key: cord-0700155-83qlovhp authors: Baruah, H. K. title: Hyper-Exponential Growth of COVID-19 during Resurgence of the Disease in Russia date: 2020-10-27 journal: nan DOI: 10.1101/2020.10.26.20219626 sha: 5f13a382550e1de1522a02654f37e45bc0b7a9a6 doc_id: 700155 cord_uid: 83qlovhp In Russia, COVID-19 has currently been growing hyper-exponentially. This type of a spread pattern was not seen during the first wave of the pandemic the world over. Indeed when the disease had first appeared, in the accelerating stage the spread pattern was observed to have followed a highly nonlinear pattern that could be said to be approximately exponential or sub-exponential. As to why in the resurgence the growth has become hyper-exponential is another matter. But this has been happening in Europe and how long this would continue cannot be predicted. It may so happen that in the countries in which retardation has already been taking place, there may be resurgence of the disease. It was observed that in the World as a whole, retardation was on the threshold during the second half of September. But if the resurgence happens to follow the hyper-exponential growth pattern in different countries, there may be resurgence in the World as a whole. When COVID-19 had first appeared, it was observed that in the accelerating stage the disease had been spreading following a highly nonlinear pattern which was approximately exponential which might be termed as sub-exponential. In Europe, the spread of the disease almost came to a halt months ago. However, in Europe it has reappeared and this time unlike in the first wave the growth rate has been much faster than what was observed earlier. At the sub-exponential stage, forecasting about retardation was not difficult. In this article, we are going to show that the growth pattern in Russia is hyper-exponential, and it is not going to be easy to forecast when the growth would start retarding. Regarding hyper-exponential pattern we would now like to discuss a few things in short. The Malthusian model [1] was the initial work on population dynamics in which it was stated that a population when unchecked grows exponentially following the pattern is the population at time ‫ݐ‬ and ‫ݎ‬ is the exponential rate of growth of the population. This can equivalently be written as . 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 (which was not certified by peer review) The copyright holder for this preprint this version posted October 27, 2020. This in turn can be equivalently written as This is what was referred to hyper-exponential growth of a population. Fisher's theorem was indeed the limit theorem of population dynamics which can be seen as a generalization of the Malthusian model. In Fisher's model it is inherently presumed that the reason behind such a time dependent positive rate of growth is caused by genetic variations causing an additive effect with reference to time. Therefore in Fisher's model, the rate of change of the natural logarithm of ܰ ሺ ‫ݐ‬ ሻ is of first degree in ‫ݐ‬ . Accordingly, the population growth in this model is much faster than the exponential growth model due to Malthus. It may be noted that all biological populations grow according to exponential law, whereas the human population grows following the hyper-exponential law which essentially moves ahead of the exponential one. A hyper-exponential equation that allows describing dynamics of human population was in fact obtained in [3] . In this article, we are going to demonstrate that in Russia currently the rate of growth instead of decreasing with time is actually increasing. This would show that currently in Russia the growth is hyper-exponential. The classical epidemiological models [4, 5, 6] assume exponential growth in the accelerating stage of the spread of an epidemic. Indeed, in the accelerating stage an exponential growth pattern should be only theoretically possible, for if the increase is actually exponential the epidemic would never come to a halt. Therefore, an epidemic growth can only be approximately exponential or sub-exponential with the rate of growth decreasing in time. This decreasing rate of growth would be reflected in the estimated values of the growth parameter ‫ݎ‬ with respect to time. In this study, we shall use data from Worldometers.info [7] . We would like to mention here that the data in this source get edited after publication, which results in minor changes in the data. . 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 27, 2020. ; https://doi.org/10.1101/2020.10.26.20219626 doi: medRxiv preprint Therefore there may be seen some minor changes in the data that we are going to tabulate in this article. We are going to demonstrate that the current growth rate of COVID-19 in Russia is following the hyper-exponential law. For that we would define is the cumulative total number of COVID-19 cases in Russia at time ‫ݐ‬ . We shall first assume that the spread is approximately following 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 27, 2020. was negative, during the next 20 days it was insignificant and in the last 20 days the regression coefficient was negative again. Using the equation fitted from data from August 2 to August 21, a forecast was made in [11] that starting from August 2, within no more than 78 days retardation would start in India. That forecast could be seen to be true. In for Russia for the period of 20 days from 2 October to 21 October. In Fig.1 for Russia for the period from 2 October to 21 October has been found to be . 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 27, 2020. , we have found that the calculated value of t is equal to 12.526309 which is far greater than the two sided theoretical value of t (= 2.10092), at 5% probability level of significance for 18 degrees of freedom. Therefore we conclude that the null hypothesis is to be rejected, and that there is a significant linear relationship between ∆ ‫ݖ‬ ሺ ‫ݐ‬ ሻ and t. This is equivalent to concluding that the value of the regression coefficient (= + 0.000170338) is significantly different from 0.0. Indeed we would like to mention at this point that the theoretical value of ‫ݐ‬ for 18 degrees of freedom at 1% probability level of significance is 2.87844, and at 0.1% probability level of significance it is 3.9651. This means that we can be 99.9% sure that the population correlation coefficient is very highly significantly different from zero. 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 27, 2020. ; https://doi.org/10.1101/2020. 10.26.20219626 doi: medRxiv preprint In Table-2 we have shown the forecasts of the cumulative total number of cases for a short period from 22 October to 5 November. We have depicted the forecasts in Fig. 2 . ort en . 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 27, 2020. Thus we have demonstrated that Fisher's population dynamics model is at work in Russia with reference to spread of COVID-19. Hyper-exponential growth is very unusual in this case because when the disease had first appeared it was following the Malthusian model of population dynamics which states that the growth is exponential. Whatever be the reason behind, the spread of COVID-19 in Russia is not following the pattern that it followed when it had first appeared in the World. During the accelerating stage, the growth was sub-exponential everywhere in the first wave, but in this second wave, the disease has started to grow much faster than that in the first wave. In the first wave, predictions could be made regarding around what time the retardation would start. But in the second wave the growth has been confirmed to be hyper-exponential. Therefore it is now difficult to say when the retardation would restart. . 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 27, 2020. ; https://doi.org/10.1101/2020.10.26.20219626 doi: medRxiv preprint An Essay on the Principle of Population The Genetical Theory of Natural Selection The Hyperexponential Growth of the Human Population on a Macrohistorical Scale 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 Russia, Publishing Date A Simple Method of Finding an Approximate Pattern of the COVID-19 The current COVID-19 spread pattern in India Nearly Perfect Forecasting of the Total COVID-19 Cases in India: A Numerical Approach The Uncertain COVID-19 Spread Pattern in India: A Statistical Analysis