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Edited by: Zisis Kozlakidis, International Agency For Research On Cancer (IARC), France

Reviewed by: Gui-Quan Sun, North University of China, China; Jinfeng Xu, The University of Hong Kong, Hong Kong

This article was submitted to Infectious Diseases - Surveillance, Prevention and Treatment, a section of the journal Frontiers in Public Health

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Countries around the globe have implemented unprecedented measures to mitigate the coronavirus disease 2019 (COVID-19) pandemic. We aim to predict the COVID-19 disease course and compare the effectiveness of mitigation measures across countries to inform policy decision making using a robust and parsimonious survival-convolution model. We account for transmission during a pre-symptomatic incubation period and use a time-varying effective reproduction number (_{t}) to reflect the temporal trend of transmission and change in response to a public health intervention. We estimate the intervention effect on reducing the transmission rate using a natural experiment design and quantify uncertainty by permutation. In China and South Korea, we predicted the entire disease epidemic using only early phase data (2–3 weeks after the outbreak). A fast rate of decline in _{t} was observed, and adopting mitigation strategies early in the epidemic was effective in reducing the transmission rate in these two countries. The nationwide lockdown in Italy did not accelerate the speed at which the transmission rate decreases. In the United States, _{t} significantly decreased during a 2-week period after the declaration of national emergency, but it declined at a much slower rate afterwards. If the trend continues after May 1, COVID-19 may be controlled by late July. However, a loss of temporal effect (e.g., due to relaxing mitigation measures after May 1) could lead to a long delay in controlling the epidemic (mid-November with fewer than 100 daily cases) and a total of more than 2 million cases.

The COVID-19 pandemic is currently a daunting global health challenge. The novel coronavirus was observed to have a long incubation period and highly infectious during this period (

Various infectious disease models (

SEIR models can incorporate mechanistic characteristics and scientific knowledge of virus transmission to provide useful estimates of its temporal dynamics, especially when individual-level epidemiological data are available through surveillance and contact tracing. However, these sophisticated models may involve a large number of parameters and assumptions about individual transmission dynamics. They may thus be susceptible to perturbation of parameters and prior assumptions, yielding wide confidence intervals especially when granular individual-level data are not available. In contrast to infectious disease models, alternative statistical models are proposed to predict summary statistics such as deaths and hospital demand under a non-linear mixed effects model framework (

In this work, we propose a parsimonious and robust population-level survival-convolution model that is based on main characteristics of COVID-19 epidemic and observed number of confirmed cases to predict disease course and assess public health intervention effect. Our method models only key statistics (e.g., daily new cases) that reflect the disease epidemic over time with at most six parameters, and it may therefore be more robust than models that rely on individual transmission processes or a large number of parameters and assumptions. We constructed our model based on prior scientific knowledge about COVID-19 instead of

We aim to achieve the following goals. The first goal is to fit observed data to predict daily new confirmed cases and latent pre-symptomatic cases, the peak date, and the final total number of cases. The second goal is to assess the effect of nationwide major interventions across countries (e.g., mitigation measures) under the framework of natural experiments [e.g., longitudinal pre-post quasi-experimental design, (

We used data from a publicly available database that consolidates multiple sources of official reports (World Meters[

Let _{0}(_{j} denote the time when individual _{j} = ∞ if never infected), and let _{j} be the duration of this individual remaining infectious to any other individual and in the transmission chain. Let _{0} be the unknown calendar time when the first patient (patient zero) is infected. Therefore, at time _{0}, _{1}) with _{1} as the maximum incubation period (i.e., 21 days for SARS-CoV-2) and _{0}(_{j}. On the other hand, right after time _{j} = (_{j}). The total number of these individuals is

Therefore, (

Note that _{t} is the instantaneous time-varying effective reproduction number (

Models (1) and (2) provide a robust dynamic model to characterize COVID-19 epidemic. Equation (2) gives a convolution update for the new cases using the past numbers, while equation (1) gives the number of cases out of transmission chain at time _{0}, of the first (likely undetected) case in the epidemic, the survival function of time to out of transmission,

We model the transmission rate

First, with parameters estimated from data and assuming that the future transmission rate remains the same trend, we can use models (1) and (2) to predict future daily new cases, the peak time, expected number of cases at the peak, when _{t} will be reduced to below 1.0 and the epidemic will be controlled (the number of daily new cases below a threshold or decreases to zero). Furthermore, our model provides the number of latent cases cumulative over the incubation period at each future date, which can be useful to anticipate challenges and allocate resources effectively.

Second, we can estimate the effects of mitigation strategies, leveraging the nature of quasi-experiments where subjects receive different interventions before and after the initiation of the intervention. The longitudinal pre-post intervention design allows valid inferences, assuming that pre-intervention disease trend would have continued had the intervention not taken place and local randomization holds (whether a subject falls immediately before or after the initiation date of an intervention may be considered as random, and the “intervention assignment” may thus be considered to be random). Applying this design, the intervention effects will be estimated as the difference in the rate of change of the transmission rate function before and after an intervention takes place.

Third, we can study the impact of an intervention (e.g., lifting mitigation measures) that changes the epidemic at a future date. Using permutations, we can obtain the joint distribution of the parameter estimators and construct confidence intervals (CI) for the projected case numbers and interventions effects.

For China, the transmission rate _{t} decreases quickly from 3.34 to below 1.0 in 14 days (

Model estimated parameters in each country.

^{*} |
|||
---|---|---|---|

China | _{0}( |
Jan 3 (17) | (12, 21)^{**} |

Training data: Jan 20 to Feb 4 | _{0} |
0.793 | (0.68, 1.02) |

Testing data: Feb 5 to May 10 | _{1} |
-0.693 | (-1.13, -0.42) |

Duration | 44 | (39, 55) | |

End date | Mar 4 | (Feb 28, Mar 15) | |

Total | 58,415 | (42,516, 133,083) | |

South Korea | _{0}( |
Feb 11 (4) | (1, 7) |

Training data: Feb 15 to Mar 4 | _{0} |
1.363 | (1.03, 1.98) |

Testing data: Mar 5 to May 10 | _{1} |
-1.496 | (-2.39, -0.96) |

Duration | 39 | (37, 43) | |

End date | Mar 25 | (Mar 23, Mar 29) | |

Total | 7,977 | (7,307, 10,562) | |

Italy | _{0}( |
Feb 10 (10) | (4, 11) |

Training data: Feb 20 to Apr 29 | _{0} |
0.789 | (0.73, 1.10) |

Testing data: Apr 30 to May 10 | _{1} |
-0.358 | (-0.68, -0.26) |

_{2} |
-0.372 | (-0.46, -0.31) | |

_{3} |
0.061 | (0.02, 0.12) | |

_{4} |
-0.057 | (-0.12, -0.01) | |

Duration | 123 | (103, 179) | |

End date | Jun 22 | (Jun 2, Aug 17) | |

Total | 223,410 | (216,848, 257,710) | |

United States | _{0}( |
Feb 15 (6) | (1, 4) |

Training data: Feb 21 to May 1 | _{0} |
0.410 | (0.34, 0.62) |

Testing data: May 2 to May 10 | _{1} |
0.526 | (0.23, 0.72) |

_{2} |
-1.031 | (-1.24, -0.86) | |

_{3} |
-0.042 | (-0.06, -0.03) | |

Scenario 1: Continue current^{†} |
Duration | 156 | (139, 188) |

End date | Jul 26 | (Jul 9, Aug 27) | |

Total | 1,626,950 | (1,501,036, 1,918,602) | |

Scenario 2: 50% slower | Duration | 188 | (163, 233) |

after May 1 | End date | Aug 27 | (Aug 2, Oct 11) |

Total | 1,731,992 | (1,563,122, 2,113,294) | |

Scenario 3: 75% slower | Duration | 226 | (190, 289) |

after May 1 | End date | Oct 4 | (Aug 29, Dec 5) |

Total | 1,832,291 | (1,616,574, 2,324,552) | |

Scenario 4: 100% slower | Duration^{‡} |
272 | (201, 448) |

after May 1 | Control date^{‡} |
Nov 19 | [Sep 9, May 13 (2021)] |

Total^{‡} |
2,084,235 | (1,728,028, 3,094,518) |

_{0} is the estimated date of the first undetected community infection; d is the estimated gap days between the first undetected case and the first reported case; a_{0} is the transmission rate before the reported first case; a_{1}, a_{2}, and a_{3} are rates of change of a(t) in each period measured as change per 21 days; “Duration” is the number of days from the date of the first reported case to “End date”; “End date” is the date when predicted new case decreases to zero; and “Total” is the total number of predicted cases by the “End date.”

_{3}) after May 1; Scenarios 2–4 assume the relaxation of quarantine measures after May 1 will lead to a slower decrease of transmission rate by 50, 75, and 100% (complete loss of temporal effect over time).

Observed and predicted daily new cases and 95% confidence interval (shaded).

Effective reproduction number _{t} for each country computed as the average number of secondary infections generated by a primary case at time

For South Korea,

For Italy, we model _{t} is not significantly different before and 2 weeks after the lockdown (_{t} stops decreasing and remains close to 1.0 until April 16. The slope of _{2} and _{3} for Italy). This is consistent with a relatively flat trend of observed daily new cases during this period (

In the US, we fit a three-piece model for _{t} increases during the early phase but decreases sharply after the declaration of national emergency (_{t} decreases at a much slower rate. If this trend continues, the end of epidemic date is predicted to be July 26 (scenario 1,

United States: observed and predicted daily new cases, 95% confidence intervals under four scenarios that assume relaxation of mitigation measures occurs after May 1. Scenario 1: transmission rate _{t}.

The estimated number of latent cases present on each day (i.e., including pre-symptomatic patients infected

In this study, we propose a parsimonious and robust survival convolution model to predict daily new cases of the COVID-19 outbreak and use a natural quasi-experimental design to estimate the effects of mitigation measures. Our model accounts for major characteristics of COVID-19 (long incubation period and highly contagious during incubation) with a small number of parameters (up to six) and assumptions, directly targets prediction accuracy, and provides measures of uncertainty and inference based on permuting the residuals. We allow the transmission rate to depend on time and modify the basic reproduction number _{0} as a time-dependent measure _{t} to estimate change in disease transmission over time. Thus, _{t} corrects for the naturally impact of time on the disease spread. Our estimated reproduction number at the beginning of the epidemic ranges from 2.81 to 5.37, which is consistent with the _{0} reported in other studies (

Comparing the effective reproduction numbers across countries, _{t} decreased much more rapidly in South Korea and China than Italy (_{t} decreased until almost reaching 1.0 on March 25 but remained around 1.0 for 3 weeks. The US followed a fast decreasing trend during a 2-week period after declaring national emergency (_{2} = −1.031), which is faster than the first 2 weeks in China (_{1} = −0.693), but its _{t} decreased at a much slower rate (_{3} = −0.042) afterwards and was below 1.0 on May 5.

Comparing mitigation strategies across countries, the fast decline in _{t} in China suggests that the initial mitigation measures put forth on January 23 (lockdown of Wuhan city, traffic suspension, home quarantine) were successful in controlling the transmission speed of COVID-19. Additional mitigation measures were in place after February 2 (centralized quarantine and treatment) but did not seem to have significantly changed the disease course. In fact, our model assumes the same transmission rate trajectory after February 2 fits all observed data up to May 10. A recent analysis of Wuhan's data (_{t} closely matches with our estimates. However, their analyses were based on self-reported symptom onset and other additional surveillance data, where we used only widely available official reports of confirmed cases. Another mechanistic (

South Korea did not impose a nation-wide lockdown or closure of businesses but, at the very early stage (when many cases linked to patient 31 were reported on February 20), conducted extensive broad-based testing and detection (drive through tests started on February 26), rigorous contact tracing, isolation of cases, and mobile phone tracking. Our results suggest that South Korea's early mitigation measures were also effective.

Italy's initial mitigation strategies in the most affected areas reduced _{t} from 3.73 to 1.92 in 20 days. To estimate the effect of the nation-wide lockdown as in a natural experiment, we require local randomization and the continuity assumption. The former requires that characteristics of subjects who are infected right before or after the lockdown are similar. Since, in a very short time period, whether a person is infected at time _{t} has remained around 1.0 for about 2–3 weeks before it starts to decrease again.

For the US, _{t} was as high as 4.50 before the declaration of national emergency on March 13 but declines rapidly over a 2-week period after March 13. Although the disease trend and mitigation strategies vary across states in the US, since the declaration of national emergency, many states have implemented social distancing and ban of large gathering. The large difference before and 2 weeks after March 13 is likely due to states with large numbers of cases that implemented state-wide stay-at-home orders (e.g., New York and New Jersey), which indicates that these measures may be effective. Our model estimated a continued decrease in _{t} from March 27 to May 1 but at a much slower rate ( 95.9% slower; _{2} and _{3} for the US) when it approached 1.0. In China, centralized quarantine and treatment were implemented when _{t} was around 1.0 (_{t} to zero and final control of the epidemic. If the trend in US continues after May 1, the first wave of epidemic will be controlled by July 26 (CI: July 9, August 27). However, after May 1, many states enter a re-opening phase. If the guidelines on quarantine measures are relaxed in order for the temporal effect of quarantine measures to be completely lost, the predicted total number of cases is more than 2 million, with a long delay in controlling the epidemic (less than 100 cases by November 19 and no new case by May, 2021). In an updated analysis that includes additional observed data in May, the recent _{t} is near a constant between 1.1 and 1.2 from April 11 to May 29, and the confidence interval suggests some possibility of an uptake of new cases (

Other studies reported transmission between asymptomatic individuals (

Despite these limitations, our study offers several implications. Implementing mitigation measures earlier in the disease epidemic reduces the disease transmission rate at a faster speed (South Korea, China). Consequently, for regions at the early stage of disease epidemic, mitigation measures should be introduced early. Nation-wide lockdown may not further reduce the speed of _{t} reduction compared to regional quarantine measures as seen in Italy. In countries where disease transmissions have slowed down, lifting of quarantine measures may lead to a persistent transmission rate delaying control of epidemic and thus should be implemented with caution and close monitoring.

Publicly available datasets were analyzed in this study. This data can be found here:

DZ and YW conceived the study. QW and DZ implemented the codes. SX, QW, and YW made figures and tables. All authors interpreted the results, contributed to writing the article, and approved the final version for submission.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This manuscript has been released as a pre-print at MedRxiv [(

The Supplementary Material for this article can be found online at: