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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, N≈109

Italy 24/Feb/20, N0= 150 N=150 ×e0.31×10e0.15×22 ≈89,421 9/May/20, N≈107

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(t−t0)= 31 ×e0.14×(57−35) ≈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×(34−24) ≈3,329, which is close to exact data 3,089

and next, on 27th March or day 56, N= 3300 ×e0.15×(56−34) ≈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) = e∆λt 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 =λ0Ns1−Ns

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

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[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

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