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A Numerical Study of the Current COVID-19 Spread Patterns in
India, the USA and the World
Hemanta K. Baruah
Department of Mathematics, The Assam Royal Global University
Guwahati, Assam, India
E-mail: hemanta@gauhati.ac.in, hkbaruah@rgu.ac
Abstract: 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.
Key Words: Pandemic, infectious disease, epidemiological modeling.
Introduction
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
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2
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.
Methodology
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
exp
,
, 0,
0
,
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3
where
is the cumulative total number of cases at time
, should fit the concerned data.
Write
. We shall now proceed to study the values of
∆
, the first order
differences of
.
It was observed [5, 6, 7, 8, 9] that from
∆
we can extract an important information. When
the pandemic continues to grow exponentially,
∆
would continue to decrease linearly in
time. If it continues to be very nearly constant for a sufficiently long duration after continuing to
decrease linearly earlier, that should be taken as a signal that the pattern might start to change to
a nearly logarithmic one soon after [10]. We shall now show how
∆
can lead us to conclude
about the current situations in India, in the USA and in the World.
Analysis and Discussions
In Tables-1, 2 and 3, we shall now show the values of
,
and
∆
of the World, of the
United States of America and of India respectively for 15 days from September 17 to October 1.
The data have been taken from the Woldometers.info [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.
Table-1: World: Values of
,
and
∆
from 17 September to 1 October
Dates
∆
Oct. 1 34469967 17.3556 0.009332
Sept. 30 34149803 17.34627 0.009287
Sept. 29 33834125 17.33698 0.008554
Sept. 28 33545940 17.32843 0.00691
Sept. 27 33314931 17.32152 0.00778
Sept. 26 33056748 17.31374 0.009125
Sept. 25 32756467 17.30461 0.009774
Sept. 24 32437864 17.29484 0.009552
Sept. 23 32129499 17.28529 0.009673
Sept. 22 31820223 17.27561 0.008554
Sept. 21 31549195 17.26706 0.0074
Sept. 20 31316586 17.25966 0.008172
Sept. 19 31061697 17.25149 0.009653
Sept. 18 30763308 17.24183 0.010545
Sept. 17 30440619 17.23129 0.010366
Sept. 16 30126687 17.22092 ---
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4
Table-2: USA: Values of
,
and
∆
from 17 September to 1 October
Dates
∆
Oct. 1 7494671 15.8297 0.006343
Sept. 30 7447282 15.82336 0.005511
Sept. 29 7406353 15.81785 0.005989
Sept. 28 7362126 15.81186 0.005096
Sept. 27 7324706 15.80676 0.004631
Sept. 26 7290867 15.80213 0.005945
Sept. 25 7247651 15.79619 0.00743
Sept. 24 7194003 15.78876 0.006348
Sept. 23 7148483 15.78241 0.005862
Sept. 22 7106700 15.77655 0.00506
Sept. 21 7070834 15.77149 0.005187
Sept. 20 7034253 15.7663 0.004778
Sept. 19 7000725 15.76152 0.006274
Sept. 18 6956937 15.75525 0.007445
Sept. 17 6905332 15.7478 0.006766
Sept. 16 6858767 15.74104 ----
Table-3: India: Values of
,
and
∆
from 17 September to 1 October
Dates
∆
Oct. 1 6391960 15.67055 0.012863
Sept. 30 6310267 15.65769 0.013842
Sept. 29 6223519 15.64385 0.013019
Sept. 28 6143019 15.63083 0.011406
Sept. 27 6073348 15.61942 0.013722
Sept. 26 5990581 15.6057 0.01497
Sept. 25 5901571 15.59073 0.014588
Sept. 24 5816103 15.57614 0.014883
Sept. 23 5730184 15.56126 0.015776
Sept. 22 5640496 15.54548 0.014355
Sept. 21 5560105 15.53113 0.013488
Sept. 20 5485612 15.51764 0.016058
Sept. 19 5398230 15.50158 0.017332
Sept. 18 5305475 15.48425 0.017644
Sept. 17 5212686 15.46661 0.018743
Sept. 16 5115893 15.44786 ----
In [5] it was shown that in the World outside China, the values of
∆
were around 0.13705
from March 21 to March 28, around 0.09287 from March 29 to April 4, around 0.05916 from
April 5 to April 11, around 0.0407 from April 12 to April 18, and around 0.03323 from April 19
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5
to April 25. It was clear that within the period from March 21 to April 25, the values of
∆
were steadily reducing. We have found that the average of
∆
reduced to
0.008978
during
the period from September 17 to October 1.
In [6] it was shown that in the United States, the average of
∆
was 0.02165 during the period
from May 3 to May 8. During the period from September 17 to October 1, it was found to be
0.005911.
In [7] it was shown that in India, the average of
∆
during May 11 to May 24 was 0.051716,
and it came down to 0.045584 during May 25 to May 31. It was seen that [8] during June 1 to
June 9 it became 0.04063. When the matter was studied with data for 60 consecutive days [9],
from June 23 to July 12, the average value of
∆
was 0.034576, from July 13 to August 1 it
was 0.034458, and from August 2 to August 21 it was 0.026449. So we have seen that as time
progressed, the values of
∆
were reducing. During the period from September 17 to October
1, it was found to be 0.014846
.
The values of
∆
have been depicted in Fig. 1. In the figure, Series-1 represents India, Series-
2 represents the USA and Series-3 represents the World. We have not gone for a statistical fit of
regression equations of
∆
on
because it looks obvious from the Figure that
∆
is
actually showing a decreasing trend for India, while for the USA and the World the values are
very nearly constant. It may further be observed that there may be cycles of length 8 days from
one trough to another in all three cases.
Fig. 1: Values of
∆
: Series 1- India, Series 2- USA, Series 3- World
From Fig, 1, we can clearly see the following things. In India, as the values of
∆
are small
enough and are still in a decreasing trend, the spread pattern is approximately exponential and
that very soon the first order differences of
would start to show approximate constancy,
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
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7
-
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p
1
8
-
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e
p
1
9
-
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e
p
2
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-
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p
2
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p
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c
t
Series1
Series2
Series3
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after which the situation in India would start improving. We can see further that the values of
∆
in the case of the United States are very small and are very nearly constant, and the current
exponential pattern would soon change to a nearly logarithmic pattern. In the case of the World
as a whole, the smallness and the nearly constant values of
∆
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
∆
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.
The averages of
∆
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
exp17.3556 0.008978
,
1, 2, … . . , 15
(1)
where
exp17.3556
= 34469967, the cumulative total number of cases in the World on October
1. This will give us the forecasts from October 2 onwards for the next 15 days.
For the United States, the equation we shall apply is
exp
15.8297
0.005911
,
1,2,…..,15
(2)
where
exp15.8297 7494671,
the cumulative total number of cases in the United States on
October 1. This will give us the forecasts from October 2 onwards for the next 15 days.
For India, the equation we shall apply is
exp
15.67055
0.014846
,
1, 2, … . . , 15
(3)
where
exp15.67055 6391960,
the cumulative total number of cases in India on October 1.
This will give us the forecasts from October 2 onwards for the next 15 days.
In Table-4, we have shown the forecasts for the World, the United States and for India. We
would like to mention at this point that the forecasts may be overestimates for India in particular
because the values of
∆
were decreasing in the period from September 17 to October 1 and
we have taken help of the average of
∆
for the forecasts.
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7
Table-4: Forecasts of the Cumulative Total Number of Cases
Dates World USA India
2-Oct 34780848 7539103 6487563
3-Oct 35094533 7583798 6584595
4-Oct 35411047 7628759 6683079
5-Oct 35730415 7673986 6783035
6-Oct 36052664 7719481 6884487
7-Oct 36377820 7765245 6987456
8-Oct 36705907 7811281 7091965
9-Oct 37036954 7857590 7198037
10-Oct 37370987 7904174 7305696
11-Oct 37708032 7951034 7414965
12-Oct 38048116 7998171 7525868
13-Oct 38391268 8045588 7638430
14-Oct 38737515 8093286 7752676
15-Oct 39086885 8141267 7868630
16-Oct 39439406 8189532 7986319
Conclusions
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.
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