# No. 3229: THE MATHEMATICS OF HIV INFECTIONS

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Today, we talk about the mathematics of HIV infections and treatment. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Reading popular math books and columns by Cornell professor Steven Strogatz is always a delight. Strogatz has a way of making abstract mathematics accessible. His most recent book on the history, and beauty of calculus is a case in point. Here Strogatz brings to life the millenia long development of calculus, and the enormous impact it had on society and technology.

I've taught calculus often, but still learned much from this book. The mathematics behind the development of the drug cocktail that halted the devastation of the AIDS epidemic is a great example. In the early 90s scientists started finding the first effective treatments for the disease. However, the progress of HIV infections remained a puzzle that scientists were eager to crack. Often a better understanding of how a disease develops can lead to better treatments.

HIV is different from many other viruses. Infected patients first suffer flu-like symptoms, but then get better after a couple of months. After this initial stage patients may live without any signs of infection for years. But eventually the virus gains the upper hand. The last stage of the infection is known as acquired immunodeficiency syndrome or AIDS. It is this stage that we associate with the disease, as it has devastated the lives of millions.

But why do patients get better after the first infection, and then live free of symptoms for years? Is the virus just biding its time, waiting for a chance to strike? A team lead by physician David Ho and mathematician Alan Perelson set out to solve this puzzle in the early 90s. To do so they tested the effect of protease inhibitors, drugs that are effective in fighting HIV infections.

Left: Physician David Ho. Image courtesy of Wikipedia Commons.

Right: Mathematician Alan Perelson. Image courtesy of the University of Washington

Their results were surprising: Mathematical modeling showed that as the drug helped the patient fight the disease, the virus did not become dormant. Rather the body was clearing HIV particles at a furious rate, as they kept reproducing. Thus even in the absence of symptoms, a ferocious battle was raging between the immune system and the invading virus.

The mathematical modeling did far more: Since the virus was active during the entire infection, the team found that drugs should be administered as soon as an infection is detected. Also, the virus mutates quickly, and can evolve to beat a single drug, or even a combination of two drugs. But mathematics showed that the virus is unlikely to get around a cocktail of three or more drugs. This suggested a way of treating the disease with a cocktail of drugs which we still use today.

After reading this amazing story, I went back to the early history of calculus, back to the Greeks who were trying to find the volumes of abstract geometrical objects and shortest paths between points. These seemingly frivolous questions lead to the development of ideas that more than two millennia later, provided us with a way to fight invisible killers within our bodies.

This is Krešo Josić at the University of Houston, where we're interested the ways inventive minds work.

Here is an interesting discussion of dynamic medicine, that is treatments that take into account nonconstant responses that can be described by differential equations.

I am referring to the excellent book entitled Infinite Powers by Steven Strogatz which you can find here.

Here is a nice description of what else you can find in the book.

This episode was first aired on March 3, 2020