Research of Global Attractability of Solutions and Stability of the Immunosensor Model Using Difference Equations on the Hexagonal Lattice
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.
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Mosinska L, Fabisiak K, Paprocki K, Kowalska M, Popielarski P, Szybowicz M, et al. Diamond as a transducer material for the production of biosensors. Przemysl Chemiczny. 2013;92(6):919-23.
Adley C. Past, present and future of sensors in food production. Foods. 2014;3(3):491-510. DOI: 10.3390/foods3030491
Kłos-Witkowska A. Enzyme-based fluorescent biosensors and their environmental, clinical and industrial applications. Polish J Environ Stud. 2015;24:19-25. DOI: 10.15244/pjoes/28352
Burnworth M, Rowan S, Weder C. Fluorescent sensors for the detection of chemical warfare agents. Chemistry – A European Journal. 2007;13(28):7828-36. DOI: 10.1002/chem.200700720
Mehrotra P. Biosensors and their applications – a review. J Oral Biol Craniofacial Res. 2016;6(2):153-9. DOI: 10.1016/j.jobcr.2015.12.002
Martsenyuk VP, Klos-Witkowska A, Sverstiuk AS. Study of classification of immunosensors from viewpoint of medical tasks. Medical Inform Eng. 2018;1:13-9. DOI: 10.11603/mie.1996-1960.2018.1.8887
Martsenyuk VP, Klos-Witkowska A, Sverstiuk AS, Bihunyak TV. On principles, methods and areas of medical and biological application of optical immunosensors. Medical Inform Eng. 2018;2:28-36. DOI: 10.11603/mie.1996-1960.2018.2.9289
Bihuniak TV, Sverstiuk AS, Bihuniak KO. Some aspects of using immunosensors in medicine. Medychnyi Forum. 2018;14:8-11.
Moina C, Ybarra G. Fundamentals and applications of immunosensors. In: Chiu N, editor. Advances in Immunoassay Technology. InTech; 2012:65-80. DOI: 10.5772/36947
Kłos-Witkowska A. The phenomenon of fluorescence in immunosensors. Acta Biochimica Polonica. 2016;63(2):215-21. DOI: 10.18388/abp.2015_1231
Red Blob Games: Hexagonal Grids [Internet]. Redblobgames.com. 2019 [cited Jun 2019]. Available from: https://www.redblobgames.com/grids/hexagons/
McCluskey CC. Complete global stability for an SIR epidemic model with delay – distributed or discrete. Nonlinear Analysis Real World Applications. 2010;11(1):55-9. DOI: 10.1016/j.nonrwa.2008.10.014
Nakonechny A, Marzeniuk V. Uncertainties in medical processes control. Lect Notes Econom Math Syst. 2006;581:185-92. DOI: 10.1007/3-540-35262-7_11
Marzeniuk V. Taking into account delay in the problem of immune protection of organism. Nonlinear Analysis Real World Applications. 2001;2(4):483-96. DOI: 10.1016/S1468-1218(01)00005-0
Prindle A, Samayoa P, Razinkov I, Danino T, Tsimring LS, Hasty J. A sensing array of radically coupled genetic 'biopixels'. Nature. 2011;481(7379):39-44. DOI: 10.1038/nature10722
Hofbauer J, Iooss G. A hopf bifurcation theorem for difference equations approximating a differential equation. Monatshefte für Mathematik. 1984;98(2):99-113. DOI: 10.1007/BF01637279
Martsenyuk V, Kłos-Witkowska A, Sverstiuk A. Stability, bifurcation and transition to chaos in a model of immunosensor based on lattice differential equations with delay. Electron J Qual Theory Differ Equ. 2018;27:1-31. DOI: 10.14232/ejqtde.2018.1.27
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