Mathematics AI SL

Sample Internal Assessment

Table of content

Rationale

Aim

Background

Data collection and processing

Conclusion

Reflection

Bibliography

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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.

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.

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.

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

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 = 15.82*x*^{2 }- 761.4*x* + 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*) = 15.82*t*^{2 }- 761.4*t* + 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

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* = 1001e^{0.046x }...(*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*) = 1001*e*^{0.046t}...(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

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* = 2.38*x*^{2} + 7.737*x* + 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*) = 2.38*t*^{2} + 7.737*t* + 1088...(*iii*)

Now, considering the total population N to be 100000000 i.e., 10^{8}, 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*) = \(\frac{2.38t^2+7.737t+1088}{10^8}\)

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.

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.

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*(*t*)×*I*(*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*(*t*)×*i*(*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.

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.

- 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 1
^{st }April, 2020 to 3^{rd}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.

- 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.

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.

*Coronavirus.*https://www.who.int/westernpacific/health-topics/coronavirus. Accessed 3 Oct. 2021.- “Susceptible Individual.”
*Wikipedia*, 23 June 2021.*Wikipedia,*https://en.wikipedia.org/w/index.php?title=Susceptible_individual&oldid=1030024274. - “Cartesian Coordinate System.”
*Wikipedia*, 30 Aug. 2021.*Wikipedia*, https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system&oldid=1041463197. - “Regression Analysis.”
*Wikipedia,*12 July 2021.*Wikipedia,*https://en.wikipedia.org/w/index.php?title=Regression_analysis&oldid=1033186605. CloseDeleteEdit - James, Emily.
*What Is Correlation Analysis? Definition and Exploration.*https://blog.flexmr.net/correlation-analysis-definition-exploration. Accessed 11 Sept. 2021. - “#IndiaFightsCorona COVID-19.”
*MyGov.In,*16 Mar. 2020, https://mygov.in/covid-19/. - “#IndiaFightsCorona COVID-19.”
*MyGov.In,*16 Mar. 2020, https://mygov.in/covid-19/. - “#IndiaFightsCorona COVID-19.”
*MyGov.In*, 16 Mar. 2020, https://mygov.in/covid-19/.