ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Shabunin A. V. Spatial and temporal dynamics of the emergence of epidemics in the hybrid SIRS+V model of cellular automata. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 3, pp. 271-285. DOI: 10.18500/0869-6632-003042, EDN: PBXBCY

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
Article type: 
517.9, 621.372

Spatial and temporal dynamics of the emergence of epidemics in the hybrid SIRS+V model of cellular automata

Shabunin Aleksej Vladimirovich, Saratov State University

Purpose of this work is to construct a model of infection spread in the form of a lattice of probabilistic cellular automata, which takes into account the inertial nature of infection transmission between individuals. Identification of the relationship between the spatial and temporal dynamics of the model depending on the probability of migration of individuals.

Methods. The numerical simulation of stochastic dynamics of the lattice of cellular automata by the Monte Carlo method.

Results. A modified SIRS+V model of epidemic spread in the form of a lattice of probabilistic cellular automata is constructed. It differs from standard models by taking into account the inertial nature of the transmission of infection between individuals of the population, which is realized by introducing a "carrier agent" into the model, which viruses act as. The similarity and difference between the dynamics of the cellular automata model and the previously studied mean field model are revealed.

Discussion. The model in the form of cellular automata allows us to study the processes of infection spread in the population, including in conditions of spatially heterogeneous distribution of the disease. The latter situation occurs if the probability of migration of individuals is not too high. At the same time, a significant change in the quantitative characteristics of the processes is possible, as well as the emergence of qualitatively new modes, such as the regime of undamped oscillations.

  1. Bailey NTJ. The Mathematical Approach to Biology and Medicine. London: John Wiley and Sons; 1967. 296 p. DOI: 10.2307/2982529.
  2. Marchuk GI. Mathematical Models in the Immunology: Simulation Methods and Experiments. Moscow: Nauka; 1991. 276 p. (in Russian).
  3. Hethcote HW. The mathematics of infectious diseases. SIAM Review. 2000;42(4):599–653. DOI: 10.1137/S0036144500371907.
  4. Anderson RM, May R. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991. 768 p.
  5. Serfling RE. Methods for current statistical analysis of excess pneumonia-influenza deaths. Public Health Reports. 1963;78(6):494–506. DOI: 10.2307/4591848.
  6. Burkom HS, Murphy SP, Shmueli G. Automated time series forecasting for biosurveillance. Statistics in Medicine. 2007;26(22):4202–4218. DOI: 10.1002/sim.2835. 
  7. Pelat C, Boelle PY, Cowling BJ, Carrat F, Flahault A, Ansart S, Valleron AJ. Online detection and quantification of epidemics. BMC Medical Informatics and Decision Making. 2007;7:29. DOI: 10.1186/1472-6947-7-29.
  8. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A. 1927;115(772):700–721. DOI: 10.1098/rspa.1927.0118.
  9. Bailey NTJ. The Mathematical Theory of Infectious Diseases and Its Applications. 2nd edition. London: Griffin; 1975. 413 p.
  10. 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;25(9): 2447–2461. DOI: 10.1088/0305-4470/25/9/018.
  11. Sirakoulis GC, Karafyllidis I, Thanailakis A. A cellular automaton model for the effects of population movement and vaccination on epidemic propagation. Ecological Modelling. 2000;133(3): 209–223. DOI: 10.1016/S0304-3800(00)00294-5.
  12. Shabunin AV. SIRS-model with dynamic regulation of the population: Probabilistic cellular automata approach. Izvestiya VUZ. Applied Nonlinear Dynamics. 2019;27(2):5–20 (in Russian). DOI: 10.18500/0869-6632-2019-27-2-5-20.
  13. Shabunin AV. Synchronization of infections spread processes in populations interacting: Modeling by lattices of cellular automata. Izvestiya VUZ. Applied Nonlinear Dynamics. 2020;28(4):383–396 (in Russian). DOI: 10.18500/0869-6632-2020-28-4-383-396.
  14. Hamer WH. Epidemic disease in England – the evidence of variability and persistence of type. The Lancet. 1906;1:733–739.
  15. Gopalsamy K. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Dordrecht: Springer; 1992. 502 p. DOI: 10.1007/978-94-015-7920-9.
  16. Perevaryukha AY. A continuous model of three scenarios of the infection process with delayed immune response factors. Biophysics. 2021;66(2):327–348. DOI: 10.1134/S0006350921020160.
  17. Perevaryukha AY. Modeling of adaptive counteraction of the induced biotic environment during the invasive process. Izvestiya VUZ. Applied Nonlinear Dynamics. 2022;30(4):436–455. DOI: 10.18500/0869-6632-2022-30-4-436-455.
  18. Shabunin AV. Hybrid SIRS model of infection spread. Izvestiya VUZ. Applied Nonlinear Dynamics. 2022;30(6):717–731. DOI: 10.18500/0869-6632-003014.
  19. Kobrinskii NE, Trahtenberg BA. Introduction to the Theory of Finite Automata. Moscow: Fizmatgiz; 1962. 405 p. (in Russian).
  20. Toffoli T, Margolus N. Cellular Automata Machines: A New Environment for Modeling. Cambridge: MIT Press; 1987. 259 p.
  21. Vanag VK. Study of spatially extended dynamical systems using probabilistic cellular automata. Phys. Usp. 1999;42(5):413–434. DOI: 10.1070/PU1999v042n05ABEH000558.
  22. Provata A, Nicolis G, Baras F. Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model. J. Chem. Phys. 1999;110(17):8361–8368. DOI: 10.1063/1.478746.
  23. Shabunin AV, Baras F, Provata A. Oscillatory reactive dynamics on surfaces: A lattice limit cycle model. Phys. Rev. E. 2002;66(3):036219. DOI: 10.1103/PhysRevE.66.036219.
  24. Tsekouras G, Provata A, Baras F. Waves and their interactions in the lattice Lotka–Volterra mode. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(2):63–71.
  25. Boccara N, Cheong K. Critical behaviour of a probabilistic automata network SIS model for the spread of an infectious disease in a population of moving individuals. Journal of Physics A: Mathematical and General. 1993;26(15):3707–3717. DOI: 10.1088/0305-4470/26/15/020.
  26. Benyoussef A, HafidAllah NE, ElKenz A, Ez-Zahraouy H, Loulidi M. Dynamics of HIV infection on 2D cellular automata. Physica A. 2003;322:506–520. DOI: 10.1016/S0378-4371(02)01915-5.
  27. Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics. 1983;69(1):32–47. DOI: 10.1143/PTP.69.32.
  28. Yamada T, Fujisaka H. Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach. Progress of Theoretical Physics. 1983;70(5):1240–1248. DOI: 10.1143/ PTP.70.1240.
Available online: 
Уважаемые участники конференции "Нелинейные дни"! Если вы не видите свою статью в тематическом №5 за этот год, то не расстраивайтесь, значит ваша статья просто туда не уместилась (Роскомнадзор ограничивает нас 132 страницами в выпуске) и будет опубликована в №1 за 2024 год. С уважением, редакция журнала "Известия вузов. Прикладная нелинейная динамика".