The researchers from Stanford School of Medicine and ShangaiTech University show that the growth of a coronavirus disease (COVID-19) outbreak does not behave in accordance with an exponential growth law, but instead slows down exponentially with time from the very first days. Their thought-provoking findings can be currently found in the medRxiv* preprint server.
The ongoing COVID-19 pandemic, caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), resulted in a significant number of deaths and disrupted health care systems around the world. Early predictions of case numbers and deaths in any epidemic/pandemic are pivotal for making informed decisions to contain the pathogen and optimize resource allocation.
Novel Coronavirus SARS-CoV-2 Colorized scanning electron micrograph of an apoptotic cell (pink) heavily infected with SARS-COV-2 virus particles (green), isolated from a patient sample. Image captured at the NIAID Integrated Research Facility (IRF) in Fort Detrick, Maryland. Credit: NIAID
Consequently, various research groups have tried to devise reliable predictions of SARS-CoV-2 diffusion, basing their approaches on a wide array of mathematical and statistical models, different types of data (disease data, mobility details, demographic information), as well as the impact of interventions (social distancing, mask usage, hand hygiene).
The problem is that such variables may differ from country to country, and the criteria to detect COVID-19 cases and deaths sometimes vary even for states and provinces within the same country. All of these factors complicate the development of the universal approach to fit and predict COVID-19 trajectories.
Nobel prize-winning scientist Prof Michael Levitt and Dr. Andrea Scaiewicz from Stanford School of Medicine in the US, together with Dr. Francesco Zonta from ShanghaiTech University in China, decided to tackle this issue with a comprehensive mathematical approach and showed that the trajectory of cases or deaths in any outbreak could be actually converted into a straight line.
Graphs and mathematical models
This group of scientists began working on COVID-19 during the last week of January 2020, using the data released by several sources in the US, China, and India. Encouraged by the initial results, they have started an Excel spreadsheet to follow the daily progression of the disease – encompassing a total of 3,546 locations worldwide.
Each day, the researchers developed graphs encompassing four simple measures. The first three were rather obvious: the total case number, the total death number, and their ratio (i.e., the death rate). The fourth was incidental and less obvious, and expressed as the ratio of the total cases (or deaths) for today divided by the same ratio of yesterday – also known as the 'fractional change function'.
Further mathematical modeling allowed the researchers to describe the spread of SARS-CoV-2 in different countries consistently. They also were able to simplify the task of fitting inconsistent data sets to the fitting of a straight line, for which quality controls and extrapolations are trivial.
This allowed them to automate data fitting, quality assessment and extrapolation – all simultaneously and astonishingly quick (i.e., less than one hour of CPU time for all the outbreaks in the world). An essential methodological step in this study was also cleaning and curating the data stemming from a myriad of countries.
COVID-19 behaving according to the Gompertz function
This study demonstrated that the progression of the COVID-19 epidemic did not follow an exponential growth law even in the very beginning, but instead, its growth is slowing down exponentially with time. More specifically, the results irrevocably show that COVID-19 cases grew in accordance with the Gompertz function, and not the sigmoid function.
The main difference is that the sigmoid function starts off growing exponentially (it has a constant exponential growth factor) and then slows down. At the same time, the Gompertz function is never exponential, but instead exhibits a growth rate that decreases exponentially from the very first confirmed case.
"The most important result of this study is that the Gompertz function can be transformed into a straight line, provided one knows the final plateau value of total counts of either cases or deaths," emphasize study authors in their medRxiv paper.
The authors also introduced a new method named Best-Line Fitting, which entails a straight-line facilitation extrapolation necessary for any robust predictions. This method is shown to be exceedingly fast and amenable to additional optimization.
By using this technique, the study has found that in certain locations, the entire infection/disease trajectory can be predicted early. In contrast, others may take much longer to follow the aforementioned simple, functional form. If the predictions show a stable plateau of total cases and/or deaths, the Best-Line Fitting method can then be utilized to show whether they are likely correct.
Explaining and quantifying sub-exponential growth
What can we make of these interesting observations? Individuals who are mildly symptomatic and, hence, not counted as confirmed cases may be considered 'invisible' and explain the observed non-exponential behavior of COVID-19. Likewise, the known cases are then not able to easily find people to infect since the hidden cases have already infected them.
Other factors may play a role as well, most notably the structure of the human interaction network that can lead to a sub-exponential growth curve. In any case, a relatively simple functional form of the Gompertz function allowed the researchers to develop an efficient computer code to fit data in diverse locations rather consistently.
"Initial sub-exponential growth is not a unique feature of COVID-19, but has been observed in previous viral outbreaks and needs to be taken into account to produce accurate predictions", highlight study authors. "Our method provides a quick way to analyze early epidemic data and identify and also quantify sub-exponential growth," they add.
In conclusion, this study provides important tools for analyzing the behavior of this pandemic in many countries worldwide. Future studies have to address how and why are the detailed trajectories from various locations different, and elucidate population fatality ratios when the infection runs its course.
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medRxiv publishes preliminary scientific reports that are not peer-reviewed and, therefore, should not be regarded as conclusive, guide clinical practice/health-related behavior, or treated as established information.