key: cord-171703-n22tr8f2
authors: Hanmo, Li; Mengyang, Gu
title: Robust estimation of SARS-CoV-2 epidemic at US counties
date: 2020-10-22
journal: nan
DOI: nan
sha: 
doc_id: 171703
cord_uid: n22tr8f2

The COVID-19 outbreak is asynchronous at US counties. Mitigating the COVID-19 transmission requires not only the state and federal level order of protection measures such as social distancing and testing, but also public's awareness of the time-dependent risk and reactions at county and community levels. We propose a robust approach to estimate the heterogeneous progression of SARS-CoV-2 at all US counties having no less than 2 COVID-19 associated deaths, and we develop the daily probability of contracting (PoC) SARS-CoV-2 for a susceptible individual to quantify the risk of SARS-CoV-2 transmission in community. We found that shortening only $5%$ of the infectious period of SARS-CoV-2 can reduce around $39%$ (or $78$K, $95%$ CI: $[66$K $, 89$K $]$) of the COVID-19 associated deaths in the US as of 20 September 2020. Our findings also indicate that the reduction of infection and deaths by shortened infectious period is more pronounced for areas with the effective reproduction number close to 1, suggesting that testing should be used along with other mitigation measures, such as social distancing and facial mask wearing, to reduce the transmission rate. Our deliverable includes a dynamic county-level map for local officials to determine optimal policy responses, and for public to better understand the risk of contracting SARS-CoV-2 on each day.

The outbreak of new coronavirus 2019 (COVID-19) has caused nearly 200,000 deaths in the US, and among those, there are 2,277 counties with no less than 2 associated deaths as of 20 September 2020 1 . The ongoing COVID-19 pandemic has led to unprecedented non-pharmaceutical interventions (NPIs), including travel restrictions, lockdowns, social distancing, facial masks wearing, and quarantine to reduce the spread of SARS-CoV-2 in the US. The COVID-19 outbreak is prolonged and asynchronous across regions. It is thus of critical importance to estimate the dynamics of COVID-19 to determine appropriate protective measures before the availability of effective vaccine.

A non-negligible proportion of SARS-CoV-2 infectious individuals is asymptomatic or have mild symptoms 2 . We term the individuals the active infectious individuals who can transmit the disease to others, but may not be diagnosed yet. Identifying the number of active infectious individuals is crucial to monitor the transmission in a community. Another important timedependent quantity is the expected number of secondary cases resulted from each active infectious individual, or effective reproduction number. In this article, we estimate these two time-dependent quantities for all US counties with no less than 2 COVID-19 associated deaths; the population of some counties that falls within this category is even less than ten thousand. Furthermore, based on these two time-dependent quantities, we develop a more interpretable measure, called the daily probability of contracting (PoC) SARS-CoV-2 for an individual at county-level. The fine-grain estimation of disease progression characteristics allows public to understand the risk of contracting COVID-19 on a daily basis.

Predictive mathematical models are useful for analyzing an epidemic to guide policy responses 3 . The epidemiology compartmental models such as SIR, SEIR, SIRD, and their extensions 4-7 , stochastic agent based models 8, 9 , branching processes 10 , and network analysis 11 have advanced our understanding of transmission rates and incubation period of SARS-CoV-2, which are connected to the traffic flow and mobility during the COVID-19 outbreaks at different regions 12, 13 . The disease progression characteristics, such as the transmission rate, are often estimated based on the daily death toll 4, 7-9 . The increase of death associated with COVID-19 in small regions, however, is often temporally sparse, making it challenging to robustly estimate the progression of the epidemic in most US counties. Meanwhile, the COVID-19 observed confirmed cases (henceforth, observed confirmed cases) from the test data may significantly underestimate the population that have contracted the SARS-CoV-2. It was found in 14 that around 9.3% of the US individuals (or roughly 30 million) may have contracted the COVID by July 2020 based on serology tests, whereas less than 4.8 million COVID-19 positive cases have been confirmed in the US prior to August 2020 1 . The significant difference between the estimated prevalence and the number of observed confirmed cases in the US thus requires an estimate of the number of individuals who contracted COVID-19 but have not been tested positive. The focus herein is on integrating the death toll and test data to obtain a robust estimation of the disease Figure 1 . a, The SIRDC model and the data used for analysis. b, c 7-day death toll forecast and 21-day death toll forecast against the held-out truth in 2,277 US counties with no less than 2 deaths as of 20 September 2020. Each dot is a cumulative death toll for one county at one held-out day. The counties from the same state are graphed with the same color.

progression characteristics of COVID-19 at county and community levels.

One critical quantity to evaluate an infectious disease outbreak is the time-dependent transmission rate, based on which one can compute the basic reproduction number and the effective reproduction number of the disease. Various approaches were proposed to estimate this parameter. The transmission rates are modeled as a decreasing function of the time in 4 , a function of NPIs in 8 and a geometric Brownian motion in 15 . Unlike the outbreak in China or other northeastern countries in Asia, the transmission rates of the COVID-19 progression in the US does not monotonically decrease due to the prolonged duration of the outbreak, and it is challenging to determine a suitable parametric form of this parameter in terms of time. In 7 , the transmission rate parameter was related to the initial values of infectious cases, resolving cases and up to two derivatives of the daily death toll. This method provides a flexible way to estimate the time-dependent transmission rate from the death toll and its derivatives; however, the method is unstable for a county with a moderate or small population size, as the numerical estimation of the derivatives of daily death toll is often unstable.

In this work, we propose a robust approach of integrating test data and death toll to estimate COVID-19 transmission characteristics by a Susceptible, Infectious, Resolving (but not infectious), Deceased and reCovered (SIRDC) model initially studied in 7 . We illustrate that the transition between different stages of disease progression in the SIRDC model in part a of the Figure 1 . First, a part of the population are infected by the active infectious individuals each day, depending on the transmission rate parameter (β t ). After γ −1 day, an active infectious individual is expected to be no longer infectious, denoted by the resolving compartment, meaning that this individual will not transmit COVID-19 to others as a result of hospitalization or self-quarantine. We term the average length of an active infectious individual the infectious period. A resolving case is expected to be resolved (either recovered or deceased) after θ −1 day. The proportion of deaths from the number of resolved cases is controlled by the fatality rate parameter δ .

The innovation of our approach is on solving the compartmental models using a midpoint rule with a step size of 1 day, discussed in the method section, as the confirmed cases and death toll are updated daily in most US counties. Our estimate of transmission rates and reproduction numbers is robust and accurate to reproduce the number of death toll and other compartments for counties with medium to small population sizes (Figure 3 and Extended data Figure 1 ). The simulated studies also suggest that our approach is more robust than the solution in 7 (Extended Data Figure 2 ), as our solution does not require estimating derivatives of the daily death toll. Instead, the test data and death toll are used for estimation. Only two parameters, the initial values of the number of active infectious individuals and the number of resolving cases, are needed to be estimated numerically for each county, whereas the time-dependent transmission rates and all other compartments can be solved subsequently. Since only two parameters are estimated for each county, our estimation rarely depends on the initial values we choose for the optimization. A summary of the main findings, limitations and policy implication are given in Table 1 . The transmission of SARS-CoV-2 is heterogeneous and asynchronous in US counties. It is thus important to assess the risk before lifting or replacing any mitigation measure in the community. We have developed a novel approach to integrate test data and death toll to estimate probability of contracting COVID-19, as well as the time-dependent transmission rate and the number of active infectious individuals at the county level in the US. Main findings and limitations National level order of protection measures reduce the transmission rate and active number of infectious individuals for most US counties in April, whereas the risk of contracting SARS-CoV-2 rebounded between late June and early July, as the protection measures were relaxed. We found that when the infectious period of SARS-CoV-2 is shortened by 5% and 10%, the number of deaths can be reduced from 199K to 120K (95% CI: [109K, 132K ]) and 80K (95% CI: [72K, 89K]) as of 20 September 2020, respectively, when other protection measures were kept the same. The reduction of infectious period can be achieved by extra testing in addition to the ongoing protection measures. Our model relies on the existing knowledge of the COVID-19 and model assumptions. Other information, such as demographic profiles, mobility and serology test data can be used to calibrate the model parameters and assumptions at the community level.

Our model indicates that extra testings, along with the current NPIs, can significantly reduce the number of deaths associated with COVID-19. The estimated probability of contracting COVID-19 can be used as an interpretable risk factor to guide policy responses in community. 

We first verify our model performance by forecast at the county level. The 7-day and 21-day death projections for 2, 277 US counties using data by 20 September 2020, for instance, are close to the held out test death toll in these counties, shown in part b and part c of Figure 1 . The R 2 and Pearson correlation coefficient (ρ) are larger than 0.999 for both 7-day and 21-day forecast. The 21 day forecast of each considered county in Florida and California using observations by 20 September 2020 is provided in Extended Data Figure 6 and 7, respectively. The forecast of the death toll based on our model is accurate for most US counties, and around 95% of the held out test data is covered by nominal 95% predictive interval (Table S1 in supplementary materials), indicating that the uncertainty assessment is accurate. Based on the robust estimation of transmission rates, we derived the county-level estimation of daily PoC SARS-CoV-2 on each day. We classify the daily PoC SARS-CoV-2 in a community into 5 levels listed in Table 2 . On 20 September, out of the 2,277 US counties, only 60 counties were at the controllable level and 311 counties were at the moderate level, whereas 1906 counties were at the either alarming, strongly alarming or hazardous level. The daily PoC SARS-CoV-2 measures the average probability to contract SARS-CoV-2 for a susceptible individual in a community, and the risk varies from individuals to individuals. Nonetheless, the PoC SARS-CoV-2 is an interpretable measure for public understanding of the average risk of contracting SARS-CoV-2 in a community on a given day.

We graph the estimated PoC SARS-CoV-2 of an individual at US counties on 20 April and 20 September in Figure 2 . On 20 April, the PoC SARS-CoV-2 is large in northeastern regions, and in some southern states such as Arizona, New Mexico and New Orleans. On 20 September, the PoC SARS-CoV-2 is large in many inland states, for instance, Montana, North Dakota, Mississippi and Alabama. Although the PoC SARS-CoV-2 on 20 September in northeastern regions is substantially lower than that on 20 April, the probability of contracting SARS-CoV-2 for an individual is large in most other states on 20 September, suggesting that the relaxation of protection measures can lead to more population contracting COVID-19, and consequently more deaths at a rate no slower than that in late April.

The daily PoC SARS-CoV-2 can be used by officials to determine where mitigation policies can be lifted or replaced by other measures for different regions. The probability of contracting COVID-19 in many counties in Texas on 20 September, for example, is larger than those in Washington (part (a) and (d) in Figure 3 ), indicating that more protective measures should be undertaken in Texas to reduce the risk. The nationwide lockdown order and social distancing in spring effectively reduced the PoC SARS-CoV-2 in 4 out of 5 counties in Washington, while the PoC SARS-CoV-2 of all counties increases in late June and early July, as some of the nonpharmaceutical interventions (NPIs) were lifted (part b in Figure 3) . Part (c) shows that the model fit to death toll. With only two parameters estimated numerically for each county, the fit is reasonably good for these counties at a wide range of dates. In comparison, though the outbreak of 5 counties in Texas started in early summer, the PoC SARS-CoV-2 in these Texas counties is much higher on 20 September than the one in Washington (part (e) in Figure 3 ). Our model also fits the death toll of the counties in Texas relatively well (part (f) in Figure 3 ).

The effectiveness of protection measures were studied to reduce the transmission rate 5, 6, 8, 9, 11, 16 , whereas the efficacy of these measures depends on the reactions from the public, which is likely to vary from region to region. Another simultaneous effort to mitigate the spread of COVID-19 outbreak is through testing and contacting tracing, which reduces the infectious period, and consequently, the number of active infectious individuals. For Washington and Texas, we simulate the model output with infectious period reduced by 5% (or equivalently 4.75 days in total), while the transmission rate (β t in SIRDC model) is March 2020 to 20 September 2020 with infectious period assumed to be 5 days (blue), 4.75 days (green) and 4.5 days (red). c, the estimate overall death toll in the US. The time period and interpretation of c are aligned with a and b, except that the black dots in c stand for the observed death toll in the US. held the same. We found that the PoC SARS-CoV-2 is reduced by 5 times for 12 counties out of 28 considered counties in Washington and 6 counties out of 209 considered counties in Texas, as shown in the Extended Data Figure 3 . Furthermore, when we reduce the infectious period by 10% (or equivalently 4.5 days in total), while the transmission rate (β t in SIRDC model) is held the same, the PoC SARS-CoV-2 is reduced by 5 times for 26 counties in Washington and 146 counties in Texas, shown in Extended Data Figure 4 .

We graph the estimated effective reproduction number, the number of active infectious individuals and cumulative death toll in the US, along with the simulated values when the average infectious period is reduced to 4.75 days and 4.5 days, from 5 days in Figure 4 . First we found that mitigation measures in March effectively reduce the effective reproduction number to below 1, whereas the value rebounded in summer after some of these measures were relaxed in different regions. Consequently, US has experienced two waves of the outbreak in terms of the number of active infectious individuals (part (b) in Figure 4 ). The high test positive rate at the beginning of the epidemic (Extended Data Figure 5) indicates that a substantial number of active infectious individuals were not diagnosed in April due to the lack of diagnostic tests. According to our estimates, the peak of the first wave in April is larger than that of the second wave in July in terms of the number of active infectious individuals, whereas the peak of the daily observed confirmed cases in April is smaller than that of the second wave in July (Extended Data Figure 5 ).

Second, the simulated results suggest that a shortened infectious period of SARS-CoV-2 by 5% and 10% can reduce the total deaths from 199K to 120K (95% CI: [109K, 132K]) and 80K (95% CI: [72K, 89K]), respectively, as of 20 September 2020, when other protection measures were held as the same (part c in Figure 4 ). Note that since we held the transmission rate parameter (β t ) to be the same (a scenario that the public adhere to the protection measure same as the reality), the effective reproduction number changes very little (part a in Figure 4 ). However, the slightly shortened infectious periods of SARS-CoV-2 can reduce the active infectious individuals substantially (part b in Figure 4 ), as the number of active infectious individuals decreases.

We found that a shortened infectious period substantially reduces the number of active infectious individuals and fatality in the second wave, whereas the change is smaller in the first wave, since the effective reproduction number in the second wave is smaller than that in the first wave ( Figure 4 ). The county level estimation also validates this point (Extended Data Figure 3 and 4). This finding indicates that the efforts of shortening the infectious period of SARS-CoV-2 should not replace the other protection measures, such as social distancing and facial mask wearing to reduce the transmission rate.

Diagnostic tests can be used to shorten the length of infectious period of an active infectious individual. Drastically reducing the infectious period may not be possible without contact tracing, which is challenging when a large number of infection pairs exist. Reducing infectious period by around 5%, in comparison, can be achieved by periodically diagnostic tests every 20 day for each susceptible individual. A more efficient way is to test susceptible individuals with a high risk of contracting or spreading SARS-CoV-2, such as individuals with more daily contacts or have contacts with vulnerable population, e.g. workers from senior living facilities. Our estimation of contracting SARS-CoV-2 can be used as a response to develop regression models using covariates including demographic information and mobility to elicit personalized risk of contracting SARS-CoV-2 for susceptible individuals. Furthermore, efforts on reducing the length of infectious period should not replace other protection measures for reducing transmission rates of SARS-CoV-2, as the number of active infectious individuals and death toll can only 5/18 be effectively reduced, if the effective reproduction number is not substantially larger than 1.

Our study have several limitations. First, our findings are based on available knowledge and model assumptions, as with all other studies. One critical parameter is the death rate, assumed to be 0.66% on average 17 , whereas this parameter can vary across regions due to the demographic profile of the population and available medical resources. The studies of prevalence of SARS-CoV-2 antibodies based on serology tests 14 can be used to determine the size of population that have contracted SARS-CoV-2, and thus provides estimates on death rate, as the death toll is observed. Besides, we assume the infected population can develop immunity since recovery at least for a few months, which is commonly used in other models. The exactly duration of post-immunity, however, remains unverified scientifically. Third, we assume that the number of susceptible individuals, and, consequently, the number of individuals that contracted SARS-CoV-2 can be written as a function of the number of observed confirmed cases and test positive rates, calibrated based on the death toll. More information such as the proportion of population adhere to the mitigation measures, mobility and demographic profile can be used to improve the estimation of susceptible individuals in a region.

Our results can be used to mitigate the ongoing pandemic by SARS-COV-2 and other infectious disease outbreak in future. The estimated daily PoC SARS-CoV-2 at the county level, for example, is an interpretable measure to understand the risk of contracting COVID-19 on a daily basis, and a surveillance marker to determine appropriate policy responses. Further studies of this measure relative to different mobility, demographic information and social economic status of individuals can provide more precise guidance for local officials to protect vulnerable population from contracting SARS-CoV-2, when an effective vaccine is not available.

The SIRDC model for the jth county in the ith state in the US is described below:

where S i, j (t), I i, j (t), R i, j (t), D i, j (t) and C i, j (t) denote the number of individuals at these 5 compartmental groups on day t, respectively, and N i, j denotes the number of individuals in the j county at the ith state for i = 1, 2, ..., k, j = 1, 2, ..., n i with n i being the number of counties of the ith state considered in the analysis and t = 1, 2, ..., T i, j . The time-dependent transmission rate parameter is denoted by β i, j (t) and the inverse of average number of days an infectious individual can transmit the COVID-19 is denoted by γ. The inverse of the average number of dates for a case to get resolved (i.e. deceased or recovered) is denoted by θ and the proportion of deceased cases (i.e. death rate) is denoted by δ . The parameters (γ, θ , δ ) were invariant over time and held fixed in this study. Following 16 , we assume the infectious period to be 5 day on average and a case is expected to resolve after 10 days. The average death rate is assumed to be 0.66% 17 . Additional verification of these assumptions and a sensitivity analysis of these parameters are provided in the supplementary materials.

To determine the characteristics of the SARS-CoV-2 epidemic at US counties, we define the time-dependent effective reproduction number, i.e. the average number of secondary cases per primary cases as R i, j e f f (t) = R i, j 0 (t)S i, j (t)/N i, j , where the R i, j 0 (t) = β i, j (t)/γ denotes the basic reproduction number on day t. When R i, j e f f (t) < 1, it means that the number of the active infectious individuals will decrease (and vice versa, if R i, j e f f (t) > 1). The effective reproduction number was often used to quantify whether or not the disease is under control 18 . The effective reproduction number, however, does not directly quantify risk of contracting SARS-COV-2 for a susceptible individual, as the number of active infectious individuals in a region was not taken into consideration. We compute the average probability of contracting (PoC) SARS-CoV-2, denoted as P i, j (t) = R i, j e f f (t)I i, j (t)γ/(S i, j (t)) = β i, j (t)I i, j (t)/N i, j , which quantifies the risk of a susceptible individual in county j from state i to catch SARS-CoV-2 on day t. Here the risk is in an average sense among all susceptible individuals in a region.

The most critical parameter of the SIRDC is the transmission rate parameter, β i, j (t), as a function of time, based on which we have reproduction numbers on day t. To estimate the time dependent transmission rate for communities with small 6/18 population sizes, we derive a more robust estimation of transmission rate of each county based on death toll and testing data, discussed below.

Closed-form expressions of time-dependent transmission rates. Since the observations such as death toll and confirmed cases are typically updated daily, we approximate the solution of the ODEs of the SIRDC model in (1) by the midpoint rule of the integral with a step size of 1 day. For day t ∈ N + , the approximation is described below:

Further by assuming the transmission rate parameter β i, j (t) is day-to-day invariant (i.e. a step function with step size 1). Based on equations (2) and (3), we obtain β i, j (t + 0.5) from t = 1 to T i, j − 1, iteratively, based on the sequence of susceptible individuals {S i, j (t)} T i, j t=1 and the initial number of active infectious individuals I i, j (1) described in algorithm 1.

S 1 = S 2 ; S 2 = S i, j (t + 1); I 1 = I i, j (t + 1); end Algorithm 1: Iterative approach for estimating transmission rate β i, j (t + 0.5). After we get the number of active infective individual (I i, j (t)) on each day, the sequences of the resolving, deceased and recovered compartments can be solved subsequently, following the same manner using equation (4)-(6), after specifying their initial values. Expressing the time-dependent transmission rate by the number of susceptive and infective cases is the key to integrate death toll and testing data for estimation.

In Extended Data Figure 1 and 2, we demonstrated that the solution by our approach is more accurate and more robust to solve the ODEs of SIRDC model than the method in 7 for both simulated and real scenarios. Other more accurate methods (such as the Runge-Kutta method) exist to solve the ODEs of SIRDC model, whereas the time-dependent transmission rates are not easily expressed as a function of the death toll and the number of active infectious individuals as the way they are in our solution.

Estimation of the number of susceptible individuals. Note that we have S i, j (t) + c o i, j (t) + c u i, j (t) = N i, j for any t, where c o i, j (t) and c u i, j (t) are the number of cumulative observed confirmed cases and unobserved confirmed cases, respectively. Estimating the number of susceptible individuals is equivalent to estimating the number of unobserved confirmed cases c u i, j (t), because the number of observed confirmed cases c o i, j (t) and the population N i, j are known. Here we combine them with the positive test rates to estimate c u i, j (t), as large positive test rates typically indicate a large number of unobserved confirmed cases. We assume that the total number of confirmed cases is equal to the observed confirmed cases, adjusted by the state level test positive rate p i (t), a power parameter α i and a weight parameter ω i , leading to the following formula of the susceptible population:

where ∆c o i, j (t) is the observed daily confirmed cases on day t, for t = 1, 2, ..., T i, j , i = 1, 2, ..., k and j = 1, 2, ..., n i . Since the positive test rates are only available at the state level, the power parameter α i ∈ [0, 2] is estimated by the state-level observations. According to equation (7), the time-invariant weight ω i, j can be expressed below:

where I i, j (1), R i, j (1), D i, j (1) and C i, j (1) are the number of active infectious, resolving, deceased and recovered cases on day 1, respectively.

Estimation of initial values of infectious and resolving cases. We define day 1 of a county as the more recent date between 21 March 2020 and the date that the county has 5 observed confirmed cases for the first time. Since all counties were at an early stage of the epidemic on the starting day, we let the initial value of death toll D i, j (1) to be the observed death toll on day 1, and the initial value of the recovered cases to be 0. This assumption is not likely going to influence our analysis, as the number of recovered cases is only a negligible proportion of the susceptible individual on the starting day, if not zero. The only parameters to estimate are the number of infectious individuals I i, j (1) and the number of resolving cases R i, j (1) on day 1 for the jth county in the ith state, after the power parameter α i is estimated using the state-level observations to minimize the same loss function below:

where the upper bound U i, j is chosen to guarantee the estimated number of the susceptible cases S i, j (t) to be larger than 0:

Iterative updates for the fitted compartments and reproduction rates. After the initial values of infectious and resolving cases are estimated, we obtain the estimation of the susceptible cases from equation (7), and the infectious cases and transmission rates on each date for each county from Algorithm 1. The resolving cases, deaths and recovered cased can be derived subsequently from equation (4)-(6), respectively. The estimated basic and effective reproduction rates can be derived by the fitted time-dependent transmission rate, and the estimated probability of contracting for an individual COVID-19 can be computed based on transmission rate and number of infectious individuals for each county on each day.

Forecast. Our method can also be used as a tool for forecasting compartments (e.g. death toll), reproduction numbers and the probability of contracting COVID-19 at each county for a short time period. We extrapolate the transmission rate based on Gaussian processes implemented in RobustGaSP R package 19 with robust parameter estimation 20, 21 . Based on the extrapolated transmission rates, the compartments can be solved iteratively based on equation (2)- (6) .

We also found the forecast will generally be improved by assuming the cumulative deaths are modeled as D i, j (t) = D i, j (t) + z i, j (t), where z i, j is a zero-mean Gaussian process. The details of the forecast method, the comparison and uncertainty assessment of the forecast are presented in the supplementary materials.

Supplementary materials for robust estimation of SARS-CoV-2 epidemic at US counties The supplementary materials contain two parts. In the first part, We discuss the details of model parameter specification and conduct a sensitive analysis. The forecast algorithm and uncertain assessment are introduced in the second part.

We discuss the choice of model parameter and its sensitivity analysis. The following parameters of the SIRDC model were specified based on previous studies.

• The death rate or the infection fatality ratio (δ ) measures the proportion of death among all infected individuals. We assume δ = 0.66% following 17 .

• The inverse of the number of days an infectious individual can transmit the COVID-19 (γ). The average time of a COVID-19 patient to transmit disease is assumed to be 5 days in 16 , indicating that γ = 0.2. Another evidence comes from the study of incubation period. The latent period (exposed but not contagious) for COVID-19 is found to be 3.69 days on average 2 and the mean incubation period (time from infection to onset of symptoms) is 5.2 days 22 , meaning that infectious period being around 1.5 days before the onset of symptom. The diagnosis test could take less than one day to up to week. Assuming 3.5 days to get the result of a diagnosis test on average. The total infectious period is around 5 day.

• The inverse of the number of dates for resolving case to get resolved (θ ). According to the CDC report 23 , for mild and moderate symptom, the replication-competent virus has not been recovered after 10 days following symptom onset, indicating the individuals remains infectious no longer than 10 days after symptom onset. The infectious period could be as long as 20 days for patients with more severe illness from COVID-19 infection. Since the majority of the COVID-19 infection is mild to moderate, we assume the infectious period to be 13.5 days and after reducing 3.5 days from onset of the symptom to become resolving (after quarantine or hospitalized), it takes around 10 days for a resolving case to resolved on average.

We conduct a sensitive analysis to examine the change of the estimation in 4 different configuration.

• (Configuration 1) (γ, θ , δ ) = (0.2, 0.1, 0.0066), the default parameter set.

• (Configuration 2) (γ, θ , δ ) = (0.14, 0.1, 0.0066). The average length of infectious period changes from 5 days to 1 0.14 ≈ 7 days, whereas other parameters are held unchanged.

• (Configuration 3) (γ, θ , δ ) = (0.2, 0.067, 0.0066). The average length of resolving period changes from 10 days to 1 0.067 = 15 days, whereas other parameters are held unchanged.

• (Configuration 4) (γ, θ , δ ) = (0.2, 0.1, 0.0075). The infection fatality ratio changes from 0.66% to 0.75%, whereas other parameters are held unchanged.

After specifying the parameters (γ, θ , δ ), the transmission rate β (t) can be obtained from algorithm 1. Figure S1 gives result of the sensitive analysis. First we found the estimated death toll for 4 scenarios is almost the same (part d in Figure  S1 ). Extending the infectious period from 5 to 7 days (Configuration 2) increases the number of active infectious individuals and effective reproduction number shown in part a and part c in Figure S1 , respectively. Consequently, the peak of average daily PoC SARS-CoV-2 slightly increases in the first wave, whereas the scale of increase is smaller than the change in the effective reproduction number and active infectious individuals. The average daily PoC SARS-CoV-2 has almost no change for other periods, indicating that the length of the average infectious period has almost no influence of our estimation of PoC SARS-CoV-2 for most of the days.

Second, when the average length of resolving period changes from 10 to 15 days, the peak of PoC SARS-CoV-2, effective reproduction number and the number of active infectious individuals increases in the first wave, whereas these quantities remain largely unchanged for the rest of the days (part a-c in Figure S1 ). The result indicates that the average length of resolving period also barely affects the estimated characteristics of COVID-19 progression for most of the days.

When the death rate increases from 0.66% to 0.75%, the effective reproduction number seems to have almost no change (part a in Figure S1 ), whereas the PoC SARS-CoV-2 and the number of active infectious individuals (figure S1 part b-c) both reduce. This is because when the death rates increases, the number of individuals infected decreases, as the death toll is observed (and thus fixed). The death rate is a key parameter to calibrate and the studies of prevalence of SARS-CoV-2 antibodies based on serology tests 14 can be used to estimate the death rate in each state.

In conclusion, the parameter values of the average length of infectious period and the average length of resolving period do not change the COVID-19 progression characteristics for most of the days, including the fitted death toll. On the other hand, even the effective reproduction number and the fitted death toll has almost no change when the death rate changes, the number of active infectious individuals and the daily PoC SARS-CoV-2 depend critically on the death rate parameter. Further studies of prevalence would be useful for estimating the death rate parameter in different regions.

The epidemiological SIRDC model lead to a robust fit for most counties (considering only two parameters to estimate for each county in the model). The performance of forecast can be improved by adding a Gaussian Process (GP) to model the residual between the death toll and the estimate from the SIRDC model. GP is widely used to model temporal or spatio-temporal correlated observations. One advantage of a GP model is the internal assessment of the uncertainty of the forecast from the predictive distribution, which is of crucial importance. The aggregated model that combines the SIRDC model and the GP model for the jth county in the ith state in the US is described as follows.

where D i, j (t) and F i, j (t) denote the observed death toll and estimated death toll via the SIRDC model, respectively; The noise follows independently as a Gaussian distribution ε i, j,t ∼ N(0, σ 2 i, j,0 ) with variance parameter σ 2 i, j,0 . The latent temporal process z i, j (t) is modeled by a zero-mean GP, meaning that for time points {1, 2, . . . , T i, j }, z i, j = (z i, j (1), . . . , z i, j (T i, j )) T follows a multivariate normal distribution:

where the (l, m) entry of Σ Σ Σ i, j is parameterized by a covariance function σ 2 i, j K i, j (l, m) for 1 ≤ l, m ≤ T i, j . Here σ 2 i, j is the variance parameter and K i, j (·, ·) is a one-dimensional correlation function. We use the power exponential correlation function:

where a is the roughness parameter fixed to be 1.9 as in other studies 24, 25 , to avoid possible singularity in inversion of the covariance matrix using the Gaussian correlation (a = 2), and b i, j is a range parameter for each county estimated from the data.

Data: c o i, j , D i, j , p i , c o i , and D i . Result: Estimates of county-level epidemiological compartmentsβ β β i, j ,Ŝ i, j ,Î i, j ,R i, j ,Ĉ i, j , forecastD * i, j , wherê D * i, j := D i, j (T i, j + 1), . . . ,D i, j (T i, j + T * ) T , and the uncertainty assessment of the compartments.

Step 1 Conduct a three-parameter constrained optimization treating state-level power parameter α i unknown to minimize the loss function in equation (9) using p i , c o i and D i .

Step 2 For each county, set initial values I i, j (1) = R i, j (1) = 1, 000, C i, j (1) = 0 and D i, j (1) to be the observed death toll on day 1. Find the optimized values of I i, j (1) and R i, j (1) to minimize equation (9) .

Step 3 Simulate S = 500 samples of the observed confirmed cases sampled from the predictive distribution of a GP model of the observed confirmed cases. For each sample, obtain the other compartments and time-dependent transmission rate by equation (1)-(5) and algorithm 1 using the estimate of the initial values.

Step 4 Extrapolate the time-dependent transmission rate parameters from a GP model for each sample and obtain S = 500 samples of output death toll of the SIRDC at the forecast period.

Step 5 Sample the residuals from the predictive distribution (S2) at the forecast period and obtain S = 500 samples of the ensemble forecast for the death toll. Compute the mean for forecast and 95% predictive interval to quantify the uncertainty.

Algorithm S1: Ensemble forecast and uncertainty assessment.

We define the nugget parameter η i, j = σ 2 i, j,0 /σ 2 i, j . The range parameter b i, j , and the nugget parameter η i, j in equation (S1) are estimated based on the marginal posterior mode estimation using the rgasp function in the package RobustGaSP available on CRAN 20 .

After marginalizing out the variance parameter by the reference prior p(σ 2 i, j ) ∝ 1/σ 2 i, j , for any t * , the predictive distribution of z i, j (t * ), conditional on the observations, range parameter b i, j and nugget parameter η i, j , follows a non-central Student's t-distribution with degrees of freedom T i, j

by plugging in the estimated range parameter b i, j and nugget η i, j . The predictive meanẑ i, j (t * ) for forecast the death toll of the jth county in the ith state at a future day t * , and the predictive interval can be computed based on the quantiles and percentiles of the Student's t distribution.

An overview of our algorithm for the forecast the uncertainty assessment is given in algorithm S1, where the inputs are the county-level observed cumulative number of confirmed cases To evaluate the performance of our approach, we implement 7-day and 21-day forecasts using our approach and other approaches on 2,277 US counties with training period from 21 March 2020 to 20 September 2020, and with the forecast period starting from 21 September 2020. To compare the prediction performance of different methods, we computed the rooted mean square error (RMSE), proportion of the observations covered in the 95% predictive interval (P CI (95%)) and length of the 95% Table S1 . 7-day and 21-day forecast for 2,277 US counties with training period from 21 March 2020 to 20 September 2020 and with prediction period starting from 21 September 2020. Four methods are compared. Our proposed approach that combines the SIRDC model and a zero-mean GP to model the residuals is denoted as SIRDC+GP. Second the death forecast by SIRDC model is denoted as SIRDC, which contains Step 1 and 2 in the algorithm S1 and provides point projection of the death toll. Third a GP with constant mean function is denoted as GP without linear trend, which equivalently replaces the SIRDC model of a constant mean parameter estimated by the data for each county. The fourth model is the same as the third method, except that the mean of GP contains constant mean and a linear trend of time with two linear coefficient parameters estimated from the data (denoted as GP with linear trend). The method of the best performance by each criterion is highlighted.

confidence interval (L CI (95%)), defined as follows:

where t * := (T i, j + 1, . . . , T i, j + T * ), T * = 7 and T * = 21 for the 7-day forecast and 21-day forecast, respectively. A model with small RMSE, P CI (95%) close to the nominal 95% and small L CI (95%) are precise for forecast and uncertainty assessment. The comparison between our approach and the other three approaches are recorded in Table S1 . Our approach (denoted in SIRDC+GP) has the lowest RMSE among 4 methods considered herein. Approximately 95% of the held-out death toll are covered by the 95% predictive interval by our approach, indicating our uncertainty assessment is accurate. Although other approaches produce shorter length of the predictive interval, the number of held-out observations contained in the 95% predictive interval is smaller than ours. Therefore, combining the SIRDC model and GP for modeling the residuals can improve the predictive accuracy for forecasting COVID-19 associated death toll at US counties, compared to the one using the SIRDC model or the GP model alone.

Extended data figures for robust estimation of SARS-CoV-2 epidemic at US counties Extended Data Figure 1 . a-c, Comparisons between the estimation COVID-19 progression characteristics for Santa Barbara, CA as of 20 September 2020 by our algorithm 1 (blue solid curves) and the method F&J 7 (red dash curves) . The shaded area represents 95% confidence intervals. The black solid curve in part c is the observed cumulative death toll in Santa Barbara. d-f, Results for Imperial, CA as of 20 September 2020, which have the same interpretation as a-c. The transmission rate estimated from the method F&J is truncated to be within [0,10]. Figure 2 . a-c, Simulated comparison with noise-free observations. The black circles are the solution of the ODEs of the SIRDC model via the default numerical solver Isoda in the function ode in deSolve R package. The green solid and dash curves are the numerical solutions from Runge-Kutta method with the 4th order integration and step size being 1 and 0.1, respectively. The Blue solid curves are the robust estimation from algorithm 1 and red dash curves are the estimation in 7 . In the simulation with noise-free observations, we let time duration be T = 100 days, the population size N = 10 7 , the initial values of 5 compartments chosen as (S(1), I(1), R(1), D(1),C(1)) = (N − 2000, 1000, 1000, 0, 0) and the transmission rate

, results of the simulation with noisy observations, which have the same interpretation as a-c. In this simulation, we set the transmission rate β (t) = exp −0.7( 9 T −1 (t − 1) + 1) + ε, for 1 ≤ t ≤ T and ε ∼ N(0, 0.04), and the other parameters are held the same as in the noise-free simulation. The transmission rates estimated from the method F&J are truncated to be within [0,10]. The solution from our robust estimation approach, the Isoda, the Runge-Kutta method with the 4th order and step size being 0.1 overlap for both scenarios.

Extended Data Figure 3 . a-f, the simulated results of COVID-19 progression in Washington (the first row) and in Texas (the second row) that have the same interpretation as a-f in Figure 3 with the infection period changed from 5 days, to 4.75 days, whereas other parameters are held the same. Figure 4 . a-f, the simulated results of COVID-19 progression characteristics in Washington (the first row) and in Texas (the second row) that have the same interpretation as a-f in Figure 3 with the infection period changed from 5 days to 4.5 days, whereas other parameters are held the same. 9.20310330840854, 9.34076484197512, 9.47495674925274, 9.60598265936292, 9.73441266884615 Apr May Jun Jul Aug Sep Oct 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 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10.257975959 Apr May Jun Jul Aug Sep Oct 18.1256062258264, 18.1643037285639, 18.2053706350613, 18.248516331611, 18.2935193943954, 18.3402007696657, 18.3884081702236, 18.4380066019006, 18.4888724605496, 18.5408898071103, 18.5939480172543, 18.6479403200922, 18.7027629216853, 18.7583145169991, 18.8144960603796, 18.8712107070915, 18.9283638659032, 18.9858633213106) Extended Data Figure 6 . The 21-day prediction in 67 Florida counties with death toll no less than 2 as of 20 September 2020. The training period is from 21 March 2020 to 20 September 2020, whereas the prediction starts from 21 September 2020. The red curves are the cumulative observed death toll from 21 September 2020 to 11 October 2020 and the blue line indicates the forecast for the same period. The shaded area represents the 95% predictive intervals of the forecast for each analyzed county in Florida.

Apr May Jun Jul Aug Sep Oct 0 200 500 Alameda, population=1.67M 8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 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18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520, 18 Apr May Jun Jul Aug Sep Oct 0 40 80 Madera, population=0.16M 3, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504 98, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 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18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474 Mendocino, population=0.09M 368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459 Merced, population=0.28M 8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 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18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475 8.96665339614193, 9.07965619843174, 9.19477525124454, 9.31205406562166, 9.43153360293837, 9.5532519807012, 9.67724452498403, 9.80354397127564 Apr May Jun Jul Aug Sep Oct 0 10 20 Shasta, population=0. 18M   368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18429, 18430, 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18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 8.30557817245775, 8.55149664314108, 8.78962166393395, 9.01966373587258, 9.24134886748988, 9.45442213618 Apr May Jun Jul Aug Sep Oct Extended Data Figure 7 . The 21-day prediction in 50 California counties with death toll no less than 2 as of 20 September 2020. The training period is from 21 March 2020 to 20 September 2020, whereas the prediction starts from 21 September 2020. The red curves are the cumulative observed death toll from 21 September 2020 to 11 October 2020 and the blue line indicates the forecast for the same period. The shaded area represents the 95% predictive intervals of the forecast for each analyzed county in California.

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This research is supported by the UCSB Office of Research COVID-19 seed grant program.

H.L. analyzed data, developed the model, derived mathematical results, wrote computer code, collected results and participated in manuscript writing. M.G. conceptualized the project, analyzed data, developed the model, derived mathematical results, wrote computer code, analyzed results, led manuscript writing.