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

Reviewed by: Sérvio Pontes Ribeiro, Universidade Federal de Ouro Preto, Brazil; Enahoro Iboi, Spelman College, United States

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.

The COVID-19 outbreak was first declared an international public health, and it was later deemed a pandemic. In most countries, the COVID-19 incidence curve rises sharply over a short period of time, suggesting a transition from a disease-free (or low-burden disease) equilibrium state to a sustained infected (or high-burden disease) state. Such a transition is often known to exhibit characteristics of “critical slowing down.” Critical slowing down can be, in general, successfully detected using many statistical measures, such as variance, lag-1 autocorrelation, density ratio, and skewness. Here, we report an empirical test of this phenomena on the COVID-19 datasets of nine countries, including India, China, and the United States. For most of the datasets, increases in variance and autocorrelation predict the onset of a critical transition. Our analysis suggests two key features in predicting the COVID-19 incidence curve for a specific country: (a) the timing of strict social distancing and/or lockdown interventions implemented and (b) the fraction of a nation's population being affected by COVID-19 at that time. Furthermore, using satellite data of nitrogen dioxide as an indicator of lockdown efficacy, we found that countries where lockdown was implemented early and firmly have been successful in reducing COVID-19 spread. These results are essential for designing effective strategies to control the spread/resurgence of infectious pandemics.

The outbreak of the COVID-19 disease caused by a novel pathogenic coronavirus (SARS-CoV-2), which began in Wuhan, China, in December 2019, is a global challenge for the healthcare, economy and the society (

The COVID-19 disease can spread in a population through infected symptomatic/asymptomatic individuals who come into contact directly or indirectly (

The COVID-19 incidence curve of total confirmed cases for many countries initially demonstrates a gradual increase near the start of the epidemic and is often followed by a sudden shoot or a transition to a supercritical state (

EWSs are hallmarks of critical slowing down (CSD) of a system as it approaches a catastrophic/non-catastrophic transition. The phenomenon of CSD is caused by to the loss of resilience in the system such that even small disturbances can invoke an often irreversible transition to an alternative stable state (

Model based epidemiological investigations predict the phenomenon of CSD preceded by the epidemic transitions (

To mitigate the epidemic, China strictly restricted public movement and followed with measures of quarantine and symptomatic isolation 24 days after (i.e., January 23) the arrival of the first reported case. The total reported cases (confirmed) at the time of the lockdown were nearly 623 (accounting for ~4.4732 × 10^{−7} of the total population). The daily increase in the number of confirmed cases in China was saturated in mid-March, hence flattening the incidence curve of the total confirmed cases. European countries adopted different non-pharmaceutical measures to intervene in the disease transmission. The spread began later in Italy compared to China; however, the strict interventions were initiated on March 9, which marks a gap of nearly 40 days from the first reported case with ≈ 1.22 × 10^{−4} proportion of the cases. Spain, which is continued to suffer severely by the virus, reported its first infected case on February 1 and took nearly 45 days when the proportion of affected cases was more than 9.05 × 10^{−5}, to put the country into lockdown (see ^{−7} of its population (COVID-19-infected cases), while this proportion was more than 1.7 × 10^{−4} in the US. Therefore, it is essential to understand how prolonged gaps between the arrival of the epidemic and non-pharmaceutical interventions, such as quarantining/social distancing can influence public health and the environment at a national as well as a global scale. Of greater interest is outlining whether the EWSs can be useful to stifle the spread of an epidemic.

In this work, we analyze how the timing of strict controlling strategies influence the COVID-19 incidence curve of the total confirmed cases in different countries. We first use the “change in the gradient” analysis (for details see

We obtain the datasets of the cumulative number of the COVID-19 cases from the date of reporting of the first affected person up to April 29, 2020, for India, China, South Korea, the United States (US), Singapore, Germany, Italy, the United Kingdom (UK), and Spain (for the data source see Materials and Methods).

Time series constructed as cumulative number of infected cases in nine different countries across the world from the onset of the epidemic in the respective countries up to April 29.

To estimate statistical indicators anticipating the upcoming shifts in each country, we consider the data of the cumulative daily number of COVID-19 cases before a transition is detected in the incidence curve of the epidemic (shaded regions in

Statistical estimates used to analyze the signals of a forthcoming transition in the COVID-19 incidence curve.

The timing of intervention measures varies among the countries. Notice that, apart from China and South Korea, in other countries, EWS analyses are carried out using the data before the implementation of social distancing measures. China was the first country to take the containment measures, nearly 24 days after the beginning of the epidemic, while Italy took around 40 days, and other countries followed even later. As a consequence, the COVID-19 incidence curve in China flattened after nearly 20–25 days of implementing the intervention measures. Like China, South Korea adopted different combinations of controlling measures around mid-February (in the time window of 20–25 days since the epidemic began there). This measure was accompanied by a drop in the number of cases, and the curve followed the pattern observed for China (

The scenario is quite different in the case of India. The EWSs weakly signal the behavior of CSD within the initial 40 days of the disease emergence (

Statistical analysis to measure the indicators of CSD for the dataset of India up to the date of the nationwide lockdown.

Another important aspect is to consider the reported proportion of a population affected at the time of the implementation of intervention measures. So far, Germany, which accounted for one of the largest outbreaks in Europe around mid-March, had visible signals of the forthcoming transition (^{−4} and 1.2 × 10^{−4}, respectively) compared to India −2.36 × 10^{−7}. Thus, the incidence curve projected a significant rise in these two countries, whereas the rise in the number of cases in India is relatively slow and is expected to follow a similar response owing to the effectiveness of these interventions. Overall, our analyzes suggest that delayed interventions (depending upon the signals of CSD) along with the fraction of the affected population can influence the country-wide variation in the daily number of rising cases.

The choices made to remove/filter out non-stationarities in the time series datasets using Gaussian detrending can also influence the trends observed. Thus, it is necessary to test the robustness of the estimated trends toward the choice of rolling window size and the filtering bandwidth. Here, we employ sensitivity analysis for the variance (see

The sensitivity of the choice of the rolling window size and the filtering bandwidth to estimate the EWSs.

The sensitivity of the choice of the rolling window size and the filtering bandwidth to estimate the EWSs.

We find that the observed trends in the variance are robust to the choice of parameters and does not vary between the datasets of most countries. High bandwidths reveal the opposite outcome of the variance for the datasets of South Korea (

The lower number of data points available for the analysis can lead to feeble trends and influence the probability of occurrence of the increased signals of CSD by chance. Further, due to undocumented patients, there is always a chance of stochasticity in the number of reported cases. Thus, we studied the likelihood of coincidence in the occurrence of trends in the variance and the ACF(1) observed in our original datasets by investigating the indicators in the surrogate time series (see Materials and Methods). The surrogate time series is generated to follow similar distribution (mean and variance) of the data time series before the episode of a sudden rise in the number, denoted by shaded regions in ^{*} ≤ τ). The probability of, by chance, obtaining similar trend statistic varies from country to country, depicting significant estimates for changes in the variance, except for South Korea (

The probability distribution of Kendall-τ test statistic on a set of 1,000 surrogate time series generated by bootstrapping and shuffling (with replacement) the residual time series of the original data.

Probability of, by chance, obtaining the observed trend statistic of the original data for the set of 1, 000 surrogates having similar distribution (mean and variance) as the original datasets.

China | 0.04 | 0.09 |

South Korea | 0.21 | 0.01 |

US | 0.001 | 0.1 |

Singapore | 0.48 | 0.09 |

Germany | 0.08 | 0.20 |

UK | 0.001 | 0.05 |

Spain | 0.001 | 0.21 |

Overall, we find a low probability of randomness in both the ACF(1) and the variance estimates for most of the cases. However, the observations are more significant for the variance. This analysis suggests the robustness of the variance as an EWS in predicting the signals of CSD.

The rigor of social distancing/intervention strategies can be measured by atmospheric data, as the lockdown periods have witnessed better air quality across the globe (_{2} is emitted predominantly at the surface from transportation activities, industries, and power plants. NO_{2} emitted has a short lifetime and can be transported up to a few hundred meters during the day. Therefore, NO_{2} is expected to be a profound indicator of the efficiency of lockdown measures enforced by the countries. Thus, we first obtain time series of triads of the population-weighted total-column NO_{2} (molecules/cm^{2}) density over the length of the study period for the nine countries considered in this work (the circular points in _{2} data source see Materials and Methods). The solid curve in each subfigure of _{2} decline concurs with the spread of the virus and the onset of pragmatic lockdown in a country may be hypothesized by the reversal (or break) in the trend of NO_{2}. In China (_{2} is evident from January to February; after that, it starts increasing which is coincident with the dynamics of the spread of COVID-19 disease. In India (_{2} coincides with time of the rapid spread in the virus (_{2} decreased by 26.6% in China and 55.6% in Italy compared to the pre-lockdown period. Spain, USA, and India have also seen a significant decrease after the lockdown was enforced in these countries by 33, 22.9, and 11.8%, respectively. It increased in the UK and Germany by 18 and 32%, respectively, however, even after the initiation of lockdown, which indicates an inefficient closure of anthropogenic activities (like road and rail transport, industries, and power plants). The spatial distribution of total-column NO_{2} for all the triads from Dec 28, 2019, to May, 10, 2020, can be visualized _{2} over the period of our study (

_{2} (molecules/cm^{2}) density over the length of the study period for the nine countries considered in this work (depicted by the circular points). The solid curve in each subfigure represents a 10-triad moving window average of the time series.

We propose a minimal kinetic model for the short-term prediction of the spreading of COVID-19 disease. Suppose that the only processes are infection and recovery. The processes can be described as
_{i} and _{r} are rate constants for infection and recovery. The first equation shows that if _{i}; and the second equation indicates that _{r}. A minimal kinetic model can be formulated as ordinary differential equations for the population of

We develop a master equation for the infected population by considering the two elementary processes (Equation 1). The transition probability at which the number of infected population increases from _{i}_{r}

To simulate the system (Equation 3), we first obtain the parameters from the cumulative time series data of confirmed cases for India, China, and South Korea. In the datasets, we fitted the below logistic function (which is a solution of Equation 2):
_{i}, _{r} and _{0} is the initial infected population. We list those parameters below:

_{i} |
_{r} |
_{0} |
||
---|---|---|---|---|

India | 0.608 | 0.486 | 11,722,830 | 2 |

China | 1.235 | 0.988 | 419,880 | 27 |

S. Korea | 0.9075 | 0.726 | 54,200 | 4 |

Then the above parameters are used to solve the Master equation (Equation 3), and we perform Monte-Carlo simulation to get stochastic trajectories up to April 15. We present the simulated stochastic trajectories in

Stochastic trajectories (marked with pink, yellow, gray, green, and blue curves) of the infected population generated using the Gillespie algorithm for:

The problem of predicting the spreading of COVID-19 is a complex one and depends on many factors like social distancing, an early detection of the disease, the detection of major hubs of the disease, etc. Here, we have provided a minimal kinetic model that uses the trends of the available data and may work only for short term prediction.

The COVID-19 pandemic revealed an exponential rise in the reported number of cases and has affected the public health, ranging from mild to severe conditions. Countries across the world are combating the spread of the coronavirus through various social distancing/intervention measures, such as the closure of schools and universities, banning of public events and large gatherings, isolation of symptomatic COVID-19 cases, implementation of mass quarantines, etc. For national as well as international control of public health, it is crucial to understand the significance of the onset timing of such measures (

The World Health Organization lately reported new cases being detected in several new countries across the globe (

We observe that the initial time window from the arrival of the first case in each country signaled an impending transition. An increase in the ACF(1) of the data as well as variance, before an actual rise in the number of cases, indicates the phenomenon of critical slowing down. Our work suggests that while non-pharmaceutical interventions are necessary to mitigate such an epidemic, the timing of initiation of concerned actions can strongly influence the outcome of the situation. Owing to the time lag in the detection of symptomatic cases, the statistical indicators suggest that a time period of 2–3 weeks before an impending transition is crucial to suppress the loss of public health. The controlled response of the epidemic incidence curve for China and South Korea can be associated with the time distance between implementation of interventions and the transition point. Both these countries initiated interventions before the visible signals of CSD in the incidence curve. Timely interventions were thus important factors to suppress the fluctuations in the number of cases and shape the curve. The analysis of EWSs analysis is crucial while defining the onset of the interventions and suppress the rise in daily cases. Importantly, another crucial aspect is the proportion of affected cases in each country, i.e., a measure of the fraction of the country's population, and not the absolute numbers, which is infected at the time of interventions, such as a strict lockdown. As probability of the propagation of disease can be thought of as mostly similar or equal amongst individuals across the globe, it depends upon the fraction of infected cases in each country during the beginning of interventions. For instance, the EWS analysis anticipated the upcoming rise in the incidence curves for both India as well as Italy, and, interestingly, both the countries imposed individual nationwide lockdown near the situation close to the transition (see ^{−7} cases of the total population, while Italy, with a relatively smaller population density, crossed 1.22 × 10^{−4} cases of their total population. Thus, even with imposition of the public health measures near the signals of CSD, the outcome for both the countries can vary dramatically. The variation is the consequence of the proportion of affected cases when visible signals of EWSs are observed and at the time of interventions. This suggests that the proportion of affected population during visible signals of CSD is key to shaping the disease incidence curve. The strength of the signals can alter the duration and scale of the interventions needed. Furthermore, the disruptive situation in the US is indicated by EWSs, as the EWSs indicators show significant trends for the US in addition to a large fraction of population being affected at that time. A sharp rise in the number of cases for the country is a consequence of both the delay in effective social distancing interventions as well as a significant proportion of affected cases. Overall, our work suggests that, in almost all the countries, an imminent sharp rise in the incidence curve can be seen using statistical measures prior to the actual transition.

Another issue the infectious coronavirus raises is the quality of air pollution in countries where social distancing/lockdown is enforced. NO_{2}, which is majorly emitted from anthropogenic activities like land transportation, industries, and energy sectors, was estimated to decrease in consequence to lockdown measures implemented by the government of respective countries. Population-weighted average column NO_{2} was found to decrease with amplification in a number of cases across most of the countries. Apart from this, NO_{2} column quantities may be used as a proxy to estimate the effectiveness of a lockdown on air quality. We find that NO_{2} column quantities started following a decreasing trend during the last week of February in Italy and the US, which indicates a partial unofficial closure of anthropogenic activities, taking into consideration that the official COVID-19-induced lockdown was enforced on March 09 and around March 25 in Italy and the US, respectively. In the UK, however, an increasing trend in the NO_{2} column up until May 10 indicates no such public awareness to restrict anthropogenic activities (the government declared the lockdown from March 23). We acknowledge that the reduction in NO_{2} is also associated with the compliance of the population of the individual nation to abide by the lockdown measures.

Furthermore, we suggest that the interventions employed by India may not come at a time when the curve is very far from reaching the transition; however, the smaller number of affected cases may be the determining factor in limiting the disease spread in India. Implementation of a nationwide lockdown in India may have better prepared the country for taking measures to control the epidemic spread and bend of the curve. However, our analysis also suggests that the period beyond the signals of CSD also needs efficient monitoring. The results of our minimal stochastic model predicted that, on May 20, the number of infected people could go up to ~109,262. Thus, an extended period of such measures is needed and likely to be effective (

We envision that it is fundamental to identify the situation of such a crisis across the world and make use of the lead time. The EWSs can keep track of the changes in the trend statistics in the number of reported cases and warn when a threshold is reached. The statistical tools used can be beneficial to identify whether the features of shift in a system are suppressed by the intervention strategies being adopted. In particular, while different combinations of strategies are adopted to overcome such a crisis, the information of an upcoming transition and its threshold is important to formulate the degree of such interventions. However, special care should be taken in the choice of rolling window size and the filtering bandwidth while estimating the signals of slowing down. Inappropriate choices may give weak and/or diminished signals of an imminent transition, which may deviate from understanding the urgency of the situation. Another aspect to consider is that the varying extent of testing for COVID-19 across the countries may have affected the total number of reported cases; thus, our results here hold specifically for the number of reported cases.

We have used the COVID-19 dataset provided by the European Centre for Disease Prevention and Control (ECDC): An agency of the European Union (available from

We use the available time series to test the predictability of an upcoming transition for each country. The generic indicators are examined using the time series segments before the transition in the number of cases of the epidemic (see

Often, non-stationarities in the data lead to false indications of impending transitions. To overcome this, we obtain the residual time series by subtracting a Gaussian kernel smoothing function from the empirical time series (

The fluctuations in the time series reveal different novel phenomena, such as sudden transition, flickering, stochastic switching, etc. It is established that followed by a perturbation, the rate of return of the system slows down near an impending transition or a tipping point. This phenomenon of slow return rate or recovery from a perturbation in the vicinity of a sudden transition is known as critical slowing down (CSD). We capture the signals of CSD by estimating changes in the short-term autocorrelation (at lag-1) and variance of the time series. CSD increases the short-term memory of the time series, which is observed through the correlation structure of the time series before a transition. We compute autocorrelation at lag-1 by fitting an autoregressive model of order 1 (of the form _{t+1} = α_{1}_{t} + ϵ_{t}) using an ordinary least-squares fitting method. The time series analysis has been performed using the “Early Warning Signals Toolbox” (

The predictability of each of the indicator depends upon the datasets investigated as well as the choices made for processing the data. Thus, it is essential to check the efficacy of our results to such choices. In particular, we analyze the sensitivity of our observations to the choice of rolling window size and degree of smoothing (filtering bandwidth) used during the calculation of indicators and detrending/filtering the datasets, respectively. We estimate the CSD indicators using window sizes ranging from 40 to 90% of the time series length in an increment of 1 point and for bandwidths ranging from 5 to 100% with the increment of 1 point. We quantify the robustness of the outcomes toward the range of window sizes and bandwidth using the distribution of the Kendall-τ test statistic.

To test the significance of our statistical analysis, we estimate Kendall rank correlation-τ test statistics for both the generic indicators. We generate 1,000 surrogate time series of the same length as the analyzed real datasets to test the likelihood of obtaining the computed trends by chance. The surrogate records are obtained on bootstrapping the real datasets by shuffling the original residual time series and sampling the data with replacement. This method generates the surrogate time series with a similar distribution of the original time series (^{*} ≤ τ) to calculate that the observed test statistic is by chance. We also generate surrogate time series using phase randomization method (for details see

Worldwide, the lockdown response to the onset and spread of COVID-19 caused a decrease in daily and economic activities, which in turn is expected to cause a reduction in ambient air pollution. This can also be used as an indicator to determine whether government policies of lockdowns/restricted human movements are successful or not. To further examine this, we use the Ozone Monitoring Instrument (OMI) retrieved total column NO_{2} (available from ^{2} (_{2} column was satisfactorily validated against surface spectrometer measurements in recent studies (

All datasets presented in this study are included in the article/

SC, SKS, MJ, and PD designed the research. TK, SS, SC, SKS, and PD performed the research. TK, SS, SC, SKS, MJ, and PD analyzed the data. TK, SC, SKS, MJ, and PD wrote the paper. All authors contributed to the article and approved the submitted version.

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.

SS acknowledges the financial support from the Department of Science & Technology (DST), Government of India, under the scheme DST-Inspire (Grant No: IF160459). PD acknowledges financial support from the Science & Engineering Research Board (SERB), Government of India (Grant No: CRG/2019/002402). This manuscript has been released as a pre-print at medRxiv (

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

_{2}standard product

_{2}, SO

_{2}and HCHO products using MAX-DOAS observations from 2011 to 2014 in Wuxi, China: investigation of the effects of priori profiles and aerosols on the satellite products

_{2}column retrievals using in situ and surface-based NO

_{2}observations