Automata network models for the spread of an infectious disease in populations of moving individuals

Probabilistic automata network SIR and SIS models for the spread of an infectious disease in populations of moving individuals are studied. The local transition rule consists of two subrules. The first subrule, applied synchronously, models infection and removal (or recovery). It is a probabilistic cellular automaton rule. The second one, applied sequentially, describes the motion of the individuals. The spatial correlations created by the application of the first subrule are partially destroyed according to the degree of mixing of the population which follows from the application of the second subrule. The emphasis is on the influence of the degree of mixing of individuals which follows from their motion.; For SIR models, time evolutions of an epidemic for different values of the degree of mixing are studied for different types of motion and removal, and it is correctly predicted by the mean-field approximation when the degree of mixing tends to infinity. The asymptotic behaviors for very small and very large degree of mixing are determined for different types of motion and removal. Two-population models that individuals belonging to one population may be infected only by individuals belonging to the other population are studied.; For SIS models, in the infinite-time limit, the system is either in the disease-free state or in the endemic state. It goes from the one state to the other through a transcritical bifurcation similar to a second-order phase transition characterized by a nonnegative order parameter, whose role is played by the stationary density of infected individuals. The phase diagram and the critical behavior in the vicinity of the phase transition are studied as a function of the degree of mixing. According to whether individuals perform short- or long-range moves, it is found that the parameters characterizing the transition have a qualitatively different behavior as the degree of mixing varies. When the degree of mixing is very large, the behavior of the system is correctly predicted by the mean-field approximation, but when it is not large, this assumption is no more correct.