DOI: https://doi.org/10.20535/ibb.2019.3.1.157644

Research of Global Attractability of Solutions and Stability of the Immunosensor Model Using Difference Equations on the Hexagonal Lattice

Andriy Sverstiuk

Abstract


Background. An important stage in the design of immunosensor systems is the development and research of their mathematical models that adequately would reflect the important aspects of the spatial structure of immunopicles, which are important in terms of the research tasks. After all, the quality of the mathematical model of the immunosensor determines the effectiveness of its processing methods in measuring systems. Designing immunosensor devices involves the selection of parameters that would ensure its operational stability. Such a task, in particular, arises in the development of an immunosensor, which includes a three-dimensional array of immune pixels, and which consists in finding appropriate parameters describing immunological and diffusion processes. This problem can be studied by studying the global attractability of the solutions and the stability of the corresponding dynamic model on the hexagonal lattice. The results of the immunosensor model study using differential equations will enable the development of highly selective sensory systems for rapid and accurate measurements in the food industry, with the control of environmental parameters, defense industry and medicine.

Objective. The aim of the paper is to investigate the global attractability of the solutions and stability of the immunosensor model using the system of difference equations on the hexagonal lattice, taking into account the presence of colonies of antigens and antibodies that are localized in pixels, as well as the diffusion of colonies of antigens between pixels.

Methods. The paper studies the global attractability of solutions and stability of a model of immunosensor, based on a system of difference equations on a hexagonal lattice using packet R. A class of lattice variance equations with delay in time is introduced for modeling the interaction of "antigen-antibody" in pixels of an immunosensor. The model is based on a number of biological assumptions about the interaction of colonies of antigens and antibodies, as well as the diffusion of antigens. To describe colonies discrete in the space localized in the corresponding pixels, the apparatus of difference equations on a hexagonal lattice is used.

Results. The results of numerical simulation of a model of an immunosensor based on a system of difference equations on a hexagonal lattice showed that the qualitative behavior of the system significantly depends on the time of the immune response r. In particular for r £ 16, there are trajectories that correspond to a stable focus for all pixels. For r = 17 the Hopf bifurcation occurs, and the following trajectories correspond to the stable ellipsoidal boundary cycles for all pixels. For values r ≥ 22, the behavior of the model under study becomes chaotic.

Conclusions. The paper studied the global attractability of the solutions and stability of the immunosensor model using a system of difference equations on a hexagonal lattice, which takes into account the presence of colonies of antigens and antibodies localized in pixels, as well as the diffusion of colonies of antigens between pixels. Based on the results of numerical modeling of the immunosensor for different time of the immune response r, one can conclude that the qualitative behavior of the model under study depends heavily on its significance. The obtained results of the study of a model of an immunosensor using difference equations on a hexagonal lattice can be used for the design of immunosensor devices with the ability to control parameters that would ensure their operational stability.

Keywords


Biosensor; Immunosensor; Difference equations; Hexagonal lattice

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