ON ONE NUMERICAL METHOD OF IMMUNOLOGY PROBLEMS MODELING

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), boikov@pnzgu.ru
Zakharova Yuliya Fridrikhovna, Candidate of physical and mathematical sciences, associate professor, sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru
Dmitrieva Alla Arkad'evna, Senior lecturer, sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru

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Background. Research of mathematical models of immunology at the present time is an actively developing field, located in the junction of medicine, biology and mathematics. There have been suggested multiple models of development of immune system reactions to various external influences, among which the closest to clinical practice are the Marchuk models and generalizations thereof. The models are described by systems of regular differential equations of high order with various delays, and solution thereof in the analytical form is impossible. Therefore, the development of numerical methods for solving nonlinear differential equation systems with various delays in nonlinear operators is topical relevant.
Materials and methods. Computing schemes are based of the exponential representation of solution, suggested in the article, allowing to form the iteration method with nonnegative approximations at each step.
Results. The authors suggested the iteration method for solving nonlinear regular differential equation systems with delays, modeling immune reactions to viral and bacterial diseases. The researchers studied the ways of implementing various therapies by the example of the basic (simple) model.
Conclusions. The researchers developed the approximate method of researching mathematical models of immunology, featuring nonnegative approximation at each step of the iteration process. The method may be used in the research of similar models in engineering, ecology and economy (Volterra type models).

mathematical methods of immunology, numerical methods, iteration method.

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