Mathematics and HIV

World AIDS Day

There is a link between mathematics and HIV that goes beyond the geometric structure of the virus, based on the icosahedron. Denise Kirschner describes this relationship very well in Using Mathematics to Understand HIV Immune Dynamics:

Since the early 1980s there has been a tremendous effort made in the mathematical modeling of the human immunodeficiency virus (HIV), the virus which causes AIDS (Acquired Immune Deficiency Syndrome). The approaches in this endeavor have been twofold; they can be separated into the epidemiology of AIDS as a disease and the immunology of HIV as a pathogen (a foreign substance detrimental to the body).⁽¹⁾

The paper focuses on HIV immunology:

Our goal then is to better understand the interaction of HIV and the human immune system for the purpose of testing treatment strategies.⁽¹⁾

The behavior of the immune system is schematized in this way:

In the diagram, the cells that the HIV virus uses to reproduce itself are CD4⁺ T, that is, the cells that play a key role in the immune system's response against invasions. In summary, this means that any type of treatment must interfere with the transcription mechanism of the virus' RNA into the host cell's DNA. In this way it is not possible to find definitive cures, but simply treatments that block the progression of the disease. The search for a treatment that is ever closer to being a cure is therefore an important step in the fight against AIDS and mathematics could play an important role, and not just with simple statistical studies, but with the creation of mathematical models, such as the deterministic approach proposed by Kirschner.

Continuous dynamical systems, whether ordinary or partial differential equations, are lending new insights into HIV infection. Population models are most commonly used, and, given hypotheses about the interactions of those populations, models can be created, analyzed, and refined.⁽¹⁾

The differential equations used modeling the dynamics within blood cells, both of the diseased and the healthy ones. The equations used are three: The first equation represents the source of new T cells from the thymus.

Since it has been shown that virus can infect thymocytes, we choose a function describing the decreasing source as a function of viral load; assuming that the uninfected T cell populations are reduced by half.⁽¹⁾

It is therefore necessary to model the stimulus to proliferation induced in T cells by the presence of the virus and finally the infection itself. The second equation describes the changes within the infected cell population.

(...) infected cells are lost either by having finite life span or by being stimulated to proliferate. They are destroyed during the proliferation process by bursting due to the large viral load.⁽¹⁾

Finally, in the third equation, we focus on virus and parameters such as the source of the virus itself or its growth rate.
It is very interesting to note how, starting from a series of three numerically solved differential equations, we can also arrive at the formulation of a treatment: it is actually more to show the approach and its possible effectiveness that Kirschner is interested in item. And we can see that mathematics is playing an increasingly important role in the field of medicine, as in some ways Avner Friedman tries to show in What Is Mathematical Biology and How Useful Is It?

Viewing the present trends in mathematical biology, I believe that the coming decade will demonstrate very clearly that mathematics is the future frontier of biology and biology is the future frontier of mathematics.⁽²⁾

In 2012 Rao, Thomas, Kurapati, and Bhat⁽³⁾, based on Indian data, tried to create a model that was able to predict that, in the future, we would use the two different antiretroviral therapies present in India. And even in this case the equations used are just differential equations!

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1. Denise Kirschner (1996). Using Mathematics to Understand HIV Immune Dynamics, Notices of the American Mathematical Society, 43 (02) 191-202 (pdf) ↩︎ ↩︎ ↩︎ ↩︎ ↩︎
2. Avner Friedman (2010). What Is Mathematical Biology and How Useful Is It?, Notices of the American Mathematical Society, 57 (07) 851-857 (pdf) ↩︎
3. Rao A.S.R.S., Thomas K., Kurapati S. & Bhat R. (2012). Improvement in Survival of People Living with HIV/AIDS and Requirement for 1st- and 2nd-Line ART in India: A Mathematical Model, Notices of the American Mathematical Society, 59 (04) 560-562. doi:10.1090/noti835 ↩︎