For citation:
Shabunin A. V. Synchronization of infections spread processes in populations interacting: Modeling by lattices of cellular automata. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 4, pp. 383-396. DOI: 10.18500/0869-6632-2020-28-4-383-396
Synchronization of infections spread processes in populations interacting: Modeling by lattices of cellular automata
Purpose. Study of synchronization of oscillations in ensembles of probabilistic cellular automata that simulate the spread of infections in biological populations. Method. Numerical simulation of the square lattice of cellular automata by means of the Monte Carlo method, investigation of synchronization of oscillations by time-series analisys and by the coherence function. Results. The effect has been found of synchronization of irregular oscillations, similar to the phenomenon of synchronization of chaos in dynamical systems. It is shown that the coupled lattices of cellular automata can demonstrate the effect of phase locking, as well as the tuning of the main frequencies in the oscillation spectra; at strong coupling they also can demonstrate an almost complete synchronization regime. Discussion. The most interesting result of the work is the revealed similarity of the phenomenon of chaos synchronization, which is well known for deterministic systems, with synchronization of irregular oscillations in interacting stochastic ensembles, whose behavior is determined exclusively by probabilistic laws. At the same time, the interaction algorithm, which has been used in the modeling is too idealized, becouse it does not consider the finite speed of diffusion processes. Taking the last into account will probably lead to more complex types of synchronization that has been found in the research.
1. Stratonovich R.L. Topics in the Theory of Random Noise. London: Gordon and Breach, 1967.
2. Neiman A. Synchronization like phenomena in coupled stochastic bistable systems // Phys. Rev. E. 1994. Vol. 49. P. 3484.
3. Shulgin B., Neiman A., Anishchenko V.S. Mean switching frequency locking in stochastic bistable systems driven by a periodic force // Phys. Rev. Lett. 1995. Vol. 75. P. 4157.
4. Gan Q. Exponential synchronization of stochastic neural networks with leakage delay and reaction-diffusion terms via periodically intermittent control // Chaos. 2012. Vol. 22. 013124.
5. Pan L., Cao J., Al-Juboori U.A., Abdel-Aty M. Cluster synchronization of stochastic neural networks with delay via pinning impulsive control // Neurocomputing. 2019. Vol. 366. P. 109.
6. Kobrinsky N.E., Trahtenberg B.A. Introduction to the Theory of Finite Automata. Moscow: Fizmatgiz, 1962 (in Russian).
7. Toffoli T., Margolus N. Cellular Automata Machines: A New Environment for Modeling. Cambridge: MIT Press Cambridge, 1987.
8. Vanag V.K. Study of spatially extended dynamical systems using probabilistic cellular automata. Physics-Uspekhi, 1999, vol. 42, no. 5. p. 413.
9. Provata A., Nicolis G., Baras F. Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model // J of Chem. Phys. 1999. Vol. 110. P. 8361–8368.
10. Shabunin A., Baras F., Provata A. Oscillatory reactive dynamics on surfaces: A lattice limit cycle model // Physical Review E. 2002. Vol. 66, no. 3. 036219.
11. Tsekouras G., Provata A., Baras F. Waves and their interactions in the lattice Lotka–Volterra model // Изв. вузов. ПНД. 2003. Т. 11, № 2. С. 63–71.
12. Wood K., Van den Broeck C., Kawai R., Lindenberg K. Universality of synchrony: Critical behavior in a discrete of stochastic phase-coupled oscillators Model // Physical Review Letters. 2006. Vol. 96, no. 14. P. 145701.
13. Efimov A., Shabunin A., Provata A. Synchronization of stochastic oscillations due to long-range diffusion // Physical Review E. 2008. Vol. 78, no. 5. 056201.
14. Kouvaris N., Provata A. Synchronization, stickiness effects and intermittent oscillations in coupled nonlinear stochastic networks // Eur. Phys. J. B. 2009. Vol. 70. P. 535–541.
15. Kouvaris N., Provata A., Kugiumtzis D. Detecting synchronization in coupled stochastic ecosystem networks // Physics Letters A. 2010. Vol. 374. P. 507–515.
16. Kouvaris N., Kugiumtzis D., Provata A. Species mobility induces synchronization in chaotic population dynamics // Physical Review E. 2011. Vol. 84. 036211.
17. Bailey N. The Mathematical Approach to Biology and Medicine. London: John Wiley and Sons, 1967.
18. Hethcote H.W. The mathematics of infectious diseases // SIAM Review. 2000. Vol. 42, no. 4. P. 599–653.
19. Anderson R., May R. Infectios Deseases of Humans – Dynamics and Control. Oxford: Oxford University Press, 1991.
20. Kermack W., McKendrick A. A contribution to the mathematical theory of epidemics // Proc. R. Soc. 1927. Vol. A115. P. 700–721.
21. Boccara N., Cheong K. Automata network SIR models for the spread of infectious diseases in populations of moving individuals // Journal of Physics A: Mathematical and General. 1992. Vol. 25, no. 9. P. 2447.
22. Sirakoulis G.C., Karafyllidis I., Thanailakis A. A cellular automaton model for the effects of population movement and vaccination on epidemic propagation // Ecological Modelling. 2000. Vol. 133, no. 3. P. 209–223.
23. Shabunin A.V. SIRS-model with dynamic regulation of the population: Probabilistic cellular automata approach. Izvestiya VUZ, Applied Nonlinear Dynamics, 2019, vol. 27, no. 2, pp. 5–20 (in Russian).
24. Verhulst P. Notice sur la loi que la population suit dans son accroissement // Corr. Math. et Phys. 1838. Vol. 10. P. 113–121.
25. Shabunin A., Astakhov V., Kurths J. Quantitative analysis of chaotic synchronization by means of coherence // Phys. Rev. E. 2005. Vol. 72. 016218.
26. Anishchenko V.S., Vadivasova T.E., Postnov D.E., Safonova M.A. Forced and mutual synchronization of chaos. Radiotekhnika i Elektronika, 1991, vol. 36, no. 2, pp. 338–351 (in Russian).
- 1389 reads