**Mathematical models are a key tool in the study of how autoimmune diseases appear and how our bodies can fight them.**

**T type lymphocytes** are cells that are part of the immune system of the human body.

Its processes of creation and maturation are especially delicate, since any failure can lead to serious problems for the individual. Those problems include leukemia and a number of autoimmune diseases.

In recent years, differential equations have proven to be key in mathematical models that help study and understand the immune system and how it deals with autoimmune diseases.

T type lymphocytes participate in a process called “**adaptive immune response**”, which is the second stage of action of the immune system directed to protecting the body from infections caused by viruses, bacteria and other kinds of pathogens.

T type lymphocytes are created in the bone marrow, from the bodies stem cells. These cells become precursors of T type lymphocytes through **thymic selection**, a process of cell differentiation that lasts approximately three weeks and takes place in a part of body know as the **thymus**.

At each instant of the process, each of the cells can take a different route: It dies, divides or produces two new cells, or instead it produces a different cell.

It is very important to understand where and when each thymocyte receives a signal that indicates the option to follow.

These signals depend both on the **epithelial cells** of the thymus, in particular on a type of molecules known as antigens, that they have on their cell membrane, and on the type T receptor that the thymocyte displays on its surface.

It is precisely the interaction between the T receptors of a thymocyte and the antigens of the epithelial cells what determines its future.

If the interaction is of great biochemical affinity, the thymocyte must die by a process known as apoptosis, or the programmed cell death.

If the affinity is very small or null, death is due to “negligence”.

In the case of intermediate affinities, the thymocyte undergoes a process of differentiation and continues maturation.

The process of quantifying the kinetics of thymic selection is carried out in part by measuring death rates, or the frequency with which a thymocyte receives a death signal, as well as the rates of differentiation or proliferation, the frequency with which it receives a signal of differentiation or cell division.

Knowing these rates would allow scientists to predict, for example, the average time a thymocyte spends in each phase of the thymic maturation process.

However, it is not for scientists to determine these parameters by conducting experiments, since it would require observing the trajectory of each T type pre-lymphocyte in the thymus of the individual studied, and current microscopy techniques only allow this observation for a relatively short period.

Mathematics provides precise tools to describe cell populations and their changes over time, through something called **deterministic population models**. In essence, these models describe the temporal evolution of the population.

If the initial population is assumed to consist of a certain number of individuals, the equation describes how many there will be a little later, or if the population changes by migration, by death or by birth of new individuals.

Each population model depends on its own migration mechanisms, for example, a constant flow or not of individual cells, their death and/or birth.

Ordinary differential equations (EDOs) are key to these models, which allow the number of thymocytes in each maturation stage to be described over time time, and the mathematical models of populations that incorporate their death, differentiation and division rates, which characterize the thymic selection process.

The parameters that are used to obtain trustworthy measurements are: the influx into the thymus from the bone marrow, the death rates of each population, the differentiation rates, proliferation rates, and the rate of migration to the blood of the lymphocytes that have survived the entire process.

Experimental data required by the model to determine these parameters include the number of thymocytes of each population at different times.

By joining experimental data with a mathematical model, scientists determined the rates of the thymic selection process.

A preliminary study allows scientists to conclude that less than 9% of the pre-T type lymphocytes that begin the process of thymic maturation actually manage to reach the end.

This is just one of the many examples of applications of mathematics to immunology, but much remains to be done.

A major challenge today is to understand both the dynamics and the molecular mechanisms of immune responses in tumors and thus improve existing immune therapies against cancer, for example.

For mathematicians, an important challenge is to learn how to describe biological processes at the individual cell level, instead of at the population level.