key: cord-0971005-fqu54j4h
authors: Zhang, Feng; Zhang, Jinmei; Cao, Menglan; Zhang, Yong; Hui, Cang
title: Exponential damping: The key to successful containment of COVID-19
date: 2020-03-27
journal: nan
DOI: 10.1101/2020.03.22.20041111
sha: 729a9faa0dd40ae0392c7590418583096673a5b4
doc_id: 971005
cord_uid: fqu54j4h

Due to its excessively high capacity for human-to-human transmission, the 2019 novel coronavirus disease (COVID-19), first reported in Wuhan in China, spread rapidly to the entire nation and beyond, and has now been declared a global public health emergency. Understanding the transmission pattern of the virus and the efficacy of transmission control measures is crucial to ensuring regional and global disease control. Here we propose a simple model based on exponential infectious growth, but with a time-varying, largely damping, transmission rate. This model provides an excellent fit to the existing data from the 102 countries and regions which have reported cases for more than 6 days, and, we think, has largely captured the transmission patterns of the COVID-19 outbreak under a variety of intervention and control measures. We found that the damping rate, defined as the rate of the exponential decline in transmission rate, ranged from -0.125 to 0.513 d-1 globally (a negative damping rate represents acceleration in spread). The estimated peak time (when the fastest spread occurs) and the final number of infections were found to be greatly affected by the damping rate. Successful control measures, such as those implemented in China and South Korea, have resulted in a clear pattern of exponential damping in the viral spread (also shown during the 2003 outbreak of Severe Acute Respiratory Syndrome, SARS). The damping rate, therefore, could be used as an indicator for the efficacy of implemented control measures. Our model suggests that the COVID-19 outbreak is currently accelerating worldwide, especially rapidly in certain countries (e.g. USA and Australia) where exponential damping is yet to emerge. Consistent with the message from the World Health Organisation (WHO), we thus strongly suggest all countries to take active measures to contain this global pandemic. Slight increments in the damping rate from additional control efforts, especially in countries showing weak or no exponential damping in COVID-19 transmission, could lead to a radically more positive outcome in the fight to contain the pandemic.

The 2019 novel coronavirus , which can cause acute pneumonia, was first reported in Wuhan in December 2019, the capital of Hubei Province in central China (1, 2) . Due to the excessively high rate of human-to-human transmission, the virus has quickly spread across all provinces of China and all countries of the world (3). In order to contain this outbreak, governments and healthcare authorities across the globe have taken a series of strict public health measures. Wuhan and all major cities in Hubei, for instance, were sealed off, human movement and traffic prohibited, quarantine imposed on all potentially exposed people, makeshift hospitals quickly built to receive and cure for infected patients. After implementation for just one month, these control measures effectively contained the spread of this highly infectious novel coronavirus in China (4), and were considered therefore highly efficient by the World Health Organization (WHO)(5). As the first wave of the pandemic has passed beyond China, COVID-19 now begins to rage worldwide, sweeping across all continents except Antarctica (6). For effective monitoring and containment of the pandemic, it is crucial to understand the patterns of its rapidly changing and localised transmission and promptly evaluate whether the currently implemented control measures are adequate to 'flatten the curve'.

Traditional epidemiological models, such as the SIR and SEIR models, explain the rapid increase in the number of infections by the presence of a large susceptible population exposed to infection, and the decline of infection by the depletion of the susceptible population (7). Such a model structure is questionable for capturing the spread of COVID-19 due to the massive size of regional and global susceptible populations (easily running into tens or hundreds of millions of residents in a region).

The relatively limited infection, albeit excessively high when focused solely on the sheer number of infections, as well as the resultant mortality, have rather small effects on the demography of regional and global populations, unless a large fraction of the population eventually contracts the virus. In addition, the parameterisation of such models is also unreliable for a novel virus where its pathology and transmission pathways remain unclear with little data support. As such, we here propose a population ecology model with a time-varying infection rate to capture the transmission patterns of COVID-19. The All rights reserved. No reuse allowed without permission. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.03.22.20041111 doi: medRxiv preprint advantage of this phenomenological model is that it does not rely on detailed pathology, yet can still provide an accurate and rapid assessment of COVID-19 transmission patterns under implemented control measures. The rate of exponential damping in transmission rate, as will be shown, provides a real-time evaluation of the efficacy of any implemented control measures.

Assuming the population is large yet the outbreak limited, so that its impact on the demographic dynamics of the population itself is negligible, we could capture the number of infected cases ( ) over time using an ordinary differential equation,

the carrying capacity of the number of infections (set as 70% of the entire population, but please note, in most cases the final number of infections is much lower than , so we have essentially ignored its effect on the outbreak). We estimated the transmission rate as ( + 1/2) = ln� ( + 1)� − ln( ( )), where is measured in days. Notably, ( )/ > 0 represents the acceleration of the epidemic spread, while ( )/ < 0 the deceleration and damping dynamics. We define the damping rate ( ) as the rate of the exponential decline in the transmission rate ( ); that is, ( ) = − + . An effective control measure should, arguably, result in the deceleration of the spread at a high damping rate (large positive ), while inadequate control measures could lead to a low damping rate (small positive close to zero) and even the acceleration of the spread ( < 0). The solution to the above differential equation is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is Using the daily infection numbers from 20 January to 16 March 2020, from the WHO website (www.who.int), we analysed the dynamics of the COVID-19 outbreak in the three worst affected countries (South Korea, Iran and Italy). We found that the transmission rates of COVID-19 in these three countries were all declining exponentially over time, with South Korea experiencing the highest damping rate, similar to that of China (around 0.16 d -1 ), while the other two countries lagged behind at a much lower damping rate, especially Italy (Fig.2 ). Using the current exponential function of the transmission rate, we estimated the peak of COVID-19 spread at 18 March 2020 (95% We further calculated the damping rate for all 102 countries and regions that have reported 7 or more days of infection, from which we also estimated the peak time since All rights reserved. No reuse allowed without permission. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.03.22.20041111 doi: medRxiv preprint first local infection, and the final number of infections (Table S1 ). Results suggest a large variation in the damping rate of COVID-19 transmission across the world (Fig.3a) , from effective control (a>0.14 d -1 ) in 14 countries, to outbreak acceleration (a<-0.01 d -1 ) in India, USA, Canada, Australia, Singapore and Thailand (Table S1) . April 2020 and the final number of infections of 3,829,338 people.

The data from China and South Korea show that it is possible to contain the spread of COVID-19; this is signalled by the exponential damping of the transmission rate ( Fig.1; the two top panels of Fig.2) . Such exponential damping is also evident from the data of the 2003 SARS outbreak in mainland China, Hong Kong and the entire world (data from All rights reserved. No reuse allowed without permission. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.03.22.20041111 doi: medRxiv preprint the website of WHO, Fig.S1 ). This implies that exponential damping in disease transmission could be a universal pattern of successful infectious disease containment.

The damping rate of virus transmission reflects the effectiveness of implemented control measures over the natural infection rate of the disease, and its variation across countries therefore reveals whether the current implemented local/regional measures are adequate (see Fig.1, Fig.2 & Fig.3) . By estimating the time-varying transmission rate and its damping rate, our model provides a simple theoretical framework for monitoring the spread of an outbreak and assessing the efficacy of implemented control measures in real time. This is important for regional decision-makers and global governance to reflect upon, in order to modify any implemented control measures and practices in time.

Our analysis shows that, at the moment, the pandemic is accelerating exponentially around the world (Fig.2) and an overall damping pattern is yet to emerge. Theoretically, All rights reserved. No reuse allowed without permission. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.03.22.20041111 doi: medRxiv preprint lines on the right panels are the corresponding predictions using the fitted timedependent transmission rate (r(t)). Circles indicate real data, and green circles indicate cases imported from other countries but were not considered in the regression. All rights reserved. No reuse allowed without permission. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is respectively from a 1‰ and 5‰ increase in the observed damping rate (Table S1 ).

All rights reserved. No reuse allowed without permission. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.03.22.20041111 doi: medRxiv preprint 13 Supplementary File: Table S1 : Global rankings of the damping rate and its lower and upper bounds of 95% confidence interval (a, LB and UB, respectively), peak time, the estimated final number of infections (FNI), and the number of daily transmission rates (n) used in the model fitting. Estimated for 102 countries and regions with at least 6 days of transmission until 24 March 2020, with K representing 70% of its total population. With 1‰ and 5‰ increase in the observed damping rate, the peak time is expected to shift by PTS days (+1 indicates a delay of 1 day), while the final number of infections is expected to decline by a number of FNID people. All rights reserved. No reuse allowed without permission.

the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.03.22.20041111 doi: medRxiv preprint

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