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# Abstract

# Media Summary

# 1. Introduction

## 1.1. (Limitations of) Data on COVID-19

## 1.2. Our Modeling Approach

# 2. Related Work

# 3. Model

## 3.1. Fixed Model Parameters

## 3.2. Estimated Model Parameters

# 4. Computation

# 5. Results

## 5.1. Model Fit and Projections

#### Table 1. Estimated posterior mean of $\mathbf{R_{0}}$ , with 95% posterior credible interval shown in parentheses.

## 5.2. Undercount Estimates

#### Table 2. Estimated posterior mean of undercount, with 95% posterior credible interval shown in parentheses.

## 5.3. Effects of Executive Orders for Social Distancing

#### Table 3. Estimated posterior mean of $\boldsymbol{\rho}_{\mathbf{T}}$ , with 95% posterior credible interval shown in parentheses.

## 5.4. A Prior on $p$

#### Table 4. $\mathbf{R_0}$ , $\boldsymbol{\rho}_\mathbf{T}$ , and the estimated undercount (95% posterior credible intervals) calculated by averaging samples over a prior on $\mathbf{p}$ .

# 6. Sensitivity Analysis

## 6.1. Alternative $\theta$

## 6.2. Excess Mortality-Inflated Death Counts

## 6.3. Alternative Results

#### Table 5. Estimated posterior mean of $\mathbf{R_{0}}$ , with 95% posterior credible interval shown in parentheses.

#### Table 6. Estimated posterior mean of $\boldsymbol{\rho}_{\mathbf{T}}$ , with 95% posterior credible interval shown in parentheses.

#### Table 7. Estimated posterior mean of undercount, with 95% posterior credible interval shown in parentheses for alternative scenarios: Deaths adjusted according to excess mortality multipliers or using the alternative $\boldsymbol{\theta}$ .

# 7. Conclusion

# Acknowledgments

# Disclosure Statement

# References

# Appendices

## Appendix A. Markov Chain Monte Carlo Diagnostics

#### Table A1. Gelman-Rubin convergence diagnostic point estimate (upper 95% confidence interval) for parameters of model calculated with five independent runs of Markov Chain Monte Carlo.

#### Table A2. Multivariate Gelman-Rubin potential scale reduction factor.

## Appendix B. Simulation Study

#### Table A3. Estimated posterior mean and 95% posterior credible intervals for each simulation scenario.

Estimating the Number of SARS-CoV-2 Infections and the Impact of Mitigation Policies in the United States

Published onNov 23, 2020

Estimating the Number of SARS-CoV-2 Infections and the Impact of Mitigation Policies in the United States

Knowledge of the number of individuals who have been infected with the novel coronavirus SARS-CoV-2 and the extent to which attempts for mitigation by executive order have been effective at limiting its spread are critical for effective policy going forward. Directly assessing prevalence and policy effects is complicated by the fact that case counts are unreliable. In this article, we present a model for using death-only data—in our opinion, the most stable and reliable source of COVID-19 information—to estimate the underlying epidemic curves. Our model links observed deaths to a susceptible-infected-removed (SIR) model of disease spread via a likelihood that accounts for the lag in time from infection to death and the infection fatality rate. We present estimates of the extent to which confirmed cases in the United States undercount the true number of infections, and analyze how effective social distancing orders have been at mitigating or suppressing the virus. We provide analysis for four states with significant epidemics: California, Florida, New York, and Washington.

**Keywords:** COVID-19, SARS-CoV-2, compartmental model, SIR model, Bayesian

Our analysis offers two findings. First, we estimate that the true number of COVID-19 infections is likely 6 to 10 times larger than the number of confirmed, officially reported cases as of late March. Second, we find that the first round of executive orders for social distancing had mixed effects across states. While in New York the effective reproductive number (a measure of the spread of the virus) dropped low enough to suppress the spread of the virus, this was less clear in states like Florida, where our estimates suggest that the executive orders caused the infection rate only to plateau rather than decrease.

Our model offers two technical improvements, in addition to these findings. First, we use what we believe is the most reliable, or least unreliable, source of data about the pandemic: reported deaths. Next, we build a Bayesian model that accounts for the time lag between infection and death.

We model the transmission of the virus (creating new infections) by a susceptible-infected-removed (SIR) model, a mechanistic description of the spread of infectious disease. Based on prior research, we assume an infection fatality rate of 1%. The true infection fatality rate is still contested, even among experts, so we provide results for infection fatality rates covering a range of plausible scenarios.

Our work is motivated by the problem that official data about confirmed COVID-19 infections drastically understates the true extent of the disease. Many people who have been infected have not sought testing or have not succeeded in being tested due to restrictive policies delineating who is eligible for testing. As testing capabilities in the United States have ramped up, larger proportions of infected individuals have been recorded as infected in the data. This has even made comparative statements across time based on case counts unreliable. Subsequent declines in testing further complicate the use of the reported cases.

These findings offer insights into the true scale of the pandemic. Furthermore, with estimated infection rates that correct for underreporting bias, these findings enable comparisons among regions and over time. A measure of the magnitude and pattern of the infections is an essential input to health care planning, public policy about distancing, and public communication about the progress of the pandemic.

The coronavirus SARS-CoV-2, which causes the disease COVID-19, has already changed the lives of billions of people globally. Many people have been ordered to stay in their homes, and economies worldwide have largely come to a halt. Thousands have died and at least several million have been infected. In the United States, social distancing policies began being implemented in mid-March, as states such as Washington, California, and New York saw a sharp rise in the number of hospitalizations attributable to the virus. Government response has been hindered by an insufficient supply of materials needed for testing, and by the large proportion of infected individuals who are asymptomatic and therefore are unlikely to seek testing even if it is available. These factors make it difficult to know the true size of the infected population. Because effective surveillance has not been possible, policymakers have instead turned to social distancing policies as the best available tool to slow the spread of the virus.

Here, we seek to address two key questions: (1) How many people are actually infected or have ever been infected with SARS-CoV-2?; and (2) Are the social distancing policies currently in place effective at suppressing the virus? That is, can they be expected to lead to a decrease in the number of infections? Effectively addressing these questions requires innovative modeling due to severe limitations in commonly used sources of data for tracking the spread of the virus (Angelopoulos et al., 2020).

Ideally, to address these questions we would use data on the number of confirmed cases to understand the prevalence of the disease and assess policy measures. However, we view confirmed case counts for COVID-19 to be unreliable and ill-suited to this type of analysis for a number of reasons. Media reports have made clear that testing is more available in some regions than others, and so case counts are primarily an indication of where testing is most comprehensive.1 In fact, there may be as much as a 20-fold difference in case detection between countries, leading to incomparable numbers (Golding et al., 2020).

Evidence for the insufficiency of case counts within the United States can also be seen in the large observed difference in test-positive rates—the proportion of positives among all tests performed—across states. For example, as of May 17, 2020, California has conducted 1.24 million tests and enumerated about 80,000 confirmed cases, for a test positive rate of about 6.5%. By contrast, New York has conducted 1.41 million tests and enumerated about 350,000 confirmed cases, for a test positive rate of close to 25%. For case counts to represent an accurate count of the number of infections in both locations, one would need to believe that testing policies and test-seeking behavior in New York are much more likely to identify individuals who have the disease while simultaneously not missing infected people at a higher rate than in California, a highly dubious prospect. One way such large discrepancies in test positive rates could arise is through random testing. If people were tested at random in both states, we would expect to see higher test positive rates in locations with higher prevalence, such as New York. While the prevalence likely does vary between the states, outside of limited designed experiments accounting for a tiny fraction of the total number of tests, testing is not administered to a random sample of people. Instead, tests are administered mainly to symptomatic individuals and people who seek testing. Rather, the most likely explanation for this disparity in test positive rates is that in locations like New York—where the epidemic is large—the number of people infected has persistently outstripped the capacity to test, while in California—where the epidemic is much smaller—this is less true. Thus, the case counts likely do not accurately reflect even the *relative* size of the infected population across states. It is even more questionable that they could accurately reflect the *absolute* size of the infected population across states.

Furthermore, even within a single administrative boundary, case counts from earlier in the epidemic cannot meaningfully be compared with more recent case counts. Early in the epidemic, tests were administered only to people meeting strict criteria that likely excluded many sick individuals (Johnson & McGinley, 2020, March 7). In some cases, these criteria were chosen due to shortages of testing resources, such as reagents (Cavitt, 2020, April 12). Over time, shortages of key components of testing have slowly resolved, and the administrative rules and guidelines have themselves changed. For example, according to the Centers for Disease Control’s (CDC) website, several revisions to the official guidance on which patients should be tested have been made, including a March 4, 2020, revision that modified the criteria for testing to expand the pool of eligible people (Centers for Disease Control and Prevention, 2020). Thus, some of the trends in cases we observe over time within a given location are likely attributable to changes in test availability and criteria for test eligibility rather than purely to changes in prevalence of the disease. This makes case data of questionable value for modeling, even if model results are not compared across different locations.

Another, potentially more reliable, data source is hospitalizations. Unfortunately, these data have become universally available only fairly recently. For example, the Johns Hopkins COVID-19 database provides hospitalization data by state beginning only on April 14, 2020—more than two months after the epidemic began in New York, Washington, and California. High-quality data on the size of the epidemic in the early days are critical to fitting epidemiological models, which tend to be sensitive to initial conditions, and thus a partial time series lacks information that is critical for obtaining the best estimates. Moreover, even if the data source as it currently exists had been available since the beginning of the outbreak, it is a measure of *total* hospitalizations by day. Without supplementary information on the length of stay for each patient, it is not possible to calculate the number of *new* hospital admissions over time, which is much more useful from the perspective of modeling the total number of infected individuals. Thus, while this could be useful as an additional data source, particularly in the future as the length of the available time series grows, at the moment it is not adequate for our purposes.

This brings us to data on deaths attributable to COVID-19. Death data have been recorded since the earliest days of the epidemic and are universally available across states. Because gravely ill patients who die from severe disease are more likely than the average infected person to be hospitalized and tested, we believe death data are the most complete and representative data source available on a timely basis that could be used to estimate the number of SARS-CoV-2 infections. Even so, death data are imperfect. Excess mortality calculations suggest that the death-only data will miss some people who died of COVID-19 but whose cause of death was not listed as such on their death certificates (Weinberger et al., 2020). Why, then, not use excess mortality data? Excess mortality estimates offer an invaluable approach to evaluating the disease’s overall impact. However, excess mortality almost certainly overcounts actual deaths directly attributable to COVID-19, since some of the excess mortality is a result of people with other health conditions avoiding hospitals and clinics, cancellation of procedures to reduce hospital census, and so forth. Such second-order effects of the disease don’t fit into traditional models of the spread of infectious disease that are built around the assumption that people who die from the disease were at some point themselves *infectious*. Thus, while approaches based on excess mortality are incredibly useful for revealing the total effect of the epidemic on mortality, we find them less suitable for fitting models of the spread of the virus. As a result, we fit our model to the data that we believe are best suited for modeling the spread of the virus and have a reasonable chance of being accurate, or—more precisely—are likely the *least inaccurate* of available measures of the extent of COVID-19 infections: death data. Notably, several other influential COVID-19 modeling efforts have come to the same conclusion and have turned to death data to fit or calibrate their models (Altieri et al., 2020; Ferguson et al., 2020, March 16; Flaxman et al., 2020; Golding et al., 2020).

A common approach in the statistical epidemiological literature focuses on fitting or calibrating ordinary differential equation (ODE) models to observed data, and this underpins our approach as well (Hethcote, 2000). However, because we build our model using death-only data, we propose modifications to the standard approaches that rely on case data. In order to do this, the observed deaths need to be linked to the underlying state variables of the ODE model via a sensible, scientifically motivated likelihood. We do this by means of a distribution for the time from infection to death, and an assumption about the infection fatality rate (IFR), that is, the proportion of infected individuals who will eventually die of COVID-19. Using death-only data in the early stage of the epidemic, these parameters and the parameters of the underlying ODE model are not separately identifiable. For this reason, the time to death distribution and IFR are *assumptions* of our model.

There exist external estimates of the time to death distribution based on high-quality data. For the IFR, precisely because of the difficulty outlined above in establishing the true number of infections, there remains considerable uncertainty about its true value. Because the IFR is such an important assumption of our model, we perform analysis for five scenarios with different values of the IFR that span the (relatively small) range of plausible values. Our model assumes that the IFR is constant over the time period considered. While we believe this is a useful approximation to the truth, in reality there may be drift over time. For example, as physicians and researchers gain more insight into the most efficacious treatments for the disease, outcomes for seriously ill patients may improve over time, leading to a lower rate of death given infection later in the epidemic. It has also been speculated that the demographics of who becomes infected may change the IFR.2 However, because our focus is on the first months of the pandemic during which we believe the IFR was likely close to constant, we do not incorporate a time-varying IFR in our model.

Conditional on these assumptions and a sampling model for deaths given infections, we fit the parameters of an underlying ODE model of epidemic dynamics to the observed death data. We emphasize that our analysis should be read as ‘if the infection fatality rate is X, then the following would be true,’ rather than primarily as advocating for a particular value of the IFR over others.

As a side-effect of fitting epidemic curves and evaluating the impact of interventions, our model allows us to make projections for the likely trajectory of infections and deaths. However, exact projections is not the primary goal of this work. A model focused more on precise predictions would likely include location-specific measures of containment and treatment efficacy, as well as age- and comorbidity-specific infection fatality rates. In order to minimize the prediction error, it would also account for reporting artifacts, such as delays in recording of deaths in quickly updated records over weekends or holidays. It would also account for increased mortality if the sick overwhelm hospital systems. It would also emphasize finding external data sources that provide strong leading indicators of future deaths. This kind of information could be included in a more prediction-optimized version of the model. Our focus here is to outline a modeling approach using minimal but relatively reliable data that are described by a likelihood and priors that incorporate our understanding of the data-generating process, is fitted to data, and is underpinned by a widely used epidemiological model that is designed to approximate the real dynamics of disease spread. It also turns out that this model predicts reasonably well over a two-week time horizon—the only prediction that we assess here.

The remainder of this article is organized as follows. In section 2 we review related work. In section 3 we introduce our model and explain how the model can be used to infer the number of infections and the effect of executive orders for social distancing. In section 4 we describe an Markov chain Monte Carlo (MCMC) algorithm for fitting our model. In section 5 we give results. In section 6 we conduct a sensitivity analysis. Section 7 concludes.

Several previous studies have attempted to model the dynamics of the pandemic in various geographic locations and with varying goals. Several of these have provided some analysis or estimate of the amount by which confirmed cases undercount the true number of infections. For example, R. Li et al. (2020) propose that in the first month of the epidemic in China, 82–90% of infections were undocumented. Riou et al. (2020) use a susceptible-exposed-infected-removed (SEIR) model, an epidemiological ordinary differential equation (ODE) model, and calibrate their model to the time series of reported deaths and reported infections. By modeling the underreporting of symptomatic cases, and by assuming that approximately half of infections lead to symptomatic cases, they estimate the infected population in Hubei, finding that approximately 30% of infections were documented. Perkins et al. (2020) estimate directly that in the United States, more than 90% of infections have been undocumented by tests using Chinese data and initial reports in the United States. The first wave of a random sampling design in Indiana conducted by researchers at the University of Indiana and the Indiana State Department of Health concluded in a preliminary report that the number of positive tests undercounted the number of people ever infected by a factor of 9.6 (Menachemi et al., 2020).

Ferguson et al. (2020, March 16) model the effect of transmission between susceptible and infectious individuals using a microsimulation model built on synthetic populations designed to mimic the populations of the United Kingdom and United States. They assume a fixed time-to-onset and a range of

The CHIME app (Weissman et al., 2020) is an online tool created by researchers at the University of Pennsylvania to help hospitals anticipate the number of incoming COVID-19 patients and their needs. The CHIME model uses the current number of COVID-19 hospitalizations to ‘back out’ the total number of cases based on a user-provided hospitalization rate conditional on infection. Similar to our work, their model does not rely on the case counts. They make forward projections for the number of hospital admissions, ICU admissions, and ventilators needed over the coming weeks. They allow the user to specify the parameters of their underlying epidemiological model as inputs in terms of the doubling time for the infected population.

The model given in Murray (2020) has a goal similar to CHIME (hospital use planning). This model takes a different approach by fitting parametric curves to observed cumulative death rates. They use a hierarchical model on the parameters of the parametric curve. The model essentially projects that the future course of the cumulative death rate curve in the United States will follow a path similar to that observed in other locations that are farther along in the course of their epidemics. There is no underlying model of epidemic dynamics, but there is a sampling model for death rates, which allows them to give confidence intervals. The research team developing this model has updated the details of their estimation procedure several times since their model was made public. Instead of an arbitrary curve-fitting procedure, the current version as of writing fits an underlying SEIR model (Institute for Health Metrics & Evaluation, 2020). The *New York Times* online tool allows the user to specify inputs to understand how those inputs affect likely infections, hospital loads, and deaths; infections are a side-effect of the rest of the model.

Flaxman et al. (2020) take a similar approach to ours to evaluate the effect of social distancing orders across countries in Europe. They define a likelihood for death-only data and incorporate mitigation efforts in the model. They build a hierarchical model to estimate the number of infections and effects of mitigation across countries in Europe. The overall approach is similar, but their analysis is done for Europe, whereas here we consider the United States. Lewnard et al. (2020) similarly evaluate the effect of mitigation efforts. They observe reductions in estimates of the effective reproduction number for patients in three hospital systems in Northern California, Southern California, and Washington State as a consequence of the implementation of nonpharmaceutical interventions, like social distancing. Song et al. (2020) describe a statistical software package that can be used to estimate the impact of quarantine protocols on the spread of COVID-19, and they apply their method to data from China.

Let *new* infections on day

$(3.1) \ \ \ \ X(t,t') \mid p, \theta, \nu_t \sim \mathop{\mathrm{Poisson}}(p \nu_t \theta_{(t'-t)}).$

The observed deaths on day

$(3.2) \ \ \ \ D(r) = \sum_{t=1}^r X(t,r),$

the sum over all previous days of the number of individuals infected on that day who went on to die on day

$(3.3) \ \ \ \ D(r) \mid p, \theta, \nu \sim \mathop{\mathrm{Poisson}}\left(p \sum_{t=1}^r \nu_t \theta_{(r-t)}\right),$

and for

The observed number of deaths

$(3.4) \ \ \ \
\begin{aligned}
\frac{d s_t}{d t} &= \begin{cases} -\beta s_t i_t & t < T_1 \\ -\phi \beta s_t I_t & t \ge T_1 \end{cases} \\
\frac{d i_t}{d t} &= \begin{cases} \beta s_t i_t - \gamma i_t & t < T_1 \\ \phi \beta s_t i_t - \eta \gamma i_t & t \ge T_1 \end{cases} \\
\frac{d r_t}{d t} &= \begin{cases} \gamma i_t & t < T_1 \\ \eta \gamma i_t & t \ge T_1 \end{cases},
\end{aligned}$

where

Up to time

The SIR model is a stylized mathematical model of the spread of infectious disease, and—like any model of a complex process—is an approximation of a more complicated reality. In particular, there are several ways in which its compartments only approximately represent distinct subpopulations in the real world. For example, although the

These limitations notwithstanding, we elect to use the SIR model to underpin our likelihood because it is among the simplest options that generate epidemic simulations incorporating the key facets of an epidemic. Specifically, the SIR model manifests herd immunity, rapid growth in the early phase, and is constrained to produce a finite number of infections (Hethcote, 2000). The main purpose of incorporating a compartmental model into our method is to have a realistic, low-dimensional generative model of the new infection curves,

Figure 1 shows one realization of our SIR model. The top panel shows one draw of the daily number of deaths. Notice that this is not a smooth curve. When fitting our model, these *new* infections. The series of

Due to identifiability constraints, we do not estimate all of the parameters of the model. Rather, we fix those parameters for which there exists relatively higher quality information on reasonable values applicable in the context studied here. By and large, this includes those parameters pertaining more to the biology of the disease than to the social dynamics of its spread, which we expect to vary more widely across cultural and geographic contexts.3 These include

For

Because of its importance in the undercount analysis and the considerable uncertainty surrounding the value of the infection fatality rate

We base the lower bound of values considered on data from the United States. As of May 18, 2020, the New York City Health website reported 20,806 deaths for which the deceased tested positive for COVID-19 or COVID-19 was listed as the cause of death on their death certificate. Compared with an estimated population of about 8.4 million people, this gives an *overall* mortality rate of about 0.25%. Given that each day there are more reported COVID-19 deaths in New York City, and that it is unlikely that every person in New York City has already been infected, this offers a compelling lower bound on the infection fatality rate in the United States.

This range of values is consistent with other external sources of information. One high-quality source is reported case fatality rates in countries that have done aggressive testing and contact tracing, including testing asymptomatic individuals. In these places, the case numbers may approach the true number of infections, so case fatality information in these locations provides a reasonably tight upper bound on infection fatality rates. Here, we look to South Korea, which has done the most testing per capita. As of a May 17, 2020 press release from the Korea Centers for Disease Control and Prevention (Korea Centers for Disease Control and Prevention, 2020, May 17), there were 11,050 confirmed positive cases and 262 deaths in South Korea. This gives a naive estimate of the case fatality rate of 2.4%.

For each of these data sources, differences in underlying age and income structure, comorbidities, and other risk factors could limit the generalizability of these estimates to the United States at large. For example, the age distribution on the Diamond Princess cruise ship skews considerably older than the U.S. population in general with about 58% of its passengers 60 years of age or older, while only about 16% of the U.S. population is 62 years of age or older.4 The Diamond Princess IFR might be higher than the IFR for the U.S. population in general, as older age is a known risk factor for COVID-19 (Zhou et al., 2020). However, the fact that the people on the Diamond Princess were able to travel may indicate that these people suffer fewer other comorbidities on average than their similarly aged U.S. cohort. IFR estimates from other geographic regions, such as South Korea, may be limited by similar demographic and risk-factor differences relative to the United States. Additionally, the IFR is influenced by the quality and availability of medical care. Data from locations where medical care is unavailable or inadequate, or where the health care system has been overwhelmed by a surge of COVID-19 patients, may overestimate the IFR relative to situations where quality care is available to anyone who needs it. Such factors are difficult to adjust for, since data on risk factors is preliminary and limited and health care availability can change dramatically if the number of infections in an area explodes.

The differences between the context from which we draw our estimates of

With all of these qualifications, we cannot be certain that any given value of

The parameters we estimate in our model are *R* compartment while still infected with the virus—we use these as rough guidance. We choose the prior

We place a *some* effect, as there was a significant increase in the proportion of the population staying home. Furthermore, based on this, we also believe that transmissions were not completely eliminated, as there still remain a significant proportion of the population interacting outside of their homes.

Finally, we use a

We carry out computation by MCMC using the adaptive Metropolis algorithm (Haario et al., 2001). The algorithm produces samples from the Bayesian posterior distribution of the parameters of our model,

$(4.1) \ \ \ \ \xi^* \sim N(\xi, c_0 \Sigma_0),$

where

$(4.2) \ \ \ \ \xi^*_t \sim N(\xi, c_1 \hat \Sigma_t),$

where

$(4.3) \ \ \ \
\ell(\nu(\xi), \xi) = \sum_{t=1}^T p(D(t) \mid \nu) + \log(\pi(\xi)),$

where

To obtain the initial value of the proposal covariance

We fit our model to the daily number of deaths for several states in the United States: California, Florida, New York, and Washington. We use the state-level data compiled by the *New York Times*. The last date in our training data is April 30, 2020. We base the state populations on 2018 US state population estimates from *World Population Review*, which pulls data from the U.S. Census. We take the date of implementation of social distancing to be the first day on which restaurants and schools were both closed statewide by executive order, as recorded in a GitHub repository maintained by researchers at the University of Washington. This definition is of course somewhat arbitrary, but given limitations of the available data, it is difficult to conceive of a richer model that would allow the effects of social distancing to phase in gradually without making similarly strong assumptions about how much each type of measure is ‘worth’ compared to the eventual statewide lockdowns that were implemented everywhere.

Figure 3 shows model fit for each of the states for

Figure 3 includes data through May 13, 2020, the last 13 days of which were not used for training. The values shown after April 30, 2020 (marked with a vertical line in Figure 3) are projections from the model and associated 95% pointwise posterior credible intervals. For Florida, New York, and Washington, the projections are fairly accurate over this 2-week time span and the moving average largely falls within the intervals. For California, the model predicts a slowly moving increase in deaths, whereas the data in that time period appear to have reached a plateau. This discrepancy could occur if the true time to death distribution were slightly longer than that indicated by our chosen

Table 1 shows posterior means and 95% posterior credible intervals of preintervention

CA | FL | NY | WA | |
---|---|---|---|---|

.002 | 2.99 (1.91,3.83) | 3.25 (2.59,3.77) | 3.96 (3.87,4.00) | 3.06 (2.37,3.39) |

.005 | 2.85 (1.45,3.79) | 3.19 (2.45,3.68) | 3.91 (3.70,4.00) | 2.86 (2.13,3.20) |

.01 | 3.13 (1.82,3.88) | 3.19 (2.32,3.68) | 3.89 (3.63,4.00) | 2.72 (2.05,3.06) |

.015 | 3.12 (2.06,3.83) | 3.21 (2.56,3.70) | 3.88 (3.65,4.00) | 2.66 (1.99,2.99) |

.025 | 3.11 (1.99,3.82) | 3.21 (2.49,3.69) | 3.81 (3.52,3.99) | 2.56 (1.94,2.88) |

One quantity estimated by our model is the cumulative number of SARS-CoV-2 infections over time. Comparing these estimates with the reported number of confirmed cases by state allows us to estimate the extent by which confirmed cases of COVID-19 undercount the number of infections. We define the undercount at each time point to be our estimated number of cumulative infections as of that time point divided by the cumulative number of confirmed cases at that time. That is, the undercount is the multiplicative factor by which the recorded number of confirmed cases underestimates the true number. Figure 4 shows the undercount for each state across time for each value of

CA | FL | NY | WA | |
---|---|---|---|---|

.002 | 43.30 (39.29,47.72) | 31.92 (28.04,35.88) | 44.90 (44.30,45.46) | 33.77 (30.87,36.92) |

.005 | 17.60 (15.95,19.45) | 12.89 (11.16,14.49) | 19.25 (18.89,19.59) | 13.59 (12.29,14.88) |

.01 | 8.96 (8.05,10.02) | 6.47 (5.68,7.30) | 9.78 (9.58,9.96) | 6.86 (6.23,7.52) |

.015 | 5.99 (5.36,6.67) | 4.30 (3.77,4.84) | 6.55 (6.41,6.68) | 4.59 (4.16,5.04) |

.025 | 3.58 (3.23,3.97) | 2.59 (2.26,2.93) | 3.93 (3.84,4.01) | 2.77 (2.51,3.05) |

These estimated undercounts—particularly those for

New York State also recently conducted a seroprevalence study. In this study, they found that about 21.2% of New York City residents tested positive for having had the disease. Applying this to New York City’s population of 8.4 million people, this results in an estimate of about 1.8 million infections in the New York City compared to the number of reported cases at the time of about 153,000. This implies an undercount factor of a little over 11. Compared with around 11,460 reported deaths at the time of the release of this study, this also implies a raw IFR of about

These kinds of estimates can also help inform discussions about herd immunity. For example, in New York as of April 30, there had been about 311,000 cases. In our

Executive orders were issued and social distancing policies began taking effect in parts of the United States around March 15, and thus as of the writing of this article, enough time has now passed since the orders in at least some states that the effects will be visible in the deaths data. Recalling the time from infection to death distribution from Figure 2, changes in the dynamics of new infections should begin to be visible in death data around two weeks following the onset of the policy change, and should be mostly visible by around four weeks. This makes it now appropriate to attempt to estimate the effect of such social distancing orders on infection dynamics. In the SIR model, the number of infected individuals will grow when the rate of new infections is higher than the rate of removal and will decrease otherwise. In other words, the rates are equal when the time derivative of

$(5.1) \ \ \ \ \begin{aligned}
\phi \beta s_t i_t &= \eta \gamma i_t \\
\frac{\phi \beta s_t}{\eta \gamma} &= 1 \equiv \rho_t.
\end{aligned}$

We thus focus on estimating the quantity

Table 3 shows estimates of the posterior mean of

CA | FL | NY | WA | |
---|---|---|---|---|

.002 | 1.06 (1.01,1.13) | 1.00 (0.86,1.14) | 0.44 (0.43,0.44) | 0.71 (0.63,0.79) |

.005 | 1.07 (1.02,1.16) | 1.01 (0.87,1.16) | 0.63 (0.62,0.64) | 0.73 (0.64,0.82) |

.01 | 1.10 (1.04,1.20) | 1.02 (0.88,1.17) | 0.68 (0.67,0.69) | 0.75 (0.66,0.83) |

.015 | 1.10 (1.04,1.19) | 1.02 (0.88,1.17) | 0.69 (0.68,0.71) | 0.75 (0.67,0.84) |

.025 | 1.10 (1.04,1.18) | 1.03 (0.88,1.18) | 0.70 (0.69,0.72) | 0.77 (0.68,0.85) |

Our model could be sensitive to various types of misspecification, including incorrect specification of

We have so far presented results for different values of

We assign a prior on

To obtain results, we run the MCMC algorithm described in section 4 for

Undercount | |||
---|---|---|---|

CA | 3.09 (1.89,3.87) | 1.09 (1.03,1.19) | 9.00 (6.33,13.29) |

FL | 3.20 (2.46,3.68) | 1.02 (0.88,1.17) | 7.00 (4.54,9.75) |

NY | 3.89 (3.64,4.00) | 0.68 (0.65,0.70) | 10.00 (7.02,14.00) |

WA | 2.74 (2.08,3.09) | 0.75 (0.66,0.83) | 7.00 (4.89,10.20) |

In this section we address sensitivity of our findings to the value we have chosen for

One of the first major interventions outside of China to stem the spread of COVID-19 was the lockdown in Italy on March 9, 2020. Around 20 days later, the number of daily deaths in Italy showed a sustained downward trend. We adjust the

To create a

Using the raw death count data assumes that deaths due to COVID-19 have been accurately recorded. Recent studies on excess mortality suggest that, in some states, there has been underreporting of COVID-19 deaths. For example, Weinberger et al. (2020) calculate excess mortality due to pneumonia and influenza (P&I) from February 9, 2020, to March 28, 2020. They note that in California during this time period, while only 101 COVID-19 deaths were reported, there were 399 excess deaths due to P&I during that same time period. If we interpret all of those deaths to be COVID-related, this implies that that only about one quarter of COVID-19 deaths were reported as such. Similar analysis for Florida and Washington showed reporting rates of about 30% and 100%, respectively. Though New York did not show underreporting when comparing only to pneumonia and influenza deaths, due to the large volume of cases there, the authors noted a reporting rate of 60% in New York City and 40% in New York State (excluding New York City) based on all-cause excess mortality. We assume a rough estimate of underreporting for New York State in its entirety is around 50%.

Based on these findings, we inflate the number of observed deaths on each day by a factor of four, three, two, and one for California, Florida, New York, and Washington, respectively. By inflating the data in this way, this implicitly assumes that excess mortality during this time period is all due to COVID-19 infections. While many of the excess deaths during this time period may be the result of the environment caused by COVID-19 (e.g., not going to the hospital for other illnesses out of fear of contracting COVID-19, reduction in medical services available to non COVID-19 patients, and saturation of emergency medical services), it is unlikely that all are directly attributable to infection itself. This calculation also assumes that the rate at which COVID-related deaths were not attributed to COVID-19 in official records is constant across time. As testing capacity has expanded over time and the estimates we use are based on data only through March 28, this is unlikely to be true. Because not all excess mortality is due to COVID-19 disease, these two cases (raw death counts presented as our main results and the excess mortality-inflated death counts presented here) likely bracket the true number of deaths. However, more updated estimates of P&I excess mortality do not seem to be available. Even all-cause excess mortality estimates from the CDC are lagged by a few weeks.

Results from applying our model with the alternative

CA | FL | NY | WA | |
---|---|---|---|---|

multiplier | 2.98 (1.73,3.74) | 3.38 (2.93,3.66) | 3.94 (3.81,4.00) | 2.72 (2.00,3.06) |

alternative theta | 3.26 (2.00,3.97) | 3.43 (2.40,3.98) | 3.58 (2.87,3.99) | 2.58 (1.72,3.04) |

^{Note}^{: Rows correspond to two alternative scenarios considered: death counts adjusted using an excess mortality multiplier and alternative value of }^{ used. In both cases, }^{.}

Table 6 gives estimates of

CA | FL | NY | WA | |
---|---|---|---|---|

multiplier | 1.06 (1.02,1.11) | 1.07 (0.97,1.16) | 0.63 (0.63,0.64) | 0.75 (0.66,0.84) |

alternative theta | 1.09 (1.05,1.14) | 1.02 (0.91,1.19) | 0.78 (0.77,0.80) | 0.83 (0.75,0.91) |

^{Note}^{: Rows correspond to two alternative scenarios considered: death counts adjusted using an excess mortality multiplier and alternative value of }^{ used. In both cases, }^{.}

Table 7 shows the estimated factor by which current testing undercounts the actual number of infections as estimated by our model. This is analogous to Table 2 in the main text. Unsurprisingly, we found that inflating the number of COVID-19 deaths results in higher estimates of the total infected and, consequently, a larger undercount factor (relative to results for

CA | FL | NY | WA | |
---|---|---|---|---|

multiplier | 34.69 (32.93,36.79) | 19.97 (18.58,21.31) | 19.31 (19.07,19.56) | 6.85 (6.22,7.50) |

alternative theta | 8.51 (7.84,9.19) | 6.20 (5.46,7.01) | 8.87 (8.72,9.03) | 6.93 (6.28,7.64) |

^{Note}^{: For both alternative scenarios, }^{.}

In all but one of the calculations under alternative scenarios, we find that the alternative parameters do not substantively change the results. The only exception to this is the undercount factors, which—unsurprisingly—exhibit an increase when we add a multiplier to the time series to account for possible undercounting. However, early evidence suggests that accounting for under-reporting would change the death counts by at most a factor of 4, and this is likely a significant overestimate for the reasons outlined above. Since increasing the death counts with

We have built a model for the transmission of SARS-CoV-2 using only information that has a reasonable chance of being measured correctly: the observed number of daily deaths, timing of containment measures, and information on the clinical progression of the disease. In contrast to models that use a wider range of information that may be less precisely measured, we believe the main strengths of our approach are that it prioritizes simplicity, interpretability, and identifiability. Because not all parameters one might want to estimate in this setting are separately identifiable, we have carefully specified our model so that all estimated parameters are identifiable conditional on quantities for which we have high-quality auxiliary information. Our model also has a proper likelihood and is fit to data in the Bayesian paradigm, rather than relying on ad hoc calibration of model parameters to produce trajectories resembling the observed data. This allows us to formally account for uncertainty in all of our estimates by posterior credible intervals. Our model is underpinned by an SIR model of infection dynamics to link observed deaths to the underlying unobserved infections. It would not be difficult with our approach to substitute some other model in place of the SIR. In particular, any other compartmental model, such as an SEIR model could be used, or the model could be elaborated to incorporate more change points in the parameters of the compartmental model to account for fine grained policy analysis.7

We estimate that official case counts substantially undercount the number of infections. This is not a surprise. Despite recent increases in testing capabilities, our analysis suggests that testing continues to undercount the number of infections by a factor between 6 and 20 for

Seroprevalence studies done in two states in the United States result in similar conclusions with respect to undercount. Such studies have only recently become available and do not give snapshots of undercount over time. We believe that the correspondence between our estimates and serological studies that have since been done points to the utility of such a model for estimating key quantities of interest (such as the extent of undercounting), especially early in the epidemic when testing capabilities are yet built up.

While the seroprevalence studies using random design are just now reporting results, our initial estimates were first posted in a preprint in late March. This suggests that approaches similar to what we’ve described here could be useful in providing indications of the undercount and the total number of infections of emerging diseases in the early days of its spread. During this time, when the number of cases is relatively low and the rate of spread is nearly exponential, accounting for the time lag between infection and death as we do here can lead to substantial differences in estimates of the total number of cases relative to naive methods, such as simply dividing the total number of deaths by the IFR.

Our model also suggests wide variability in the initial transmissibility of the disease prior to intervention policies as well as wide variability in the effects of those policies. For example, we estimate that New York had the highest

All of these projections are conditioned on the current policies staying in place and their effects on transmission remaining constant. Tightening or relaxing mitigation policies, or ‘quarantine fatigue’ leading to a decrease in compliance by the public, would impact the trajectory of the spread. Estimates of

We thank Alexander D’Amour for finding a misstated equation in an earlier version. We thank Amy Herring and David Dunson for comments on an early draft.

This work was supported by grants to the Human Rights Data Analysis Group by the John D. and Catherine T. MacArthur Foundation and the Oak Foundation. Kristian Lum and James Johndrow acknowledge funding from National Institute of Environmental Health Sciences (NIEHS) grant 3R01ES028804-03S.

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Figure A1 shows representative trace plots of model parameters for

California | Florida | New York | Washington | |
---|---|---|---|---|

T0 | 1.013 (1.035) | 1.002 (1.003) | 1.001 (1.001) | 1.001 (1.002) |

R0 | 1.011 (1.028) | 1.002 (1.004) | 1.001 (1.001) | 1.001 (1.002) |

1/Gamma | 1.012 (1.03) | 1.003 (1.004) | 1 (1.001) | 1.001 (1.002) |

Phi | 1.004 (1.008) | 1.004 (1.006) | 1.001 (1.002) | 1.001 (1.002) |

Eta | 1.006 (1.012) | 1.012 (1.023) | 1.001 (1.002) | 1.005 (1.01) |

MPSRF | |
---|---|

California | 1.03 |

Florida | 1.01 |

New York | 1.00 |

Washington | 1.00 |

We conduct a simulation study to assess sensitivity of the method to various assumptions. We analyze five simulation cases. In the first four cases, the daily new infections are obtained from the two-period susceptible-infected-recovered (SIR) model in (3.4), but we explore several ways in which other assumptions can be violated. The parameters of the SIR model are:

The model is properly specified.

We use the alternative (incorrect) value of

$\theta$ displayed in Figure 5 when statistically inferring.We assume that

$T_1$ is one week too early (corresponding to March 11 instead of the true value of March 18) when statistically inferring.We assume that

$T_1$ is one week too late (corresponding to March 25 instead of the true value of March 18) when statistically inferring.The true new infection series

$\nu$ is generated by a susceptible-exposed-infected-removed (SEIR) model with two time periods rather than an SIR model with two time periods, but otherwise the model is correctly specified. The SEIR model is a popular alternative to the SIR model for modeling SARS-CoV-2. We choose the parameters of the SEIR model to give a death series similar to that for the other four cases.

Figure A2 shows the simulated data used in Cases 1–4 (left column) and Case 5 (right column). The top panels show deaths, the middle row of panels shows new infections, and the bottom row shows the state variables of the ODE model (SIR in the left column and SEIR in the right column) through a period of 121 days (corresponding to the dates January 1, 2020, through April 30, 2020).

Total Infections | |||
---|---|---|---|

1 | 3.54 (2.82,3.98) | 0.72 (0.70,0.74) | 11.57 (11.23,11.91) |

2 | 3.40 (2.30,3.98) | 0.83 (0.82,0.86) | 10.70 (10.51,10.90) |

3 | 1.77 (1.61,2.14) | 0.92 (0.92,0.92) | 10.63 (10.44,10.81) |

4 | 2.87 (2.63,3.33) | 0.48 (0.45,0.51) | 10.79 (10.54,11.06) |

5 | 3.38 (2.45,3.98) | 0.74 (0.72,0.77) | 12.19 (11.85,12.54) |

We report estimates for

Overall, the results suggest that, although some types of misspecification can lead to poor estimates of

©2020 James Johndrow, Patrick Ball, Maria Gargiulo, and Kristian Lum. This article is licensed under a Creative Commons Attribution (CC BY 4.0) International license, except where otherwise indicated with respect to particular material included in the article.