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Start Differential Equations Against HIV
26 February 2015

Differential Equations Against HIV

Estimated reading time Time 5 to read

Every two years, the American Institute of Mathematical Sciences organizes the international conference on “Dynamical Systems, Differential Equations and Applications”, which was held in Madrid in 2014. Among the numerous subjects addressed in the talks at the conference, there were many devoted to the modeling and dynamical analysis of complex patterns in biological systems, where it was shown that differential equations provide a tool for the fight against one of the pandemics of our century, HIV.

Although aids was first detected decades before it ravaged different areas of Africa, the first clinical cases of this disease were recorded in the early 1980s in North America before sowing panic right across the world.

At that time, little was known about this disease that slowly undermined the immune system of those infected, except that it was transmitted by bodily fluids and that its cause, the human immunodeficiency virus, better known as HIV, was indestructible. Those who caught the disease were well aware of their fate; their bodily defenses would gradually collapse before other infectious diseases whose effects would eventually reduce them to skin and bone, with no known cure to save them from their inevitable end.

To date, no known treatment has been found to wipe out this lethal virus, which in recent decades has claimed more than thirty million lives. However, progress in research has enabled aids to be controlled by substantially improving the quality of life of sufferers thanks to the administration of drugs that attack the reproductive process of the virus at different points, thereby blocking its activity.

One of the great challenges facing the scientific community is to find an effective vaccine against this agent that is so lethal that, according to data from the World Health Organization, last year alone it infected 2.3 million people throughout the world. While research goes on to find a possible vaccine, alternative treatments currently consist of pharmacotherapy capable of slowing down the effect of the virus so that it causes no resistance to the medication in the patient.

Doctor Nicoleta E. Tarfulea is involved in this fight against HIV, but using an approach that is somewhat different from that employed in conventional laboratories. She is a professor in the Department of Applied Mathematics at the University of Wisconsin (USA), and the weapons she is using to tackle HIV are nothing less than differential equations.

Her mathematical developments enable very specific experimental situations to be reproduced numerically and therefore at a much reduced cost. These situations consist of testing the effect of a combination of drugs, observing their short- and long-term effect of interrupting the treatment on the rate of mutation of the virus, comparing the effectiveness of early and late treatment, and even studying the effect of a potential vaccine on infected patients.

She does this by means of mathematical models for studying the dynamics of the virus and the dosage in the treatment with antiretroviral drugs.  She presented her latest results last June at the American Institute of Mathematical Sciences (AIMS) Conference on Dynamical Systems, Differential Equations and Applications, which she has attended regularly in recent years. As this mathematician explains, “there’s a great deal of information about the individual processes involved in the dynamics, evolution and treatment of HIV, but there’s still a lot more to do. Integrating these results into a qualitative model in order to describe how the body reacts when it is infected by HIV can help us to tackle the virus more effectively.”

Tarfulea’s research work is focused on the characteristic speed with which the virus mutates and becomes resistant, one of the most puzzling issues that HIV poses to science, since it constitutes the main obstacle for the development of an effective vaccine. Her models enable the dynamics of HIV and the evolution of resistance it sets up in the body to be explored, as well as studying the different possible strategies of treatment for the virus.

Resistance takes algebraic forms

Tarfulea points out that, “resistance to treatment has its origin in different factors, among which is failing to take the medication exactly as it was prescribed.” Skipping a dose or a poor absorption of the medication leads to a reduction in the antiretroviral drugs found in blood samples and allows the virus to reproduce and mutate more freely.

While the appearance of each new mutation of the virus in the organism means stopping one treatment and trying another, in the experiments with mathematical models it is not necessary to change the whole initial formula, but rather add it to all the mutations as if it were forming an increasingly longer tail to the equation. As Tarfulea points ouT.

“From the moment when a mutation is introduced and we have observed its behavior, we can deduce the dynamics of the second one, and so on successively.”

This researcher goes on to say that, “I first introduce into the models a mutation into the system, bearing in mind the wild-type strains so that the original drug is assigned to them, and after that the mutant strains. By adjusting the variables and the concentrations, I can also investigate the different strategies in the treatment.”

In addition to providing an infinite variety of scenarios in which to analyze the behavior of the virus, the mathematical models enable ethical questions to be addressed that would not be possible using real patients. As Tarfulea explains, “HIV is a very delicate subject, especially in the United States. It’s very difficult to find volunteers willing to subject themselves to experimental therapies with drugs. That’s the advantage of our models from all points of view. Thanks to these models a large amount of money and resources can be optimized, as well as solving many of the technical problems that arise in most biomedical studies in general, and particularly in HIV.”

Life in equations

In Madrid she spoke about her latest achievements. “We’ve improved on the existing models and arrived at some very relevant conclusions.” One of the main questions that has been studied is the role played by cells in the immunological system when responding to the virus. “We’ve shown that their presence reduces the concentration of infected T cells – those that coordinate the immune response and which are attacked specifically by the virus – with the result that concentrations of the virus stabilize at the lowest level and enable the healthy cells to increase.”

The most ambitious part of her project consists in setting up a low-cost virtual laboratory capable of handling large quantities of data in order to continue exploring new models that may improve our understanding of the dynamics of HIV and its treatment. [A1] [U2] Says Tarfulea, “I’m optimistic, and believe that this research will lead to a better understanding of the dynamics of the disease as well as bringing us closer to the development of new strategies of treatment. But I’m also aware that we still have a long way to go and that a cure for aids will not be found in the immediate future.”

Her research work also forms part of her teaching duties. “This kind of research in biology shows young scientists the highly important applications of mathematics and the challenges it provides for them. That’s why I always have at least two projects in which my students can participate.” Her models cover practically all fields of life sciences, ranging from studies on salmonella to population dynamics and the monitoring of fish migrations.


Instituto de Ciencias Matemáticas (ICMAT)

Mathematical research center set up by the Consejo Superior de Investigaciones Científicas (CSIC – Higher Council for Scientific Research) and three universities in Madrid: Autónoma University (UAM), Carlos III University (UC3M) and Complutense University (UCM).

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