Cyber physical systems (CPS) include a lot of high complexity computing such as dynamic analysis and verification of continuous dynamic property, analysis and verification of real- time property, analysis and verification of spatial property, scheduling and fault tolerance. In this paper, some of the research directions that we are taking toward addressing some of the challenges involved in building cyber physical systems have been described. Taking into account the features of the cyber-physical sensor systems, the basic model has been modified. Lattice images in biopixels have been modified according to the laws of discrete dynamics. The developed models take into account the interaction of biopixels with each other by antigen diffusion. The comparative analysis of CPS models on rectangular and hexagonal lattices using differenсе equations has been considered in the work. 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 cyber-physical biosensor systems for antibody populations relative to antigen populations have been received in the paper. The comparative analysis of the results of numerical modeling of mathematical models of cyber-physical biosensor systems on rectangular and hexagonal lattices using lattice difference equations with delay has been considered.
[1] Lee, E.: Cyber physical systems: Design challenges. In: International Symposium on Object/Component/Service- Oriented Real-Time Distributed Computing, pp. 10 (2008).
https://doi.org/10.1109/ISORC.2008.25
[2] Lee, J., Bagheri, B., Kao, H.: A cyber-physical systems architecture for industry 4.0-based manufacturing systems. Manufacturing Letters, vol. 3, 18-23 (2015).
https://doi.org/10.1016/j.mfglet.2014.12.001
[3] Krainyk, Y., Davydenko, Y., Tomas, V.: Configurable Control Node for Wireless Sensor Network. In: 3rd International Conference on Advanced Information and Communications Technologies (AICT), pp. 258-262. IEEE Press, Lviv, Ukraine (2019).
https://doi.org/10.1109/AIACT.2019.8847732
[4] Kim, K.-D., Kumar, P.: Cyber-physical systems: A perspective at the centennial. In: Proceedings of the IEEE, vol. 100, pp. 1287-1308, (2012).
https://doi.org/10.1109/JPROC.2012.2189792
[5] Platzer, A.: Differential dynamic logic for hybrid systems. Journal of Automated Reasoning, 2(41), 143-189 (2008). [6] Platzer, A.: Logical Foundations of Cyber-Physical Systems. Springer International Publishing, pp. 1-24, (2018).
https://doi.org/10.1007/s10817-008-9103-8
[7] Adley C.: Past, present and future of sensors in food production. Foods, 3(3), pp. 491-510, (2014).
https://doi.org/10.3390/foods3030491
[8] Bahadır E., Sezgintürk M.: Applications of commercial biosensors in clinical, food, environmental, and biothreat/biowarfare analyses. Analytical Biochemistry, vol. 478, pp. 107-120 (2015).
https://doi.org/10.1016/j.ab.2015.03.011
[9] Martsenyuk V., Sverstiuk A., Gvozdetska I. Using Differential Equations with Time Delay on a Hexagonal Lattice for Modeling Immunosensors. Cybernetics and Systems Analysis, vol. 55 (4), p. 625-636 (2019).
https://doi.org/10.1007/s10559-019-00171-2
[10] Liu L., Liu Z.: Asymptotic behaviors of a delayed nonautonomous predator-prey system governed by difference equations. Discrete Dynamics in Nature and Society, vol. 2011, pp. 1-15, (2011).
https://doi.org/10.1155/2011/271928
[11] Berger C., Hees A., Braunreuther S., Reinhart G.: Characterization of Cyber-Physical Sensor Systems. Procedia CIRP, vol. 41, pp. 638-643, (2016).
https://doi.org/10.1016/j.procir.2015.12.019
[12] Internet-ressource: https://www.redblobgames.com/grids/hexagons/.
[13] McCluskey, C.: Complete global stability for an SIR epidemic model with delay - distributed or discrete. Nonlinear Analysis: Real World Applications, 1 (11), 55-59, (2010).
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, vol. 581, pp. 185-192 (2006).
https://doi.org/10.1007/3-540-35262-7_11
[15] Piotrowska, M.: An immune system-tumour interactions model with discrete time delay: Model analysis and validation, Communications in Nonlinear Science and Numerical Simulation, vol. 34, pp. 185-198, (2016).
https://doi.org/10.1016/j.cnsns.2015.10.022
[16] Prindle, A., Samayoa, P., Razinkov, I., Danino, T., Tsimring, L., Hasty, J.: A sensing array of radically coupled genetic 'biopixels', Nature, 481(7379), 39-44 (2011).
https://doi.org/10.1038/nature10722
[17] Martsenyuk, V., KÅos-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, 27, pp. 1-31 (2018).
https://doi.org/10.14232/ejqtde.2018.1.27
[18] Prindle A., Samayoa P., Razinkov I., Danino T., Tsimring L., Hasty J.: A sensing array of radically coupled genetic 'biopixels'. Nature, vol. 481(7379), pp. 39-44 (2012).
https://doi.org/10.1038/nature10722
[19] Liu L., Liu Z.: Asymptotic behaviors of a delayed nonautonomous predator-prey system governed by difference equations. Discrete Dynamics in Nature and Society, vol. 2011, pp. 1-15, 2011.
https://doi.org/10.1155/2011/271928
[20] Letellier C., Elaydi S., Aguirre L., Alaoui A.: Difference equations versus differential equations, a possible equivalence for the Rossler system. Physica D: Nonlinear Phenomena, vol. 1-2(195), pp. 29-49, (2004).
https://doi.org/10.1016/j.physd.2004.02.007
[21] Hofbauer, J., Iooss, G.: A Hopf bifurcation theorem for difference equations approximating a differential equation, Monatshefte fur Mathematik, 2(98), pp. 99-113 (1984).
https://doi.org/10.1007/BF01637279
[22] Burlachenko, I., Zhuravska, I., Davydenko, Y., Savinov, V.: Vulnerability Analysis and Defense Based on MAS Method in Fast Dynamic Wireless Networks. In: 4th International Symposium on Wireless Systems within the International Conferences on Intelligent Data Acquisition and Advanced Computing Systems (IDAACS-SWS), pp. 98-102. IEEE Press, Lviv, Ukraine (2018).
https://doi.org/10.1109/IDAACS-SWS.2018.8525692
[23] Fisun, M., Dvoretskyi, M., Shved, A., Davydenko, Y.: Query parsing in order to optimize distributed DB structure. In: 9th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), pp. 172-178. IEEE Press, Bucharest, Romania (2017).
https://doi.org/10.1109/IDAACS.2017.8095071