Automata network SIR models for the spread of infectious diseases in populations of moving individuals

Automata network SIR models for the spread of infectious diseases are studied. The local rule consists of two subrules. The first one, applied sequentially, describes the motion of the individuals, the second is synchronous and models infection and removal (or recovery). The spatial correlations created by the application of the second subrule are partially destroyed according to the degree of mixing of the population which follows from the application of the first subrule. One- and two-population models are considered. In the second case, individuals belonging to one population may be infected only by individuals belonging to the other population as is the case, for example, for the heterosexual propagation of a venereal disease. It is shown that the occurrence of the epidemic in one population may be triggered by the occurrence of the epidemic in the other population. The emphasis is on the influence of the degree of mixing of the individuals which follows from their diffusive motion. In particular, the asymptotic behaviours for very small and very large mixing are determined. When the degree of mixing tends to infinity the correlations are completely destroyed and the time evolution of the epidemic is then correctly predicted by the mean-field approximation.