   Mathematics AI SL
Sample Internal Assessment

Table of content
Rationale
Aim
Background
Data collection and processing
Conclusion
Reflection
Bibliography

# To investigate the spread of the pandemic COVID-19 in new delhi, india by studying the variations of the susceptible, infected and recovered population with respect to time over a period of three months and determination of correlation and regression between them 6/7  0 Words
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## Rationale

Mathematics is an essential and versatile subject to analytically interpret various real life scenarios. I have always been fascinated about the application of Mathematics in the real world. It has always intrigued me to go beyond the academic curriculum and think about various real life scenarios from a mathematical point of view. Recently, the outbreak of the COVID-19 pandemic has attracted my attention towards it. I got to know about the concept of mathematical modeling of a pandemic, how it spreads and based on that study important measures are taken in controlling the pandemic.

Although, the world is suffering, the pandemic offered me an opportunity to stir up my interests. Since, the concept of mathematical modeling of an epidemic has occupied my mind, I thought of doing my own mathematical analysis and constructing one such model to study the spread of this disease COVID-19. I took the data of Susceptible, Infected and Recovered population over a period of three months and three days. In this study, the concepts of maxima, minima, correlation and regression really helped me arrive at some important conclusions.

The study would help more enthusiasts like me to get an idea of how mathematics is extensively used in the study of spread of a pandemic. It would also help to get an idea of how mathematical modeling is necessary to make important decisions regarding controlling of the pandemic among the population.

## Aim

To investigate the spread of the pandemic COVID-19 in New Delhi, India by studying the variations of the Susceptible, Infected and Recovered population with respect to time over a period of three months and determining the extent of correlation and regression between them.

## Background

### Susceptible, infected and recovered population

In epidemiology, Susceptible population is the population which is at a risk of becoming infected by the disease. Infected population includes the individuals who are infected by the disease. Recovered population consists of the inviduals who were once infected by the disease but recovered from it.

## Data collection and processing

Year
Month
Index
Susceptible population
2020
April
3
1022
2020
April
6
1346
2020
April
9
1712
2020
April
12
2089
2020
April
15
2591
2020
April
18
2943
2020
April
21
3432
2020
April
24
3946
2020
April
27
4465
2020
April
30
5087
2020
May
33
5738
2020
May
36
6526
2020
May
39
7169
2020
May
42
7622
2020
May
45
8312
2020
May
48
8867
2020
May
51
9299
2020
May
54
9769
2020
May
57
11351
2020
May
60
16512
2020
June
63
17136
2020
June
66
19005
2020
June
69
28601
2020
June
72
36650
2020
June
75
43884
2020
June
78
50283
2020
June
81
55689
2020
June
84
59453
2020
June
87
64887
2020
June
90
68797
2020
July
93
76438
Figure 1 - Table On Variation Of Susceptible Population (Taken Per Three Days) From 1st April, 2020 to 3rd July, 2020 Figure 2 - Variation Of Susceptible Population (Taken Per Three Days) From 1st April, 2020 to 3rd July, 2020

After plotting the values of Susceptible population in the y axis with respect to time (in number of days), taken in the x axis, we get the expression for Susceptible population as a function of time. The equation is given by, y = 15.82x- 761.4x + 9369...(i), where y = S (say) denotes the Susceptible population or the number of susceptible individuals and x  = t (say) denotes the time (in number of days).

S(t) = 15.82t- 761.4t + 9369...(i)

Year
Month
Index
Infected population
2020
April
3
822
2020
April
6
1146
2020
April
9
1512
2020
April
12
1889
2020
April
15
2291
2020
April
18
2743
2020
April
21
3232
2020
April
24
3646
2020
April
27
4265
2020
April
30
4587
2020
May
33
5138
2020
May
36
5526
2020
May
39
6169
2020
May
42
6822
2020
May
45
7612
2020
May
48
8167
2020
May
51
8799
2020
May
54
9169
2020
May
57
10351
2020
May
60
12512
2020
June
63
14136
2020
June
66
16005
2020
June
69
22601
2020
June
72
30650
2020
June
75
39884
2020
June
78
47283
2020
June
81
50689
2020
June
84
54453
2020
June
87
61887
2020
June
90
63797
2020
July
93
68438
Figure 3 - Table On Variation Of Infected Population (Taken Per Three Days) From 1st April, 2020 to 3rd July, 2020 Figure 4 - Variation Of Infected Population (Taken Per Three Days) From 1st April, 2020 To 3rd July, 2020

After plotting the values of Infected population in the y axis with respect to time (in number of days), taken in the x axis, we get the expression for Infected population as a function of time. The equation is given by, y = 1001e0.046...(ii), where y = I (say) denotes the Infected population or the number of infected individuals and x = t (say) denotes the time (in number of days).

I(t) = 1001e0.046t...(ii)

Year
Month
Index
Recovered population
2020
April
3
315
2020
April
6
866
2020
April
9
1013
2020
April
12
1262
2020
April
15
1765
2020
April
18
2076
2020
April
21
2689
2020
April
24
3043
2020
April
27
3577
2020
April
30
3844
2020
May
33
4268
2020
May
36
4981
2020
May
39
5369
2020
May
42
5822
2020
May
45
6712
2020
May
48
7567
2020
May
51
8299
2020
May
54
8669
2020
May
57
9151
2020
May
60
9812
2020
June
63
10436
2020
June
66
11105
2020
June
69
12066
2020
June
72
12742
2020
June
75
13884
2020
June
78
15683
2020
June
81
16989
2020
June
84
18553
2020
June
87
19787
2020
June
90
21297
2020
July
93
24738
Figure 5 - Table On Variation Of Recovered Population (Taken Per Three Days) From 1st April, 2020 To 3rd July, 2020 Figure 6 - Variation Of Recovered Population (Taken Per Three Days) From 1st April, 2020 To 3rd July, 2020:

After plotting the values of Recovered population in the y axis with respect to time (in number of days), taken in the x axis, we get the expression for Recovered population as a function of time. The equation is given by, y = 2.38x2 + 7.737x + 1088...(iii), where y = R (say) denotes the Recovered population or the number of recovered individuals and x = t (say) denotes the time (in number of days).

R(t) = 2.38t2 + 7.737t + 1088...(iii)

Now, considering the total population N to be 100000000 i.e., 108, we define our second set of dependent variables which are,

s(t)$$\frac{s(t)}{N},$$ the susceptible fraction of the population

i(t)$$\frac{I(t)}{N},$$ the infected fraction of the population

r(t)$$\frac{R(t)}{N},$$ the recovered fraction of the population

The two sets of dependent variables i.e., S(t), I(t), R(t) and s(t), i(t), r(t) are proportional to each other. So, studying either set of the dependent variables will give us similar information and spread or progress of the pandemic.

s(t) = $$\frac{15.82t^2-761.4t+9369}{10^8}$$

i(t) = $$\frac{1001e^{0.046t}}{10^8}$$

and, r(t) = $$\frac{2.38t^2+7.737t+1088}{10^8}$$ Figure 7 - Variation Of Susceptible Fraction Of The Population With Respect To Time

We sketched the graph of the variation of susceptible fraction of population i.e., of the equation

s(t) = $$\frac{15.82t^2-761.4t+9369}{10^8}$$

Comparing the graphs of the variation of the Susceptible population vs. time and the variation of susceptible fraction of population vs. time, we observe that the increase rate of the Susceptible population is much greater than the increase rate of the susceptible fraction of population. Thus, the growth of the susceptible fraction of population is very slow with respect to time. Figure 8 - Variation Of Infected Fraction Of The Population With Respect To Time

We sketched the graph of the variation of infected fraction of population i.e., of the equation

i(t) = $$\frac{1001e^{0.046t}}{10^8}$$

Comparing the graphs of the variation of the Infected population vs. time and the variation of infected fraction of population vs. time, we observe that the increase rate of the Infected population is greater than the increase rate of the infected fraction of the population between time t = 70 to t = 100 and the increase rate of the infected fraction of the population is greater than the Infected population between time t = 200 to t = 400. Figure 9 - Variation Of Recovered Fraction Of The Population With Respect To Time

We sketched the graph of the variation of recovered fraction of population i.e., of the equation

r(t)$$\frac{2.38t^2+7.737t+1088}{10^8}$$

Comparing the graphs of the variation of the Recovered population vs. time and the variation of recovered fraction of population vs. time, we observe that the increase rate of the Recovered population is much greater than the increase rate of the recovered fraction of the population. Thus, the growth of the recovered fraction of population is very slow with respect to time.

Now, notice that the Infected population depends on the Susceptible population and the Recovered population depends on the number of Infected individuals. The only way the susceptible number of individuals is affected i.e., an individual leaves the susceptible group only when the individual gets infected. So, we can assume that the time rate of change of Susceptible population S(t) depends on the number of susceptible individuals, number of infected individuals and the amount of contact between these two group of individuals.

If we assume that an infected individual has n (a fixed number) of contacts per day with the population. The fraction of these contact made by the infected individual with the susceptible individuals is s(t). Hence, an infected individual generates n×s(t) number of new infected individuals per day.

Therefore, the time rate of change of the Susceptible population S(t) is given by,

$$\frac{dS}{dt}$$ = - n × s(t) × I(t)

One infected individual produces n×s(t) number of new infected individuals per day so, the total number of infected individuals I(t) at a given time t produces n×s(tI(t) new infected individuals. In the above differential equation, the negative sign indicates the loss of the infected individuals from the total susceptible population at a time t.

Therefore, the time rate of change of the susceptible fraction of the population is given by,

$$\frac{ds}{dt}$$ = - n×s(ti(t)

Now, we further assume that a fixed fraction h of the group of infected individuals recovers per day. For example, if the infection lasts for 7 days in an individual then, one-seventh of the Infected population recovers per day.

## Conclusion

According to the data taken of the variation of Susceptible population, Infected population and Recovered population and the mathematical analysis made studying it, the following assumption can be made:

• There exists a very strong correlation between the Susceptible population and the Infected population. This indicates that there is a very strong tendency that almost the entire exposed population, who are at a risk of the infection, is going to get infected.
• Graphically, neither the Susceptible population nor the Infected population reaches a peak within our considered time frame and it is very less likely that a peak would be reached anytime soon.
• There exists a moderately strong correlation between the Infected population and the Recovered population. Graphically, there is a clear indication that the rate of infection is higher than the rate of recovery. This indicates that either the death rate is increasing or the infection takes a much longer time to be healed.
• There is a very weak possibility of a herd immunity to be reached in the near future.

## Reflection

### Strength

• The data of the variations of Susceptible population, Infected population and Recovered population has been taken over a period of 3 months and 3 days, from 1st April, 2020 to 3rd July, 2020, following the standard SIR model. The variations and the correlation between them have been studied and important conclusions have been made. This makes the study coherent.
• A wide range of data has been taken and studied. This makes our study relevant to an extent to arrive at important conclusions.
• The data has been taken from the official website of WHO (World Health Organization), which makes it authentic, error free and could be readily considered to do a critical scientific analysis.

### Weakness

• The data of the variations of the Susceptible, Infected and Recovered has been taken per three days, over a period of 3 months and studied. The data would have been more accurate if the values were taken per one day instead of per three days.
• The study would have been more reliable if the number of deaths and the variation of the deaths with respect to time were included.
• The study would be more scientific and reliable to actually use in real life if higher mathematical and statistical tools like differential equations, Time series analysis, Continuum models, spatial models, Markov chain models etc.

### Further scope

Mathematics is very much essential and needed to analyze the spread of a pandemic (here COVID-19). To take crucial decisions in order to cut the spread of a raging pandemic among a larger population, a mathematical analysis is necessary. The above study has been done without using some really important factors such as the data on the daily count of the number of deaths. As an extension to this analysis, one could study the variation of number of daily deaths and its relation with the Infected population using correlation, regression and calculus. Another study could be done in order to predict and take crucial decisions accordingly using Time series analysis. In order to make the study more analytical, scientific, accurate and reliable to use for making important decisions in real life, one could use higher tools like differential equations, Markov chain models, spatial models, continuum models, complex network models etc.