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Fear of exponential growth in Covid19 data of India and future sketching

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Abstract

We have attempted to interpret existing n-cov positive data in India with respect to other countries-Italy, USA, China and South Korea. We have mainly zoomed in the exponential growth in a particular zone of time axis, which is well followed in the data profile of India and Italy but not in others. A deviation from exponential growth to Sigmoid function is analyzed in the data profile of China and South Korea. Projecting that pattern to time dependent data of total number and new cases in India, we have drawn three possible Sigmoid functions, which saturate to cases 10 4 , 10 5 , 10 6. Ongoing data has doubtful signal of those possibilities and future hope is probably in extension of lock-down and additional imposition of interventions.
Fear of exponential growth in Covid19 data of India and
future sketching
Supriya Mondal1, Sabyasachi Ghosh2
1Aadarsh Nursing Institute, Raipur, Chhattisgarh 492015, India
2Indian Institute of Technology Bhilai, GEC Campus, Sejbahar, Raipur 492015,
Chhattisgarh, India
Abstract
We have attempted to interpret existing n-cov positive data in India with respect to
other countries - Italy, USA, China and South Korea. We have mainly zoomed in the
exponential growth in a particular zone of time axis, which is well followed in the data
profile of India and Italy but not in others. A deviation from exponential growth to
Sigmoid function is analyzed in the data profile of China and South Korea. Projecting
that pattern to time dependent data of total number and new cases in India, we have
drawn three possible Sigmoid functions, which saturate to cases 104, 105, 106. Ongoing
data has doubtful signal of those possibilities and future hope is probably in extension
of lock-down and additional imposition of interventions.
1 Introduction
Presently, people of entire world is in fear from Covid19 spreading which started from wuhan,
China on December 2019. Within a three months time period, the spreading from China to
entire world become so violent that WHO declared it as a pandemic disease on 11th March,
2020 [1]. In present draft, we are just going to describe the existing n-cov data of India
with respect to other countries where we have zoomed in the exponential growth and its
corresponding time axis zone. Then, its deviation from exponential growth is discussed and
we have sketched different possible deviated curves, which may be or may not be expected
from present lock-down schenario of our country (India).
2 Identification of Exponential growth
Without any artificial immunization, the virus infection can spread exponentially [2]. A
recent time example is the spreading of Covid19 infection in different countries, whose
real-time documentation can be seen in world-meter data [3]. This growth alarm already
gave a threat on survival probability of human civilization and force them to think many
interventions to fight against this virus. These interventions can make a deviation from
exponential curve, which will be reflected in real and raw data of different countries.
1
0 10 20 30 40 50 60 70 80
Days
100
101
102
103
104
105
Number
India
Italy
USA
SK
China
20 30 40 50 60
Days
0
0.05
0.1
0.15
0.2
0.25
λ (Day-1)
Italy
India
USA
SK
China
Figure 1: Left panel: No of + ve cases in India (black circles), Italy (red squares), USA
(blue triangles), SK (brown stars), China (green pluses) with days. Right panel: Assuming
exponential growth of positive ncov cases, λvs days (keeping N0= 1).
Here we have attempted to analyze the raw data of India with comparison to other
countries and tried to understand the growth pattern. Taking the data of ncov positive
cases for India, Italy, USA, South Korea (SK), China from the wikipedia links [4, 5, 6, 7, 8],
we have plotted in the left panel of Fig. (1), where x axis is denoting as days and y axis is
denoting the number of cases. The day, from when first cases is detected, is considered as
day one. The first case detected in India, Italy, USA, SK, are 30th January, 31st January,
20th January, 20th January in 2020 respectively and for China date of first detection is
not very clear but wuhan CDC admitted about a cluster of cases on 31st December, 2019.
Though detection dates of first case is different for different countries but they are considered
as same initial point in the graph. The y axis is taken in logarithm scale to cover rapidly
growing numbers of +ve cases with days. If we assume an exponential growth of +ve cases
N=N0eλt ,(1)
where N0is number of case in day one, then one expect a constant value of λ, which can be
obtained from the relation
λ=1
tlog(N/N0).(2)
Using the N(t), given in left panel of Fig. (1), we have obtained λfrom the Eq. (2), which is
plotted in the right panel of Fig. (1). In principle λshould be constant if the curve follows
exact exponential growth but in reality it may not follow. Therefore, we get non monotonic
curve of λas a function of days instead of a horizontal curve. Another important point, as
a simplified model, we have considered N0= 1 for each data points to see the approximate
deviation in growth of cases.
If we critically analyze the left panel of Fig. (1), then we can identify a threshold date
of different countries, after which a rapid growth is started. Here we see the day 32, 21,
37, 29, 17 may be roughly considered as the threshold day for India, Italy, USA, SK, and
China respectively. At that point of time, their respective number of cases were 3, 3, 14,
33, 46. However if we are interested to identify the exponential growth then we have to
focus on particular time axis zone, where a linear growth in our logarithmic figure (left
2
Table 1: Identifying exponential growth via different time and number of cases for India
and Italy.
Name of countries t0,N027/March, N=N0×eλ×tt,N
India 5/March/20, N0= 31 N= 31 ×e0.14×22 674 6/July/20, N109
Italy 24/Feb/20, N0= 150 N=150 ×e0.31×10e0.15×22 89,421 9/May/20, N107
panel of Fig. (1)) is observed. For India and Italy, we notice this linear growth in log scale
from t0=day 35 and 24. Right panel of Fig. (1) support this exponential growth by showing
approximately constant values of λin the range day 35-57 (5th - 27th March) and day 24- 56
(24th Feb - 27th March) for India and Italy respectively. On day 35 for India-data and day 24
for Italy-data the number of cases are 31 and 150, which are considered as N0of exponential
growth function, described in Eq. (1). For India, Italy, we can find the exponential relation
with λ= 0.11,0.22 by starting time axis from 5/March/20 and 24/Feb/20 respectively. If we
noticed the right panel of Fig. (1), then we can see that after 5/March/20 and 24/Feb/20 i.e.
after day 35 and 24 the λremain constant (0.11 and 0.22) for India and Italy. Although,
we should remember that approximate constant λ= 0.11, 0.22 within day 35-58, 24-56
for India, Italy are obtained when we crudely consider N0= 1. Actually we have to do
exponential fitting within day 35-58, 22-56 with initial values N0= 31, 150 as shown in left
panel of Fig. (2). For India, N0= 31, λ= 0.14 can able to fit data within day 35-58. Hence,
on 27th March or day 57, we will get N=N0eλ(tt0)= 31 ×e0.14×(5735) 674, which is
close to exact data 694. On the other hand, Italy-data within day 24-56 can be fitted by
two sets - N0= 150, λ= 0.31 in day 24-34 and N0= 3500, λ= 0.15 in day 34-56. So, on
5th March or day 34, N= 150 ×e0.31×(3424) 3,329, which is close to exact data 3,089
and next, on 27th March or day 56, N= 3300 ×e0.15×(5634) 89,471, which is close to
exact data 86,498. These information are briefly tabulated in Table (1). Extrapolating the
last exponential growth of India and Italy is really a danger alert, but we should be fear on
that extrapolation possibility until or unless the interventions, taken from those countries,
will defeat this exponential growth. To cover Indian population 109, required time scale
can be calculated as
t=t0+1
λlogN
N0
= 35 + 1
0.14log1.2×109
31 158day ,for India
= 34 + 1
0.15log6×107
3300 99day ,for Italy .(3)
158 day means around 1st week of July for India and 99 day means around 1st week of May
for Italy. It means that next each dates are vital to us and we should adopt all possible
interventions, by which we can transform this exponential growth to a very stable and mild
growth function like Sigmoid function [9]. Recent time Refs. [10, 11, 12] might be a guiding
points about that kind of studies. Next section will focus on that kind of discussions.
3
25 30 35 40 45 50 55 60 65
Days
102
103
104
105
Number
N0=23.5, λ=0.15
N0=32, λ=0.14
N0=3300, λ=0.15
Italy
India
N0=150, λ=0.31
0 10 20 30 40 50 60 70 80 90
Days
100100
101101
102102
103103
104104
105105
106106
107107
108108
109109
1010 1010
Number
exp
SK
China
S
China Population
Jan Feb March April
SK Population
Figure 2: Left panel: Identifying the time zone - day 35 57 for India and day 24-56 for
Italy, exponential fitting has been done. Right panel: Fitting the data of China (pluses) and
SK (stars) by Sigmoid-type function and drawing their possible exponential growth, from
where a fruitful deviation has been made due to their undertaken interventions.
3 Deviation from Exponential growth and Future sketch-
ing
The deviation of exponential function is really necessary to prevent this massive infection
spread. The λis main controlling parameter, which is a proportional constant for the relation
dN
dt N, which means no. of new cases total no cases. In real data this proportional
relation does not hold for entire time axis, therefore we find a time dependent λinstead
of a constant value. However, we can identify different time zones, within where λbecome
constant and from one zone to other this constant value changes. Positive changes is not
a good news for us but negative changes is a hope for possibility of transformation this
exponential function to Sigmoid-type function.
Let us assume that our real data, carry a time dependent λ(t), which may be splitted
into time-independent (λ0) and dependent part (∆λ(t)) as
λ(t) = λ0+ ∆λ(t).(4)
So real data (RD) follow the relation
NRD(t) = N0eλ0t×f(t),(5)
where f(t) = eλt is an important function, which can suppress the exponential growth for
negative values of ∆λ(t). If ∆λ= 0, then we will get exactly exponential function
Nexp(t) = N0eλ0t.(6)
Through proper interventions, there is a possibilities of getting the time dependent functions,
f(t) = 1/eλ0t
a+ 1(7)
4
i.e.
λ(t) = logeλ0t
a+ 1,(8)
for which our exponential function will transform to a Sigmoid function
NS(t) = N0eλ0t/eλ0t
a+ 1.(9)
Here ais very important parameter which fix the maximum number of cases Nmax =N0×a,
where the sigmoid curve will saturate.
As we found that China and SK data already reached to a Sigmoid-type function, so
they may be used as good example where transformation of exponential curve to sigmoid
functions has been achieved. In right panel of Fig. (2) we have attempted to fit the data
of China and SK via sigmoid function (solid line), given in Eq. (9), and we obtained the
parameters - λ0= 0.21, a= 177.8 and λ0= 0.29, a= 46.6 respectively. We know that
for small t, sigmoid function cannot be distinguishable from exponential function, because
f(t)1, when eλ0t<< a, or in other word - when tis very small. Therefore, we draw the
eλ0tcurve(dotted line) for China and SK data in the right panel of Fig. (2) and we noticed
the merging of exponential and Sigmoid function in low tzone. Pointing out China and SK
populations by red arrows, the exponential growth might reach within April-May.
The countries, which have not achieved the sigmoid type function pattern and still
continuing the exponential growth, should follow proper interventions to turn into sigmoid
function (NS) from exponential function (Nexp ). For India, projecting the total number
of cases data (black triangles) we have drawn exponential curve (solid orange line) with
N0= 31, λ0= 0.14 and different possible Sigmoid functions - S1(blue dash line), S2
(red dotted line), S3(green dash-dotted line) in upper left panel of Fig. (3) whose zoomed
version is repeated in it’s upper right panel. The S1,S2,S3are designed by restricting the
saturate values of number of cases - Nmax =N0×a= 104,105,106respectively. Similarly,
projecting the new cases data (black circles), we have drawn time derivative of different
Sigmoid functions in lower left panel of Fig. (3) whose zoomed version is repeated in it’s
lower right panel.
In Fig. (3), data are taken from 5th March from when exponential growth is started.
Within 1st week of July this exponential growth might cover the entire Indian population
(109), pointed by red arrow in upper left panel of Fig. (3). Till now Covid-19 containment
is not possible because vaccine is not available to stop the viral spreading. So we have to
emphasize on mitigation (measures taken to slow it’s spreading) measures/ interventions like
lock-down; screening, testing and isolation of mass population; hand hygiene; using mask
etc. Indian Govt declared lock down for 21 days from 23rd March to 14th April which is
denoted by pink box in Fig. (3). In the upper right panel of Fig. (3) we can see that data
is till not achieved any sigmoid functions so it may be considered as part of exponential
function or low tlimit of sigmoid function. To achieve S1function with Nmax = 104is
quite challenging task to our country. Within the lock-down box, if we can see a rapid
suppression then it is possible, otherwise we have to hope on S2and S3with Nmax = 105
and 106, whose rapid deviation from exponential curve will be seen after lock-down period.
It is only possible after extending the lock-down with more additional interventions. Hence
coming data are very important to us and more critical analysis is required on those data.
5
0 30 60 90 120 150
Days
101
102
103
104
105
106
107
108
109
1010
1011
Number
Exp
S2
S1
S3
April May June July Aug
March
Indian Populations
5th 4th 4th 3rd 3rd 2nd
Present
Lockdown
15 20 25 30 35 40 45 50 55 60
Days
102
103
104
105
Number
April May
Present Lockdown
March
20th 14th 4th
0 30 60 90 120 150
Days
101
102
103
104
New cases
dS3/dt
dS2/dt
dS1/dt
April May June July Aug
March
Present
5th 4th 4th 3rd 3rd 2nd
Lockdown
15 20 25 30 35 40 45 50 55 60
Days
102
103
New Cases
April May
Present Lockdown
March
20th 14th 4th
Figure 3: (a) Upper left panel: To match total case data of India, the exponential function
(brown solid line), and different possible sigmoid functions S1,2,3(blue dash, red dotted and
green dash-dotted lines). (b) Upper right panel: Zoom version of (a) to see the association
of present lock-down (denoted by box). (c) Lower left panel: To match new case data of
India, time derivative of different possible sigmoid functions S1,2,3(blue dash, red dotted and
green dash-dotted lines). (d) Lower right panel: Zoom version of (c) to see the association
of present lock-down (denoted by box).
6
Another important quantity is new cases, which is plotted in lower panel of Fig. (3).
Approximately time derivative of total cases will give new cases as a function of time. For
exact exponential growth, given in Eq. (6), we will get time derivative:
dNexp
dt =λ0Nexp ,(10)
while for Sigmoid function, given in Eq. (9), we will get time derivative:
dNs
dt =λ0Ns1Ns
aN0.(11)
So our expectation is to transform the Eq. (10) to (11) in new cases data. The zoomed version
of new cases graph, shown in right panel of Fig. (3), is supporting our earlier conclusion
obtained for total cases data, discussed above.
4 Summary and Conclusion
In summary, we have attempted to identify growth pattern of covid19 cases in India, where
other country like Italy, USA, China and South Korea are also considered for reference
point of our understanding. Realistic data of different countries carry their own complexity,
but still all of them has common trend that initially they follow very mild growth and
then suddenly a rapid growth. After that rapid growth, they have a tendency to follow
exponential pattern, which is well maintained in the data of India and Italy but the data of
USA, China and South Korea follow very dynamical growth. Present article is intended to
zoom in the straight forward alarm that exponential growth can cover entire population of
India (Italy) within July (May) if it will be continued. In this regards, a positive hope can
be found from the data profile of China and South Korea, whose transformation pattern
from exponential growth function to a stable Sigmoid function is sketched and analyzed. As
a positive hope from present interventions, mainly lock-down, considered by India Govt.,
we have sketched three possible Sigmoid functions, which can be saturated in either 104
or, 105or, 106. Within the lock-down period, we have not found any deviation trend from
exponential to Sigmoid function till now. A rapid, intermediate and mild reduction can
create a possibility of saturation within 104, 105and 106but all of them never be achieved
probably without extending lock-down. Apart from them, we might have to think about
additional interventions for getting deviated from exponential to Sigmoid-type growth.
Acknowledgment: SM and SG thank to their daughter Adrika Ghosh for allowing
time for this investigation during lock-down period.
References
[1] Wikipedia : Pandemic
[2] Wikipedia : Exponential growth
[3] World-meter
7
[4] Wikipedia : India covid19
[5] Wikipedia : Italy covid19
[6] Wikipedia : USA covid19
[7] Wikipedia : South Korea covid19
[8] Wikipedia : China covid19
[9] Wikipedia : Sigmoid function
[10] M. Batista, Estimation of the final size of the second phase of the coro-
navirus epidemic by the logistic model medRxiv 2020.03.11.20024901. doi:
https://doi.org/10.1101/2020.03.11.20024901
[11] M. Batista, Estimation of the final size of the COVID-19 epidemic, medRxiv
2020.02.16.20023606. doi: https://doi.org/10.1101/2020.02.16.20023606
[12] C. Pongkitivanichkul, D. Samart, T. Tangphati, P. Koomhin, P. Pimton, P. Dam-o,
A. Payaka, P. Channuie, Estimating the size of COVID-19 epidemic outbreak, doi:
https://doi.org/10.13140/RG.2.2.29866.36808
8
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The whole world is facing a big crisis due to the spreading of newly detected novel corona virus 2019 (COVID-19). A huge number of people have already been infected since last 4 months. People are thinking about the prevention of infected individuals, vaccine, medical treatment, and other precautions. The governments of most countries including India have already taken several measures like lockdown, social distancing, closure of schools, colleges, religious gatherings etc., to reduce its spreading. India is a developing country and most of the people are having below the standard income. So the lockdown in India has aff ected the poor and middle income group people. In this article, we will discuss in detail on the societal eff ects in India due to COVID-19 pandemic. The eff ects of health, essential commodities, Indian economy, domestic violence, politics, and psychology on society due to COVID-19 will be elaborated in detail. The aim of this research is to have a clear understanding of the present societal scenario during lockdown, which may help the government for bett er management and prevention of the disease.
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In this work, we analyze the epidemic data of cumulative infected cases collected from many countries as reported by WHO starting from January 21 st 2020 and up till March 21 st 2020. Our inspection is motivated by the renormalization group (RG) framework. Here we propose the RG-inspired logistic function of the form αE(t) = a 1 + e −c(t−t 0) −n as an epidemic strength function with n being asymmetry in the modified logistic function. We perform the non-linear least-squares analysis with data from various countries. The uncertainty for model parameters is computed using the squared root of the corresponding diagonal components of the covariance matrix. We carefully divide countries under consideration into 2 categories based on the estimation of the inflection point: the maturing phase and the growth-dominated phase. We observe that long-term estimations of cumulative infected cases of countries in the maturing phase for both n = 1 and n = 1 are close to each other. We find from the value of root mean squared error (RMSE) that the RG-inspired logistic model with n = 1 is slightly preferable in this category. We also argue that n determines the characteristic of the epidemic at an early stage. However, in the second category, the estimated asymptotic number of cumulative infected cases contain rather large uncertainty. Therefore, in the growth-dominated phase, we focus on using n = 1 for countries in this phase. Some of them are in an early stage of an epidemic with an insufficient amount of data leading to a large uncertainty on parameter fits. In terms of the accuracy of the size estimation, the results do strongly depend on limitations on data collection and the epidemic phase for each country.
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In the note, the logistic growth regression model is used for the estimation of the final size and its peak time of the coronavirus epidemic in China, South Korea, and the rest of the World.
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In the note, the logistic growth regression model and the SIR model are used for the estimation of the final size and its peak time of the coronavirus epidemic. PS. Today (28.2.2020) it is more or less clear that the predictions of the article apply only to China because, by February 20, 99% of the case was from China. The linear trend in data from Feb 20 onward meant a decreasing number of infected in China and increasing infected elsewhere in the world. In other words, in China, the epidemic is slowing down, however, it is now developing elsewhere in the world. We note that the forecasting methods used in this article are inapplicable in the early stages of an epidemic.