Abstract
Mathematical models have been widely used to understand the dynamics of the ongoing coronavirus disease 2019 (COVID-19) pandemic as well as to predict future trends and assess intervention strategies. The asynchronicity of infection patterns during this pandemic illustrates the need for models that can capture dynamics beyond a single-peak trajectory to forecast the worldwide spread and for the spread within nations and within other sub-regions at various geographic scales. Here, we demonstrate a five-parameter sub-epidemic wave modeling framework that provides a simple characterization of unfolding trajectories of COVID-19 epidemics that are progressing across the world at different spatial scales. We calibrate the model to daily reported COVID-19 incidence data to generate six sequential weekly forecasts for five European countries and five hotspot states within the United States. The sub-epidemic approach captures the rise to an initial peak followed by a wide range of post-peak behavior, ranging from a typical decline to a steady incidence level to repeated small waves for sub-epidemic outbreaks. We show that the sub-epidemic model outperforms a three-parameter Richards model, in terms of calibration and forecasting performance, and yields excellent short- and intermediate-term forecasts that are not attainable with other single-peak transmission models of similar complexity. Overall, this approach predicts that a relaxation of social distancing measures would result in continuing sub-epidemics and ongoing endemic transmission. We illustrate how this view of the epidemic could help data scientists and policymakers better understand and predict the underlying transmission dynamics of COVID-19, as early detection of potential sub-epidemics can inform model-based decisions for tighter distancing controls.
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Acknowledgments
Funding: GC was supported by grants NSF 1414374 as part of the joint NSF-National Institutes of Health NIH-United States Department of Agriculture USDA Ecology and Evolution of Infectious Diseases program; UK Biotechnology and Biological Sciences Research Council [grant BB/M008894/1] and RAPID NSF 2026797.
Author contributions: GC conceived the study. KR and AT contributed to data analysis. All authors contributed to the interpretation of the results. GC and RR wrote the first draft of the manuscript. All authors contributed to writing subsequent drafts of the manuscript. All authors read and approved the final manuscript.
Competing interests: Authors declare no competing interests.
Data and materials availability: All data are publicly available.
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Appendix
Appendix
The calibration performance metrics across five countries are uniformly better for the overlap** sub-epidemic models (for MAE, MSE, and MIS, smaller is better; for % covered, larger is better)
The calibration performance metrics across five hotspots in the USA are uniformly better for the overlap** sub-epidemic models (for MAE, MSE, and MIS, smaller is better; for % covered, larger is better)
The best fit of the sub-epidemic model to the COVID-19 epidemic in Italy. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different colors
The best fit of the sub-epidemic model to the COVID-19 epidemic in France. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different colors
The best fit of the sub-epidemic model to the COVID-19 epidemic in the United Kingdom. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different colors
The best fit of the sub-epidemic model to the COVID-19 epidemic in Louisiana, USA. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different colors
The best fit of the sub-epidemic model to the COVID-19 epidemic in Georgia, USA. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different colors
The best fit of the sub-epidemic model to the COVID-19 epidemic in Arizona, USA. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different colors
The best fit of the sub-epidemic model to the COVID-19 epidemic in Washington. The sub-epidemic wave model successfully captures the multimodal pattern of the COVID-19 epidemic. Further, parameter estimates are well identified, as indicated by their relatively narrow confidence intervals. The top panels display the empirical distribution of r, p, K, and q. Bottom panels show the model fit (left), the sub-epidemic profile (center), and the residuals (right). Black circles correspond to the data points. The best model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. Cyan curves are the associated uncertainty from individual bootstrapped curves. Three hundred realizations of the sub-epidemic waves are plotted using different color
The sub-epidemic decline function across countries and USA states based on results presented in Table 1
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in Italy. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons except for 2-day ahead forecasts based on the MAE and the MSE
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in Spain. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons, but the MSE and MAE reached similar values at longer forecasting horizons
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in France. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in the UK. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons except for 2-day ahead forecasts for which the Richards model reached somewhat better performance
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in New York. The sub-epidemic model outperformed the Richards model across all forecasting horizons based on the PI Coverage and the MIS except for 2-day ahead forecasts. However, the Richards model more frequently outperformed the sub-epidemic wave model based on the MAE and MSE
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in Louisiana. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in Georgia. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in Arizona. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons
Mean performance of the sub-epidemic wave and the Richards models in 2–20 day ahead forecasts conducted during the epidemic in Washington. The sub-epidemic model outperformed the Richards model across all metrics and forecasting horizons
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in the USA. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in Italy. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in France. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in Spain. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in the UK. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in New York State. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sub-epidemic profiles of the sequential 20-day ahead forecasts for the COVID-19 epidemic in New York. Different colors represent different sub-epidemics of the epidemic wave profile. The aggregated trajectories are shown in gray and black circles correspond to the data points. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in Louisiana. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in Georgia. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in Arizona. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
Sequential 20-day ahead forecasts of the sub-epidemic wave model for the COVID-19 epidemic in Washington. Black circles correspond to the data points. The model fit (solid red line) and 95% prediction interval (dashed red lines) are also shown. The vertical line separates the calibration period (left) from the forecasting period (right). The sequential forecasts were conducted on March 30, April 6, April 13, April 20, April 27, and May 4, 2020
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Chowell, G., Rothenberg, R., Roosa, K., Tariq, A., Hyman, J.M., Luo, R. (2022). Sub-epidemic Model Forecasts During the First Wave of the COVID-19 Pandemic in the USA and European Hotspots. In: Murty, V.K., Wu, J. (eds) Mathematics of Public Health. Fields Institute Communications, vol 85. Springer, Cham. https://doi.org/10.1007/978-3-030-85053-1_5
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