key: cord-1027952-zgvsns26
authors: Baruah, H. K.
title: A Numerical Study of the Current COVID-19 Spread Patterns in India, the USA and the World
date: 2020-10-06
journal: nan
DOI: 10.1101/2020.10.05.20206839
sha: a0e461758b117f77545079e54cbc0bacc3b88409
doc_id: 1027952
cord_uid: zgvsns26

In this article, we are going to study the current COVID-19 spread patterns in India and the United States. We are interested to show how the daily increase in the total number of cases in these two countries is affecting the COVID-19 spread pattern in the World. For the study, we have considered the cumulative total numbers of cases in India, the United States and the World. We have found that the situation in the United States is already on the threshold of a change towards retardation. In the World as a whole also we have observed that a similar conclusion can be made. In India, the situation can be expected to move towards betterment soon, and once that happens the situation in the World as a whole would start improving. We shall demonstrate that as long as the rate of change of the logarithm of the cumulative total number of cases with respect to time in a pandemic continues to reduce, the pattern of growth would continue to remain nearly exponential, and as soon as it is seen that the rate of change starts to become nearly constant the growth can be expected to start to change towards a nearly logarithmic pattern.

A look at the daily increase in the cumulative totals of the COVID-19 cases in India, the United States of America and the World gives us an apparently clear picture how much India is contributing to the daily increase in the total number of cases in the World, and in comparison how much the USA is currently contributing towards that. For example, on October 1 the increase in the World total were nearly 320,000 out of which India alone contributed nearly 81,000 which are about 25% of the increase in the World total, while on that day the USA contributed around 47,000 which are about 15% of the increase in the World total.

In what follows, we are going for a numerical study of the rate at which India is contributing towards the daily increase in the cumulative total number of COVID-19 cases in the world in comparison to the rate at which the USA is contributing. For this we need not go for any compartmental epidemiological model [1. 2, 3] . Indeed, for the World as a whole and for India too, the compartmental models may not probably be very suitable because of geographical and economic heterogeneity of the susceptible population. Further, we are going to study the current situation only for which we do not really need to take resort to the compartmental epidemiological models. In our eyes, unless the data regarding recovered cases and deaths are absolutely reliable, the epidemiological models such as SIR (Susceptible-Infectious-Recovered), SIS (Susceptible-Infectious-Susceptible) and SIRD (Susceptible-Infectious-Recovered-Dead) cannot be criticized. The population exposed to the disease may not have economic homogeneity, and in that situation the SEIR (Susceptible-Exposed-Infectious-Recovered) model similarly cannot be criticized for unacceptable results. Such mathematical models were established under certain assumptions. If those assumptions are not valid it is obvious that the results returned would be unacceptable.

In our present study we shall use data of the cumulative total number of cases only. As a matter of fact, in the initial stage of the pandemic this number was of the population that showed some symptoms of the disease. Later on, when planned testing for COVID-19 positivity got started, those who were tested included people without any symptoms too. So it is obvious that whereas in the initial stage those with symptoms were included in the total number of cases, later on this figure became dependent on the number of tests performed. If due to some reason lesser number of tests was performed in a particular region on any particular day, the increase in the total number of effected cases would automatically be less than that seen on any other day in which more tests were performed. Therefore it has to be agreed upon that the cumulative total number of cases is indeed situation dependent in the sense that the randomness that was inherent in the initial stage of the pandemic is absent in the current number of cases. This may have an adverse effect even in our kind of a study in which only the cumulative total number of cases are being used. Indeed, this would automatically affect the results returned by application of the compartmental epidemiological models too.

We shall use a simple numerical procedure of our own for this study using data from Worldometers.info [4] . Regarding our method of forecasting without using the standard epidemiological methods [5, 6, 7, 8, 9] , it was observed that our numerical method does work very well in forecasting of the spread pattern during the nearly exponential phase. In this article, we shall put forward a way to compare the rates of growth of the natural logarithm of the cumulative total of the cases of the pandemic in India, the USA and the World as a whole. Our objective is to demonstrate that as long as the rate of change of the logarithm of the cumulative total number of cases with respect to time in a pandemic continues to reduce, the pattern of growth would continue to remain nearly exponential, and as soon as it is seen that the rate of change starts to become nearly constant the growth would start to change towards a nearly logarithmic pattern.

It is apparent from the graphs published by Worldometers.info [4] that the spread patterns are still approximately exponential in India, in the USA and in the World. Therefore the function

, . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 6, 2020. 

In Tables-1, 2 [4] dated October 2. It should be mentioned here that the data source Woldometers.info gets edited sometimes creating some small changes in the figures thereby. This is why we have cited the source with the date concerned. However, such changes are very insignificant with respect to the largeness of the numbers. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 6, 2020. . https://doi.org/10.1101/2020.10.05.20206839 doi: medRxiv preprint . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 6, 2020. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 6, 2020. assert that the current exponential pattern of growth of the pandemic would soon change towards a nearly logarithmic pattern soon. We can say further that the growth would retard in the United States first to be followed by the World as a whole. Finally, when ∆ ‫ݖ‬ ሺ ‫ݐ‬ ሻ would become very nearly constant in India which is currently the largest contributor towards the growth of the cumulative total number of cases, the situation in the World would automatically change towards a logarithmic increasing pattern which is the last phase of growth of a pandemic. A nearly logarithmic pattern of growth in the retarding phase of a pandemic we have observed in the case of Italy [10].

We now proceed to find the forecasts from October 2 to October 16 using the following method. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 6, 2020. . https://doi.org/10.1101/2020.10.05.20206839 doi: medRxiv preprint . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 6, 2020. . https://doi.org/10.1101/2020.10.05.20206839 doi: medRxiv preprint

A Contribution to the Mathematical Theory of Epidemics

Contributions to the Mathematical Theory of Epidemics II. The Problem of Endemicity

Contributions to the Mathematical Theory of Epidemics III. Further Studies of the Problem of Endemicity

Total corona virus cases in India, Publishing Date

A Simple Method of Finding an Approximate Pattern of the COVID-19

On Reliability of the COVID-19 Forecasts

from September 17 to October 1 were found to be 0.008978 for the World, 0.005911 for the USA and 0.014846 for India.We shall apply the following equation for forecasts of the cumulative total in the World where 

We conclude that the growth of the pandemic in the United States is in the threshold already to take a change from exponential to nearly logarithmic. As far as the situation in India is concerned, we can conclude that the pattern of growth is still approximately exponential and that very soon the situation in India would enter into the retarding phase of the pandemic. For the situation in the World as a whole too we can conclude that the growth of the pandemic is already in the threshold to take a change from exponential to logarithmic. However till India continues to be in the nearly exponential phase, the retardation in the World as a whole would have to wait.