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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. 14 20 30 20 20 10 10 10 0 0 0.5 1 0 -0.6 0 -0.4 -0.2 0 0 0.5 1 0 0.5 1 C 15 15 10 10 5 5 60 40 20 0 0 0 0.2 0.4 0.6 0.8 0 0 1 2 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 4 20 3 6 2 2 4 10 1 0 -1 -0.5 0 0 -1 -0.8 -0.6 -0.4 0 -3 -0.2 2 0 -2 -1 0 0 1 2 3 4 C 0.4 MEI 2 6 positive test rate 4 0.2 1 2 0 0 -0.2 0 0.2 0.4 -2 -1 0 0 30 1 35 40 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 New Cases (NY Parameters) 0.04 0.03 0.02 0.01 0 0 100 200 New Cases (TX Parameters) 0.1 0.05 0 300 0 100 days Mobility and Engagement Index 300 -0.5 -1 Positive Test Rate 0.3 fraction index 0 200 days 0.2 baseline 0.1 high social distancing high testing 0 0 100 200 300 0 100 days 200 300 days 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 New Cases New York California Texas 15000 15000 15000 10000 10000 10000 5000 5000 5000 0 Apr May Jun Jul Aug Sep 0 Mar 0 Apr May Jun Jul Aug Sep Apr May Jun Jul Aug Sep Aug Sep Aug Sep Aug Sep New Deaths New York California 1000 Texas 600 1500 400 1000 200 500 800 600 400 200 0 Apr May Jun Jul Aug Sep 0 Mar 0 Apr May Jun Jul Aug Sep Apr May Jun Jul Mobility and Engagement Index New York California Texas 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 Apr May Jun Jul Aug Sep Mar Apr May Jun Jul Aug Sep Apr May Jun Jul Positive Test Rate New York California Texas 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 Apr May Jun Jul Aug Sep 0 Mar 0 Apr May Jun Jul Aug Sep Apr May Jun Jul 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 References Almagro, Milena and Angelo Orane-Hutchinson (2020), “The Determinants of the Differential Exposure to COVID-19 in New York City and Their Evolution over Time.” Covid Economics: Vetted and Real-Time Papers. Arroyo Marioli, Francisco, Francisco Bullano, Simas Kučinskas, and Carlos Rondón-Moreno (2020), “Tracking R of COVID-19: A New Real-Time Estimation Using the Kalman Filter.” Working paper. Atkeson, Andrew (2020), “What Will Be the Economic Impact of COVID-19 in the U.S.? Rough Estimates of Disease Scenarios.” NBER Working Paper 26867, National Bureau of Economic Research. Atkeson, Andrew, Karen A. 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Wiley. 28 A Additional Figures and Tables Mobility and Engagement Index 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 Mar Apr May Jun Jul Aug Jul Aug Positive Test Rate 1 0.8 0.6 0.4 0.2 0 Mar Apr May Jun Figure 9: MEI and positive test rate for all U.S. states. 29 Alabama Alaska 4000 Arizona Arkansas 300 3000 10000 3000 8000 200 2000 6000 2000 4000 100 1000 1000 2000 0 0 Apr May Jun Jul Aug Sep 0 Apr May California Jun Jul Aug Sep 0 Apr May Colorado Jun Jul Aug Sep Apr May Connecticut Jul Aug Sep Aug Sep Aug Sep Aug Sep Aug Sep Aug Sep Aug Sep Delaware 2500 500 2000 400 1500 300 1000 200 200 500 100 0 0 15000 Jun 1000 800 10000 600 400 5000 0 Mar Apr May Jun Jul Aug Sep District of Columbia 400 Apr 2.5 300 10 May 4 Jun Jul Aug Sep Florida 0 Apr May Jun Jul Aug Sep Georgia 10000 Apr May 8000 800 1.5 6000 600 1 4000 400 0.5 2000 200 Jul Hawaii 1000 2 Jun 200 100 0 0 Apr May Jun Jul Aug Sep 0 Apr May Idaho Jun Jul Aug Sep 0 Apr May Illinois Jun Jul Aug Sep Apr May Jun Jul Indiana 2500 5000 2500 2000 4000 2000 1500 3000 1500 1000 2000 1000 500 1000 500 Iowa 1500 1000 500 0 0 Apr May Jun Jul Aug Sep Kansas 0 Apr May Jun Jul Aug Sep Kentucky 2500 0 Apr May Jun Jul Aug Sep 5000 4000 1500 3000 400 1000 2000 200 500 1000 800 600 May Jun Jul Maine 150 1000 2000 Apr Louisiana 100 50 0 0 Apr May Jun Jul Aug Sep Maryland 2000 0 Apr May Jul Aug Sep Massachusetts 5000 1500 Jun 0 Apr May Jun Jul Aug Sep Michigan 5000 Apr May 4000 2000 3000 3000 1500 2000 2000 1000 1000 1000 500 0 0 Jul Minnesota 2500 4000 Jun 1000 500 0 Apr May Jun Jul Aug Sep Mississippi 5000 Apr May Jun Jul Aug Sep Missouri 3000 0 Apr May Jul Aug Sep Montana 800 4000 Jun May Jun Jul Nebraska 1000 800 600 2000 3000 Apr 600 400 2000 400 1000 200 1000 0 0 Apr May Jun Jul Aug Sep 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