# Mathematics to curb HIV infection | Science

After the discovery of human immunodeficiency virus (HIV-1 and HIV2) in the early eighties, more than 70 million people worldwide have been infected and almost half of them have died from AIDS. Currently, continuous treatment with combination antiretroviral therapy (cART) can keep the virus below detectable blood levels, and thus prevent the development of the disease. However, HIV remains as a latent provirus in the genome of host cells that were initially infected, being undetectable by the immune system and by current medications, and therefore, the infection can not be cured. When cART stops, HIV quickly re-spreads into the blood and exerts its destructive effect on the immune system. Would it be possible to control the virus better? Mathematics is bringing a new approach to address the issue.

As with any complex problem, the first step in solving it is to understand the underlying system. In this case, it is necessary to understand which are the interconnected elements in the advance of the infection of the virus and in the attempts of protection of the organism mediated by the immune system. At this point mathematicians, in collaboration with experimental biologists, can formalize and link all the important elements of the system, and thus generate predictive mathematical models.

Since it infects a new cell, HIV takes about 24 hours to replicate. Therefore, to block its spread the immune cells must find and kill the infected cells in this period of time. Infection control can be transformed into a mathematical problem based on elements such as the number and spatial distribution of HIV-infected cells, as well as on the properties of the cells. cytotoxic cells responsible for eliminating them (how they move, search and find infected cells).

Based on certain assumptions, mathematicians have been able to predict, for example, that a certain minimum frequency of cytotoxic cells is needed to permanently inhibit the expansion of the virus. This deduction has to be confirmed in the laboratory, in an iterative process in which the experimental data and the quantitative models advance until the predictions have biological value. At that point you can use the models to make predictions of therapy.

The first mathematical models of HIV infections date back to 1995, when Alan Perelson Y Martin Nowak they joined immunologists and virologists, and estimated the rates of HIV replication and death in infected patients. However, attempts to understand the factors that drive the pathogenesis of HIV and the dynamics of infection can be traced back to the studies by Simon Wain-Hobson Y by Zvi Grossman among many others. In recent years, with the development of research techniques, our understanding of the interaction between HIV and the host organism has grown enormously.

In parallel, the Mathematical modeling approaches evolved from simple designs based on ordinary derivative equations to hybrid multi-scale frames. This reflects an important shift towards a high resolution model, which is needed to replicate the complexity of systems immunology results. The use of these mathematical models based on information from each patient could establish a therapeutic strategy to neutralize the infection in a personalized way. This approach would help in the implementation of new therapeutic options that are in development, such as the so-called inhibitors of immune control points and the drugs anti-fibrotic.

However, physiological parameters vary from one patient to another and, therefore, personalized treatments require the development of Parameter estimation methods robust and efficient, capable of assimilating the individual data in the models. The authors of this text we work on computational algorithms for the identification of parameters of this type. In combination with clinical and experimental studies, mathematical modeling offers a new paradigm of medical care known as systemic medicine.

In addition, given that the search for HIV-infected cells may not be very different from the search for cancer cells, it is expected that these same models of the immune system will provide an important advance on how to customize anti-tumor medical treatments.

Irina Gainova is a researcher at the Sobolev Institute of Mathematics of the Russian Academy of Sciences

Larisa Beilina is a professor at the Chalmers University of Technology and at the University and Gothenburg (Sweden)

Jordi Argilaguet is a researcher at Pompeu Fabra University

Andreas Meyerhans is an ICREA research professor at Pompeu Fabra University

Gennady Bocharov is a professor at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences

Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".

Editing and coordination: Agate Rudder (ICMAT)