004:94:53:616-073

cyber-physical system, biosensor, continuous dynamics, dynamic logic, differential equations.

The article deals with the comparative analysis of the results of numerical modeling of mathematical models of cyber-physical biosensor systems on hexagonal and rectangular lattices using lattice differential equations. The main attention is given to the mathematical description of the discrete population dynamics in combination with the dynamic logic of the studied models. The lattice differential equations with delay are proposed to simulate antigen-antibody interaction within hexagonal and rectangular biopixels. Appropriate spatial operators have been used to model the interaction between biopixels similar to the phenomenon of diffusion. The paper presents the results of numerical simulations in the form of phase plane images and lattice images of the probability of antigen to antibody binding in the biopixels of cyberphysical biosensor systems for antibody populations relative to antigen populations. The obtained experimental results make it possible to carry out a comparative analysis of the stability of mathematical models of cyberphysical immunosensory systems on hexagonal and rectangular lattices using lattice differential equations. It is concluded that at a constant delay value  for the model on the hexagonal lattice and  when using a rectangular lattice, respectively, the solutions of the mathematical models studied tend to non-identical endemic states, which in this case are stable foci. The results of the phase diagrams of antigen populations, antibodies and lattice images of the likelihood of antigen binding to antibodies in the biopixels of cyberphysical biosensor systems conclude that at a constant delay value  (in the case of a hexagonal lattice) and  (in the case of a rectangular lattice), Hopf bifurcation occurs and all subsequent trajectories correspond to stable boundary cycles for all pixels. The obtained experimental results make it possible to perform a comparative analysis of the stability of mathematical models of cyberphysical biosensor systems on hexagonal and rectangular lattices using lattice differential equations.

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1. Lee E. A. Cyber physical systems: Design challenges. Center for Hybrid and Embedded Software Systems. URL: https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-8.pdf.
2. Lee J., Bagheri B., Kao H.-A. A cyber-physical systems architecture for industry 4.0-based manufacturing systems. Manufacturing Letters. Vol. 3. P. 18–23. 2015. ISSN: 2213-8463. DOI: https://doi.org/10.1016/j.mfglet.2014.12.001. URL: http://www.sciencedirect.com/science/article/pii/S 221384631400025X.
3. Kim K.-D., Kumar P. R. Cyber-physical systems: A perspective at the centennial. Proceedings of the IEEE. Vol. 100. No. Special Centennial Issue. 2012. P. 1287–1308. DOI: 10.1109/jproc.2012.2189792. URL: https://doi.org/10.1109/jproc.2012.2189792.
4. Platzer A. Differential dynamic logic for hybrid systems. J. Autom. Reas. Vol. 41. No. 2. P. 143–189. 2008. ISSN: 0168-7433. DOI: 10.1007/s10817-008-9103-8.
5. Logical Foundations of Cyber-Physical Systems. Springer International Publishing. 2018. DOI: 10.1007/978-3-319-63588-0. URL: https://doi.org/10.1007/ 978-3-319-63588-0.
6. Martsenyuk V. P., Klos-Witkowska A., Sverstiuk A. S., Bihunyak T. V. On principles, methods and areas of medical and biological application of optical immunosensors. Medical informatics and engineering. 2018. № 2 (42). Р. 28–36. DOI: https://dx.doi.org/10.11603/mie.1996-1960.2018.2.9289.
7. Martsenyuk V., Andrushchak I., Zinko P., Sverstiuk A. On Application of Latticed Differential Equations with a Delay for Immunosensor Modeling. Journal of Automation and Information Sciences 2018. Volume 50. Issue 6. Р. 55–65.
8. Jiang X., Spencer M. G. Electrochemical impedance biosensor with electrode pixels for precise counting of CD4+ cells: A microchip for quantitative diagnosis of HIV infec- tion status of AIDS patients. Biosensors and Bioelectronics. Vol. 25. No. 7. P. 1622–1628. 2010. DOI: 10.1016/j.bios.2009.11.024. URL: https://doi.org/10. 1016/j.bios.2009.11.024.
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10. Berger C., Hees A., Braunreuther V., Reinhart G. Characterization of cyber-physical sensor systems. Procedia CIRP. 2016. Vol. 41. P. 638–643. DOI: 10.1016/j.procir.2015.12.019. URL: https://doi.org/ 10.1016/j.procir. 2015.12.019.
11. Soulier P., Li D., Williams J. R., A survey of language- based approaches to cyber-physical and embedded system development. Tsinghua Science and Technology. 2015. Vol. 20. No. 2. P. 130–141.
12. URL: https://www.redblobgames.com/grids/hexagons/.
13. McCluskey C. C. Complete global stability for an SIR epidemic model with delay – distributed or discrete. Nonlinear Analysis: Real World Applications. 2010. Vol. 11. No. 1. P. 55–59. DOI: 10.1016/j.nonrwa.2008.10.014. URL: https: //doi.org/10.1016/j.nonrwa.2008.10.014.
14. Nakonechny A., Marzeniuk V. Uncertainties in medical processes control. Lecture Notes in Economics and Mathematical Systems. 2006. Vol. 581. P. 185–192. DOI: 10.1007/3-540-35262-7_11. URL: https://www.scopus.com/inward/record.uri?eid=2-s2.0-53749093113&doi=10.1007%2f3-540-352627_11 &partnerID=40&md5=03be7ef103cbbc1e94cacbb471daa03f.
15. Marzeniuk V. Taking into account delay in the problem of immune protection of organism. Nonlinear Analysis: Real World Applications. 2001. Vol. 2. No. 4. P. 483–496.
16. Prindle A., Samayoa P., Razinkov I., Danino T., Tsim- ring L. S., Hasty J. A sensing array of radically coupled genetic biopixels. Nature. 2011. Vol. 481. No. 7379. P. 39–44. DOI: 10.1038/nature10722. URL: https://doi.org/10.1038/nature10722.
17. Martsenyuk V., Klos-Witkowska A., Sverstiuk A. Stability, bifurcation and transition to chaos in a model of immunosensor based on lattice differential equations with delay. Electronic Journal of Qualitative Theory of Differential Equations. 2018. No. 27. Р. 1–31.
18. Hofbauer J. A., Iooss G. A hopf bifurcation theorem for difference equations approximating a differential equation. Monatshefte fur Mathematik. 1984. Vol. 98. № 2. Р. 99–113.