It is the first quantitative study published in literature on the phenomenon of tolerance mediated by T cells, starting from technical biomathematics. This technique type, based on the construction of mathematical formalisms and its simulation in computers, is indispensable for the understanding of the immunologic phenomena, as the science is overcoming the reductionist approaches of the 20th century, and substituting them by analysis at a complex systems level, where the system's properties (tolerance and immunogenicity in this case) are not tried to be reduced to properties of the component cells, but rather the dynamic properties from the whole system are studied. Intuition (even the best trained) and quantitative analyses, usually fail in this level of complexity; and tools such as those built in this work, become essentials for the understanding of the phenomena and for the design of new experiments.

In this work the authors:

• 1. Build a mathematical formalism based on differential equations in order to describe the interactions among cells bearers, effector lymphocites and regulator lymphocites,
• 2. Extend the model to incorporate the permanent exit effect of new cellular clones of thymus,
• 3. Establish four possible models for tricellular interaction. They obtain the planes of phases and they make the bifurcation analysis of each model. The first two models are discarded for not being adjusted to the experimental data,
• 4. Predict, according to the models 3 and 4 that the population of regulator T cells should expand in their interaction with the effector cells, and they obtain experimental data that demonstrate that this proliferation occurs (model 3),
• 5. Obtain a completely novel explanation for the positive selection and the negative selection in thymus, which diverges from conventional vision, when establishing that positive selection is necessary to assure a sufficient number of auto-reactive cells and to make tolerance possible; while negative selection of the very high affinities is required in order to avoid the appearance in periphery of regulator cells of high affinity that would make impossible the reaction against strange antigens,
• 6. Predict the inverse epidemiologic correspondance between the incidence of autoimmune diseases and the incidence in several common infections; which didn't have a clear theoretical explanation until the moment. They predict the conditions in which a "vaccination" against cancerous cells is possible, including some "contra-intuitive" conditions, such as the necessity of combining vaccines with treatment of reduction of regulator lymphocites.