Persistence, extinction and spatio-temporal synchronization of SIRS spatial models

Spatially explicit models are widely used in today's mathematical ecology and epidemiology to study persistence and extinction of populations as well as their spatial patterns. Here we extend the earlier work on static dispersal between neighboring individuals to the mobility of individuals as well as multi-patch environments. As is commonly found, the basic reproductive ratio is maximized for the evolutionarily stable strategy for disease persistence in mean field theory. This has important implications, as it implies that for a wide range of parameters the infection rate tends to a maximum. This is opposite to the present result obtained from spatially explicit models, which is that the infection rate is limited by an upper bound. We observe the emergence of tradeoffs of extinction and persistence for the parameters of the infection period and infection rate, and show the extinction time as having a linear relationship with respect to system size. We further find that higher mobility can pronouncedly promote the persistence of the spread of epidemics, i.e., a phase transition occurs from the extinction domain to the persistence domain, and the wavelength of the spirals increases with the mobility ratio enhancement and will ultimately saturate at a certain value. Furthermore, for the multi-patch case, we find that lower coupling strength leads to anti-phase oscillation of the infected fraction, while higher coupling strength corresponds to in-phase oscillation.