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How To Go Viral: A COVID-19 Model with
Endogenously Time-Varying Parameters
WP 20-10
Paul Ho
Federal Reserve Bank of Richmond
Thomas A. Lubik
Federal Reserve Bank of Richmond
Christian Matthes
Indiana University
How To Go Viral:
A COVID-19 Model with
Endogenously Time-Varying Parameters∗
Paul Ho
Thomas A. Lubik
Federal Reserve Bank of Richmond†
Federal Reserve Bank of Richmond‡
Christian Matthes
Indiana University§
August 21, 2020
Abstract
We estimate a panel model with endogenously time-varying parameters for COVID19 cases and deaths in U.S. states. The functional form for infections incorporates
important features of epidemiological models but is flexibly parameterized to capture
different trajectories of the pandemic. Daily deaths are modeled as a spike-and-slab
regression on lagged cases. Our Bayesian estimation reveals that social distancing and
testing have significant effects on the parameters. For example, a 10 percentage point
increase in the positive test rate is associated with a 2 percentage point increase in the
death rate among reported cases. The model forecasts perform well, even relative to
models from epidemiology and statistics.
JEL Classification: C32, C51
Key Words: Bayesian Estimation, Panel, Time-Varying Parameters
∗
We thank seminar participants at the Federal Reserve Bank of Richmond for helpful comments. James
Geary and James Lee provided exceptional research assistance. This research was supported in part through
computational resources provided by the Big-Tex High Performance Computing Group at the Federal Reserve
Bank of Dallas. The views expressed herein are those of the authors and not necessarily those of the Federal
Reserve Bank of Richmond or the Federal Reserve System.
†
Research Department, P.O. Box 27622, Richmond, VA 23261. Email: paul.ho@rich.frb.org.
‡
Research Department, P.O. Box 27622, Richmond, VA 23261. Email: thomas.lubik@rich.frb.org.
§
Wylie Hall, 100 South Woodlawn Avenue, Bloomington, IN 47405. Email: matthesc@iu.edu.
1
1
Introduction
A new form of coronavirus, SARS-CoV-2, which causes the respiratory disease COVID-19,
appeared in the U.S. in January 2020.1 Since then, the U.S. has seen over 5 million cases and
170,000 deaths as of mid-August.2 Any policy response to the pandemic crucially depends
on understanding how the virus spreads, how the disease evolves over time, what its effects
on mortality rates are, and how factors such as increased testing and measures such as social
distancing affect outcomes. We contribute to this effort from a statistical perspective that
pays heed to prior epidemiological research.
To that end, we develop and estimate a time series model for the number of cases and the
number of deaths in U.S. states that has three key features: (i) it exploits the panel dimension
of the data without forcing dynamics to be the same across states, (ii) it is a statistical model
that, while using some insights from epidemiological models, is more flexible than common
models in epidemiology, and (iii) it features time variation in parameters tied directly to
fluctuations in observable predictors to account for the fact that as the pandemic grew,
citizens and governments changed their behavior.3 The model produces accurate forecasts for
COVID-19 cases and deaths in the U.S., outperforming many competing models’ forecasts
for new and cumulative deaths. Our estimates show that increased social distancing and
testing are associated with lower numbers of cases, but this association does not hold in all
states. In addition, increased testing is associated with lower death rates among reported
cases. We estimate the model using Bayesian methods, which allows us to quantify the
uncertainty in our forecasts and estimates explicitly.
Our model for the number of infections is based on the observation that the time path of
infections during an epidemic follows a typical pattern. When a pathogen enters a population
that is susceptible to infection, the number of infected cases is initially low. However, the
growth rate of new infections is high and tends to rise sharply at an exponential rate since
each infected person creates a chain of new infections. At some point, however, the pathogen
runs out of susceptible hosts because they are already infected, immune, or simply not
physically present because of health policies such as social distancing. At this inflection
point, the growth rate of infections falls until it eventually declines to zero. We replicate
these broad patterns of an epidemic by specifying a flexible functional form that describes
the path of infections over time as depending on the current and the lagged levels of the
number of infections. In contrast to theoretical epidemiological models, our specification
1
https://en.wikipedia.org/wiki/COVID-19_pandemic
https://coronavirus.jhu.edu
3
The usefulness of time-varying parameter models during times of policy changes was first noted by
Robert Lucas in his original work on the Lucas Critique (Lucas (1976)).
2
2
has more leeway to go where the data tell it to and is not constrained by precise theoretical
relationships that may be specified incorrectly.
Since deaths from COVID-19 fundamentally arise from infections, we model the number
of deaths as depending on lagged cases. In particular, we use a spike-and-slab regression
model (Mitchell and Beauchamp (1988); Ishwaran et al. (2005)), in which the number of
deaths on a given day in a particular state depends on the lagged number of daily new cases
in that state. Due to the long lag between the time COVID-19 patients test positive and
the time they may die, our specification includes 35 lags, introducing a large number of
coefficients relative to the length of the sample period. The spike-and-slab structure shrinks
the regression coefficients in order to improve forecast performance.
We adapt our empirical specification to account for the fact that over time and across
states, there has been heterogeneity in how the pandemic has evolved and how states have
responded. First, we introduce endogenous time-varying parameters (TVP). The parameters
for the model of infections depend on social distancing and testing, and the death rate
depends on testing. We measure social distancing using geolocation data from 16 to 20
million mobile devices and measure the intensity of testing using the ratio of infections to
tests conducted. The model thus captures how these predictors alter the predicted path of
infections and deaths while providing additional flexibility to match different trajectories in a
way that is tightly disciplined by data. Second, we utilize the panel structure of the data. In
particular, our estimation allows the data to determine the correlation in parameters across
states and imposes that social distancing and testing have the same effect on parameters in
all states. Exploiting the panel structure sharpens estimates and forecasts in the presence
of a short sample period by leveraging on data from all states to inform the estimates for a
given individual state.
The model forecasts from May and June at horizons of up to 4 weeks are generally
corroborated by the data. In particular, we check the empirical frequency at which the data
realizations fall below various quantiles of our model forecasts. The forecasts match the
empirical realizations for daily new cases except during the sharp rise at the end of June and
start of July. However, by mid-July, the parameter estimates update and produce forecasts
that largely match the data in the second half of July. The density forecasts for daily new
deaths also match the empirical realizations well, especially at the upper quantiles. Our
forecasts for new and cumulative deaths perform favorably relative to a repository of models
from leading teams of epidemiologists and statisticians.
The endogenous TVP allow us to consider how social distancing and testing drive the
model-implied paths for cases and deaths. Due to the nonlinearity of our model, we find
that the effect of increased social distancing and testing on the predicted number of cases
3
differs across states even though the parameters determining this dependence are fixed across
states. For instance, under the median estimates for Texas, quantitatively plausible increases
in social distancing or testing are associated with a reduction in the number of cases by up to
50%. Under the median estimates for New York, the peak number of cases is early and sharp,
and neither social distancing nor testing substantially changes the model-implied path for
cases. We also find that more testing is associated with a lower death rate since the reported
infections are likely to include more asymptomatic or mild cases.
Epidemiologists have long studied the spread of infectious diseases, using both increasingly complex theoretical models and also more purely empirical frameworks. We contribute
to the latter by utilizing the toolkit prevalent in the analysis of economic data. In that
respect, our work is similar to Harvey and Kattuman (2020), Li and Linton (2020), and Liu
et al. (2020), who also use statistical models to forecast the pandemic. Our work is closest
to Liu et al. (2020), who similarly use a panel structure. They use a linear time-trend model
that allows for an exogenous break, whereas our model is a nonlinear autoregressive model
whose parameters are connected to observable predictors. Our functional form shares similarities with the generalized logistic curve used by Harvey and Kattuman (2020) to model the
number of cases. Both are flexible models for monotone progress from an initial condition
toward a saturation point. In contrast, Li and Linton (2020) use a polynomial time trend
for the logarithm of cases that is less flexible. Both Harvey and Kattuman (2020) and Li
and Linton (2020) focus on locality-by-locality estimation.
Our paper also connects with recent work that enriches structural models from epidemiology, primarily the so-called Susceptible-Infected-Recovered (SIR) framework. The structural
nature of the SIR model allows for the analysis of policy and counterfactual scenarios (e.g.,
Atkeson (2020); Fernández-Villaverde and Jones (2020); Hornstein (2020)). A hybrid approach is taken by Atkeson et al. (2020), who fit data on daily deaths to a mixture of
Weibull functions, then use the model-implied mixture of Weibulls to obtain a time-varying
reproduction rate for an SIR model. However, Koroloev (2020) and Kopecky and Zha (2020)
highlight identification issues in SIR frameworks, which pose a challenge to accurate forecasting and quantification of uncertainty.
With the growing data on the COVID-19 pandemic, numerous attempts have been made
to study the connection between different variables and the spread of the disease. One approach is to incorporate the SIR model into a choice-theoretic framework (e.g., Eichenbaum
et al. (2020); Farboodi et al. (2020); Bognanni et al. (2020)) in order to model the feedback
between individual or policy decisions and the transmission of the virus. Our reduced form
approach seeks to minimize assumptions and gives the data a greater role in informing the
researcher. A second approach is to estimate a SIR model with exogenous TVP (e.g., Ar4
royo Marioli et al. (2020); Buckman et al. (2020); Dandekar and Barbastathis (2020)), then
check the correlation of the parameters with various observables, such as social distancing
or quarantine measures, ex post. In contrast, we estimate the dependence of parameters
on these predictors jointly with the rest of the model. Finally, numerous papers have used
microeconometric methods that make use of differences across localities (e.g., Almagro and
Orane-Hutchinson (2020); Desmet and Wacziarg (2020); Glaeser et al. (2020)). By incorporating the panel structure, we similarly utilize variation across both states and time to
determine how social distancing and testing affect the path of the virus.
The paper is structured as follows. In Section 2, we introduce our model specification
which we use to capture the evolution of infections and deaths over the course of an epidemic.
We describe the data and estimation procedure in Section 3 and present the results in Section
4. Section 5 concludes.
2
A Panel Model for Estimating and Forecasting Pandemics
We now introduce and specify our empirical modeling framework for estimating and forecasting infections and deaths over the course of a pandemic. We formally introduce the
model setup before highlighting the distinctive features of our specifications.
2.1
Model Setup
We begin by modeling the number of infections independently since it is the key variable in
any theoretical or empirical model that studies the evolution of an epidemic. The number
of subsequent deaths is a function of the number of infections, which we consequently model
as a function of the lagged number of new cases.
2.1.1
Number of Cases
We specify the following model for the reported number of infections.4 Given states i =
1, ..., N and time periods t = 1, ..., T , denote the cumulative number of reported cases nor4
We model the number of reported cases directly since it is the most common approach in the literature.
Testing in the U.S. has increased substantially since the onset of the pandemic, which we capture by allowing
the parameters to depend on the positive test rate. Therefore, the estimates and projections should not be
interpreted as capturing the unobserved true number of cases in the population.
5
malized by population by Ci,t . We assume that Ci,t follows:
∆ log Ci,t = log(1 + γi,t )
φ(Ci,t−1 ; αi,t , ζi,t , ηi,t )
exp(uC
i,t )
φ(10−5 ; αi,t , ζi,t , ηi,t )
φ(C; α, ζ, η) ≡ exp[−C −α − (ζ η − C η )−2 ]
C
C
uC
i,t = ρi ui,t−1 + εi,t ,
(1)
(2)
(3)
2
C 2
−5
where εC
i,t ∼ N (0, (1 − ρi )(σi,t ) ). The normalization by φ(10 ; αi,t , ζi,t , ηi,t ) ensures that
when a fraction 10−5 of the population has been infected, the growth rate in the absence of
shocks or time-variation in parameters is γ. The AR(1) processes uC
i,t allow for potentially
persistent deviations from the deterministic trend. We assume these shocks are stationary.
In what follows, we describe the role of each of the parameters in giving flexibility to the
model-implied path for number of cases, before describing how the parameters vary across
time and states.
Functional Form. A key feature of the model is the functional form for φ in equation (2)
and the resulting range of trajectories implied by equation (1). We choose φ so that the
model with fixed parameters follows the general pattern of infections in a pandemic, with
an initial sharp increase as the disease spreads, followed by a leveling off and decline due to
public health policies or herd immunity. On the other hand, we ensure that the functional
form is flexible enough to match a wide range of such paths.
Figure 1 plots the trajectories for a range of time-invariant parameter values, illustrating
the role of each parameter. Each path begins with the same initial condition, and each panel
shows the effect of changing one parameter while leaving the rest unchanged. The functional
form allows for different rates of increase and subsequent decrease in the number of new
cases, different peaks, and different asymptotic numbers of cumulative cases. Importantly,
there is no fixed relationship between the different stages of the pandemic, imposing neither
the tight structure of an SIR model nor the symmetry of functional forms such as quadratic
trends.
Identification of the model parameters is based on the growth rate and changes in the
growth rate of infections, with the different parameters associated with distinct phases of the
epidemic. Initially, the rate of growth is approximately exponential. The effect of increasing
α, shown in the top-right panel of Figure 1, is to increase the curvature of the number of
new cases, ∆Ct , in the early phase of the epidemic, which captures the appearance of large
clusters or the effects of social distancing measures. As the stock of susceptible hosts starts
getting smaller, the rise in the growth rate decelerates until it reaches a peak. Afterwards,
6
New Cases (100
0.2
= 0.4
= 0.3
= 0.2
0.08
0.06
0.04
0.02
percent of population
0.1
percent of population
C t)
0
= 0.10
= 0.05
= 0.00
0.15
0.1
0.05
0
20
40
60
80
20
days
60
80
days
0.1
= 0.22
= 0.20
= 0.18
0.08
0.06
0.04
0.02
percent of population
0.1
percent of population
40
0
= 1.5
= 1.0
= 0.5
0.08
0.06
0.04
0.02
0
20
40
60
80
20
days
40
60
80
days
Figure 1: Model-implied daily new cases with time-invariant parameters and no shocks.
Gray lines are identical across all panels. Each panel shows change in model-implied number
of new cases associated with a change in one parameter.
the growth rate of new infections declines.
The parameters η and ζ determine the long-run number of cumulative cases and the
speed at which a population converges to that number, which could depend on factors
such as demographics or policies.5 In particular, the bottom panels of Figure 1 show that
increasing ζ or η leaves the initial path of Ct unchanged but increases the number of new
cases around the peak. While ζ does not affect the overall shape of the trajectory materially,
decreasing η flattens the peak and leads to a slower decline in the number of cases.
Panel Structure and Time-Varying Parameters. Denote a generic parameter by
θ ∈ {γ, α, ζ, η, σ C , ρ}. We assume that the parameters depend on a vector of observables
Xi,t , which could include demographic variables, social distancing metrics, or the amount of
testing:
g(θi,t ) = g(θi ) + κ0θ Xi,t
g(θi ) ∼ N (µθ , ωθ2 ),
5
(4)
(5)
We also estimated a model that replaces the exponent of 2 on the second term in (2) with a freely
estimated parameter. We fix the exponent because it is not well-identified separately from ζ.
7
where the function g is chosen to map the appropriate support of θ to the real line:
θ
supp(θ) = (−∞, ∞)
g(θ) = log(θ)
.
supp(θ) = (0, ∞)
log( 1 − 1) supp(θ) = (0, 1)
1−θ
The model (4) assumes that time variation in the parameters within a state can arise
only through time-varying predictors but does not allow for exogenous time variation in
parameters. We allow for differences across states through the fixed effect θi . The joint
distribution of parameters across states is determined by the hyperparameters µθ and ωθ ,
which we estimate.
2.1.2
Number of Deaths
In addition to modeling infections, we also consider the mortality rate. Not all infections are
fatal, and an observed death is the outcome of a process that can vary over time. We thus
assume that the number of deaths on any given day depends on the lagged number of cases,
but allow the data to determine the rate at which infections translate to deaths at different
horizons and which lags are most important.
In particular, we consider an extension of the spike-and-slab regression for the number
of new deaths ∆Di,t as a function of lagged new cases ∆Ci,t−` :
L
1 X
ιi,` λ(λi,` , δ; Xi,t−` )∆Ci,t−` + εD
i,t
ι
` i,` `=1
(6)
L
1 X
ιi,` ∆Ci,t−` × (σiD )2 ),
` ιi,` `=1
(7)
∆Di,t = P
εD
i,t ∼ N (0, P
where ιi,` ∼ Bernoulli(p` ) is a variable selection indicator. In the absence of shocks, the setup
nests a deterministic SIR model, in which infections lead to deaths at a Poisson rate, and the
values of λ will fall geometrically with ` at the recovery rate. We provide greater flexibility
by allowing the coefficient λ to vary freely across lags. In addition, we include shocks whose
variance scales with the number of lagged cases. The scaling captures the trade-off between
a lower variance due to a larger number of cases over which to average and a higher variance
due to a larger number of expected deaths.
The variable selection parameter ιi,` shrinks small coefficients to zero, which can improve
forecast precision since there are a large number of coefficients relative to observations. On
one hand, the parameter L is relatively large because COVID-19 patients who do not survive
8
the illness have a relatively long lag time between testing positive for the virus and dying.
On the other hand, COVID-19 is a recent disease for which we have a relatively short panel
of data. By making p` depend on `, the model assumes that lags, which are more important
for predicting mortality in one state, are likely important for other states as well.
The death rate λ roughly captures the fraction of infected individuals who die after a
given number of days.6 It depends on a state- and window-specific parameter λi,` and a
coefficient δ that determines the dependence of death rate on the predictors Xi,t−` . For
instance, the death rate likely decreases with the extensiveness of testing, as more mild and
asymptomatic cases are documented. Here we consider the functional form:
λ(λi,` , δ; Xi,t−` ) = λi,` (1 + δ 0 Xi,t−` ).
(8)
To allow the death rates to be correlated across states, we specify:
(σiD )−2 ∼ Γ(aσ , bσ )
λi,` | ιi,` = 1 ∼ N (µλ , (σiD )2 /υ),
(9)
(10)
where (aσ , bσ , µλ , υ) are hyperparameters to be estimated.
2.2
Discussion of Model Features
Endogenous Time Variation in Parameters. A key feature of our model is the endogenous time variation in parameters. In most models, any time variation in parameters is
exogenous. In contrast, we assume that the model-implied path of the pandemic can only
change if the observables Xi,t fluctuate. While the time variation in parameters offers the
flexibility to track a wide range of trajectories for infections and deaths, the endogeneity of
the time variation adds discipline to these fluctuations.7 By restricting the parameters to
only vary with observable data, we also rely more on the functional form in (2) to fit the
data and produce accurate forecasts.
In addition, we are able to estimate how different observables change the path for infections and deaths. This allows us to compute counterfactual trajectories for the pandemic
that condition on different paths for Xi,t . A more common approach in the literaure to
estimate the effect of observables has been to estimate a TVP model with exogenous time
variation, then to assess the correlation of the smoothed parameters with observables as a
6
This is exactly true if the number of new cases is independent across days and ιi,` = 1 for all `.
This is similar in spirit to the literature on endogenous Markov regime-switching, for instance, Diebold
and Lee (1994); Chang et al. (2017). However, in our model, the observables drive the actual parameter
values rather than the probability of moving between regimes.
7
9
second step (e.g. Arroyo Marioli et al. (2020); Buckman et al. (2020); Dandekar and Barbastathis (2020)). Our approach estimates the effect of the observables jointly with the rest
of the model, allowing for coherent quantification for both point estimates and posterior
uncertainty.
Panel Structure. Rather than estimate the model state-by-state, we consider a panel
model in which the parameters are correlated across states. This is designed to tighten
estimates for states that are in the early stages of transmission, since their state-specific
parameter estimates are informed by the data for states that are further along in the pandemic. The panel structure also aids in the estimation of κθ and δ. Since these parameters
are common across states, our panel estimates leverage the state-level heterogeneity in Xi,t ,
yielding tighter estimates of the effect of these predictors.
Statistical Model. Our models for infections and deaths are both statistical, unlike a
majority of models that are variants of the SIR model (see, for instance, Table 1 in the
Appendix for the list that we compare our forecasts against). Our model’s relative flexibility
allows us to fit the data well despite the restrictions we place on the time variation in
parameters. Nevertheless, the minimal structure that the model imposes on the rise and fall
in the number of cases helps generate tighter long-run forecasts.
3
Data and Estimation
3.1
Data
We use publicly available data on the daily number of reported COVID-19 cases and deaths
in the 50 U.S. states and Washington, D.C. from The New York Times8 from January 21,
2020 through August 11, 2020. For each state, we start the sample when the state has a
cumulative number of cases of at least 20. The data set collects the cumulative number of
infections at the end of each day reported by local government and health authorities.
We also use two predictors for the variation in the parameters: the Mobility and Engagement Index (MEI), constructed by the Federal Reserve Bank of Dallas from January 3,
2020 through August 8, 2020, and positive test rates from The Atlantic’s Covid Tracking
Project from March 1, 2020 through August 8, 2020. We allow the parameters of the model
of infections to depend on both the MEI and testing, and allow the parameters of the model
8
For full details, refer to the associated GitHub repository.
10
for deaths to depend on positive test rates only. See Figure 9 in the Appendix for the time
paths of these predictors.
The MEI summarizes the deviation from normal mobility behavior since the start of the
COVID-19 outbreak. The index is formed using principal components on seven variables
measured using geolocation data from 16 to 20 million mobile devices. Each variable is a
measure of how much individuals travel away from home, and the index is normalized so
that a higher value corresponds to greater mobility (i.e., less social distancing).9 We take a
seven-day lagged moving average to smooth out seasonal fluctuations. While all states show
a common pattern of declining mobility in March followed by an increase in mobility from
the second half of April, there is heterogeneity across states in how much and how quickly
mobility changed at different points of the pandemic cycle.
We define the positive test rate as the total number of reported cases over the past seven
days divided by the total number of tests conducted over the past seven days. A lower
positive test rate is an indication of more extensive testing. As reporting errors occasionally
lead to a positive test rate that is negative or greater than one, we truncate the positive
test rate to be within the [0, 1] interval. While the positive test rate for the U.S. declined
in aggregate as states increased their testing capacities in March and April, the path for the
positive test rates has differed greatly across states.
In what follows, we estimate the model using data through August 8, 2020, when our
samples for the MEI and testing data end. We also estimate the model using data through
every other Sunday from May 3, 2020 through June 14, 2020, and check the performance
of our forecasts at a horizon of 1 to 28 days. This covers the period during which states
were reopening and until the point when many states experienced a second wave of sharp
increases in case numbers. Finally, we also take forecasts using data through July 15, 2020,
in order to show how the estimates update around the peak of the second wave.
3.2
Estimation
We draw from the posterior of the model for the number of infections using the following
Gibbs sampler:
1. Condition on θ1:N .
(a) Draw (µθ , ωθ2 ) from a normal-inverse-gamma distribution.
(b) Draw κθ using Metropolis-Hastings.
9
See Atkinson et al. (2020) for details on the construction and comparison with other measures of social
distancing.
11
2. Draw θ1:N | κθ using Metropolis-Hastings.
Step 1(a) is standard and uses the property that the normal-inverse-gamma distribution
is a conjugate prior (see, for instance, Zellner (1971)). Step 1(b) requires computing the
likelihood contribution from equation (1) for the entire panel. Step 2 can be done stateby-state, similar to the estimation of the baseline model without time-varying parameters.
Hence, Step 2 could also be parallelized if a researcher wanted to use our model on a larger
set of locations.
To draw from the posterior of the model for mortality, we make use of the spike-and-slab
structure. In particular, we take the following steps:
1. Conditional on µλ , υ, aσ , bσ , p` ,
(a) Conditional on δ,
i. Draw ιi,` state-by-state using Metropolis-Hastings.
ii. Draw λi,` , σiD | ιi,` from a normal-inverse-gamma distribution.
(b) Draw δ | ιi,` , λi,` , σiD using Metropolis-Hastings.
2. Draw µλ , υ | λi,` , σiD from a normal-inverse-gamma distribution.
3. Draw aσ , bσ | σiD using Metropolis-Hastings.
4. Draw p` | ιi,` using the conjugate form of the beta prior.
Step 1(a)(i) uses the fact that a normal-inverse-gamma distribution is a conjugate prior for
a linear regression. In particular, for a given i, we can compute the marginal likelihood for
equation (6) given a candidate draw {ιi,` }L`=1 , integrating out λi,` and σiD . Given ιi,` , we
have a standard regression for Step 1(a)(ii). Conditional on (ιi,` , λi,` , σiD ), we can draw δ
using Metropolis-Hastings, since the likelihood is straightforward to compute. In Step 2,
it is straightforward to draw (µλ , υ) from a normal-inverse-gamma distribution, since λi,` is
distributed according to a generalized least squares regression on a constant, in which the
standard deviations of the shocks are known to be σiD . In Step 4, we utilize the fact that
the beta distribution is the conjugate prior for a binomial distribution. We pick L = 35,
allowing the number of deaths to depend on the number of new cases over a month ago. The
spike-and-slab structure allows the data to determine which lags are most important.
12
3.3
Prior
For the model for number of cases, we consider a relatively uninformative normal-inversegamma conjugate prior for the hyperparameters (µθ , ωθ2 ) for θ ∈ {γ, α, ζ, η, σ C , ρ}:
ωθ−2 ∼ Γ(1, 0.25)
µθ | ωθ ∼ N (0, ωθ2 ).
In addition, we impose a Gaussian prior for κθ for θ ∈ {γ, α, ζ, η, σ C , ρ}:
κθ ∼ N (0, 0.52 V −1 ),
where V is a diagonal matrix with the sample variances of each corresponding predictor Xi,t .
The prior thus represents the belief that each predictor contributes equally to the variance
of the transformed parameters g(θi,t ).
For the model of mortality, we similarly impose a Gaussian prior for δ:
δ ∼ N (0, 0.52 V −1 )
to match the prior on κθ . For the variance σiD of the shocks, we use the prior:
aσ ∼ Γ(2, 1)
bσ ∼ Γ(2, 3 × 10−7 ),
which is calibrated to the scale of the number of deaths. In particular, scale parameter for bσ
of 3 × 10−7 is chosen so that the mode of the prior is approximately the average state-specific
variance of the number of new deaths divided by the square root of the number of new cases,
p
P b
1
Vi ∆Di,t / ∆Ci,t . The shape parameters of 2 for aσ and bσ are chosen to make the
N
i
prior relatively uninformative. For the distribution of λi,` , we use the prior
υ ∼ Γ(1, 10−3 )
µλ | υ ∼ N (0, (0.052 × 10−3 )/υ).
The prior for ν is chosen to be relatively flat and is scaled such that the standard deviation
of a state-lag-specific coefficient λi,` is of order 10−2 . The conditional variance of µλ is scaled
13
by 10−3 to account for the scale of ν. Finally, we consider the prior
p` ∼ Uniform(0, 1)
for the probability of including a lag.
3.4
Forecasting
To forecast the number of cases and deaths in each state, we need to condition on a path
for the time-varying predictors. To that end, we estimate an AR(1) model independently
for each predictor in each state using the last 14 days of data. If the absolute value of
the predictor is declining, we assume a long-run mean of zero. If the absolute value of the
predictor is increasing, we assume the long-run mean is the maximum value of the absolute
value of that predictor in the full sample. We extrapolate from the last data point using this
AR(1) model without shocks. The path we condition on captures the general trend of the
predictor in the most recent data.
4
Results
We present two sets of empirical results. We first discuss the parameter estimates of the
panel, whereby we give an overview of the results from the 50 U.S. states and D.C. We then
show how these parameter estimates depend on measures of social distancing and testing.
In the next step, we check the forecast performance of the model, whereby we focus on three
large states that have exhibited different patterns for the evolution of the pandemic.
4.1
Parameter Estimates
We first discuss the parameter estimates based on data through August 8, 2020.
4.1.1
State-Specific Parameters
Figure 2 shows the marginal posterior distributions from the infections model for both the
state-specific components of γ, α, ζ, η, σ C , and ρ, as defined in equation (4), and the
aggregate distribution from equation (5). A large amount of heterogeneity across states is
required to match the wide range of trajectories across states even after accounting for the
MEI and positive test rates. Nevertheless, the posteriors for the aggregate distributions of
γ, α, η, and σ C are substantially tighter than their priors, and the data are informative as
indicated by the shifts of the posteriors.
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Figure 2: Marginal posteriors for γ, α, ζ, η, σ C , and ρ. Thin gray lines: posteriors for
each state; Thick red line: aggregate distribution across states.
For the mortality model, we define λi,` ≡ λ(λi,` , δ; T1
PT
t=1
Xi,t−` ) and plot the posterior
means of ιi,` and λi,` | ιi,` = 1 in Figure 3. The former is the probability of including a lag,
while the latter captures the average death rate in a state for a given lag. We also plot the
mean of these parameters across states on the same axes. Both parameters show a clear
weekly seasonal component, potentially reflecting measurement error due to different rates
of processing test results or documenting deaths over the week. Nevertheless, the degree of
seasonality differs greatly across states.
The posterior estimates for the mortality model also indicate that the number of deaths
on a given day depends on the number of new cases up to five weeks prior. While there is
a decreasing trend in the estimates for ιi,` as ` increases, the data favors including cases at
long lags to predict future deaths. For instance, the mean estimate for ` = 35 is 0.07, which
is roughly half the mean estimate for ` = 1. The estimates for λi,` | ιi,` = 1 show a small
upward trend. In terms of magnitude, the average posterior estimates of λi,` | ιi,` = 1 across
states lie between 0.01 and 0.04 across lags, corresponding to the typical range of death rates
reported for the U.S.. These estimates reflect the relatively long lag time between infection
and death.
Notably, the lag time between infection and death stands in contrast with the assumption
of Poisson death and recovery rates in standard SIR models. This assumption is generally
made for modeling convenience. However, our coefficient estimates do not appear to be
generated from a Poisson structure. Specifically, the fatality and recovery rates used in the
15
Figure 3: Posterior mean estimates for ιi,` and λi,` | ιi,` = 1. Gray dots correspond to
posterior means for individual states. Red crosses indicate average across states.
recent COVID literature range between 0.2% - 1.4% and 1/4 - 1/14, respectively.10 An OLS
regression, in which the numbers of cases are independent across lags, would likely show
the regression coefficients decaying rapidly. Intuitively, the spike-and-slab regression likely
inherits a similar structure both for ιi,` and λi,` | ιi,` = 1. The long lag between infection
and death is further evidenced from the recent second wave of cases in the U.S., as the rise
in cases was not followed by a corresponding increase in the number of deaths until several
weeks later.
4.1.2
Dependence of Parameters on Social Distancing and Testing
Our estimates show that differences in the MEI and positive test rates are associated with
significant variation in the model parameters. In particular, Figure 4 shows that the parameters in both the models for cases and deaths are significantly connected to the MEI
and the positive test rate through (4) and (8). These correlations are statistical and do not
10
Atkeson et al. (2020) provides an overview and use baseline fatality and recovery rates of 0.5% and 1/5,
respectively.
16
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Figure 4: Marginal posteriors for dependence κθ and δ of parameters on predictors. Blue
dashed line: social distancing; Red dotted line: amount of testing.
identify causality. In general, one would expect a greater level of social distancing when cases
increase (e.g. Glaeser et al. (2020)) due to an endogenous response from both households
and governments. On the other hand, a higher number of cases mechanically increases the
positive test rate if the number of tests remains constant.
In order to give a sense of how the parameter estimates in Figure 4 for κγ , κα , κζ , and κη
map into the behavior of the nonlinear model, we compare the model-implied path of new
cases under baseline paths for the MEI and positive test rate against alternative paths with
more social distancing or testing in Figure 5. The respective paths of the MEI follow the
typical path in the data: it decreases in the first 60 days, then increases and levels off below
the initial level of zero. For testing, we consider constant positive test rates 0.1 and 0.2. To
show how social distancing and testing can affect the model-implied paths differently across
states, we consider the model-implied paths under the median parameter estimates using
New York and Texas as examples. In particular, we initialize the number of cases at 10−4 %
of the population, then simulate the model forward without shocks.
Social distancing and testing can be associated with a lower number of infections, but
this relationship depends on the underlying trajectory of cases. Under the Texas parameter
estimates, both a lower MEI and a lower positive test rate are associated with flatter curves.
In contrast, under the New York parameter estimates, the model-implied trajectories remain
relative unchanged for different MEI and positive test rates. These differences arise because
of the different trajectories that New York and Texas faced: New York had a relatively rapid
rise and fall in the number of cases, whereas in Texas infections increased only gradually at
first. Under the baseline paths for the MEI and positive test rate, the number of new cases
17
percent of population
percent of population
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Figure 5: Model-implied paths of cases for different levels of social distancing and testing,
using posterior median for (γ, α, ζ, η) for New York and Texas and posterior median for κθ .
Gray solid line: baseline; Blue dashed line: increased social distancing (lower MEI);
Red dotted line: increased testing (higher positive test rate).
under the New York parameters falls to around 70% of its peak level by the 60-day mark,
while the number of new cases under the Texas parameters continues to rise. Our result
that the underlying trajectory of cases matters is consistent with Atkeson et al. (2020), who
use an estimated SIR model to show that the effects of distancing measures depend on the
precise scenario considered.
The MEI and the positive test rate are also associated with variation in the variance
and persistence of the shock uC
i,t in (1). Lower levels of social distancing are associated with
shocks of higher variance and higher persistence, while lower levels of testing coincide with
shocks of higher variance and lower persistence. Reduced social distancing likely leads to
more clusters developing and could cause any temporary spikes in cases to last longer. On
the other hand, lower levels of testing can also result in more measurement error.
The estimate of δ, the dependence of the death rate on the positive test rate, shows
that a higher positive test rate is associated with a substantially higher death rate. At the
posterior mean for the average death rate µλ of 0.006, an increase in the positive test rate
of 10 percentage points corresponds to an increase in death rate of 2 percentage points.
Intuitively, a higher positive test rate occurs when individuals who are tested tend to have
18
a higher ex-ante probability of being infected. These individuals tend to have more severe
symptoms, leading to a higher reported death rate.
4.2
Forecasts
We now assess the forecast performance of our model. This is a critical aspect of our analysis
since the global pandemic is still ongoing with no end in sight. Moreover, the course of the
pandemic in the U.S. has proven to be very volatile and heterogeneous across the states as
reported above. The most recent aggregate U.S. data even show the appearance of a second
peak in infections. The ability to forecast well in this changing environment is a key aspect
for an empirical epidemiological model that our panel framework with endogenous TVP is
designed to accomplish.
To check the forecast performance of our model, we estimate the model using an initial
subsample of the data and compare the model forecasts to the actual realizations. We do
this every two weeks from May 3, 2020 to June 14, 2020, covering the period during which
the aggregate number of cases in the U.S. was declining after the initial peak until the sharp
spike in cases leading to the second peak. During this time, states reopened at different
rates, and the number of cases and deaths across states followed a wide range of paths. The
heterogeneity across states provides a test for whether our model is sufficiently flexible to
match the numerous possible paths for the pandemic.
4.2.1
Coverage
Figure 6 shows Q-Q plots to compare the empirical realizations to the quantiles for our
forecasts of new cases and new deaths at the 1- to 28-day horizon. In particular, for each
horizon, we check the fraction of states whose realized number of new cases or deaths fall
below the qth quantile of our forecast for that state. We also average over each week to
remove any weekly seasonality, by counting the fraction of state-horizon observations that
fall below the respective qth quantile of our forecasts.
Overall, our forecasts match the realized data well. For new cases, the posterior quantiles
of our forecasts match the empirical frequency closely, except for the forecasts from June
14, 2020. This coincides with the sharp spike in cases in numerous states. In many cases,
our model predicts a rise in cases, but one that is smaller than the eventual spike. One
likely reason for the underprediction is that we tie the time-variation in parameters solely
to the time-variation in the MEI and positive test rates. The forecasts may be improved by
the inclusion of more variables, including more detailed mobility measures or disaggregated
data. Nevertheless, later forecasts from July 15, 2020 indicate that the parameter estimates
19
Figure 6: Q-Q plots for forecasts of increase in number of cases and deaths one to 28 days
ahead, comparison with other models. Translucent markers: fraction of states whose
realized number of new cases or deaths falls below given quantile of forecast for a specific
horizon; Opaque outline markers: average over each week. Marker colors and shapes
indicate week of forecast.
20
were updated in response to the new inflow of data. Figures 10 shows that the 95% error
bands of the corresponding forecast largely contain the realized data in late July and early
August.
The forecasts for new deaths match the empirical frequencies well at the upper quantiles,
but tend to undershoot slightly at the lower quantiles. The undershooting arises largely
among states that have many days without deaths. This occurs more regularly in the early
part of the sample and in states with a low number of cases. Indeed, by June 14, 2020, the
forecasts undershoot less as the zeros no longer bias the forecasts downward as much.
Figure 7 compares the forecast performance of our model to the forecasts compiled by
the COVID-19 Forecast Hub, which collects forecasts from leading teams of epidemiologists
and statisticians that are curated to ensure overall accuracy. As the repository primarily
provides forecasts for the number of deaths, we focus our assessment on cumulative and new
deaths. Since many models only update their forecasts once a week on Sunday or Monday,
we compare our forecasts taken on Sunday to any forecasts in the COVID-19 Forecast Hub
from the same day or one day later. We plot these competing forecasts on the same Q-Q
plot as our forecasts for the one- to four-week horizon.
Our model performs relatively well compared with alternative models at the horizons and
dates considered.11 This is notable given that the competing models include both statistical
and richly specified theoretical SIR models, many estimated using more data than we have
used. Table 1 in the Appendix provides further details on these models.
4.2.2
New York, California, and Texas
For further insight into how the model forecasts adapt to the data, we plot expanding window
forecasts for New York, California, and Texas in Figure 8. We focus on these three states
not only because they have among the largest populations in the U.S., but also because
the epidemic progressed differently in each, thereby providing a template for assessing the
forecasts in the other states. We plot forecasts from May 17, June 14, and July 15, 2020.
These correspond roughly to the decline in cases after the initial wave, the increase in cases
moving into the second wave, and the peak of the second wave. While all three states have
been severely affected by the COVID-19 pandemic, they have displayed different paths for
the number of cases, number of deaths, social distancing, and testing. The model forecasts
reflect these differences.
11
Since we check the forecasts against the New York Times data, the relative performance may be attributed partly to differences in the data used by different models. However, to fully explain the wide
dispersion in performance across models, one would require implausibly large and systematic differences
across data sets.
21
Figure 7: Q-Q plots for forecasts of cumulative and new deaths one to four weeks ahead,
comparison with other models. Opaque outline markers: fraction of states whose realized
number of new or cumulative deaths falls below given quantile of forecast from this paper
at given horizons; Translucent markers: forecasts from other models on same date or one
day later for same horizon. Marker colors and shapes indicate horizon.
22
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Figure 8: Forecasts for daily new cases, new deaths, MEI, and positive test rates in New
York, California, and Texas, using data through May 17, June 14, and July 15, 2020. Top
two panels: 95% error bands; Bottom two panels: extrapolation based on AR(1) model
for last 14 days.
23
New York. While New York was one of the hardest hit states during the early part of the
pandemic, the number of cases has steadily decreased since the first half of April. By May,
the number of cases was significantly lower than its peak. This is the typical path of cases
predicted by standard SIR models, and our model is able to fit this path well. In particular,
the model forecasts match the realized gradual decline in number of cases and deaths, with
the realized data falling within or close to the 95% error bands, which are relatively tight.
California. The number of cases in California plateaued in April, began to increase in
May, and accelerated in June before beginning to stabilize at the end of July. The model
forecasts qualitatively match these patterns. In May, we forecast a relatively stable number
of cases up till September at least. The data fall mostly within the 95% error bands until
mid-June when the increase in number of cases begins to accelerate. When we forecast
the number of cases from mid-June, the model predicts a possible rise in cases for several
weeks, albeit a smaller one than what actually occurs in the second half of June. Finally,
even though the number of cases continued to rise through the first half of July, the model
forecasts a plateau in the number of cases between mid-July and early September. The data
in the second half of July corroborate this forecast, as the number of cases has fluctuated
around the upper half of the 95% error bands.
The forecasts for the number of deaths mirrors those for the number of cases, except for
the mid-July forecast. In particular, the model predicts an increase in the number of daily
new deaths. This prediction is borne out by the data, with the number of deaths rising in
late July even as the number of new cases began to decline. This reflects the result that the
number of new deaths depends on the number of new cases up to five weeks prior.
Texas. The data and forecasts for Texas are qualitatively similar to California. In midJuly, the model forecasts slightly faster growth in the number of cases in Texas than in
California. However, there is substantial uncertainty about the rate of this increase, as the
error bands are about twice as wide than those of the June forecast. The realized data in
the second half of July show a slight decrease that is comfortably within the error bands.
While the patterns of cases in California and Texas were relatively similar, the positive test
rate during the first half of July increased more rapidly in Texas than in California. The
forecasts for the two states thus condition on different projected paths for testing, leading
to the contrasting predicted trends in new cases.
24
5
Conclusion
We develop and estimate a statistical model of the COVID-19 pandemic that has three key
features. First, parameters are allowed to vary over time, but only in line with observable
variables. Second, the model has a panel structure that sharpens estimates and forecasts.
Third, the underlying functional forms for the model are flexible and able to track the typical
paths of cases and deaths in a pandemic. The model’s forecasts perform favorably relative
to alternative epidemiological and statistical models.
By allowing parameters to depend on social distancing and testing, our estimates highlight the interaction between these predictors and underlying state-specific parameters in
generating model predictions. Specifically, while both increased social distancing and more
intensive testing can be associated with lower case numbers, this does not occur when the
peak in a locality is relatively early and followed immediately by a sharp decline. In addition,
we estimate a decline in death rates associated with a lower positive test rate, accounting
for the different composition of cases reported as testing becomes more widely available.
Our functional form captures the trajectory of cases as well as connection between infections and deaths that motivate SIR models. However, our statistical approach minimizes
modeling assumptions relative to the structural SIR literature, providing estimates that can
help inform the calibration or specification of these models. The autoregressive structure of
our setup is akin to time series econometric models and is conducive to forecasting. At the
same time, we introduce a panel structure to leverage the variation across states and time
that microeconometric methods often rely on. The Bayesian estimation transparently quantifies parameter and forecast uncertainty. Our framework thus bridges a range of approaches
to provide insights into the evolution of this global pandemic.
25
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A
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200
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
Jun
Jul
Aug
Sep
Apr
May
Jun
Jul
Figure 10a: Forecasts (95% error bands) for daily new cases in U.S. states, using data through
May 17, June 14, July 15, and August 8, 2020.
30
Nevada
2000
New Hampshire
250
1500
New Jersey
5000
New Mexico
500
200
4000
400
150
3000
300
100
2000
200
50
1000
100
0
0
1000
500
0
Apr
May
Jun
Jul
Aug
Sep
Apr
May
New York
Jun
Jul
Aug
Sep
0
Apr
May
North Carolina
15000
Jun
Jul
Aug
Sep
Apr
May
Jun
North Dakota
4000
400
3000
300
2000
200
1000
100
Jul
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Ohio
3000
10000
2000
5000
1000
0
0
Apr
May
Jun
Jul
Aug
Oklahoma
2500
0
Sep
Apr
May
Jun
Jul
Aug
Sep
Oregon
600
400
1000
200
500
0
May
Jun
Jul
Aug
Jun
Aug
Sep
Apr
Pennsylvania
Apr
May
South Carolina
Jun
Jul
Aug
400
1500
300
1000
200
500
100
Sep
May
South Dakota
Jun
Jul
Aug
Sep
Apr
Tennessee
300
5
4000
Jul
10
4
10
4
May
Jun
Jul
Texas
4
3000
200
3000
Jun
Rhode Island
0
Apr
4000
May
500
0
Sep
5000
Jul
2000
0
Apr
May
2500
2000
1500
0
Apr
3
2000
2000
2
100
1000
1000
0
0
Apr
May
Jun
Jul
Aug
Sep
Utah
1000
Apr
May
Jun
Jul
Aug
Sep
0
Apr
Vermont
80
800
1
0
May
Jun
Jul
Aug
Sep
Apr
Virginia
4000
2
60
3000
1.5
40
2000
1
20
1000
0.5
0
0
May
Jun
Jul
Washington
600
400
200
0
Apr
May
Jun
Jul
Aug
Sep
Apr
May
West Virginia
400
Jun
Jul
Aug
Sep
0
Apr
May
Jun
Jul
Aug
Sep
Wisconsin
3000
Apr
May
Jun
Jul
Wyoming
100
300
80
2000
60
200
40
1000
100
20
0
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
Jun
Jul
Aug
Sep
Apr
May
Jun
Jul
Aug
Sep
Figure 10b: Forecasts (95% error bands) for daily new cases in U.S. states, using data
through May 17, June 14, July 15, and August 8, 2020.
31
Alabama
Alaska
Arizona
2.5
200
Arkansas
1500
80
2
150
60
1000
1.5
100
40
1
50
500
20
0.5
0
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
California
Jun
Jul
Aug
Sep
0
Apr
May
Colorado
Jun
Jul
Aug
Sep
May
100
Jun
Jul
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Delaware
250
600
80
200
60
80
400
Apr
Connecticut
150
60
40
100
40
200
0
0
Mar
Apr
May
Jun
Jul
Aug
Sep
District of Columbia
20
0
Apr
May
Jun
Jul
Aug
Sep
Florida
2500
0
Apr
May
Jun
Jul
Aug
Sep
Georgia
400
2000
15
20
50
20
Apr
May
Jun
Jul
Hawaii
15
300
10
1500
10
200
1000
5
5
100
500
0
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
Idaho
Jun
Jul
Aug
Sep
0
Apr
May
Illinois
50
Jun
Jul
Aug
Sep
May
Jun
Jul
Iowa
50
200
40
40
150
200
30
Apr
Indiana
300
30
100
20
20
100
50
10
0
0
Apr
May
Jun
Jul
Aug
Sep
Kansas
20
10
0
Apr
May
Jun
Jul
Aug
Sep
Kentucky
150
0
Apr
May
Jun
Jul
Aug
Sep
Louisiana
150
Apr
May
Jun
Jul
Maine
25
20
15
100
100
50
50
0
0
15
10
10
5
5
0
Apr
May
Jun
Jul
Aug
Sep
Maryland
Apr
May
Jun
Jul
Aug
Sep
Massachusetts
300
0
Apr
May
Jun
Jul
Aug
Sep
Michigan
250
Apr
May
Jun
Jul
Minnesota
80
100
200
80
200
60
150
60
40
40
100
100
20
50
20
0
0
Apr
May
Jun
Jul
Aug
Sep
Mississippi
500
0
Apr
May
Jun
Jul
Aug
Sep
Missouri
150
0
Apr
May
Jul
Aug
Sep
Montana
10
400
Jun
Apr
May
Jun
Jul
Nebraska
15
8
100
300
200
10
6
4
50
100
5
2
0
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
Jun
Jul
Aug
Sep
Apr
May
Jun
Jul
Figure 11a: Forecasts (95% error bands) for daily new deaths in U.S. states, using data
through May 17, June 14, July 15, and August 8, 2020.
32
Nevada
80
New Hampshire
20
New Jersey
2000
60
15
1500
15
40
10
1000
10
20
5
500
5
0
0
Apr
May
Jun
Jul
Aug
0
Sep
Apr
May
New York
Jun
Jul
Aug
Sep
0
Apr
May
Jun
North Carolina
Jul
Aug
Sep
Apr
May
Jun
North Dakota
Jul
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Aug
Sep
Ohio
10
150
1000
New Mexico
20
150
8
800
100
400
100
6
600
4
50
50
2
200
0
0
Apr
May
Jun
Jul
Aug
Oklahoma
50
0
Sep
Apr
May
Jul
Aug
Sep
0
Apr
Oregon
20
40
Jun
May
300
10
200
5
100
Jul
Aug
Sep
Apr
Pennsylvania
400
15
Jun
May
Jun
Jul
Rhode Island
30
20
30
20
10
10
0
0
Apr
May
Jun
Jul
Aug
0
Sep
Apr
May
South Carolina
Jun
Jul
Aug
Sep
8
300
6
200
4
100
2
0
Jun
Jul
Aug
Sep
Utah
10
Jul
Aug
Sep
Apr
May
Jun
60
1500
40
1000
20
500
Jul
Aug
Sep
Apr
May
Jun
Jul
Aug
Vermont
May
Jun
Jul
Aug
Sep
Apr
60
600
2
40
400
1
20
200
0
0
West Virginia
15
Jun
Jul
Aug
Sep
Jul
800
0
Apr
May
Wisconsin
80
Jun
1000
3
May
Jul
Washington
80
Apr
May
Virginia
100
4
Sep
Jun
0
Apr
5
0
May
Texas
0
Apr
5
15
Jun
Tennessee
0
May
May
South Dakota
400
Apr
0
Apr
Jun
Jul
Aug
Sep
4
60
3
40
2
20
1
Apr
May
Jun
Jul
Wyoming
10
5
0
0
Apr
May
Jun
Jul
Aug
Sep
0
Apr
May
Jun
Jul
Aug
Sep
Apr
May
Jun
Jul
Aug
Sep
Figure 11b: Forecasts (95% error bands) for daily new deaths in U.S. states, using data
through May 17, June 14, July 15, and August 8, 2020.
33
34
SEIR with non-linear mixed effects curve-fitting
TVP-SEIR
Auquan
Covid Act Now
S
S
M
M
M
M
S
M
06/14
Statistical dynamical growth model
TVP-SEIR
TVP-SEIR with nonlinear infection rate
Metapopulation, age structured SLIR model
SEIRX with age distribution, disease severity
SIR with data assimilation
Machine learning SEIR with unreported cases
Bayesian TVP-SEIR
Bayesian TVP-SEIR
Bayesian multilevel negative binomial regression
Machine learning SEIR with reopening
Los Alamos National Laboratory
MGH/HMS COVID-19 Simulator
MIT Covid Analytics DELPHI
Northeastern MOBS Lab GLEAM
Predictive Science Inc DRAFT
University of Arizona EpiGro
UCLA Stat. Machine Learning Lab
UMass-Amherst MechBayes
U.S. Army ERDC
UT-Austin COVID-19 Consortium
Youyang Gu ParamSearch
S
M
S
M
M
S
S
M
S
S
M
M
M
M
S
S
M
M
S
S
S
M
M
M
S
M
S
S
S
S
S
M
M
S
S
S
M
M
M
S
S
S
M
S
S
S
S
M
M
M
05/03
S
M
S
M
S
M
M
S
M
M
S
M
S
S
M
S
05/31
New Deaths
05/17
S
M
M
S
S
M
S
S
S
M
M
M
S
06/14
Table 1: Summary of models from Figure 7. ‘S’ and ‘M’ indicate that forecast is from Sunday and Monday, respectively.
County-level metapopulation SEIR
Johns Hopkins ID Dynamics
M
S
S
• Most plausible
TVP-SEIR with parameters linked to drivers
S
• Moderate transmission rate
IHME
S
• Low transmission rate
S
S
M
M
S
S
05/31
• High transmission rate
County-level metapopulation SEIR
Columbia University
M
M
M
05/17
Cumulative Deaths
S
Combined forecast from selected models
• Ensemble
05/03
• Constant contact rate
Extrapolation from most recent observations
• Baseline
COVID-19 Forecast Hub
Description
Team/Model
@FedFRASER