AIDS (Acquired Immune Deficiency) started appearing in the 1980s, killing tens of thousands of people a year in the US and more worldwide. No one then knew what it was, where it came from, and what was causing it. But its effects were clear – it weakened patients’ immune systems so severely, they became vulnerable to all kinds of infections, from pneumonia to cancer. No cure was in sight.

Treatment in the form of antiretroviral drugs began in 1987. It slowed the culprit, the Human Immunodeficiency Virus down by interfering with the process by which it hijacked the cell’s genetic machinery and co-opted it to make more viruses. But it wasn’t a cure.

Then, in 1994, a different class of drugs called Protease Inhibitors appeared. They thwarted HIV by interfering with the newly produced virus particles, keeping them from maturing and rendering them noninfectious. This was ann impressive feat, but still, it wasn’t a cure.

Soon after the Protease Inhibitor became available, a team of scientists led by Dr. David Ho (now at Columbia University) and a mathematical immunologist named Alan Perelson (Los Alamos National Laboratory) collaborated on a study that changed how doctors would treat HIV and other life-threatening viral diseases, including the current Coronavirus. Their research would not have been possible without the use of mathematics, in particular, calculus. This is their story.

When a person becomes infected with HIV, he or she displays flu-like symptoms like fever, rash, and headaches. The body’s T-cells, a key component of the immune system, plummets. A normal T-cell count is about 1,000 cells per cubic millimeter of blood; after first-stage HIV infections, T-cell count drops to the low hundreds. Meanwhile, the number of virus particles in the blood, known as the viral load, spikes and then drops as the immune system beings to fight the HIV infection. The flu-like symptoms disappear and the patient feels better.

However, at the end of this first stage, something puzzling happens. The viral load stabilizes at a level that can last for many years, as if a truce has been reached between HIV and the immune system. Doctors call this level of viral load, the set point, and this stable period, the asymptomatic period. During this period, a patient who is untreated may survive for a decade with no HIV-related symptoms. Eventually however, this “honeymoon” period ends, and AIDS sets in, marked by a further decrease in the T-cell count and a sharp rise in the viral load. Left untreated, the patient will again be vulnerable to opportunistic infections, usually leading to death within 2 to 3 years.

What was going on during the asymptomatic period? Understanding it could be the key to effective treatment. David Ho and Alan Perelson was determined to find out. They wanted to track the dynamics of the immune system as it battled HIV. They found that after each patient took the Protease Inhibitor, the number of virus particles dropped exponentially fast; half of all the virus particles in the bloodstream were cleared by the immune system every two days!

To see what was going on, Ho and Perelson turned to calculus. First, they represented the changing concentration of virus in the blood as an unknown function, , where t denotes the elapsed time since the Protease Inhibitor was administered. Then they thought about how much the concentration of virus would change, , in an infinitesimally short time interval, . Their data indicated that a constant fraction of the virus was cleared in the blood each day, so perhaps the same constancy would hold when extrapolated down to an infinitesimal time interval.

Since is the fractional change in the virus concentration, their model could be translated into symbols as follows:

This says that over a very short time interval, the viral load decreases fractionally by a constant number, c.  This equation is a simple differential equation. It relates the differential to another differential, of the elapsed time.

Integrating both sides of this equation gives the following result:

where is the initial viral load and ln is the natural logarithm. Inverting this function implies:

where e is the base of the natural logarithm (thus modelling an exponential decay in the viral load).

Finally, by fitting an exponential decay curve to their experimental data, Ho and Perelson estimated the previously unknown value, c. They then modified their equation like so:

Here, P refers to the rate of production of the new virus before the Protease Inhibitor was administered. Ho and Perelson imagined that before the administration of the drug, at every moment, infected cells were releasing new virus particles, which then infected other cells and so on. At the set point, however, the virus was produced as fast as it was cleared. Mathematically, this means that (a “steady state”) which implies that

This is the set point level of the virus load. Ho and Perelson used this simple equation to estimate the vitally important number (P) that no one had found before: the number of virus particles being cleared each day by the immune system. It turned out to be a billion virus particles a day!

That staggering number means that the asymptomatic stage is far from the calm state that its name implies. In fact, a titanic struggle was taking place during this decade-long period. Every day, the immune system was fighting hard to dispose a billion virus in an all-out war that led to a standstill.

A bit later, Ho and Perelson collected more data and refined their model to account for the time lag between the medicine’s absorption, distribution, and eventual penetration into the target cells. When they redid the experiments and fit the data, the results were even more alarming: 10 billion virus particles were being produced and then cleared from the bloodstream each day.

Ho and Perelson’s discovery was nothing short of monumental, with far-reaching implications for treatment. Until their work, doctors waited until HIV emerged from the supposed hibernation before they prescribed antiviral drugs. They thought this was a wise decision because it conserves the strength of the patient’s immune system until it really needed help. The Ho-Perelson findings, turned this pictured upside down. There was no hibernation. A colossal war was going on and the immune system needed all the help it could get during the supposed calm period. Help eventually came in the form of cocktail drugs. The idea is that HIV must be beaten back using not one drug, but a combination of different drugs.

Ho and Perelson’s mathematics gave a quantitative estimate of how many drugs had to be used. Long story short, when Ho and his colleagues tested a three-drug combination on HIV-infected patients in clinical studies, the results were astonishing. The level of virus in the bloodstream dropped about a hundredfold in two weeks. Over the following month, it became undetectable. This doesn’t mean that a cure for HIV has been found. The virus could still rebound aggressively if patients take a break from therapy. But it does signal a new and powerful way of tackling a recalcitrant enemy, changing HIV from a death sentence to a chronic condition that could be managed for those who had access to treatment. With this valuable experience behind him, Ho is now heading a new team of researchers to tackle another the massive challenge posed by the Covid-19 virus.