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Modelling the course of viral illness

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Introduction

The aim of this essay is to model the course of a viral illness and its treatment.

When viral particles of a certain virus enter the human body, they replicate rapidly. In about four hours the number of viral particles has doubled. The immune system does not respond until there are about 1 million viral particles in the body.

The first response of the immune system is fever and the rise in temperature lowers the rate at which the viral particles replicate to 160% every four hours. The immune system tough can only eliminate about 50 000 viral particles per hour. If the number of viral particles, however, reachesimage00.png, the person dies.

For a person infected with 10 000 viral particles we could determine how long it will take for the immune system response to begin. As we have an ordered set of numbers and a common ratio we could consider this to be a geometric sequence.

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Middle

image24.pngimage25.png. Then 2.5% of this is eliminated by the kidney. Therefore we will have around 22 image25.png every hour. In order to maintain the rate of elimination we should add to 22 image25.png around 3image25.png each hour. Hence the amount entering the body should be around 25image25.png per hour.

time (h)

medication (µg)

0

25

1

24,375

2

48,141

3

71,312

4

93,904

image26.png

image27.png

image28.png where image22.pngis the amount of medication entering the body every hour and image29.pngis the amount of medication at image30.pnghours.

As the kidneys will eliminate about 2.5% of this medication per hours that means that every four hours they will eliminate 10% of this medication.

Verification using the previous model:

image31.png

image32.pngimage06.pngimage33.pngimage22.png should be greater than 2.35 in order for the dosage image34.png to slightly increase rather than decrease. image06.pngimage36.png

Hence for the patient to maintain at least 90 micrograms of the medication in his system, a dosage of aboutimage37.png per four hours should be administrated.

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Conclusion

"c4">90,870

8

93,499

8

94,250

9

90,158

9

93,404

9

94,332

10

89,464

10

93,312

10

94,411

11

88,787

11

93,221

11

94,488

12

88,128

12

93,133

12

94,563

13

87,485

13

93,047

13

94,637

14

86,857

14

92,964

14

94,708

15

86,246

15

92,882

15

94,778

16

85,650

16

92,803

16

94,846

17

85,069

17

92,725

17

94,913

18

84,502

18

92,649

18

94,977

19

83,949

19

92,576

19

95,040

20

83,411

20

92,504

20

95,102

21

82,885

21

92,434

21

95,162

22

82,373

22

92,365

22

95,220

23

81,874

23

92,299

23

95,277

24

81,387

24

92,234

24

95,333

25

80,912

25

92,170

25

95,387

26

80,450

26

92,109

26

95,440

27

79,998

27

92,048

27

95,491

28

79,558

28

91,990

28

95,541

From the spreadsheet we could conclude that it is more convenient to have an amount of medication per hour greater than the amount eliminated by the kidneys.

The last possible time from the onset of infection to start the regimen of medication is approximately 46 hours. It will take about 25 hours to clear the viral particles from the patient’s system.

image19.png

image38.png

This graph shows the entire treatment regimen from the time the treatment begins until the viral particles are eliminated from the last possible time to start the regimen of medication.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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