In what way is mathematics applied in practise in the study of HIV? A first important fact is that there are measurable quantities which can be used to gauge the state of the patient. These are the densities of virus particles and CD4+ T-cells in the blood. The latter is actually used to give a practical definition of AIDS. After infection with HIV it usually takes a long time before the full symptoms of AIDS are observed, typically many years. The T-cell count falls gradually and the start of AIDS can be associated with this count falling under a certain value. It might be supposed that the long quiescent phase of the HIV infection resulted from a kind of dormant state of the virus. The new results involving the use of mathematics suggest a different picture, showing that this long asymptomatic phase is a period of evolution of virus and immune system. There is apparently still a lot to be learned about this evolution but at least some facts are known and the evidence for a dynamic as opposed to a quiescent picture is itself very important.
When some observable quantities are available for a dynamical process one way of trying to learn more about it is to perturb it. In other words, change some aspect of the situation and see what happens. An opportunity to do this in the case of AIDS was presented by the development of drugs which interfere with the development of the virus. Of particular importance were the protease inhibitors which cause the newly produced virions to be incapable of infecting new cells. Treating a patient with this type of substance leads to an exponential decay in the virus load over a period of two weeks and a rapid increase in the T-cell count during the same period. Of course a prerequisite for finding this out is measuring the relevant concentrations frequently.
The mathematical analysis is easiest in the case of a different type of drug, the reverse transcriptase inhibitor, which prevents virions, even those which existed before treatment, from productively infecting new cells. The effect on the basic model of virus dynamics as presented in the book of Nowak and May is the set one coefficient to zero and to cause two of the three equations to decouple from the remaining one. For this the simplifying assumption is made that the inhibitor is one hundred per cent effective. The resulting system is linear and can be solved explicitly. The conclusion is that the rate of decay of the virus population is controlled by the natural decay rate of free virus and the death rate of infected T-cells. The key parameter is the slower of the two rates and it is typically assumed that this is the death rate of the T-cells. Under this assumption the parameter representing this death rate can be computed from the data. This also provides information about the rate of infection of cells before treatment and shows convincingly that it is high.
The high turnover rate of the virus implies that it is able to mutate effectively to meet the challenge of a drug. It this which means that treatment with a single inhibitor only gives a short term effect. Using more than one drug simultaneously makes it more difficult for the virus to adapt. Combining three different substances and different inhibition mechanism is what has turned out to be the successful in keeping HIV under control for extended periods. There are still serious problems such as the cost of the treatment and the serious side effects. Nevertheless this was a turning point in the treatment of AIDS and theoretical analysis, and in particular mathematics, played a major role.