Research Article

The turning point and end of an expanding epidemic cannot be precisely forecast

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PNAS October 20, 2020 117 (42) 26190-26196; first published October 1, 2020; https://doi.org/10.1073/pnas.2007868117

  1. Edited by Eugene V. Koonin, National Institutes of Health, Bethesda, MD, and approved September 11, 2020 (received for review April 23, 2020)

Article Figures & SI

Figures

  • Fig. 1.
    Fig. 1.

    Diagram of the epidemic model along with the equations ruling the dynamics. Susceptible individuals (S) can enter and exit confinement (C) or become infected (I). Infected individuals can recover (R) or die (D). N is the total population. Rates for each process are displayed in the figure; q depends on specific measures restricting mobility and contacts, while p stands for individuals that leave the confinement measures (e.g., people working at essential jobs like food supply, health care, or policing), as well as for defection. We fit I to data on officially diagnosed cases, which are automatically quarantined: The underlying assumption is that the real, mostly undetected, number of infections is proportional to the diagnosed cases.

  • Fig. 2.
    Fig. 2.

    Fit to data obtained in real time for the daily number of active cases in Spain (from March 1st to March 29th) and peak forecast. (A) Despite the reasonable agreement between model and empirical observations in the growing phase, opposite predictions for the future number of active cases can be derived. The solid line represents the expression for I(t) using the median parameters for each posterior in SI Appendix, Fig. S1. The vertical arrow denotes March 11th, the day when schools and universities closed. The shaded area represents the 95% predictive posterior interval: Its increasing width implies that predictability decays exponentially fast. (A, Inset) Same data and curves with linear vertical scale. SI Appendix, Figs. S6 and S7 show how this fit and its posteriors evolve as an increasing number of days is included in the fit. An animation is included as Movie S1. (B) Posterior distribution of the time to reach the peak of the epidemic, conditioned to actually having a peak (which occurs with probability 0.26). The vertical dashed lines stand for the days when the confinement began and for the date of the last data point used in the fit.

  • Fig. 3.
    Fig. 3.

    Fit to postpeak data for the daily number of active cases in Spain. (A) Fit to data up to April 18th (peak day). (B) Fit to data at three weeks postpeak (May 9th). Open symbols represent fitted empirical data, and blue dots correspond to actual measurements until May 17th. (C) Distribution of times until the number of confirmed cases falls below 1,000 for the first time. With about two cases per million inhabitants, this threshold can define the end of the epidemic. The distribution spans about three months, centered around the end of October 2020.

  • Fig. 4.
    Fig. 4.

    Data generated through direct simulation of the system described in Fig. 1 are used as input to determine posterior distributions for parameters through a Bayesian approach. Parameter values are β=0.425, p=0.007, q=0.062, and r+μ=0.021 in the mitigation regime taken from the median of the posteriors in SI Appendix, Fig. S1 (all measured in day−1). Though the dataset is complete and noiseless, consideration of only the growing phase of the epidemic implies a remarkable uncertainty in compatible trajectories. It is worth noting that, albeit those parameters would predict that the epidemic is not controlled, variability still leaves a 3% chance that it actually is. (Inset) Same data and curves with linear vertical scale.

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The turning point and end of an expanding epidemic cannot be precisely forecast
Mario Castro, Saúl Ares, José A. Cuesta, Susanna Manrubia
Proceedings of the National Academy of Sciences Oct 2020, 117 (42) 26190-26196; DOI: 10.1073/pnas.2007868117
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