Moving Beyond a Peak Mentality: Plateaus, Shoulders, Oscillations and Other‘Anomalous’ Behavior-Driven Shapes in COVID-19 Outbreaks

Joshua S. Weitz,1, 2, 3, ∗ Sang Woo Park,4 Ceyhun Eksin,5 and Jonathan Dushoff6, 7

1 School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA, USA2 School of Physics, Georgia Institute of Technology, Atlanta, GA, USA

3 Center for Microbial Dynamics and Infection, Georgia Institute of Technology, Atlanta, GA, USA4 Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA5 Department of Industrial and Systems Engineering, Texas A&M, College Station, Texas, USA

6 Department of Biology, McMaster University, Hamilton, ON, Canada7 DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton, ON, Canada

(Dated: May 1, 2020)

The COVID-19 pandemic has caused more than 200,000 reported deaths globally, of which morethan 50,000 have been reported in the United States. Public health interventions have had signif-icant impacts in reducing transmission and in averting even more deaths. Nonetheless, in manyjurisdictions (both at national and local levels) the decline of cases and fatalities after apparent epi-demic peaks has not been rapid. Instead, the asymmetric decline in cases appears, in some cases, tobe consistent with plateau- or shoulder-like phenomena. Here we explore a model of fatality-drivenawareness in which individual protective measures increase with death rates. In this model, epidemicdynamics can be characterized by plateaus, shoulders, and lag-driven oscillations after exponentialrises at the outset of disease dynamics. We also show that incorporating long-term awareness canavoid peak resurgence and accelerate epidemic decline. We suggest that awareness of the severity ofthe short- and long-term epidemic is likely to play a critical role in disease dynamics, beyond thatimposed by intervention-driven policies.

The spread of COVID-19 has elevated the importanceof epidemiological models as a means to forecast bothnear- and long-term spread. In the United States, theInstitute for Health Metrics and Evaluation (IHME)model has emerged as a key influencer of state- andnational-level policy [1]. The IHME model includes adetailed characterization of the variation in hospital bedcapacity, ICU beds, and ventilators between and with-in states. Predicting the projected strains on underlyinghealth resources is critical to supporting planning efforts.However such projections require an epidemic ‘forecast’.The IHME’s epidemic forecast differs from conventionalepidemic models in a significant way – IHME assumesthat the cumulative deaths in the COVID-19 epidemicfollow a symmetric, Gaussian-like trajectory. For exam-ple, the IHME model predicts that if the peak is 2 weeksaway then in 4 weeks cases will return to the level ofthe present, and continue to diminish rapidly. But, epi-demics need not have one symmetric peak – the archaicFarr’s Law of Epidemics notwithstanding (see [2] for acautionary tale of using Farr’s law as applied to the HIVepidemic).

Conventional epidemic models represent populationsin terms of their ‘status’ vis a vis the infectious agent,in this case SARS-CoV-2 (e.g., [3–9]), e.g., susceptible,exposed, infectious, hospitalized, and recovered. Newtransmission can lead to an exponential increases in cas-es when the basic reproduction number R0 > 1 (the

∗Electronic address: jsweitz@gatech.edu; URL: http://

ecotheory.biology.gatech.edu

basic reproduction number denotes the average numberof new infections caused by a single, typical individualin an otherwise susceptible population [10]). Subsequentspread, if left unchecked, would yield a single peak – intheory. That peak corresponds to when ‘herd immunity’is reached, such that the effective reproduction number,Reff = 1. The effective reproduction number denotes thenumber of new infectious cases caused by a single infec-tious individual in a population with pre-existing circu-lation. But, even when herd immunity is reached, therewill still be new cases which then diminish over time,until the epidemic concludes. A single-peak paradigm isonly robust insofar as the disease has spread sufficientlyin a population to reach and exceed ‘herd immunity’. Theconverse is also true in the case of COVID-19 – as longas a population remains predominantly immunologicallynaive, then the risk of further infection has not passed.

The Imperial College of London (ICL) model [3] is oneof the most influential of epidemiogical models shapingpublic health responses to COVID-19. The ICL modelis an example of a ‘conventional’ epidemic model thatshows the benefits of early intervention steps in reducingtransmission and preserving health system resources vs. a‘herd immunity’ strategy. The ICL model assumes thattransmission is reduced because of externalities, like lock-downs, school closings, and so on. As a result, the ICLmodel suggests that lifting of large-scale public healthinterventions could be followed by a second wave of cas-es. Yet, for a disease that is already the documentedcause of more than 50,000 deaths in the United States,we posit that individuals are likely to continue to modi-fy their behavior even after lockdowns are lifted. Hence,here, we use a simple model to ask the question: what is

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the anticipated shape of an epidemic if individuals mod-ify their behavior in direct response to the impact of adisease at the population level? In doing so, we buildupon earlier work on awareness based models (e.g. [11–14]) with a simple, initial assumption: individuals reduceinteractions when death rates are high and increase inter-actions when death rates are low.

To begin, consider a SEIR like model

S = − βSI[1 + (δ/δc)

k] (1)

E =βSI[

1 + (δ/δc)k] − µE (2)

I = µE − γI (3)

R = (1− fD)γI (4)

D = fDγI (5)

where S, E, I, R, and D denote the proportions ofsusceptible, exposed, infectious, recovered, and deaths,respectively. The awareness-based distancing is con-trolled by the death rate δ ≡ D, the half-saturation con-stant (δc > 0), and the sharpness of change in the force ofinfection (k ≥ 1). Since δ is proportional to I, this modelis closely related to a recently proposed awareness-baseddistancing model [14]. Note that the present model con-verges to the conventional SEIR model as δc →∞.

Typically, epidemics arising in SEIR models have a sin-gle case peak, corresponding to the point where γI = βSIsuch that S = 1/R0, equivalent to when the herd immu-nity level proportion of individuals 1 − 1/R0 have beeninfected. However, when individuals decrease transmis-sion in relationship to awareness of the current severity ofthe disease, δ(t), then the system can ‘peak’ when levelsof infected cases are far from herd immunity, specificallywhen

γI =βSI[

1 + (δ/δc)k] . (6)

When δc is small compared to the death rate of infectiousindividuals (γfD) we anticipate that individual behaviorwill respond quickly to the disease outbreak. Hence, wehypothesize that the emergence of an awareness-basedpeak can occur early, i.e., S(t) ≈ 1, consistent with aquasi-stationary equilibrium when the death rate is

δ(q) ≈ δc (R0 − 1)1/k

(7)

and the infection rate is

I(q) ≈ δcfD

(R0 − 1)1/k

. (8)

These early onset peak rates should arise not because ofherd immunity but because of changes in behavior.

We evaluate this hypothesis in Figure 1 for k = 1,k = 2, and k = 4 given disease dynamics with β = 0.5

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FIG. 1: Infections and deaths per day in a death-awarenessbased social distancing model. Simulations have the epidemi-ological parameters β = 0.5 /day, µ = 1/2 /day, γ = 1/6/day,and fD = 0.01, with variation in k = 1, 2 and 4.

/day, µ = 1/2 /day, γ = 1/6 /day, fD = 0.01, N = 107,and Nδc = 50 /day. As is evident, the rise and declinefrom peaks are not symmetric. Instead, increasing non-linearity of awareness a leads to shoulders where inci-dence decreases very slowly after a peak. We interpretthis finding to mean that as the awareness exponentk increases, individuals become less sensitive to fatal-ity rates where δ < δc and more sensitive to fatalityrates where δ > δc. The shoulders and plateaus emergebecause of the balance between relaxation of awareness-based distancing (which leads to increases in cases anddeaths) and an increase in awareness in response toincreases in cases and deaths.

These results suggest a generic outcome: first fatalitieswill grow exponential before plateauing near to the fatal-ity awareness level δc. In the event that δc/(γfD) is suf-ficiently high then susceptible depletion will lead to thedecline of cases and fatalities. Figure 2 shows the resultsof dynamics given δc values over a range equivalent to 5

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FIG. 2: Dynamics given variation in the critical fatalityawareness level, Dc for awareness k = 2. Panels showdeaths/day (top) and the susceptible fraction as a functionof time (bottom), the latter compared to a herd immunitylevel when only S = 1/R0 remain. These simulations sharethe epidemiological parameters β = 0.5 /day, µ = 1/2 /day,γ = 1/6 /day, and fD = 0.01.

to 500 deaths/day given a population of 107 for k = 2 (wenote that results for k = 1 and k = 4 lead to similar find-ings, and are included in the github repository). We findthat fatalities can be sustained at near-constant levels forlow vallues of δc (top) even as the population remainssusceptible at levels far above herd immunity (bottom).We also observe that as k increases, then fatalities mayovershoot the plateau. Overshoots arise because individ-uals initiate protective measures closer to when a criticalfatality rate has been reached. These overshoots maylead to oscillatory dynamics when there are larger lagsbetween new cases and fatalities.

To explore the impacts of lags on dynamics, we incor-porated an additional class H, assuming that fatali-ties follow potentially prolonged hospital stays. We do

not include explicit detailed information on symptomat-ic transmission, asymptomatic transmission, hospitaliza-tion outcome, age structure, and age-dependent risk (asin [3]). Instead, consider the extended SEIR model:

S = − βSI[1 + (δ/δc)

k] (9)

E =βSI[

1 + (δ/δc)k] − µE (10)

I = µE − γI (11)

R = (1− fD)γI (12)

H = fDγI − γHH (13)

D = γHH (14)

where TH = 1/γH defines the average time in a hos-pital stay before a fatality. The earlier analysis of thequasi-stationary equilibrium in fatalities holds; hence weanticipate that dynamics should converge to δ = δ(q) atearly times. However, increased delays between cases andfatalities could lead to oscillations in both. Indeed, thisis what we find via examination of models in which THranges from 7 to 35 days, with increasing magnitude ofoscillations as TH increases (see Figure 3 for k = 2 withqualitatively similar results for k = 1 and k = 4 on thegithub).

Finally, we recognize that awareness can vary in dura-tion. In previous work, long-term awareness of cumula-tive incidence was shown to lead to substantial decreasesin final size of epidemics compared to baseline expecta-tions from inferred strength [14]. Hence, here we consideran extension of the SEIR model with lags between infec-tion and fatalities that incorporates both short-term andlong-term awareness:

S = − βSI[1 + (δ/δc)

k+ (D/Dc)

k] (15)

E =βSI[

1 + (δ/δc)k

+ (D/Dc)k] − µE (16)

I = µE − γI (17)

R = (1− fD)γI (18)

H = fDγI − γHH (19)

D = γHH (20)

where Dc denotes a critical cumulative fatality level.Note that the relative importance of short- and long-termawareness can be modulated by δc and Dc respectively.Figure 4 shows cumulative fatalities (left) and daily fatal-ities (right) for a SEIR model with R0 = 2.5, TH = 14days, and Nδc = 50 fatalities per day and critical cumu-lative fatalities ofNDc = 2, 500, 5,000, 10,000 as well as acomparison case with vanishing long-term awareness. Asis evident, long-term awareness drives dynamics towards

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FIG. 3: Emergence of oscillatory dynamics in a death-drivenawareness model of social distancing given lags between infec-tion and fatality. Awareness is k = 2 and all other parametersas in Figure 2. The dashed lines for fatalities expected quasi-stationary value δ(q).

rapid declines after reaching a peak. This decline aris-es because D monotonically increases; increasing fatal-ities beyond Dc leads to rapid suppression of transmis-sion. However, when δc rather than Dc drives dynamics,then shoulders and plateaus can re-emerge. In reality, weexpect that individual behavior is shaped by short- andlong-term awareness of risks, including the potential for‘decay’ of long-term awareness [11, 12].

In summary, we have shown how awareness of disease-induced death can reduce transmission and also lead tohighly asymmetric epidemic curves, where the epidemicdeclines slowly even as the majority of the populationremains susceptible. In these conditions, if individualsare unable to sustain social distancing policies, or beginto tolerate higher death rates, then cases could increase.Hence: passing a ‘peak’ need not imply the rapid declineof risk. These types of impacts of awareness-drivenendogenous changes in Reff are typically absent in mod-els that form the basis for public policy and strate-gic planning. Moving forward, we hope that our find-ings highlight the impacts of short-term and long-termawareness in efforts to shape information campaigns toreduce transmission after early onset ‘peaks’, particu-larly when populations remain predominantly immuno-logically naive. Although the models here are inten-

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FIG. 4: SEIR dynamics with short- and long-term awareness.Model parameters are β = 0.5 /day, µ = 1/2 /day, γ = 1/6/day, TH = 14 days, fD = 0.01, N = 107, k = 2, Nδc = 50/day (short-term awareness), with varying NDc (long-termawareness) as shown in the legend. The dashed line (top)

denotes δ(q) due to short-term distancing alone.

tionally simple, we contend that as cumulative datafrom COVID-19 outbreaks already indicate, the asym-metric post-peak dynamics of COVID-19, including slowdeclines and plateau-like behavior, may be an emergentproperty of awareness-driven epidemiological dynamics.

Data availability: All simulation and codes used in thedevelopment of this manuscript are available at https://github.com/jsweitz/covid19-git-plateaus.

Acknowledgements: Research effort by JSW wasenabled by support from grants from the Simons Foun-dation (SCOPE Award ID 329108), the Army ResearchOffice (W911NF1910384), National Institutes of Health(1R01AI46592-01), and National Science Foundation(1806606 and 1829636). JD was supported in part bygrants from the Canadian Institutes of Health Researchand the Natural Sciences and Engineering ResearchCouncil of Canada.

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