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.