#### For citation:

Shabunin A. V. Modeling of Epidemics by Cellular Automata Lattices. SIRS Model with Reproduction and Migration. *Izvestiya of Sarat. Univ. Physics. *, 2020, vol. 20, iss. 4, pp. 278-287. DOI: 10.18500/1817-3020-2020-20-4-278-287

# Modeling of Epidemics by Cellular Automata Lattices. SIRS Model with Reproduction and Migration

**Background and Objectives: **Methods of population dynamics give the possibility to analyze many biological phenomena by constructing simple qualitative models, which allow to understand their nature and to predict their behavior. This approach is used to study the spread of epidemics of infectious diseases in biological and human populations. The work considers a modified SIRS model of epidemic spread in the form of a lattice of stochastic cellular automata. The model uses dynamic population control with a limitation of the maximum number of individuals in the population and the influence of the disease on reproduction processes. The effect of migration is explored. **Materials and Methods:** 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 considered SIRS model demonstrates irregular oscillations in the number of infected individuals at certain values of the parameters. Weak diffusion produces a significant effect on the oscillations, changing their intensity, average period and constant components. The diffusion also leads to synchronization of processes on different regions of the population. **Conclusion: **The behavior of the lattice cellular automata SIRS model essentially differs from that of the ordinary differential equations. In particular, it can demonstrate the oscillatory regime, which is absent in the mean field approach, as well as the phenomenon of synchronization.

- Bailey. N. The mathematical approach to biology and medicine. London, John Wiley and Sons, 1967. 286 p.
- Hethcote H. W. The mathematics of infectious diseases. SIAM Review, 2000, vol. 42, no. 4, pp. 599–653.
- Anderson R., May R. Infectios deseases of humans – dynamics and control. Oxford, Oxford University Press, 1991. 768 p.
- Kermack W., McKendrick A. A contribution to the mathematical theory of epidemics. Proc. R. Soc., 1927, vol. A115, pp. 700–721.
- Kobrinsky N. E., Trahtenberg B. A. Introduction to the theory of finite automata. Moscow, Fizmatgiz Publ., 1962. 405 p. (in Russian).
- Toffoli T., Margolus N. Cellular automata machines: a new environment for modeling. Cambridge, MIT Press, 1987. 259 p.
- Vanag V. K. Study of spatially extended dynamical systems using probabilistic cellular automata. Physics-Uspekhi, 1999, vol. 42, no. 5, pp. 413.
- Provata A., Nicolis G., Baras F. Oscillatory dynamics in low-dimensional supports: A lattice Lotka ‒ Volterra model. J. Chem. Phys., 1999, vol. 110, pp. 8361–8368.
- Shabunin A., Baras F., Provata A. Oscillatory reactive dynamics on surfaces: A lattice limit cycle model. Physical Review E, 2002, vol. 66, no. 3, pp. 036219.
- Tsekouras G., Provata A., Baras F. Waves and their interactions in the lattice lotka-volterra model. Izvestiya VUZ, Applied Nonlinear Dynamics, 2003, vol. 11, no. 2, pp. 63–71.
- 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, pp. 145701.
- Efimov A., Shabunin A., Provata A. Synchronization of stochastic oscillations due to long-range diffusion. Physical Review E, 2008, vol. 78, no. 5, pp. 056201.
- Kouvaris N., Provata A. Synchronization, stickiness effects and intermittent oscillations in coupled nonlinear stochastic networks. Eur. Phys. J. B, 2009, vol. 70, pp. 535–541.
- Kouvaris N., Provata A., Kugiumtzis D. Detecting synchronization in coupled stochastic ecosystem networks. Physics Letters A, 2010, vol. 374, pp. 507–515.
- Kouvaris N., Kugiumtzis D., Provata A. Species mobility induces synchronization in chaotic population dynamics. Physical Review E, 2011, vol. 84, pp. 036211.
- 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, pp. 2447.
- 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, pp. 209–223.
- 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).
- Verhulst P. Notice sur la loi que la population suit dans son accroissement. Corr. Math. et Phys., 1838, vol. 10, pp. 113–121.
- Shabunin A., Astakhov V., Kurths J. Quantitative analysis of chaotic synchronization by means of coherence. Phys. Rev. E, 2005, vol. 72, pp. 016218.

- 626 reads