Near-term prediction and fractal analysis based on current data doesn't show exponential growth
The novel coronavirus disease (COVID-19) spread continues to grow in India and in most other countries across the world. We observed that the nature of the pandemic spread is fractal. A simple analogy is a tree: It starts with a stem and as it grows it gets several branches. It is well known that trees have fractal geometries.
Inspired by the character of the spread of the virus (SARS-CoV-2), we analysed the time series data (number of deaths versus time) and were intrigued to see that it follows a fractal behavior, also known as power law distribution.
Fractal concepts are widely used in many fields like finance, biology and modelling of chaotic environments in weather forecasting and extreme events like earthquakes, avalanches, cloud bursts etc.
The total number of deaths and rate of daily increase in number of deaths shown in the graphs below — from Johns Hopkins University — show data from January 22 to April 25, 2020.
The number of confirmed cases are dependent on how much testing has been done, hence we chose to analyse the reported number of COVID-19 death cases, more reliable than the number of infected cases.
Total number of COVID-19 deaths versus time reported in India so far
Source: Johns Hopkins University
COVID-19 deaths per day reported in India
Source: Johns Hopkins University
We used data shown in the graph for the total number of deaths to show the fractal analysis of COVID-19 deaths reported in India. The analysis helps us fit a suitable curve that can explain the data and provide short-term predictions.
Since our analysis is based purely on reported data and does not include constraints, including the effects of lockdowns, social distancing, hygiene maintenance and immunity of the population, it is not fair to predict the long-term behavior of the pandemic.
It is well known that the preventive measures listed above have a significant effect in arresting the spread of the pandemic. We also know, however, that the incubation period of SARS-CoV-2 is approximately two weeks.
Present day analysis of the data, hence, can give us robust predictive power for the coming two weeks. We exploited this fact and after determining optimal parameters based on fractal analysis, we generated a fit that honours available data (deaths versus time) and also enables us to predict the expected outcome for the next two weeks as shown below.
Analysis and non-linear model fitting to report COVID-19 deaths in India
Source:VP Dimri, RP Srivastava
In our model fitting, we estimate a 95 per cent confidence interval, which means there is a 95 per cent chance that range of values (in this case number of COVID-19 deaths) will be within these bounds.
The circular black points show the actual number of COVID-19 deaths in India, with the continuous red line showing our best fit to the available data, which has been extended for the next two weeks until May 10.
Dotted lines show 95 per cent of the confidence interval and the ‘I’ symbol represents the error bar in prediction interval, which represents possible minimum and maximum cases. Points falling on the red line will be the expected number of cases in the prediction interval (from April 26 to May 10).
We tabulated predicted values for coming two weeks (from April 26 to May 10), with 95 per cent confidence interval (which provides minimum and maximum possible cases) as shown in the below table.
Prediction of COVID-19 deaths expected in India in coming two weeks
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Source: VP Dimri, RP Srivastava
Our sole purpose of doing this work is to bring to the notice of government authorities and fellow citizens, that rate of increase is not at all exponential and the situation seems well within control.
In ordinary terms, exponential means very fast. For mathematicians, scientists and those working in the sphere of public health, the term exponential means a quantity that has rate of change at each point proportional to its current size.
In simple language, if the quantity is small, then exponential growth is also small. When the quantity is large, then change is rapid.
Epidemics start slowly, thanks to exponential growth. According to our analysis, the growth of COVID-19 is close to quadratic, a much smaller growth compared to exponential.
The situation around COVID-19 is very dynamic, hence we have kept this project live and would like to forecast again after some time.
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