DOI: https://doi.org/10.15588/1607-3274-2019-2-14

### COMPUTER MODELING OF CYBER-PHYSICAL IMMUNOSENSOR SYSTEM IN A HEXAGONAL LATTICE USING LATTICE DIFFERENTIAL EQUATIONS WITH DELAY

#### Abstract

mathematical and computer models, the construction of which would be based on biological assumptions to obtain appropriate

systems of differential equations of population dynamics. Mathematical modeling would allow to obtain the value of parameters that

would ensure the operational stability of immunosensory systems.

Objective. The aim of the work is to develop a mathematical and computer model of the cyber-physical immunosensory system

using lattice-delayed differential equations on a hexagonal lattice and study its stability.

Method. The mathematical and computer models of the cyber-physical immunosensory system on the hexagonal lattice are developed.

For the simulation of continuous dynamics, the system of lattice differential equations with delay was used. Dynamic logic

of the first order is used to simulate discrete events. The permanent states of the model as solutions of the corresponding algebraic

systems are described. The conclusion on stability is based on the analysis of the corresponding phase diagrams, lattice images and

signals obtained from the corresponding computer model.

Results. The analysis of the results of numerical simulation of the investigated model in the form of an image of phase planes,

lattice images of the probability of antibody bonds and an electron signal from the converter, which characterizes the number of fluorescing

pixels, is presented.

Conclusions. Mathematical and computer modeling of the cyber-physical immunosensory system was performed. It is established

that its qualitative behavior significantly depends on the time of the immune response. The conclusion on the stability of immunosensors

can be made on the basis of the grid image of the pixels that are fluorescing. An electrical signal, modeled by the number

of fluorescent immunopips, is important in the design of cyber-physiological immunosensory systems and studies of their resilience.

Limit cycle or steady focus determine the appropriate form of immunosensory electrical signal. The experimental results obtained

have made it possible to perform a complete analysis of the stability of the immunosensor model, taking into account the delay

in time.

#### Keywords

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PDF (Українська)#### References

Meissner H., Aurich J. Implications of cyber-physical production systems on integrated process planning and scheduling,

Procedia Manufacturing, 2019, Vol. 28, pp. 167–173.

Lee J., Bagheri B., Kao H.-A. A cyber-physical systems architecture for industry 4.0-based manufacturing systems,

Manufacturing Letters, 2015, Vol. 3, pp. 18–23.

Thiede S., Juraschek M., Herrmann C. Implementing cyberphysical production systems in learning factories, Procedia

CIRP, 2016, Vol. 54, pp. 7–12.

Platzer A. Differential dynamic logic for hybrid systems, Journal of Automated Reasoning, 2018, Vol. 41, No. 2, pp. 143–189.

Platzer A. Logical foundations of cyber-physical systems.

Berlin, Springer, 2018, 639 p.

Kłos-Witkowska A. The phenomenon of fluorescence in immunosensors, Acta Biochimica Polonica, 2016, Vol. 63,

No. 2, pp. 215–221.

Martsenyuk V. P., Klos-Witkowska A., Sverstiuk A. S. Study of classification of immunosensors from viewpoint of medical tasks, Medical informatics and engineering, 2018, No. 1(41), pp. 13–19.

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, No. 2 (42), pp. 28–36.

Zhu G., Yin X., Jin D., Zhang B., Gu Y., An Y. Paper-based immunosensors: current trends in the types and applied

detection techniques, Trends in Analytical Chemistry, 2019, Vol. 111, pp. 100–117.

Jiang X., Spencer M. Electrochemical impedance biosensor with electrode pixels for precise counting of CD4+ cells: A

microchip for quantitative diagnosis of HIV infection status of AIDS patients, Biosensors and Bioelectronics, 2010, Vol.

, Issue 7, pp. 1622–1628.

Berger C., Hees A., Braunreuther S., Reinhart G. Characterization of Cyber-Physical Sensor Systems, Procedia CIRP, 2016, Vol. 41, pp. 638–643.

Hexagonal grids [Electronic resurce]: Access mode: https://www.redblobgames.com/grids/hexagons/

Prindle A., Samayoa P., Razinkov I., Danino T., Tsimring L., Hasty J. A sensing array of radically coupled genetic

biopixels, Nature, 2011, Vol. 481, No. 7379, pp. 39–44.

Soulier P., Li D., Williams J. A survey of language-based approaches to cyber-physical and embedded system

development, Tsinghua Science and Technology, 2015, Vol. 20, No. 2, pp. 130–141.

Hofbauer J., Iooss G. A hopf bifurcation theorem for difference equations approximating a differential equation,

Monatshefte fur Mathematik, 1984, Vol. 98, No. 2, pp. 99–113.

#### GOST Style Citations

2. Lee J. A cyber-physical systems architecture for industry 4.0-based manufacturing systems / J. Lee, B. Bagheri, H.-A. Kao // Manufacturing Letters. – 2015. – Vol. 3. – P. 18–23.

3. Thiede S. Implementing cyber-physical production systems in learning factories / S. Thiede, M. Juraschek, C. Herrmann // Procedia CIRP. – 2016. – Vol. 54. – P. 7–12.

4. Platzer A. Differential dynamic logic for hybrid systems / A. Platzer // Journal of Automated Reasoning. – 2018. –

Vol. 41, № 2. – P. 143–189.

5. Platzer A. Logical foundations of cyber-physical systems / A. Platzer. – Berlin: Springer, 2018. – 639 p.

6. Kłos-Witkowska A. The phenomenon of fluorescence in immunosensors / A. Kłos-Witkowska // Acta Biochimica Polonica. – 2016 – Vol. 63, № 2. – P. 215–221.

7. Martsenyuk V. P. Study of classification of immunosensors from viewpoint of medical tasks / V. P. Martsenyuk, A.

Klos-Witkowska, A. S. Sverstiuk // Medical informatics and engineering. – 2018. – № 1(41). – P. 13–19.

8. Martsenyuk V. P. On principles, methods and areas of medical and biological application of optical immunosensors / V. P. Martsenyuk, A. Klos-Witkowska, A. S. Sverstiuk, T. V. Bihunyak // Medical informatics and engineering. – 2018. – № 2 (42). – P. 28–36.

9. Paper-based immunosensors: current trends in the types and applied detection techniques / [G. Zhu, X. Yin, D. Jin et al.] // Trends in Analytical Chemistry. – 2019. – Vol. 111. – P. 100–117.

10. Jiang X. Electrochemical impedance biosensor with electrode pixels for precise counting of CD4+ cells: A microchip for quantitative diagnosis of HIV infection status of AIDS patients [Text] / X. Jiang, M. Spencer // Biosensors and Bioelectronics. – 2010. – Vol. 25, Issue 7. – P. 1622–1628.

11. Characterization of Cyber-Physical Sensor Systems / [C. Berger, A. Hees, S. Braunreuther, G. Reinhart] // Procedia CIRP. – 2016. – Vol. 41. – P. 638–643.

12. Hexagonal grids [Electronic resurce]: Access mode: https://www.redblobgames.com/grids/hexagons/

13. A sensing array of radically coupled genetic biopixels / [A. Prindle, P. Samayoa, I. Razinkov et al.] // Nature. – 2011. – Vol. 481, № 7379. – P. 39–44.

14. Soulier P. A survey of language-based approaches to cyberphysical and embedded system development / P. Soulier, D. Li, J. Williams // Tsinghua Science and Technology. – 2015. – Vol. 20, № 2. – P. 130–141.

15. Hofbauer J. A hopf bifurcation theorem for difference equations approximating a differential equation / J. Hofbauer,

G. Iooss // Monatshefte fur Mathematik. – 1984 – Vol. 98, № 2. – P. 99–113.

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