A model as simple as the ones you’ve seen in this chapter was used to tell us something important about the HIV virus that causes AIDS. One intriguing thing about HIV infection is that patients infected with the virus can go many years with no symptoms. Since the amount of HIV in their bodies appeared to be relatively stable, it was tempting to conclude that the virus was just sitting around, not doing much at all. It took a simple mathematical model to show how wrong this conclusion is.

In a famous paper of 1995 (Ho et al, 1995) a group of scientists measured the turnover of HIV virus in infected humans. They did this by assuming that the viral load, L, was given by the simple differential equation dL dt = Production − Degradation.
They then assumed that degradation of the virus was proportional to L (just like radioactivity), and they then temporarily turned off production by using a drug called ABT-538. They ended up with the equation dL dt = −k L, for some constant k.

a. Solve this equation for L, assuming an initial condition of L(0) = L0.

b. Show that if you plot ln(L) against t you get a straight line with slope −k.

c. The plot of Ho et al is shown in Fig. 25.15. Using only the straight part of the graph (the points at the beginning and end aren’t relevant for measuring k because the model doesn’t work there), show that k ≈ 0.2/day. Using this information show that the half-life of the virus is approximately 3.3 days.
More recent measurements have shown that the half-life is actually considerably shorter than 3.3 days, often as short as a few hours.

d. Show that this means that, during the years for which the patient has no symptoms, far from sitting around doing nothing the HIV virus is reproducing quickly enough to replace the entire virus population in just a few days. It is being produced and degraded very quickly, but the production and degradation almost exactly balance!