Susceptible–Exposed–Infectious–Removed Model with Short-Term Awareness of Risk.

Consider a susceptible–exposed–infectious–removed (SEIR)-like model,

Fig. 2.

Schematic of an SEIR model with awareness-driven social distancing. Transmission is reduced based on short- and/or long-term awareness of population-level disease severity (i.e., fatalities).

Table 1.

Parameter descriptions and values/ranges used for simulations

Notation	Description	Values/ranges
$$ \beta $$	Transmission rate	1/2 $$ {\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$$
$$ 1/\mu $$	Mean latent period	2 d
$$ 1/\gamma $$	Mean infectious period	6 d
$$ 1/{\gamma }_H$$	Mean time in a hospital stay before a fatality	7 to 28 d
$$ f_D$$	Infection fatality probability	0.01
$$ N$$	Population size	$$ 10^7$$
$$ N{\delta }_c$$	Half-saturation constant	5 to 500
	for short-term awareness	deaths days−1
$$ N{D}_c$$	Half-saturation constant	2,500 to 10,000
	for long-term awareness	deaths
$$ k$$	Sharpness of change in the force of infection	1 to 4
$$ ϵ$$	Time scale of behavior change	1/7 $$ {\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}}^{-1}$$

Transmission rate is chosen to match $$ {\mathcal{R}}_0=3$$.

Uncontrolled epidemics in SEIR models have a single case peak, corresponding to the point where $$ \gamma I=\beta SI$$ such that the population obtains herd immunity when only a proportion $$ S=1/{\mathcal{R}}_0$$ have yet to be infected. However, in the model above, individuals decrease transmission in response to awareness of the impacts of the disease, $$ \delta \left(t\right)$$. In this case, infected cases can peak even when the population is far from herd immunity, specifically when

We evaluate this hypothesis in Fig. 3 for $$ k=1$$, $$ k=2$$, and $$ k=4$$ given disease dynamics with $$ \beta =0.5$$ per d, $$ \mu =1/2$$ per d, $$ \gamma =1/6$$ per d, $$ f_D=0.01$$, $$ N=10^7$$, and $$ N{\delta }_c=50$$ per d. As is evident, the rise and decline from peaks are not symmetric. Instead, incorporating awareness leads to dynamics where incidence decreases very slowly after a peak. The peaks occur at levels of infection far from that associated with herd immunity. Post-peak, shoulders and plateaus emerge because of the balance between relaxation of awareness-based distancing (which leads to increases in cases and deaths) and an increase in awareness in response to increases in cases and deaths. As the steepness of response $$ k$$ increases, individuals become less sensitive to fatality rates where $$ \delta <{\delta }_c$$ and more sensitive to fatality rates where $$ \delta >{\delta }_c$$. This leads to sharper dynamics. In addition, infections can overshoot the expected plateau given that awareness is driven by fatalities which are offset with respect to new infections.

Fig. 3.

Infections and deaths per day in a death-awareness–based social distancing model. Simulations have the epidemiological parameters $$ \beta =0.5$$ days⁻¹, $$ \mu =1/2$$ days⁻¹, $$ \gamma =1/6$$ days⁻¹, and $$ f_D=0.01$$, with variation in $$ k=1$$, 2, and 4. We assume $$ N{\delta }_c=50$$ per d in all cases.

Short-Term Awareness, Long-Term Plateaus, and Oscillations.

Initial analysis of an SEIR model with short-term awareness of population-level severity suggests a generic outcome: Fatalities will first increase exponentially before slowing to a plateau at a level near $$ {\delta }_c$$. Fig. 4 shows dynamics for values of $$ {\delta }_c$$ ranging from to 5 to 500 deaths per day in a population of $$ 10^7$$ (here $$ k=2$$; results for $$ k=1$$ or $$ k=4$$ are similar; SI Appendix, Fig. S2). When $$ {\delta }_c$$ is small compared to $$ \left(\gamma f_D\right)$$, fatalities can be sustained at near-constant levels for a long time. When $$ {\delta }_c$$ is higher, then the decline of cases and fatalities due to susceptible depletion is relatively fast. However, over a wide range of assumptions about critical daily fatality rates $$ {\delta }_c$$, the population remains largely susceptible, even as sustained fatalities continue for a period far greater than the time it took to reach the plateau. To explore the impacts of lags on dynamics, we incorporated an additional class $$ H$$, assuming that fatalities follow potentially prolonged hospital stays. We do not include detailed information on symptomatic transmission, asymptomatic transmission, hospitalization outcome, age structure, and age-dependent risk (as in ref. 3). Instead, we consider the extended SEIR model,

Fig. 4.

Dynamics given variation in the critical fatality awareness level, $$ {\delta }_c$$, for awareness $$ k=2$$. Shown are deaths per d (Top) and the susceptible fraction as a function of time (Bottom), the latter compared to a herd immunity level when only a fraction $$ 1/{\mathcal{R}}_0$$ remain susceptible. These simulations share the epidemiological parameters $$ \beta =0.5$$ days⁻¹, $$ \mu =1/2$$ days⁻¹, $$ \gamma =1/6$$ days⁻¹, and $$ f_D=0.01$$.

The earlier analysis of the quasi-stationary equilibrium in fatalities holds in the case of an SEIR model with additional classes before fatalities. Hence, we anticipate that dynamics should converge to $$ \delta ={\delta }^{\left(q\right)}$$ at early times. However, increased delays between cases and fatalities could lead to oscillations. Indeed, this is what we find via examination of models in which $$ T_H$$ ranges from 7 d to 28 d, with increasing magnitude of oscillations as $$ T_H$$ increases (see Fig. 5 for $$ k=2$$, with qualitatively similar results for $$ k=1$$ and $$ k=4$$ shown in SI Appendix, Fig. S3).

Fig. 5.

Emergence of oscillatory dynamics in a death-driven awareness model of social distancing given lags between infection and fatality. Awareness is $$ k=2$$, and all other parameters are as in Fig. 3. The dashed lines are for fatalities expected quasi-stationary value $$ {\delta }^{\left(q\right)}$$.

Dynamical Consequences of Short-Term and Long-Term Awareness.

Awareness can vary in duration, for example, awareness of severe acute respiratory syndrome coronavirus 2 may prepare individuals to more readily adopt and retain social distancing measures (16, 17). In previous work, long-term awareness of cumulative incidence was shown to lead to substantial decreases in final size of epidemics compared to baseline expectations from inferred strength (14). Hence, we consider an extension of the SEIR model with lags between infection and fatalities that incorporates both short-term and long-term awareness,

Fig. 6.

SEIR dynamics with short- and long-term awareness. Model parameters are $$ \beta =0.5$$ days⁻¹, $$ \mu =1/2$$ days⁻¹, $$ \gamma =1/6$$ days⁻¹, $$ T_H=14$$ d, $$ f_D=0.01$$, $$ N=10^7$$, $$ k=2$$, and $$ N{\delta }_c=50$$ per day (short-term awareness), with varying $$ N{D}_c$$ (long-term awareness) as shown in the legend. The dashed line (Top panel) denotes $$ {\delta }^{\left(q\right)}$$ due to short-term distancing alone.

Empirical Assessment of Mechanistic Drivers of Asymmetric Peaks in COVID-19 Death Rates.

The models developed here suggest that awareness-driven distancing can lead to asymmetric epidemic peaks even in the absence of susceptible depletion. To test this hypothesis mechanistically, we jointly analyzed the dynamics of fatality rates and behavior, using mobility data obtained from Google COVID-19 Community Mobility Reports (https://www.google.com/covid19/mobility/) as a proxy for behavior (see Methods for the aggregation of multiple mobility metrics via a principal component analysis [PCA]). Notably, we find that, in the bulk of states examined, aggregated rates of mobility typically began to increase before the local peak in fatality was reached (Fig. 7A). This rebound in mobility rates implies that real populations are opening up faster than our simple model could predict. Awareness-driven models, shown in Fig. 7B, show either “reversible” or “counterclockwise” dynamics. In these models, risky behavior decreases until fatalities reach their peak. Models with short-term awareness but no long-term awareness exhibit a tight link between fatality and behavior (reversible behavior, like the top curve in Fig. 7B). Models with long-term awareness exhibit counterclockwise dynamics, as risky behavior remains at low levels even as fatalities decrease; the asymmetry here is driven by the extent of long-term awareness.

Fig. 7.

Phase plane visualizations of deaths vs. mobility for (A) state-level data and (B–D) SEIR models. (A) Deaths and mobility indexes through time for the 17 analyzed states. Both data series are smoothed. Time windows as in Fig. 1. Here, the first mobility principal component represents a proxy for behavior, with positive values associated with higher mobility (see Methods). (B) Dynamics of effective behavior and death rates in an SEIR model with short- and long-term awareness. Curves denote different assumptions regarding long-term awareness, in each case $$ \beta =0.5$$ days⁻¹, $$ \mu =0.5$$ days⁻¹, and $$ \gamma =1/6$$ days⁻¹, such that $$ {\mathcal{R}}_0=3$$, with $$ k=2$$, $$ {\gamma }_H=1/21$$ days⁻¹, and $$ f_D=0.01$$. The short-term awareness corresponds to $$ N{\delta }_c=50$$ deaths per day. Thin lines denote full dynamics over 400 d; thick lines denote the dynamics near the case fatality peak. (C) Dynamics of effective behavior and death rates in an SEIR model with awareness and fatigue. The three different curves denote different assumptions regarding long-term awareness; in each case, $$ \beta =0.5$$ days⁻¹, $$ \mu =0.5$$ days⁻¹, $$ \gamma =1/6$$ days⁻¹, such that $$ {\mathcal{R}}_0=3$$, with $$ k=2$$, $$ {\gamma }_H=1/21$$ days⁻¹, $$ f_D=0.01$$, and $$ ϵ=1/7$$ days⁻¹. The short-term awareness corresponds to $$ N{\delta }_c=50$$ deaths per day. The force of infection does not include long-term changes in behavior beyond mobility, that is, $$ g\left(D\right)=1$$. (D) As in C, but the force of infection includes long-term changes in behavior, that is, $$ g\left(D\right)=1/\left(1+{\left(D/D_c\right)}^k\right)$$.

In contrast, the real data (Fig. 7A) exhibit predominantly clockwise dynamics; exceptions include New York, which has a nearly reversible (but still clockwise) pattern, and Washington, which has a counterclockwise pattern anticipated by the awareness model. We hypothesized that a combination of awareness-driven distancing and fatigue could lead to clockwise dynamics: If people become fatigued with distancing behavior, then risk could rise even as deaths were rising. We developed the following model as a proof of concept in which fatigue is driven directly by deaths (although alternatives could also be explored linked to cases, hospitalizations, deaths, and/or a combination):

In this model with fatigue, the force of infection is related to the mobility denoted by $$ \beta \left(t\right)$$ (which dictates the number of interactions per unit time) modulated by a reduction in risk per infection $$ g\left(D\right)$$. In this model, $$ \widehat{\beta }$$ denotes the baseline behavior, and $$ ϵ$$ denotes a time scale for behavior change. The level of fatigue is controlled by $$ D_c$$, such that mobility returns to a baseline $$ \widehat{\beta }$$ once $$ D\gg D_c$$. We consider two models, corresponding to $$ g\left(D\right)=1$$ such that the force of infection depends on mobility alone, and $$ g\left(D\right)=1/\left(1+{\left(D/D_c\right)}^k\right)$$ corresponding to sustained changes in the risk of infection per contact (e.g., due to mask wearing, contact-less interactions, use of personal protective equipment, etc.). As shown in Fig. 7 C and D, the dynamics switch from counterclockwise to clockwise in the $$ \delta -\beta $$ plane given the incorporation of fatigue. Deaths drive down mobility, but, eventually, decreases in $$ \beta $$ due to short-term awareness are overcome by fatigue, leading to increases in $$ \beta $$. If $$ g\left(D\right)=1$$, then the dynamics include increases in both mobility and fatalities akin to levels expected in the absence of behavior, and, eventually, levels of infection that are stopped by herd immunity, rather than by awareness (Fig. 7C). In contrast, if there is sustained behavior change such that $$ g\left(D\right)$$ decreases with increasing cumulative deaths, then there is a single peak that forms a clockwise loop, with the peak close to, but after the minimum in behavior (Fig. 7D), as observed in nearly all state-level data sets.

Epidemiological Data.

Daily number of reported deaths as of June 7, 2020 was obtained from The COVID Tracking Project (https://covidtracking.com).

Mobility Data.

Mobility data as of June 12, 2020 were obtained from Google COVID-19 Community Mobility Reports (https://www.google.com/covid19/mobility/). The dataset describes percent changes in mobility across six categories (grocery and pharmacy; parks; residential; retail and recreation; transit; and workplaces) compared to the median value from the 5-wk period January 3 to February 6, 2020. Raw mobility data are plotted in SI Appendix, Fig. S4.

PCA.

We use PCA on the mobility data to obtain a univariate index of mobility. We exclude park visits from the analysis, due to their anomalous, noisy patterns (SI Appendix, Fig. S4). Before performing PCA, we first calculate the 7-d rolling average for each mobility measure in order to remove the effects of weekly patterns. We combined mobility data from all 17 analyzed states, and standardized each measure (to zero mean and unit variance). The first principal component explains 93% of the total variance in this analysis, and the loading of the residential metric had a different sign from the other four mobility metrics. We thus used this component as our index of mobility (setting the direction so that only the residential metric contributed negatively to the index). To draw phase planes, we further smoothed our mobility index and daily reported deaths using locally estimated scatterplot smoothing (LOESS). Daily number of deaths is smoothed in log space, only including days with one or more reported deaths. LOESS is performed by using the loess function in R.