# Virus Modelling

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

• Part7. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg26
Introduction

Through this coursework I will investigate the case of a standard adult patient who is infected by a virus and produce models to explain the entire process. I will look at the replication rates of the virus at different stages. Then I will show what happens when the immune response begins and how to eliminate the viral particles. Then I will show how and when medicine can be administered and I’ll portray its effects on the patient. To help recover the patient, I will determine how much medicine is required and at what levels. I will also explain how additional dosage can help maintain the medication level. Penultimately, I will work out what is the last point when a patient can be given medicine and if it is given at the last possible time I will find out how long it will take for the patient to fully recover. Finally, I will identify reasons to why my models would have to be modified if the patient was a young child instead of an adult.

1. Model the initial phase of the illness for a person infected with 10 000 viral particles to determine how long it will take for the body’s immune response to begin

From the information given, at 0 hours there are 10 000 viral particles. Since it increases by 200% I could formulate an equation (Using Excel) that would show the number of particles after every 4 hours:

This gave me the following results:

Using this information I was then able to produce plot the points on a graph (using Autograph 3.2).

I chose the function y = a×10nx and using the Constant Controller option in Autograph I altered the constants a and n. Hence I was able to draw a curve of best fit:

Using the constant controller the value I got for constant a was 10000 and for constant n it was 0.0753. This gave me the following equation for the curve:

y=10000×100.0753x.

This gave me the following graph:

Model 1: Initial phase of the illness where there are 100000 viral particles

Using the equation found above, I was able to determine how long it would take for the immune response to begin (after there are 1 million particles):

y=10000×100.0753x

y=1000000

x=Time

1000000=10000×100.0753x

100=100.0753x

Log10100=0.0753xLog1010 (Log1010 = 1)

2=0.0753x

x=26.56 (2 d.p.)

26.56hrs = 26hrs 33mins 36secs

It would take 26hrs 33mins 36secs for there to be 1 million particles and the immune response to begin.

To re-check my solution I added the equation y=1000000 to my graph. Using the ‘point mode’ in Autograph I was able to place a point where the two functions intersected:

As we can see in the red circle y will = 1000000 (1E+06) when x = 26.56.

Therefore, from both the methods it can be seen that it would take 26.56hrs = 26hrs 33mins 36secs for the immune response to begin for a person that that was infected with 10000 viral particles.

2. Using a spreadsheet, or otherwise, develop a model for the next phase of illness when the immune response has begun but no medications have yet been administered. Use the model to determine how long it will be before the patient dies if the infection is left untreated.

The immune response begins after there are 1 million virus particles (26h 36mins).

The patient dies when there are 1012 particles.

Since it increases by 160% I could formulate an equation (Using Excel) that would show the number of particles after every 4 hours. However 50000 would be eliminated every hour so in 4 hours 200000 would be eliminated:

Note: For convenience I have started at 0 hours. There are 1000000 particles after 26.56 hrs and therefore that will be added when I do further calculations.

The table created using Excel:

Using this information I was then able to produce plot the points on a graph (using Autograph 3.2).

Note: In the graph it uses E+n. This is equivalent to ×10n.

Again I chose the function y = a×10nx and using the Constant Controller option in Autograph I altered the constants a and n. Hence I was able to draw a curve of best fit:

Using the constant controller the value I got for constant a was 1000000 and for constant n it was 0.0496. This gave me the following equation for the curve:

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