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Figure.  Epidemic Trajectory Predicted by a Susceptible-Infected-Recovered Model
Epidemic Trajectory Predicted by a Susceptible-Infected-Recovered Model

This model is based on the rates over time of persons moving from the susceptible compartment to the infected compartment to the recovered compartment. The rate of infection is equal to β times the number of individuals in the susceptible and infected compartments, and the rate of recovery is equal to γ times the number of individuals in the infected compartment.

1.
Lourenco  J, Paton  R, Ghafari  M,  et al. Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2. medRxiv. Preprint posted March 26, 2020. doi:10.1101/2020.03.24.20042291
2.
Weissman  GE, Crane-Droesch  A, Chivers  C,  et al.  Locally informed simulation to predict hospital capacity needs during the COVID-19 pandemic.   Ann Intern Med. Published online April 7, 2020. doi:10.7326/M20-1260PubMedGoogle Scholar
3.
Murray  CJL; IHME COVID-19 health service utilization forecasting team. Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilator days and deaths by US state in the next 4 months. medRxiv. Preprint posted March 30, 2020. doi:10.1101/2020.03.27.20043752
4.
Kermack  WO, McKendrick  AG.  A contribution to the mathematical theory of epidemics.   Proc R Soc Lond A. 1927;115:700-721. doi:10.1098/rspa.1927.0118Google ScholarCrossref
5.
Clancy  D, O’Neill  PD.  Bayesian estimation of the basic reproduction number in stochastic epidemic models.   Bayesian Anal. 2008;3(4):737-757.Google ScholarCrossref
6.
Newell  NP, Lewnard  JA, Jewell  BL.  Predictive mathematical models of the COVID-19 pandemic.   JAMA. Published online April 16, 2020. doi:10.1001/jama.2020.6585PubMedGoogle Scholar
7.
Zlojutro  A, Rey  D, Gardner  L.  A decision-support framework to optimize border control for global outbreak mitigation.   Sci Rep. 2019;9(1):2216.PubMedGoogle ScholarCrossref
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    Measurement Errors in Compartmental Models of Epidemics
    Michael McAleer, PhD (Econometrics), Queen's | Asia University, Taiwan
    The excellent JAMA Guide to Statistics and Methods on "Modeling Epidemics With Compartmental Models", specifically the susceptible-infected-recovered (SIR) model, is an invaluable source of information by two experts for the legion of researchers and health care professionals who rely on sophisticated technical procedures to guide them in predicting the number of patients who are susceptible to infection, are infected through transmission, and who will ultimately recovery from infection.

    As in the case of all models, stringent assumptions are imposed.  

    The SIR model is simple and straightforward, with only two parameters to analyse three mutually exclusive and sequential
    groups, or compartments, namely S, I and R, based on disease status.

    The two parameters are the effective contact rate (β), in transitioning from S to I, and the rate of recovery or mortality (γ), in transitioning from I to R, where the transmission durations are assumed to be immediate.

    The purported recovery to a noncontagious state includes both immunity from the disease and death.

    The basic reproduction number (BRN) is the ratio between β and γ, with a decrease in the BRN leading to a “flattening of the curve”.  

    The JAMA Guide lists a number of limitations of the model, including identical probabilities of community contact and social distancing, and alternative guesses (guesstimates?) of the two model parameters, leading to a range of future trajectories.

    This is a serious imperfection in SIR models as such guesses incorporate (possibly large) measurement errors that will lead to biased forecasts and their associated standard errors in obtaining interval estimates of infection rates.

    Moreover, a lack of a distributional foundation to estimate the unknown parameters, and a lack of statistical diagnostic checks of the underlying assumptions, will make it impossible to determine the robustness of the empirical estimates and associated forecasts.
    CONFLICT OF INTEREST: None Reported
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    JAMA Guide to Statistics and Methods
    May 27, 2020

    Modeling Epidemics With Compartmental Models

    Author Affiliations
    • 1Department of Emergency Medicine, Harbor-UCLA Medical Center, Torrance, California
    • 2David Geffen School of Medicine, Department of Emergency Medicine, University of California, Los Angeles
    • 3Predictive Healthcare, University of Pennsylvania Health System, Philadelphia
    JAMA. 2020;323(24):2515-2516. doi:10.1001/jama.2020.8420

    During epidemics, there is a critical need to understand both the likely number of infections and their time course to inform both public health and health care system responses. Approaches to forecasting the course of an epidemic vary and can include simulating the dynamics of disease transmission and recovery1,2 or empirical fitting of data trends.3 A common approach is to use epidemic compartmental models, such as the susceptible-infected-recovered (SIR) model.1,2

    Why Is a SIR Model Used?

    The SIR model aims to predict the number of individuals who are susceptible to infection, are actively infected, or have recovered from infection at any given time. This model was introduced in 1927, less than a decade after the 1918 influenza pandemic,4 and its popularity may be due in part to its simplicity, which allows modelers to approximate disease behavior by estimating a small number of parameters.

    Description of the SIR Model

    In compartmental models, individuals within a closed population are separated into mutually exclusive groups, or compartments, based on their disease status. Each individual is considered to be in 1 compartment at a given time, but can move from one compartment to another based on the parameters of the model. The SIR model is one of the most basic compartmental models, named for its 3 compartments (susceptible, infected, and recovered). In this model, the assumed progression is for a susceptible individual to become infected through contact with another infected individual. Following a period as an infected individual, during which that person is assumed to be contagious, the individual advances to a noncontagious state, termed recovery, although that stage may include death or effective isolation.

    In most modeled epidemics, all of a population begins in the susceptible compartment (Figure), which contains individuals who might become infected if exposed to the pathogen. This implies that no one has immunity to the disease at the beginning of the outbreak. The infected compartment is defined as individuals who have the ability to infect individuals in the susceptible compartment. As such, this compartment includes asymptomatic transmitters of the pathogen as well as hospitalized patients who require intensive levels of care. One simplification in the SIR model is that it does not consider the latent period following exposure, rather it assumes that newly infected individuals are immediately contagious.

    The rate at which susceptible individuals become infected is dependent on the number of individuals in each of the susceptible and infected compartments. At the start of an outbreak, when there are few infected individuals, the disease spreads slowly. As more individuals become infected, they contribute to the spread and increase the rate of infection. An additional factor in calculating the rate of spread is the effective contact rate (β). This parameter accounts for the transmissibility of the disease as well as the mean number of contacts per individual. Community mitigation strategies, such as quarantining infected individuals, social distancing, and closing schools, reduce this value and therefore slow the spread. Although these interventions can alter the movement of individuals from the susceptible compartment to the infected compartment, the transition from the infected to the recovered compartment is solely dependent on the amount of time that an individual is contagious, captured in the rate of recovery (γ).

    The term recovered in the SIR model can be misleading because the recovered compartment does not necessarily refer to an individual’s clinical course of the disease, but instead represents individuals who are no longer contagious. Because compartmental models assume “closed” populations (without migration), individuals who have gained immunity to the disease and those who die of the disease are both included in this compartment.

    The SIR model is defined by only 2 parameters: the effective contact rate (β), which affects the transition from the susceptible compartment to the infected compartment, and the rate of recovery (or mortality; γ), which affects the transition from the infected compartment to the recovered compartment. If the rate at which individuals become infected exceeds the rate at which infected individuals recover, there will be an accumulation of individuals in the infected compartment. The basic reproduction number R0, the mean number of new infections caused by a single infected individual over the course of their illness, is the ratio between β and γ. A decrease in the effective contact rate β through community mitigation strategies decreases R0, delaying and lowering the peak infection rate that occurs in the epidemic (ie, “flattening the curve”). However, to maintain the decrease in total infections, the decrease in R0 generally must be sustained.

    What Are the Limitations of SIR Models?

    The simplicity of the SIR model makes it easy to compute, but also likely oversimplifies complex disease processes. The model does not, for example, incorporate the latent period between when an individual is exposed to a disease and when that individual becomes infected and contagious. In the context of coronavirus disease 2019 (COVID-19), this corresponds to the time it takes for severe acute respiratory syndrome coronavirus 2 to replicate in a newly infected individual and reach levels sufficient for transmission. Extensions of the SIR model, such as the SEIR model (“E” denotes exposed but not yet contagious), account for this parameter, but additional extensions of the model would be necessary to, for example, model the time-dependent introduction of community mitigation strategies.

    The SIR model also makes several simplifying assumptions about the population. It assumes homogeneous mixing of the population, meaning that all individuals in the population are assumed to have an equal probability of coming in contact with one another. This does not reflect human social structures, in which the majority of contact occurs within limited networks. The SIR model also assumes a closed population with no migration, births, or deaths from causes other than the epidemic.

    In addition, the parameters in a traditional SIR model do not allow for quantification of uncertainty in model parameters. The parameter inputs are point estimates, which are single values reflecting the modeler’s best guess. A common strategy in predicting the course of an epidemic is to calculate the SIR model over a few possible values for each parameter. The result is a range of future trajectories, but this strategy does not formally quantify the uncertainty in the predictions. More complex models use distributions for each parameter instead of a point estimate to characterize the probability of various future trajectories.5 If the parameters are not known with any precision, these more complex models will demonstrate the uncertainty in projections. The actual effect of social distancing, for example, is often unknown. It is also possible, in more complex adaptations of the SIR compartmental framework, to incorporate observed data formally so that parameter values are estimated from the incoming data.5

    How Should SIR Models Be Interpreted?

    The SIR model is one of several types of models that can be used to model an infectious disease epidemic. During the COVID-19 pandemic, the results of SIR models have been compared with those of other modeling approaches.6 For example, some groups have used network transmission models, which use information about connectivity between individuals and groups within a population to spatially model disease transmission.7 Alternative models that are not based on the biologic mechanisms of disease have also been developed, such as the Institute for Health Metrics and Evaluation’s COVID-19 pandemic model, which is based on fitting curves to empirically observed data.3 When different modeling approaches produce qualitatively different results, it may be due to critical differences in underlying assumptions, making it imperative to determine which assumptions are more likely to be valid. Alternatively, differing results may indicate that the supporting data are simply insufficient to draw a reliable conclusion. Although no model can perfectly predict the future, a good model provides an approximation that is accurate enough to be useful for informing public policy.

    Section Editors: Roger J. Lewis, MD, PhD, Department of Emergency Medicine, Harbor-UCLA Medical Center and David Geffen School of Medicine at UCLA; and Edward H. Livingston, MD, Deputy Editor, JAMA.
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    Article Information

    Corresponding Author: Juliana Tolles, MD, MHS, Department of Emergency Medicine, Harbor-UCLA Medical Center, 1000 W Carson St, Bldg D9, Torrance, CA 90502 (jtolles@emedharbor.edu).

    Published Online: May 27, 2020. doi:10.1001/jama.2020.8420

    Conflict of Interest Disclosures: None reported.

    References
    1.
    Lourenco  J, Paton  R, Ghafari  M,  et al. Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2. medRxiv. Preprint posted March 26, 2020. doi:10.1101/2020.03.24.20042291
    2.
    Weissman  GE, Crane-Droesch  A, Chivers  C,  et al.  Locally informed simulation to predict hospital capacity needs during the COVID-19 pandemic.   Ann Intern Med. Published online April 7, 2020. doi:10.7326/M20-1260PubMedGoogle Scholar
    3.
    Murray  CJL; IHME COVID-19 health service utilization forecasting team. Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilator days and deaths by US state in the next 4 months. medRxiv. Preprint posted March 30, 2020. doi:10.1101/2020.03.27.20043752
    4.
    Kermack  WO, McKendrick  AG.  A contribution to the mathematical theory of epidemics.   Proc R Soc Lond A. 1927;115:700-721. doi:10.1098/rspa.1927.0118Google ScholarCrossref
    5.
    Clancy  D, O’Neill  PD.  Bayesian estimation of the basic reproduction number in stochastic epidemic models.   Bayesian Anal. 2008;3(4):737-757.Google ScholarCrossref
    6.
    Newell  NP, Lewnard  JA, Jewell  BL.  Predictive mathematical models of the COVID-19 pandemic.   JAMA. Published online April 16, 2020. doi:10.1001/jama.2020.6585PubMedGoogle Scholar
    7.
    Zlojutro  A, Rey  D, Gardner  L.  A decision-support framework to optimize border control for global outbreak mitigation.   Sci Rep. 2019;9(1):2216.PubMedGoogle ScholarCrossref
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