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Generalized logistic growth modeling of the COVID-19 outbreak: comparing the dynamics in the 29 provinces in China and in the rest of the world - PubMed

  • ️Sun Nov 11 2334

Generalized logistic growth modeling of the COVID-19 outbreak: comparing the dynamics in the 29 provinces in China and in the rest of the world

Ke Wu et al. Nonlinear Dyn. 2020.

Abstract

Started in Wuhan, China, the COVID-19 has been spreading all over the world. We calibrate the logistic growth model, the generalized logistic growth model, the generalized Richards model and the generalized growth model to the reported number of infected cases for the whole of China, 29 provinces in China, and 33 countries and regions that have been or are undergoing major outbreaks. We dissect the development of the epidemics in China and the impact of the drastic control measures both at the aggregate level and within each province. We quantitatively document four phases of the outbreak in China with a detailed analysis on the heterogeneous situations across provinces. The extreme containment measures implemented by China were very effective with some instructive variations across provinces. Borrowing from the experience of China, we made scenario projections on the development of the outbreak in other countries. We identified that outbreaks in 14 countries (mostly in western Europe) have ended, while resurgences of cases have been identified in several among them. The modeling results clearly show longer after-peak trajectories in western countries, in contrast to most provinces in China where the after-peak trajectory is characterized by a much faster decay. We identified three groups of countries in different level of outbreak progress, and provide informative implications for the current global pandemic.

Keywords: Epidemic modeling; Logistic growing; Novel coronavirus (COVID-19); Prediction.

© The Author(s) 2020.

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Conflict of interest statement

Conflict of interestThe authors declare that they have no conflict of interest.

Figures

Fig. 1
Fig. 1

Time dependence of the total number of confirmed cases (upper panel), the daily number of new confirmed cases (middle panel), and the daily growth rate of confirmed cases (lower panel) in the mainland China excluding Hubei province until March 1, 2020. The empirical data are marked by the empty circles. The blue and red lines in the upper, middle and lower left panels show the fits with the logistic growth model and generalized Richards model (GRM) respectively. For the GRM, we also show the fits using data ending 20, 15, 10, 5 days earlier than March 1, 2020, as lighter red lines in the upper and middle panel. This demonstrates the consistency and robustness of the fits. The lower left panel shows the daily growth rate of the confirmed cases in log scale against time. The green and cyan straight lines show the linear regression of the logarithm of the growth rate as a function of time for the period of January 25 to February 14, and the period of February 15 to March 1, respectively. The lower right panel is the daily growth rate of the confirmed cases in linear scale against the cumulative number of confirmed cases. The red and green lines are the linear fits for the period of January 19 to February 1, and the period of February 2 to March 1, respectively

Fig. 2
Fig. 2

a Daily number of new observed confirmed cases for mainland China excluding Hubei (black circles) compared with 500 scenarios built by parametric bootstrap with a negative binomial error structure on the GRM model with best fit parameters determined on the data up to the time indicated by the vertical dashed line. The last time used in the calibration is, respectively, 5, 10, 15, 20, 25, 30 days before February 22, 2020 from bottom to top. The red continuous line is the best fitted line, and the two dashed red curves delineate the 95% confidence interval extracted from the 500 scenarios. The six panels correspond each to a different end date, shown as the sub-title of each panel, at which the data have been calibrated with the GRM model. b Convergence of the four parameters from the GRM simulations shown in a. The error bars indicate the 80% prediction intervals. (Color figure online)

Fig. 3
Fig. 3

Inverse relationship found across the 29 Chinese provinces between the number of days from peak to the end and the duration from start to the peak of the epidemics. Here, the end of the outbreak is defined operationally as the day when the 5 days moving average of the growth rate becomes smaller than 1%

Fig. 4
Fig. 4

Upper panel: estimated C0 for the logistic growth model versus travel index from Hubei and Wuhan. Lower panel: total confirmed cases versus travel index from Hubei and Wuhan. The Pearson correlation between C0 and the migration index is 0.65 (p<10-3), and the correlation between the cumulative number of confirmed cases and the migration index is 0.82 (p<10-4)

Fig. 5
Fig. 5

Daily confirmed cases of Europe and of 14 countries with the best two models among the four models (1)–(4). The empirical data are indicated by empty circles. The blue, red and green lines with the error bars show the fits with the logistic growth model, generalized Richards model (GRM), and generalized logistic model (GLM), respectively. The error bars indicate 80% prediction intervals. Data are plotted every 3 days. The vertical line indicates the date (June 5) up to when the data are fed to the model. (Color figure online)

Fig. 6
Fig. 6

a Daily confirmed cases of 10 countries with the best two models among the four models (1)–(4). The empirical data are indicated by empty circles. The blue, red and green lines with the error bars show the fits with the logistic growth model, generalized Richards model (GRM) and generalized logistic model (GLM), respectively. The error bars indicate 80% prediction intervals. Data are plotted every 3 days. The vertical line indicates the date (July 24) up to when the data are fed to the model. b Daily confirmed cases of 8 countries with the best two models among the four models (1)–(4). The empirical data are indicated by empty circles. The blue, red and green lines with the error bars show the fits with the logistic growth model, generalized Richards model (GRM) and generalized logistic model (GLM), respectively. The error bars indicate 80% prediction intervals. Data are plotted every 3 days. The vertical line indicates the date (July 24) up to when the data are fed to the model. (Color figure online)

Fig. 6
Fig. 6

a Daily confirmed cases of 10 countries with the best two models among the four models (1)–(4). The empirical data are indicated by empty circles. The blue, red and green lines with the error bars show the fits with the logistic growth model, generalized Richards model (GRM) and generalized logistic model (GLM), respectively. The error bars indicate 80% prediction intervals. Data are plotted every 3 days. The vertical line indicates the date (July 24) up to when the data are fed to the model. b Daily confirmed cases of 8 countries with the best two models among the four models (1)–(4). The empirical data are indicated by empty circles. The blue, red and green lines with the error bars show the fits with the logistic growth model, generalized Richards model (GRM) and generalized logistic model (GLM), respectively. The error bars indicate 80% prediction intervals. Data are plotted every 3 days. The vertical line indicates the date (July 24) up to when the data are fed to the model. (Color figure online)

Fig. 7
Fig. 7

Violin plot of the distributions of the final total number of confirmed cases per million derived by combining the distributions of the positive and medium scenarios. The left side of each violin in cyan shows distributions obtained on July 24, while the right side of each violin in gray shows distributions obtained on June 5. The model setup in the negative scenario does not incorporate a maximum saturation number and thus cannot be used. The yellow dots indicate the median prediction for the combined distribution, while the green and red dots indicate the median of the positive and of the medium scenarios, respectively. (Color figure online)

Fig. 8
Fig. 8

7-day prediction error of the forecast performed on July 17 for the total number of deaths for various countries/regions. The horizontal line corresponds to the empirical data on July 24. The error bars are 80% prediction intervals, and the middle dots are the median predictions based on the predictions from the positive and medium scenarios. A negative value corresponds to a prediction that underestimated the true realized value

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References

    1. Dye C, Gay N. Modeling the SARS epidemic. Science. 2003;300(5627):1884–1885. doi: 10.1126/science.1086925. - DOI - PubMed
    1. Laguzet L, Turinici G. Individual vaccination as Nash equilibrium in a SIR model with application to the 2009–2010 influenza A (H1N1) epidemic in France. Bull. Math. Biol. 2015;77(10):1955–1984. doi: 10.1007/s11538-015-0111-7. - DOI - PubMed
    1. Zhao Z, Calderón J, Xu C, Zhao G, Fenn D, Sornette D, Crane R, Hui PM, Johnson NF. Effect of social group dynamics on contagion. Phys. Rev. E. 2010;81(5):056107. doi: 10.1103/PhysRevE.81.056107. - DOI - PubMed
    1. Lekone PE, Finkenstädt BF. Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study. Biometrics. 2006;62(4):1170–1177. doi: 10.1111/j.1541-0420.2006.00609.x. - DOI - PubMed
    1. Smieszek T, Balmer M, Hattendorf J, Axhausen KW, Zinsstag J, Scholz RW. Reconstructing the 2003/2004 H3N2 influenza epidemic in Switzerland with a spatially explicit, individual-based model. BMC Infect. Dis. 2011;11(1):115. doi: 10.1186/1471-2334-11-115. - DOI - PMC - PubMed

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