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    3 Comments for this article
    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.
    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

    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.
    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.
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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.