# Immune mathematics

## Our friends lymphocytes and their enemies

Guarding the health of any living organism is the immune system protecting it from viruses, bacteria and parasites that cause disease. The main weapon of the immune system — lymphocytes, blood cells that recognize foreign proteins, which produce antibodies and destroy damaged cells and pathogens. To lymphocytes could do it, they need to be as early as possible met his target. With regard to the viruses before the infected cell will make and return into the body, new virus particles.

Lymphocytes, fighting with strangers, not universal soldiers. Among them are the division of roles. Macrophages and dendritic cells are the scouts, whose task is to detect intruders. Directly from foreign and infected cells interact with T-killers (their second title of CD8+ T lymphocytes). Control the activity of lymphocytes, increasing their activity in case of danger, “officers” — T-helper cells (CD4+ cells). And as defenders of healthy cells are regulatory T-lymphocytes, reducing the strength of the immune response after elimination of the threat. Finally, b-lymphocytes produce antibodies that neutralize viruses in the extracellular environment, as well as participating in the fight against microbes and parasites.

Because different T-killer cells “tuned” to different types of threats, and such threats millions, in determining the threat of T-helper cells “raise the alarm” — a signal for reproduction of a certain kind of T-killers, trained to handle precisely this enemy. Such cells are called antigen-specific T cells, the immune system is able to “train” a new T-killer cells to respond to new threats is the mechanism of acquired immunity.

But in the 80-ies of XX century was discovered a virus that almost completely destroys the immune system. It attacks the very cells of the immune system, the lymphocytes and therefore the immune activity of infected people was reduced to the level at which they were no longer able to resist infections. In 1986, the pooling of efforts of many scientists has helped to accurately identify the virus, and it was named the human immunodeficiency virus first type (HIV-1). It seemed that before receiving vaccines was just a step, but after nearly 40 years the problem is still not resolved. Over the years the HIV virus has been studied in almost all parts, the mechanisms of transmission, mechanisms of infection, interact with cells of various types, and more. However, the main way HIV for many years remain the prevention of infection and antiretroviral therapy, aimed at controlling the propagation of viruses and, thereby, support a weakened immune system.

In the process of finding a cure for HIV one of the main directions of the research was to find ways of dealing with the virus in the first stage of infection, when the immune system is still strong and can be mobilized to fight. But for this, scientists need to understand how to work with the immune system at this point, and in what way she can help. Then based on knowledge about the behavior of immune cells, it will be possible to build models of combination antiviral and immunomodulatory therapy, and then to develop drugs.

## The decision on the level of behavior

The task of searching for target cells of the immune system similar to the search for mushrooms after the rain, on a large meadow. If you look for mushrooms those are very long or not find at all – they will escalate (the disease will progress). If the mushroom is too much – all mushrooms and not enough pickers to blows (occurs autoimmune disease). To find the optimal number of lymphocytes-“mushroom” and determine efficient routes and speeds in the “meadow” is a very complex task with many variables and random variables. Now this problem can be solved using computer simulation.

The work performed by the group, which included Dmitry Grebennikov and Gennady Bocharov Institute of numerical mathematics RAS, head of the Interdisciplinary research center “Mathematical modeling in Biomedicine” people’s friendship University Vitaly Volpert, Nikolai Bessonov from the Institute of problems of mechanical engineeringn Academy of Sciences, and the Anassa Bosnia from Sweden and Andreas Meyerhans from Spain.

They were among the first in the world to successfully United in a single model the mathematical apparatus of the probability theory, diffusion model, differential equations and other methods and can describe the conditions under which antigen-specific T-lymphocytes have time to meet and to identify HIV-infected cells before the time when the breeding of virus particles begin to spread throughout the body.

## Newton’s laws and viruses

The basic law of motion in this model is Newton’s second law linking the acceleration of a particle, its mass and the forces acting on it. Liquid — blood or lymph — cell has mechanisms of its own motion, which we can designate as the action of the force fmot, the movement of cells thus inhibits the resistance force fdis.

When cells collide with each other, they at some time come into contact and then with some probability or interlock (lymphocyte can bind with an infected cell, forming the conjugate), or repelled and move on. The time during which the infected cell and the cell is CD8+ T lymphocyte is carried out in direct contact, directly affects the probability of recognizing and destroying infected cells of lymphocyte.

The density of cells in the body is large, and there may be situations when both are in contact three or more cells. The movement of the cells is chaotic.