Phase transitions in cellular automata models of spatial susceptible--infected--resistant--susceptible epidemics |
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Authors: | Zheng Zhi-Zhen and Wang Ai-Ling |
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Institution: | College of Mechanical Engineering and Automatization, North University of China, Taiyuan 030051, China |
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Abstract: | Spatially explicit models have become widely used in today's
mathematical ecology and epidemiology to study the persistence of
populations. For simplicity, population dynamics is often analysed
by using ordinary differential equations (ODEs) or partial
differential equations (PDEs) in the one-dimensional (1D) space. An
important question is to predict species extinction or persistence
rate by mean of computer simulation based on the spatial model.
Recently, it has been reported that stable turbulent and regular
waves are persistent based on the spatial
susceptible--infected--resistant--susceptible (SIRS) model by using the
cellular automata (CA) method in the two-dimensional (2D) space
Proc. Natl. Acad. Sci. USA 101, 18246 (2004)]. In this
paper, we address other important issues relevant to phase
transitions of epidemic persistence. We are interested in assessing
the significance of the risk of extinction in 1D space. Our results
show that the 2D space can considerably increase the possibility of
persistence of spread of epidemics when the degree distribution of
the individuals is uniform, i.e. the pattern of 2D spatial
persistence corresponding to extinction in a 1D system with the
same parameters. The trade-offs of extinction and persistence
between the infection period and infection rate are observed in the
1D case. Moreover, near the trade-off (phase transition) line, an
independent estimation of the dynamic exponent can be performed, and
it is in excellent agreement with the result obtained by using the
conjectured relationship of directed percolation. We find that the
introduction of a short-range diffusion and a long-range diffusion
among the neighbourhoods can enhance the persistence and global
disease spread in the space. |
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Keywords: | cellular automata epidemiology phase transitions emergence of trade-offs |
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