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Virus Modelling

Extracts from this document...

Introduction

IB HL Maths

Modelling the Course of a Viral Illness and its Treatment

Candidate Name: Sherul Mehta

Centre Number:002144

Candidate Number: CSY 114


Contents

Introduction - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg3

Modelling infection

  • Part 1. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg4
  • Part 2. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg7

Modelling Recovery

  • Part 3. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg11
  • Part 4. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg16
  • Part 5. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg18
  • Part 6. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg20

Analysing your models - - - - - - - - - - - - - - - - - - - - - - - -pg24

Applying your model

  • Part7. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg26
    Introduction
...read more.

Middle

13

2141151.922

14

1208116.419

15

158747.5758

16

-1021459.356

17

-2348817.615

Using this information I was able to plot a graph which would show the effect of medication when there were 8 million particles:

Model 3a: The effect of medicine after there were 8 million viral particles

image17.png

As it evident from the graph, the particles go below 0 after 15 hours. Therefore if a patient took medication when he/she had 8 million viral particles then he/she would be fully cured between 15-16hrs as all the particles are eliminate then and thus the medication would be effective.

Now I went through the same process again but this time testing whether the medication would work effectively if he/she had 9 million viral particles.

The results can be seen on the following page:


9 million particles

A

B

1

Hours

No. Particles

2

0

9000000

3

1

=(B2*(1.6^0.25))-1200000

4

2

=(B3*(1.6^0.25))-1200000

5

3

=(B4*(1.6^0.25))-1200000

6

4

=(B5*(1.6^0.25))-1200000

7

5

=(B6*(1.6^0.25))-1200000

8

6

=(B7*(1.6^0.25))-1200000

9

7

=(B8*(1.6^0.25))-1200000

10

8

=(B9*(1.6^0.25))-1200000

11

9

=(B10*(1.6^0.25))-1200000

12

10

=(B11*(1.6^0.25))-1200000

13

11

=(B12*(1.6^0.25))-1200000

14

12

=(B13*(1.6^0.25))-1200000

15

13

=(B14*(1.6^0.25))-1200000

16

14

=(B15*(1.6^0.25))-1200000

17

15

=(B16*(1.6^0.25))-1200000

18

16

=(B17*(1.6^0.25))-1200000

19

17

=(B18*(1.6^0.25))-1200000

20

18

=(B19*(1.6^0.25))-1200000

21

19

=(B20*(1.6^0.25))-1200000

22

20

=(B21*(1.6^0.25))-1200000

23

21

=(B22*(1.6^0.25))-1200000

24

22

=(B23*(1.6^0.25))-1200000

25

23

=(B24*(1.6^0.25))-1200000

26

24

=(B25*(1.6^0.25))-1200000

27

25

=(B26*(1.6^0.25))-1200000

Hours

No. Particles

0

9000000

1

8922143.853

2

8834580.396

3

8736099.295

4

8625339.309

5

8500769.475

6

8360667.943

7

8203098.181

8

8025882.203

9

7826570.468

10

7602408.018

11

7350296.399

12

7066750.835

13

6747852.058

14

6389192.137

15

5985813.547

16

5532140.644

17

5021902.602

18

4448046.729

19

3802640.984

20

3076764.34

21

2260383.472

22

1342214.075

23

309564.8829

24

-851837.747

25

-2158047.135

Using the data above, I plotted a graph:

Model 3b: The effect of medicine after there were 9 million viral particles

image18.png

Again we can see from the graph that the medication would be effective when it is administered to a patient who has 9 million viral particles. Such a patient would be cured between 23-24hrs as the particles are below 0 then. Its effect would be slower than a patient who has 8 million particles but they would still be cured.

Now I did the same for a patient with 10 million viral particles.

10 million particles

Hours

No. Particles

0

10000000

1

10046826.5

2

10099491.46

3

10158722.82

4

10225339.31

5

10300261.72

6

10384525.65

7

10479295.83

8

10585882.2

9

10705758.05

10

10840580.34

11

10992212.63

12

11162750.83

13

11354552.19

14

11570267.86

15

11812879.52

16

12085740.64

17

12392622.82

18

12737767.88

19

13125946.54

20

13562524.34

21

14053535.82

22

14605767.91

23

15226853.77

24

15925378.25

25

16710996.62

A

B

1

Hours

No. Particles

2

0

10000000

3

1

=(B2*(1.6^0.25))-1200000

4

2

=(B3*(1.6^0.25))-1200000

5

3

=(B4*(1.6^0.25))-1200000

6

4

=(B5*(1.6^0.25))-1200000

7

5

=(B6*(1.6^0.25))-1200000

8

6

=(B7*(1.6^0.25))-1200000

9

7

=(B8*(1.6^0.25))-1200000

10

8

=(B9*(1.6^0.25))-1200000

11

9

=(B10*(1.6^0.25))-1200000

12

10

=(B11*(1.6^0.25))-1200000

13

11

=(B12*(1.6^0.25))-1200000

14

12

=(B13*(1.6^0.25))-1200000

15

13

=(B14*(1.6^0.25))-1200000

16

14

=(B15*(1.6^0.25))-1200000

17

15

...read more.

Conclusion

If the patient was a young child than the models would really look different because from my research I found that a child’s immune system is not completely developed until he or she is 14 years old[1] i.e. the immune system of a child is weaker than that of an adult. Therefore the models would have to be modified in several ways.

  1. It says in the description that an adult’s immune system would reduce the replication rate from 200% to 160% every 4 hours. As a child’s immune system is weaker and therefore the rate will not decrease to 160%.
  2. Also a child’s immune system will not be able to eliminate as much as 50000 particles every hour like the adult one does.
  3. In the ‘Modelling Recovery’ section it states that together with the medication and the immune system, 1.2 million particles would be eliminated every hour. Again, as a child’s system is weaker, it will not be able to eliminate 1.2 million particles every hour with medication. Hence, for a child to make a full recovery, he/she must be given medicine before the particles reach 9624434 which is the figure for the last point of medication for a standard adult.
  4. Also, since the medication would not be as effective in a child, he/she will need more than the minimum of 90 micrograms of medication.
  5. If more than 90 micrograms of medication will be required than more dosage will be required to maintain that level.
  6. If I child was infected, then he/she would have to take medication earlier than an adult would to make a full recovery.
  7. Even when the medication was administered, it would take longer for a child to recover than an adult would.

As we can see from above, several figures would have to be altered and therefore all the models in my investigation would have to be modified.


[1]http://www.whatreallyworks.co.uk/start/kidszone.asp?article_ID=559 (accessed on 28th Oct 2008)

...read more.

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