## History and applications

### HIV virus population growth and decay

In a series of papers in the 1990s, scientists used mathematical models to study HIV viruses in the human body (see the References section).

When a person contracts HIV, viruses infect white blood cells by attaching to certain proteins, and the infected cells are reprogrammed to produce viruses. After an initial acute phase, but before the onset of AIDS, the HIV viruses circulate within the body at low levels. It was previously thought that not much is happening during this phase, but a mathematical model helped to show that this is not the case.

The number $$n$$ of HIV viruses (per millilitre of extracellular fluid) after $$t$$ days can be modelled by the differential equation

$\dfrac{dn}{dt} = kn + m,$

where $$k$$ is the rate at which viruses are cleared from the bloodstream by the immune system (that is, the continuous decay rate for viruses) and $$m$$ is a constant rate of production of viruses from host cells (per day per millilitre of fluid).

We look at a simplified version of an experiment conducted by Perelson and others (see the References section). The experiment started by assuming that the level of HIV viruses had reached equilibrium. A typical value for the number of viruses in the bloodstream was $$100\ 000$$ per millilitre.

Exercise 11

Explain why $$- \dfrac{m}{k} \approx 100\,000$$.

During the experiment, an antiviral agent was administered which effectively reduced the production of new viruses to zero, that is, $$m=0$$. It was then found that the number of viruses decreased exponentially with a half-life of 0.24 days.

Exercise 12

Find the decay rate $$k$$.

From this information, several calculations can be made.

Exercise 13

1. Based on the previous exercises, compute $$m$$, the production rate of viruses per day per millilitre of extracellular fluid.
2. The average human body contains about 15 litres of extracellular fluid. Approximately how many viruses are produced per day?

The answer found by the experimenters (using a more sophisticated model than ours) was that about $$10^{10}$$ viruses are produced per day — far more than previously thought. Even when the HIV infection is in an apparently quiet phase, a huge number of viruses are being produced and destroyed every day!

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