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### MODELLING THE COURSE OF A VIRAL ILLNESS AND ITS TREATMENT

Description

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. 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.

Modelling 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.

2. Using a spreadsheet, or otherwise, develop a model for the next phase of the 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.

Modelling 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.

3. 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.

The antiviral medication is difficult for the body to adapt to, so it must initially be carefully introduced to the body over a four-hour time period of continuous intravenous dosing. This means the same amount of medication is entering the body at any given time during the first 4 hours. At the same time, however, the kidneys eliminate about 2.5 % of this medication per hour. The doctor has calculated that the patient needs at least 90 micrograms of medication to begin and maintain the rate of elimination of 1.2 million viral particles.

4. Create a mathematical model for this four-hour period so that by the end of the four hour period the

patient has 90 micrograms of medication in their body. Find the solution to your model analytically, or estimate its solution with the help of technology.

Once the level of medication has reached 90 micrograms the patient is taken off the intravenous phase and given injections every four hours. The kidneys will still be working to eliminate the medication, so the doctor must calculate the additional dosage, D accordingly. Dosage D should allow for maintenance of a minimum of 90 micrograms within the patient's bloodstream throughout the treatment regimen.

5. What dosage, D, administered every four hours from the end of the first continual intravenous phase, would allow for the patient to maintain at least 90 micrograms of the medication in his system? Make sure you take into account the kidneys' rate of elimination. Explain carefully how you came to this number.

6. Determine the last possible time from the onset of infection to start the regimen of medication. How long it will take to clear the viral particles from the patient's system? Show on a graph the entire treatment regimen from the time treatment begins until the viral particles are eliminated.