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Figure.  Projected vs Observed COVID-19 Hospitalizations Before and After Stay-at-Home Orders, March 10 Through April 28, 2020
Projected vs Observed COVID-19 Hospitalizations Before and After Stay-at-Home Orders, March 10 Through April 28, 2020

Blue lines indicate observed cumulative hospitalizations (including those currently hospitalized and those discharged) up to each day; select values are displayed for clarity. Dashed red lines begin on the first day of available reporting by each state and are the best-fit exponential curves for cumulative hospitalizations for the fitting period: first day of reporting up to and including the median effective date (panel A: y = 3.5829 exp(0.23599t), R2 = 0.9734; B: y = 7.521 exp(0.1876t), R2 = 0.96445; C: y = 18.8482 exp(0.2268t), R2 = 0.9798; D: y = 15.932 exp(0.1397t), R2 = 0.99444). Shaded regions indicate the 95% prediction bands of the exponential growth curves. Because the median incubation period of coronavirus disease 2019 (COVID-19) was reported to be 4 to 5.1 days3,4 and the median time from first symptom to hospitalization was found to be 7 days,5 it was hypothesized that any association between stay-at-home orders and hospitalization rates would become evident after 12 days (median effective date).

Table.  Cumulative Hospitalizations Due to COVID-19 in Colorado, Minnesota, Ohio, and Virginia, March 10 Through April 28, 2020
Cumulative Hospitalizations Due to COVID-19 in Colorado, Minnesota, Ohio, and Virginia, March 10 Through April 28, 2020
1.
Karaca-Mandic  P, Georgiou  A, Sen  S. Calling all states to report standardized information on COVID-19 hospitalizations. Health Affairs blog. April 7, 2020. Accessed April 17, 2020. https://www.healthaffairs.org/do/10.1377/hblog20200406.532030/full/?cookieSet=1
2.
University of Minnesota COVID-19 Hospitalization Tracking Project website. Accessed April 30, 2020. https://carlsonschool.umn.edu/mili-misrc-covid19-tracking-project
3.
Lauer  SA, Grantz  KH, Bi  Q,  et al.  The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application.   Ann Intern Med. 2020;172(9):577-582. doi:10.7326/M20-0504PubMedGoogle ScholarCrossref
4.
Guan  WJ, Ni  ZY, Hu  Y,  et al; China Medical Treatment Expert Group for Covid-19.  Clinical characteristics of coronavirus disease 2019 in China.   N Engl J Med. 2020;382(18):1708-1720. doi:10.1056/NEJMoa2002032PubMedGoogle ScholarCrossref
5.
Wang  D, Hu  B, Hu  C,  et al.  Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus–infected pneumonia in Wuhan, China.   JAMA. 2020;323(11):1061-1069. doi:10.1001/jama.2020.1585PubMedGoogle ScholarCrossref
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    3 Comments for this article
    EXPAND ALL
    Faulty Statistical Method
    Joe Gibson, PhD, MPH | Marion County Public Health Dept., Dir. of Epidemiology
    To compare two fitted curves using their confidence intervals, both curves should be based on similar sample sizes, and both curves should be fitted starting at their start. You are not comparing two fitted curves for each state, which would be the proper approach here. You are fitting one curve, and testing whether the start of that curve fits better than the end of that curve. In fact you appear to fit the curve based on its start, and then show that the start does fit better than the (un-fitted) 2nd part of the curve.

    For each state, you
    should fit one curve as you have, but end it at the date of the stay-at-home order. Let's assume that is one month later. Then start the 2nd curve at the date of the stay-at-home order, and fit it based on the same time period - i.e., one month, _and_ the same number of observations. So, for that 2nd time period, randomly select the same number of observations as you had for the pre-stay-at-home order curve, matching observations per day from day zero of that curve.

    Then you will have two curves, both with the same number of observations (so similar CI widths), which you can fit with the same method, and compare with a method that is not biased against the fit of the 2nd curve.
    CONFLICT OF INTEREST: None Reported
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    Letter to the Editor
    Arnab Ghosh, MD, MSc, MA | Weill Cornell Medical College of Cornell University
    Stay-at-home orders have been important public health tools to stop the spread of COVID-19 in the United States and abroad. Therefore we read with interest the study by Sen at al. which examined the association between COVID-19 hospitalizations and the governors’ stay-at-home orders in four states (Colorado, Minnesota, Ohio, and Virginia). The authors argue that, because cumulative hospitalizations in these states deviate from fitted exponential projections starting the day after stay-at-home orders were declared in each, those orders may be credited with the reduction in COVID caseload.

    Unfortunately, the use of exponential functions in this setting is erroneous for
    two reasons. First, fitting cumulative data in this manner is mathematically incorrect since the exponential function relates primarily to change in time of the number of susceptible individuals moving to the exposed (and then on to infected and possibly recovered) stage of illness (i.e., the incidence, not prevalence). Fitting this function to cumulative data will bias any subsequent analysis assessing persistence of such an extreme trajectory in biological systems (including human health).

    Second and more important for the point the authors suggest, even if there were an exponential relationship in the earliest days of an outbreak, that is no reason to assume that cases should continue to accrue in a log-linear relationship. A more accurate mathematical rendering would follow the classic susceptible-infected-removed (SIR) model which employs logistic growth assumptions for the COVID-19 cases (and hospitalizations) (1). In this case, the rate of new COVID-19 cases for a given population would decrease linearly with the number of new COVID-19 cases, reaching an inflection point rather than increasing exponentially, even in the absence of interventions. This potential error of using exponential functions for asymptotically constrained natural systems (i.e., fixed population sizes) was first pointed out by Verhulst in 1845 (2).

    Therefore, this present article should be regarded with caution in evaluating the impact of stay-at-home orders on COVID-19 hospitalization rates.

    Arnab K. Ghosh MD, MSc, MA, FACP
    Nathaniel Hupert MD, MPH, FACP

    References
    2. Kermack, W. O. and McKendrick, A. G. "A Contribution to the Mathematical Theory of Epidemics." Proc. Roy. Soc. Lond. A 115, 700-721, 1927.
    3. Verhulst, Pierre-François (1845). "Recherches mathématiques sur la loi d'accroissement de la population" Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles. 18: 8.
    CONFLICT OF INTEREST: None Reported
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    Response to Comments
    Soumya Sen, PhD | University of Minnesota
    The curves of all 4 states were fitted based on similar sample sizes from the start of reporting through the median effective date of the stay-at-home order. The actual hospitalization numbers in each state closely follow an exponential growth curve and then deviate from this curve within a few days of the median incubation period. In each state the deviation reflects a slower hospitalization growth rate and occurs almost after the same time since the stay at home order was implemented in that state. In an unpublished analysis, we fitted a second exponential curve for each state starting at the issuance of the stay-at-home order in that state to the end of the study period; these curves also showed a slower rate of growth in comparison to the best-fit exponential curve of the period from the start of reporting to the issuance of stay-at-home orders. Moreover, in analysis of South Dakota, a state that did not have a stay-at-home order, COVID-19 hospitalizations numbers continued to follow an exponential growth function closely throughout the study period without any deviation like those observed for the 4 states reported here. These analyses were not included due to the space constraints.

    Exponential growth rates in cumulative data from meta-population models were reported in the Western Africa Ebola virus epidemic. We do not intend to suggest that the hospitalizations will continue to grow exponentially for an infinite duration; rather, exponential or sub-exponential rates are most often observed in the initial stages of a pandemic. And as reported in the paper, we found that the exponential function fitted the actual hospitalization numbers better than linear growth models. The main association we are highlighting in this study is that in all four states, deviation from the initial exponential growth consistently occurred after approximately 12 days of issuance of stay-at-home orders (and remained below the initial rates thereafter). The publication highlighted the limitation of assigning causal interpretation to this observation.
    CONFLICT OF INTEREST: None Reported
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    Research Letter
    May 27, 2020

    Association of Stay-at-Home Orders With COVID-19 Hospitalizations in 4 States

    Author Affiliations
    • 1Department of Information and Decision Sciences, University of Minnesota Carlson School of Management, Minneapolis
    • 2Department of Finance, University of Minnesota Carlson School of Management, Minneapolis
    • 3Starkey Hearing Technologies, Eden Prairie, Minnesota
    JAMA. 2020;323(24):2522-2524. doi:10.1001/jama.2020.9176

    In analyses of the effectiveness of response measures to the outbreak of coronavirus disease 2019 (COVID-19), most studies have used the number of confirmed cases or deaths. However, case count is a conservative estimate of the actual number of infected individuals in the absence of community-wide serologic testing. Death count is a lagging metric and insufficient for proactive hospital capacity planning. A more valuable metric for assessing the effects of public health interventions on the health care infrastructure is hospitalizations.1 As of April 18, 2020, governors in 42 states had issued statewide executive “stay-at-home” orders to help mitigate the risk that COVID-19 hospitalizations would overwhelm their state’s health care infrastructure. This study assessed the association between these orders and hospitalization trends.

    Methods

    In March 2020, we began collecting data on cumulative confirmed COVID-19 hospitalizations from each state’s department of health website on a daily basis.2 Among states issuing a statewide stay-at-home order, we identified states with at least 7 consecutive days of cumulative hospitalization data for COVID-19 (including patients currently hospitalized and those discharged) before the stay-at-home order date and at least 17 days following the order date. Because the median incubation period of COVID-19 was reported to be 4 to 5.1 days3,4 and the median time from first symptom to hospitalization was found to be 7 days,5 we hypothesized that any association between stay-at-home orders and hospitalization rates would become evident after 12 days (median effective date). States included in this sample were Colorado, Minnesota, Ohio, and Virginia. Among the 4 states meeting the inclusion criteria, the earliest date with data on hospitalizations was March 10. All states were observed through April 28. We fit the best exponential growth function to cumulative hospitalization data in each state for dates up to and including the median effective date of that state’s stay-at-home order. We computed 95% prediction bands on the exponential fit line to determine if the observed number of hospitalizations fell within the interval. We then examined whether the observed cumulative hospitalizations for dates after the median effective date deviated from the projected exponential growth in cumulative hospitalizations. In an additional analysis, a linear growth function was fit to cumulative hospitalization data for dates up to and including the median effective date, and goodness of fit was assessed with an R2 comparison. All analyses were performed using Microsoft Excel version 14.1.

    Results

    In all 4 states, cumulative hospitalizations up to and including the median effective date of a stay-at-home order closely fit and favored an exponential function over a linear fit (R2 = 0.973 vs 0.695 in Colorado; 0.965 vs 0.865 in Minnesota; 0.98 vs 0.803 in Ohio; 0.994 vs 0.775 in Virginia) (Table). However, after the median effective date, observed hospitalization growth rates deviated from projected exponential growth rates with slower growth in all 4 states. Observed hospitalizations consistently fell outside of the 95% prediction bands of the projected exponential growth curve (Figure).

    For example, Minnesota’s residents were mandated to stay at home starting March 28. On April 13, 5 days after the median effective date, the cumulative projected hospitalizations were 988 and the actual hospitalizations were 361. In Virginia, projected hospitalizations 5 days after the median effective date were 2335 and actual hospitalizations were 1048.

    Discussion

    In 4 states with stay-at-home orders, cumulative hospitalizations for COVID-19 deviated from projected best-fit exponential growth rates after these orders became effective. The deviation started 2 to 4 days sooner than the median effective date of each state’s order and may reflect the use of a median incubation period for symptom onset and time to hospitalization to establish this date. Other factors that potentially decreased the rate of virus spread and subsequent hospitalizations include school closures, social distancing guidelines, and general pandemic awareness. In addition, economic insecurity and loss of health insurance during the pandemic may have also decreased hospital utilization. Limitations of the study include that these other factors could not be modeled in the analysis and that data on only 4 states were available.

    Section Editor: Jody W. Zylke, MD, Deputy Editor.
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    Article Information

    Corresponding Author: Pinar Karaca-Mandic, PhD, University of Minnesota Carlson School of Management, 321 19th Ave S, Minneapolis, MN 55455 (pkmandic@umn.edu).

    Accepted for Publication: May 13, 2020.

    Published Online: May 27, 2020. doi:10.1001/jama.2020.9176

    Author Contributions: Drs Sen and Karaca-Mandic had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.

    Concept and design: All authors.

    Acquisition, analysis, or interpretation of data: All authors.

    Drafting of the manuscript: All authors.

    Critical revision of the manuscript for important intellectual content: All authors.

    Statistical analysis: All authors.

    Obtained funding: Sen, Karaca-Mandic.

    Administrative, technical, or material support: Georgiou.

    Conflict of Interest Disclosures: Dr Karaca-Mandic reported receiving personal fees from Tactile Medical, Precision Health Economics, and Sempre Health and grants from the Agency for Healthcare Research and Quality, the American Cancer Society, the National Institute for Health Care Management, the National Institute on Drug Abuse, and the National Institutes of Health. Dr Georgiou reported receiving personal fees from HealthGrades. Dr Sen reported no disclosures.

    Funding/Support: This research uses publicly available data from the University of Minnesota COVID-19 Hospitalization Project, which is partially funded by the University of Minnesota Office of Academic Clinical Affairs and United Health Foundation.

    Role of the Funder/Sponsor: The funders had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, or approval of the manuscript; and decision to submit the manuscript for publication.

    Additional Contributions: Yi Zhu, MA, University of Minnesota Carlson School of Management, contributed to this project by launching the University of Minnesota COVID-19 Hospitalization Project website. Mr Zhu received no compensation for his contributions.

    Additional Information: A preliminary plot of Minnesota’s current hospitalization numbers (not cumulative hospitalizations) with data prior to April 15 was shared by the authors on Twitter and LinkedIn.

    References
    1.
    Karaca-Mandic  P, Georgiou  A, Sen  S. Calling all states to report standardized information on COVID-19 hospitalizations. Health Affairs blog. April 7, 2020. Accessed April 17, 2020. https://www.healthaffairs.org/do/10.1377/hblog20200406.532030/full/?cookieSet=1
    2.
    University of Minnesota COVID-19 Hospitalization Tracking Project website. Accessed April 30, 2020. https://carlsonschool.umn.edu/mili-misrc-covid19-tracking-project
    3.
    Lauer  SA, Grantz  KH, Bi  Q,  et al.  The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application.   Ann Intern Med. 2020;172(9):577-582. doi:10.7326/M20-0504PubMedGoogle ScholarCrossref
    4.
    Guan  WJ, Ni  ZY, Hu  Y,  et al; China Medical Treatment Expert Group for Covid-19.  Clinical characteristics of coronavirus disease 2019 in China.   N Engl J Med. 2020;382(18):1708-1720. doi:10.1056/NEJMoa2002032PubMedGoogle ScholarCrossref
    5.
    Wang  D, Hu  B, Hu  C,  et al.  Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus–infected pneumonia in Wuhan, China.   JAMA. 2020;323(11):1061-1069. doi:10.1001/jama.2020.1585PubMedGoogle ScholarCrossref
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