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Edited by: Zisis Kozlakidis, International Agency for Research on Cancer (IARC), France

Reviewed by: Katerina M. Marcoulides, University of Minnesota Twin Cities, United States; Fernando Fagundes Ferreira, University of São Paulo Ribeirão Preto, Brazil

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

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.

^{2}) of COVID-19 cases in real-time.

The World Health Organization (WHO) officially declared Coronavirus Disease (COVID-19) a global pandemic on March 11th 2020 (^{2}) in disease dissemination from real-time case reports can be decisive for an effective and promptly action to restrain further contagion. Here we report the development of a simple framework dedicated to the real-time analysis of COVID-19 prevalence. This framework was built using a combination of Moving Regression (MR) (

For simplicity, assume that the cumulative number of COVID-19 cases over time (i.e., the growth curve of prevalence) in a specific country or territory follows an unknown sigmoidal function (

Growth rate and acceleration in Australia and New Zealand.

We define growth rate and growth acceleration as the first and second order derivatives, respectively, of the prevalence of COVID-19 in respect to time. In our framework, we selected MR to approximate these derivatives over competing models that are frequently used to describe the behavior of sigmoidal growth curves, such as the Gompertz model (^{2} = 0.99 with smooth factor of 3) and acceleration (median ^{2} = 0.92 with smooth factor of 3) (^{2} > 0.99) (

Accuracy (^{2}) of moving regression estimates of growth rate and growth acceleration from 50,000 simulated Gompertz growth curves.

Accuracy (^{2}) of moving regression predictions of next-day COVID-19 prevalence.

Sigmoidal growth curves can be partitioned into four easily distinguishable stages (

In spite of sigmoidal curves following the four above described stages sequentially, we anticipated that the growth of COVID-19 cases may not necessarily obey this sequence in practice, since the dynamics of the disease is likely complex and highly responsive to the implementation or relaxation of public health measures. This implies that a country that has already reached a stationary stage could resume exponential growth, for example by seeding a new outbreak via importation. Likewise, decelerating countries could as well regain acceleration by relaxing prevention measures. Furthermore, some countries may face multiple cycles of acceleration and deceleration prior to reaching a stationary growth. These scenarios could produce more complex growth curves that deviate from the sigmoidal shape by mounting different arrangements of exponential, deceleration, and stationary stages. Of note, MR has sufficient flexibility to model these complex scenarios and can easily accommodate curves exhibiting arbitrary permutations of these four stages. In addition, the near-zero acceleration that is intimately related to the stationary stage in sigmoidal curves could also arise from a non-zero constant growth rate in practice. In such cases, the growth curve would exhibit a linear pattern, which can be interpreted as a fifth growth stage that is not observed in classic sigmoidal functions. Such linear pattern may appear if the deceleration stage does not form an enough deep valley prior to acceleration rising up again toward zero. Again, MR is capable of modeling these anomalous behaviors. In this study we sought to ascertain whether these five stages of growth curves could have direct implications in understanding the dynamics of COVID-19 prevalence both globally and locally. We further developed a HMM to automate the detection of transitions between stages in the growth curve using acceleration and growth rate data obtained with MR as input (see

Using MR and HMM on ECDC data frozen on May 8th 2020, we first evaluated the utility of the framework in identifying countries reaching stationary growth. Apart from Australia (

Growth rate and acceleration in China, South Korea, and Austria.

By projecting government responses recorded by the Blavatnik School of Government from the University of Oxford (

Effect of mobility restrictions on variation of COVID-19 acceleration (cases/day^{2}) during exponential growth^{a}

^{b} |
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School closing | C1 | −15.00 | 1.61 | 4.66 × 10^{−20} |

Workplace closing | C2 | −14.25 | 1.25 | 4.33 × 10^{−29} |

Cancellation of public events | C3 | −9.33 | 1.77 | 1.53 × 10^{−7} |

Restrictions on gatherings | C4 | −10.62 | 1.30 | 7.68 × 10^{−16} |

Closure of public transportation | C5 | −5.97 | 0.97 | 8.16 × 10^{−10} |

Stay at home requirements | C6 | −7.67 | 1.11 | 7.18 × 10^{−12} |

Restrictions on internal movement | C7 | −13.43 | 1.19 | 1.96 × 10^{−28} |

International travel controls | C8 | −6.05 | 1.42 | 2.31 × 10^{−5} |

In order to illustrate the utility of the framework in detecting deceleration in real-time, we decided to look more closely to data from three countries: Germany (

Growth rate and acceleration in Germany, Spain and Italy. These three countries were in deceleration as of May 8th.

The relatively rapid response to public health measures makes the acceleration curve an useful tool for policy evaluation. Much attention has been recently given to Brazil and the United States of America (USA), as these two countries are the new epicenters of the coronavirus pandemic. Together, these two countries sum up 1,392,078 cases and 84,816 deaths to date. Monitoring the acceleration curve might be helpful in these countries by enabling the assessment of the efficacy of adopted measures. Since the beginning of the exponential growth in Brazil back in early March (data not shown), growth acceleration has presented great oscillation in the country. Currently, Brazil is experiencing an acceleration decline, and could begin a deceleration process within few weeks if effective measures are implemented and rigorously followed. On the other hand, USA has started its deceleration process on April 9th but has not formed a deceleration valley yet (data not shown), which hampers the production of an expressive decline in new cases. Furthermore, as outbreaks are expected to occur in African countries in the following months, the analysis of growth acceleration could be an invaluable asset to evaluate control strategies in the continent.

To this date, the lack of combined analysis of growth rate and acceleration of the COVID-19 pandemic is to be blamed on scarce availability of tailor made, user-friendly software. To aid to the analysis of growth rate and acceleration of COVID-19 cases, we built a web application using

We deployed a simple framework for the real-time analysis of COVID-19 prevalence. We were able to demonstrate that the real-time decomposition of growth curves of COVID-19 cases into growth rate and acceleration can be a powerful tool to monitor the impact of public health measures on the spread of the disease. We also showed that restrictions to human mobility can significantly decelerate the incidence of new cases within weeks. Furthermore, we found that the prevalence of the disease is more complex and dynamic than previously appreciated. This observation will have important implications to assumptions adopted in mathematical models to predict the evolution of the pandemic.

The MR technique (_{d} and _{d} as _{d} = [_{k} _{d}], where _{k} is a

where μ_{d} is an intercept and _{d} is the estimated growth rate (cases/day) at day _{d} corresponds to an estimate of the instantaneous rate of change in the number of cases at day

After fitting Equation (1) to all _{d} as a

where _{d} is the estimated growth acceleration (cases/day^{2}) at day _{d} is an estimate of the instantaneous rate of change of the growth rate at day

In order to automate the process of growth stage classification, we built a HMM (^{2}. However, as a free parameter, _{1}, _{2}, …, _{n}) where each element _{i} takes one of the following values: “lagging,” “exponential,” “deceleration,” or “stationary.” The initial probabilities for these hidden states were set to 1, 0, 0, and 0, respectively, assuming that all growth curves start from a lagging stage. Now let

The selected values in

To test the performance of MR in approximating growth curves and their rate of change and acceleration in scenarios where these curves have been observed only partially (i.e., real-time case report), we selected a widely used sigmoidal mathematical function, namely the Gompertz model (

where ^{2}) of the regression of true values onto estimates.

We analyzed case reports that have been updated daily by the European Center for Disease Prevention and Control (ECDC). The framework was applied to that data using smooth factors ranging from

All analyses presented in this paper were performed using

The COVID-19 case data in this study were obtained from the European Center for Disease Prevention and Control (ECDC) and are publicly available at

YU conceived the study, performed simulations, coordinated the data analysis, and wrote the manuscript. AU built R code for data analysis and programmed the Shiny App Dashboard. RT, SP, MM, and JG revised growth curves for all countries/territories and pinpointed dates of measures taken by them to reduce human mobility. All authors revised and agreed with the contents of the manuscript.

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.

We would like to express our highest gratitude to all health agents and individuals around the globe who were involved in reporting cases and making COVID-19 prevalence data available to the public. This study did not receive financial support and was conducted during voluntary social isolation.