Dataverse, 2021). These data exhibit pronounced effects of government-imposed mitigation and knowledge-based adaptation of the population during spring-early summer of 2020. In contrast, there is only a modest and slow variation in the mobility from mid-July 2020 through mid-February 2021 (shaded area) across all four regions. This variation is generally consistent with regular seasonal effects and lacks any signs of the drastic and fast changes similar to those observed in the early stages of the epidemic. Hence, this time interval is optimal for testing the predictions of our theory without embarking on calibration of a full-scale active mitigation model. Furthermore, this time window also excludes the effects of mass vaccinations and the introduction of COVID-19 variants of concern (CDC, 2021), which became relevant after February/March 2021. Below we present a proof-of-principle demonstration that the progression of the COVID-19 epidemic from July 2020 until February 2021 in all four regions can indeed be well described by our theory.

Figure 6.

14-day moving average of Google Mobility Data (retail and recreation) in four US regions (China Data Lab Dataverse, 2021).

Note that the early epidemic was associated with wide and fast swings in mobility due to government-imposed mitigation and adaptive response of the population. In contrast, there is only modest and slow variation in the mobility from mid-July 2020 through mid-February 2021. This range of dates (shaded) is of interest since it allows one to directly test our theory without accounting for knowledge-based adaptation of the population.

The time dependence of daily deaths per capita (a reliable, albeit delayed measure proportional to the true attack rate) is shown in Figure 7b and c for each of the regions and fitted by our model with , days, , together with IFR assumed to be 0.5%. This IFR value was estimated by comparing reported COVID-related deaths in the USA to two independent seroprevalence surveys (Anand et al., 2020; Angulo et al., 2021). We assume that in the USA between June 2020 and February 2021 was affected primarily by seasonal dynamics. This is reflected in the simple mitigation profile shown in Figure 7 featuring a gradual seasonal increase of the reproduction number during the fall-winter period. Thus, this wave in each of the regions was triggered by the seasonal changes in transmission. According to our model, this wave was stabilized in mid-winter due to the population reaching the TCI state. There is a good agreement between our model and the empirical data for all four regions. Note that the shape of the seasonal epidemic wave is determined by the relative change of between summer and winter, or, equivalently, by the height of the peak itself. Analysis of Equation 4; Equation 5; Equation 6 shows that, for a given height, the peak is shaped by three underlying model parameters: , k0, and . Since one of them, , could not be determined from independent studies, we checked the sensitivity of our model to the choice of that timescale. It was found that the best-fit values of range from 20 to 55 days for different US regions, and that the overall agreement remains very good for any value within that range (see Appendix 5 for further details).

Figure 7.

Fitting of the empirical data on COVID-19 epidemic in Northeast (green), Midwest (blue), West (purple), and South (orange) of the USA.

The time range corresponds to the shaded region in Figure 6. The best-fit profiles of within this range (panel a) are shaped only by seasonal changes. The time dependence of daily deaths per capita for the Northeast and Midwestern regions of the USA (panel b) as well as for Southern and Western regions (panel c). Data points represent reported daily deaths per 100,000 of population for each of the regions. Solid lines are the best theoretical fits with our model (see Appendix 5 for details of the fitting procedure).

Finally, we performed a critical test of the predictive power of our theory. To do that, the empirical data in Midwest region have been fitted up to November 15, 2020, and the epidemic dynamics beyond that date has been projected by our SSA model. As shown in Figure 8, this procedure gives a very good prediction of the overall seasonal wave, based only on its onset behavior. In contrast, use of the traditional SIR model leads to an almost threefold overestimate of the height of the peak, with predicted timing about a month later than is observed. To fit the data with the standard SIR model, we forced at all times and set . The fitting procedure and the range of fitted dates for the SIR model was identical to that of the SSA model. We chose to show the Midwest region in the main text partly because a part of this region (the state of Illinois) was the subject of our previous publication (Wong et al., 2020). The fits to all four US regions are shown in Appendix 5—figure 2. The timing of the peak of the wave in all four regions is in closer agreement with the SSA model than the SIR model. The same is true for the height of the peak except for the South region, where it is somewhere in between the predictions of these two models.

Figure 8.

Test of the predictive power of the stochastic social activity (SSA) model developed in this work.

Daily deaths data in the Midwest region of the USA have been fitted up to November 17, 2020. The epidemic dynamic beyond that date has been projected by our model (blue). One observes a good agreement between this prediction and the reported data (crosses). In contrast, the classical susceptible-infected-removed (SIR) model (red) substantially overestimates the height of the peak and projects it at a much later date than had been observed. Solid lines represent the best-fit behavior for each of the models, while dotted lines indicate the corresponding 95% confidence intervals.

Appendix 5—figure 2.

Test of the predictive power of the stochastic social activity (SSA) model developed in this work.

Daily deaths data in each of four regions of the USA have been fitted up to December 13, 2020 (November 17, 2020, for the Midwest region). The epidemic dynamic beyond that date has been projected by our model (blue). One observes a good agreement between this prediction and the reported data (crosses). In contrast, the classical susceptible-infected-removed (SIR) model (red) substantially overestimates the height of the peak and projects it at a much later date than had been observed. Solid lines represent the best-fit behavior for each of the models, while dotted lines indicate the corresponding 95% confidence intervals.

Discussion

In conclusion, we have proposed a new theory integrating the stochastic dynamics of individual social activity into traditional epidemiological models. Our SSA model describes the so-called ‘zero intelligence’ limit in which there is no feedback from the epidemic dynamics to social activity, for example, mediated by the news. Hence, our approach is complementary to knowledge-based models of Epstein et al., 2008; Funk et al., 2009; Fenichel et al., 2011; Bauch, 2013; Rizzo et al., 2014; Weitz et al., 2020; Arthur et al., 2021. The SSA in our approach is described by the CIR model (Cox et al., 1985), which captures the following important properties: (i) the activity cannot be negative; (ii) for any given individual, it reverses towards its long-term average value; and (iii) it exhibits gamma-distributed short-term overdispersion (aka superspreading) (Lloyd-Smith et al., 2005; Endo et al., 2020; Sun et al., 2021). We mapped the overall epidemic dynamics featuring heterogeneous time-varying social activity onto a system of three differential equations, two of which generalize the traditional SIR model. The third equation describes the dynamics of the heterogeneity variable , driven up by the current strength of infection and relaxing back to zero due to variable social activity.

The emergent property of our theory is the new long timescale of the order of governing the relaxation towards either the herd immunity or the endemic state of the pathogen. For parameters relevant for COVID-19 epidemic, this timescale is approximately five times longer than the relaxation time constant for social activity . This emergent timescale might be of relevance to public health measures as it describes when the epidemic is reaching a sustainable plateau and for how long this plateau is expected to last.

The long-term dynamics of our model is in striking contrast to traditional epidemiological models, generally characterized by a large overshoot above the HIT leading to a likely extinction of new pathogens. Our theory provides a plausible explanation for the long plateaus observed in real-life epidemics such as COVID-19. It also provides a qualitative description of transient suppression of individual epidemic waves well below the HIT (Tkachenko et al., 2021). In particular, this mechanism explains how the winter 2020/21 waves of the COVID-19 epidemic in the USA were suppressed in the absence of a noticeable reduction in the population mobility.

Data availability

All code needed to reproduce results of our Agent Based Model and fits of the epidemic dynamics in US regions is available on Github at https://github.com/maslov-group/COVID-19-waves-and-plateaus (Maslov and Wang, 2021; copy archived at swh:1:rev:1e03ff622f16b85515e7162eab77ebd8e4efd30a).

This is an excellent and elegant example of what theory can do at its best in epidemiology: it takes a widely observed phenomenon that is an ‘embarrassment’ (my word) to current theories; proposes a parsimonious explanation that is plausible for the phenomenon by extending the existing theories in a specific way; and makes a plausible case for the importance of the mechanism in explaining key features of the data. In this case, the embarrassing phenomenon is long periods of very slowly changing incidence/prevalence, and the modification to theory is incorporation of dynamic social heterogeneity. This should stimulate much further work in the field. Congratulations to the authors.

Our editorial process produces two outputs: i) public reviews designed to be posted alongside the preprint for the benefit of readers; ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.

Decision letter after peer review:

Thank you for submitting your article "Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus and an endemic state" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Aleksandra Walczak as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Jennie Lavine (Reviewer #3).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1. From the editor: This seems to be a very important theoretical advance.

It is written like a physics paper, and will provide unnecessary (stylistic, terminologic) obstacles to infectious disease modelers trying to understand and build on it. Strongly encourage you to find someone in the field that will use this kind of work post-COVID and have them help you explain what you have done. Major examples:

– p. 6 is very hard going. What is an "immunity factor"? This is not a biology word. Do you really mean attack rate? That is not the same as the number of people infected per day, and the heterogeneity in it is just hard to understand. Please, for the sake of your citations and longevity of this paper, translate it into epidemiology with the help of someone who would be a reader in the more applied field.

2. Please clarify and strengthen the claims of explanatory power.

A 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 it is not clear this is a particularly convincing demonstration of the correctness of a model as an explanation for the observed dynamics. Official distancing policies may have monotonially 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 increasing 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 deliberate 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.

3. A further 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 is this idealization plausible, it would be valuable 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.

4. Line 48: It seems to me that the dynamic heterogeneity you incorporate does involve feedback from the current number of infections through the dependence of h(t) on J(t), which might act as a form of knowledge-based adaptation. Please explain this point and include a biological description of how you generated the h(t) term.

5. How sensitive are the qualitative results to different values of τs?

6. Line 68: DIV(2020) – this citation is not in the references. Given that you also cite this for the data you plot, please include more details on where the data come from.

7. Line 222: The emergent long time constant seems to depend only on τs and k0 – is that correct? I would have thought the relaxation might also be affected by how rapidly the disease spread (i.e., M*J). This time scale is interesting and of relevance to public health measures, as it suggests when we might be reaching a sustainable plateau. Can you explain this in more detail?

Reviewer #1 (Recommendations for the authors):

1. Formal analysis and interpretation should better link the main text and the appendix In general, I would encourage the authors to link their formal model analysis in the Appendix more explicitly to the main text results. When a result is presented in the main text, the reader should be pointed to its derivation or justification in the appendix.

Similarly, to aid the less mathematical reader, it would be nice to interpret equations like S2 somewhat more when they are stated. For S2, for example, one could point out that the C(τ)a(t) term reflects the individual's probability of infecting others conditional on having been infected τ units of time ago, while the a(t − τ)S(t − τ)J(t − τ) term reflects the probability that they were in fact infected τ units of time ago. With that in mind, I might adjust the notation slightly to highlight this by moving the J(t − τ) term into the average (though of course it can be factored out, as it does not depend on i) and grouping the two sets of terms in parentheses.

Moreover, having done the hard work of obtaining exact and/or approximate analytical results for their model, the authors should interpret these expressions more for the reader. e.g. the result about the HIT and λ in Equation 6 should be interpreted more in terms of the capacity for persistent heterogeneity to suppress the herd immunity threshold below the well-mixed case, and the contribution of even transient heterogeneity to determining the effective HIT.

2. Model definition. When introducing mathematical results and concepts in the main text, please make an explicit link to the corresponding Appendix derivations.

3. Code. In line with eLife guidelines, it would be good to provide the code used for model fitting, numerical solutions of differential equations, and stochastic simulations.

4. Undefined parameter γ Where is the parameter γ coming from in Equations S35 and subsequent? It is never defined. The other terms seem correct. Is this a holdover from a previous parametrization?

5. Model fitting. Model fitting procedures used to generate Figure 6 should be described in more detail in the appendix, and code should ideally be provided.

Reviewer #3 (Recommendations for the authors):

Line 48: It seems to me that the dynamic heterogeneity you incorporate does involve feedback from the current number of infections through the dependence of h(t) on J(t), which might act as a form of knowledge-based adaptation. Please explain this point and include a biological description of how you generated the h(t) term.

Line 68: DIV(2020) – this citation is not in the references. Given that you also cite this for the data you plot, please include more details on where the data come from.

How sensitive are the qualitative results to different values of τs?

Line 222: The emergent long time constant seems to depend only on τs and k0 – is that correct? I would have thought the relaxation might also be affected by how rapidly the disease spread (i.e., M*J). This time scale is interesting and of relevance to public health measures, as it suggests when we might be reaching a sustainable plateau. Can you explain this in more detail?

Essential revisions:

1. From the editor: This seems to be a very important theoretical advance.

It is written like a physics paper, and will provide unnecessary (stylistic, terminologic) obstacles to infectious disease modelers trying to understand and build on it. Strongly encourage you to find someone in the field that will use this kind of work post-COVID and have them help you explain what you have done. Major examples:

– p. 6 is very hard going. What is an "immunity factor"? This is not a biology word. Do you really mean attack rate? That is not the same as the number of people infected per day, and the heterogeneity in it is just hard to understand. Please, for the sake of your citations and longevity of this paper, translate it into epidemiology with the help of someone who would be a reader in the more applied field.

We now discuss the immunity factor together with other results from our previous paper (PNAS 2021) in the introduction section. All major concepts are defined so that readers don't need to refer to that earlier work. Also, to streamline the discussion on page 6 we moved the complex equation for the immunity factor to the Appendix.

We were able to get feedback on our manuscript from Neil Ferguson (Imperial College London) and addressed his comments in the revised version.

2. Please clarify and strengthen the claims of explanatory power.

A 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.

The epidemic dynamics is determined by the combination R0*M(t). The unmitigated R0 reveals itself only during the very early phases of the epidemics before any social distancing was in place. Therefore, within the time interval considered in our study we cannot separately determine R0 and M(t).

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 it is not clear this is a particularly convincing demonstration of the correctness of a model as an explanation for the observed dynamics. Official distancing policies may have monotonially 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 increasing 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 deliberate 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.

We agree with the referee that multiple factors affect the epidemic dynamic. These include: government imposed mitigation, knowledge-based adaptation of social behavior, seasonal forces, vaccination, emergence of new variants, etc. Constructing and calibrating a model taking into account all of these factors is well beyond the scope of this study focused on stochastic changes in social activity. As pointed out by the referee, a principled way of integrating effects of mitigation and knowledge-based adaptation is to use the average mobility data available from Google or other providers. By their nature, these data capture population-wide trends in social activity, while averaging out individual level stochasticity. In this work, we take advantage of the observation that the average mobility across the US (see the new Figure 6) remained remarkably steady during the period considered in our study (July 2020- February 2021). Hence, this time interval is optimal for testing the predictions of our theory without embarking on calibration of a full scale multi-parameters model.

Another major addition to our analysis in the revised version of the manuscript is the new Figure 8 (as well as Appendix 5. Figure 2) comparing the predictions of our model with the classical SIR model. As one can see from this figure, our model is capable of predicting the shape of the epidemic wave based on the data covering its early stages. This should be contrasted with the predictions of the SIR model dramatically overshooting the observed maximum of daily deaths. This is consistent with the Transient Collective Immunity proposed and analyzed in this study as well as our earlier publication (Tkachenko et al. PNAS (2021)).

3. A further 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 is this idealization plausible, it would be valuable 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 email (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 αi with the dispersion parameter κ.

4. Line 48: It seems to me that the dynamic heterogeneity you incorporate does involve feedback from the current number of infections through the dependence of h(t) on J(t), which might act as a form of knowledge-based adaptation. Please explain this point and include a biological description of how you generated the h(t) term.

As now explained in the text (see the caption to Figure 2), this feedback mechanism is due to the selective removal of susceptibles with high current levels of social activity in the course of the epidemic. Therefore, it does not involve any knowledge-based adaptation, defined as modulation of average social activity in response to the perceived danger of the current level of infection.

5. How sensitive are the qualitative results to different values of τs?

We conducted the sensitivity analysis with respect to τs and presented the results of this analysis in the main text as well as in a new section in Appendix 5 (see Appendix 5. Figure 2).

6. Line 68: DIV(2020) – this citation is not in the references. Given that you also cite this for the data you plot, please include more details on where the data come from.

Done.

7. Line 222: The emergent long time constant seems to depend only on τs and k0 – is that correct? I would have thought the relaxation might also be affected by how rapidly the disease spread (i.e., M*J). This time scale is interesting and of relevance to public health measures, as it suggests when we might be reaching a sustainable plateau. Can you explain this in more detail?

We have substantially expanded the explanation of the origin of this emergent timescale and its implications for public policy measures. Furthermore, we have better explained its derivation in Appendix 3 (see the red text in the newly formed section "Long-term plateau dynamics"). The editor is correct in pointing out that this new timescale () is affected by the basic reproduction number corrected by the mitigation: M*R0. This dependence is explicitly presented in Appendix 3 (see Equation S42). However, this dependence is relatively weak. For COVID-19, we estimated this timescale to be approximately 5* τs. Also, there is no dependence of this timescale on the current strength of the epidemic M*J, since the latter is itself exponentially decaying with the same time constant (unless waning of biological immunity is taken into account).

Reviewer #1 (Recommendations for the authors):

1. Formal analysis and interpretation should better link the main text and the appendix In general, I would encourage the authors to link their formal model analysis in the Appendix more explicitly to the main text results. When a result is presented in the main text, the reader should be pointed to its derivation or justification in the appendix.

Similarly, to aid the less mathematical reader, it would be nice to interpret Equations like S2 somewhat more when they are stated. For S2, for example, one could point out that the C(τ)ai(t) term reflects the individual's probability of infecting others conditional on having been infected τ units of time ago, while the ai(t − τ)Si(t − τ)J(t − τ) term reflects the probability that they were in fact infected τ units of time ago. With that in mind, I might adjust the notation slightly to highlight this by moving the J(t − τ) term into the average (though of course it can be factored out, as it does not depend on i) and grouping the two sets of terms in parentheses.

We corrected the text as suggested and added explanations of individual terms in Equation S2.

Moreover, having done the hard work of obtaining exact and/or approximate analytical results for their model, the authors should interpret these expressions more for the reader. e.g. the result about the HIT and λ in Equation 6 should be interpreted more in terms of the capacity for persistent heterogeneity to suppress the herd immunity threshold below the well-mixed case, and the contribution of even transient heterogeneity to determining the effective HIT.

In response to suggestion of other referees and our editor we decided to remove the Equation 6 from the main text. In addition we expanded the discussion of the results from our earlier publication (PNAS 2021) in the introduction. We specifically focused on λ and its impact on HIT.

2. Model definition. When introducing mathematical results and concepts in the main text, please make an explicit link to the corresponding Appendix derivations.

We completely restructured the Appendix by breaking it down into several independent "boxes" with titles aligned to the main text of the article. Furthermore, these separate Appendix boxes are now properly referenced in the main text greatly facilitating the search of the corresponding mathematical derivations.

3. Code. In line with eLife guidelines, it would be good to provide the code used for model fitting, numerical solutions of differential equations, and stochastic simulations.

Done. Our code for the Agent Based Model and the US regions fits were added to the public Github repository: https://github.com/maslov-group/COVID-19-waves-and-plateaus/.

4. Undefined parameter γ Where is the parameter γ coming from in Equations S35 and subsequent? It is never defined. The other terms seem correct. Is this a holdover from a previous parametrization?

The parameter γ was indeed a holdover from a previous parameterization. It has now been replaced with 1/τg.

5. Model fitting. Model fitting procedures used to generate Figure 6 should be described in more detail in the appendix, and code should ideally be provided.

Our fitting procedures are now explained in Appendix 5 and the Matlab code for generating the fits is available on Github.

Reviewer #3 (Recommendations for the authors):

Line 48: It seems to me that the dynamic heterogeneity you incorporate does involve feedback from the current number of infections through the dependence of h(t) on J(t), which might act as a form of knowledge-based adaptation. Please explain this point and include a biological description of how you generated the h(t) term.

As now explained in the text (see the caption to Figure 2), this feedback mechanism is due to the selective removal of susceptibles with high current levels of social activity in the course of the epidemic. Therefore, it does not involve any knowledge-based adaptation defined as modulation of average social activity in response to the perceived danger of the current level of infection.

Line 68: DIV(2020) – this citation is not in the references. Given that you also cite this for the data you plot, please include more details on where the data come from.

Corrected.

How sensitive are the qualitative results to different values of τs?

We conducted the sensitivity analysis by varying τs from 15 days to 90 days and presented the results of this analysis in a new supplementary figure.

Line 222: The emergent long time constant seems to depend only on τs and k0 – is that correct? I would have thought the relaxation might also be affected by how rapidly the disease spread (i.e., M*J). This time scale is interesting and of relevance to public health measures, as it suggests when we might be reaching a sustainable plateau. Can you explain this in more detail?

We have substantially expanded the explanation of the origin of this emergent timescale and its implications for public policy measures. Furthermore, we have better explained its derivation in the Appendix 3 (see the red text in the newly formed section "Long-term plateau dynamics"). The editor is correct in pointing out that this new timescale (tilde tau) is affected by the basic reproduction number corrected by the mitigation: M*R0. This dependence is explicitly presented in the Appendix 3 (see Equation S42). However, this dependence is relatively weak. For COVID-19 we estimated this timescale to be approximately 5* τs. Also, there is no dependence of this timescale on the current strength of the epidemic M*J, since the latter is itself exponentially decaying with the same time constant (unless waning of biological immunity is taken into account).

Appendix 5—table 1.

The set of fixed model parameters used throughout this study.

k 0	κ	1/γ	τ_s	τ_b	IFR	Δ
0.4	2	5 days	30 days	1 year	0.5%	30 days

Appendix 5—table 2.

The best-fit values for seven parameters in each of the four US regions.

The fits were made using MATLAB R2021a nonlinear least-squares regression function (nlinfit)

US region	j 0	R 1	t spring	R 2	R_summer	R_winter	t fall
Northeast	0.0014	1.14	26 Mar 2020	1.25	1.10	1.46	06 Oct 2020
Midwest	0.00036	0.95	20 Mar 2020	1.07	1.04	1.34	29 Sep 2020
South	0.00014	0.91	24 Mar 2020	1.43	1.04	1.33	07 Nov 2020
West	0.00015	0.97	15 Mar 2020	1.27	0.98	1.29	25 Oct 2020

Appendix 5—table 3.

The best-fit values of in each of the four US regions.

The fits were made using MATLAB R2021a nonlinear least-squares regression function (nlinfit).

	Northeast	Midwest	South	West
Best fit (days)	19	39	33	55
Lower 95% CI (days)	17	36	27	49
Upper 95% CI (days)	20	42	38	61