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Megan Tsui 12N2



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 1million viral particles in the body.

The first response of the immune system is fever. The rise in temperature lowers the rate at which the viral particles replicate to 160% every four hours, but the immune system can only eliminate these particular viral particles at the rate of about 50,000 viral particles per hour. Often people do not seek medical attention immediately as they think they have a common cold. If the number of viral particles however, reaches 1012, the person dies.

Modeling infection

  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.

Since we know that in 4 hours the viral particles would double its number. Therefore we know that the formula for 4 hours would be


However to work out the growth rate of the particles in 1 hour we would need to work out


From this we can work out that the general formula for would be:


Since we need to work out the total number of particles for each hour, we put 2 to the power of t divided by 4.

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If medication were not administered, the immune system would not be able to eliminate a lot of viral particles. Then the particles would accumulate and grow faster and faster. When the particles reach to 1012, which is between the 98th and the 99th hour, the person would die. This mean the patient would die on the 4th day if he or she has not taken any medicine.

Modeling recovery

An antiviral medication can be administered as soon as a person seeks medical attention. The medication does not affect the growth rate of the viruses but together with the immune response can eliminate 1.2 million viral particles per hour.

  1. If the person is to make a full recovery, explain why effective medication must be administered before the number of viral particles reaches 9 to 10 million.

Now let’s explore when the patient must take medication by latest in order to make a full recovery, assuming that once the particles reaches 9 to 10 million, the growth rate would maintain 1.6 even when the particles drop below 1 million.

Increasing rate = 160% per 4 hrs

Decreasing rate = 1,200,000 per hr

Increasing3 rate per hr = 1.1246826503807

All the table and graphs below will be base on this general formula:

xt+1 = (1.6)1/4xt - 1200000 if xt ≥ 1,000,000

The Patient starts taking medicine when there is 9,500,000 viral particles inside his body


No. of Viral Particles



No.of Viral Particles




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In question 4 we assume that the patient take the medicine per hour, so that we could calculate how many viral particles the kidneys would eliminate in every hour.

The model should be modified if the patient were not an adult but a child. Since the children’s immune system would be weaker. The immune system of the adult would be 50000 but the children would smaller than 50000. The medicine that the children take should be less than the adults’ proportion as the children’s body is not capable to take in too much medicine. Therefore it might means that the children might need to administrate effective medication before 9,000,000 to 10,000,000, in order to survive.

Weakness is that we had made assumptions and therefore we cannot from time to time check the growth factor of the particles. Another weakness is that we estimate by the graphs, therefore it may be in terms of the hours, but not minutes, therefore it is not accurate to the patient. The reliability is also not as reliable because the units are in hours which is very general and it is not precise enough. In order to solve this problem we might need to change the units to minutes to seconds, so that we can estimate precisely what exact time the patient would die. To be more precise I can also work out the exact value of the particles that would cause the death of the patient.

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

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