Critical behaviour of a probabilistic automata network SIS model for the spread of an infectious disease in a population of moving individuals

A probabilistic automata network SIS (susceptible-infective-susceptible) model for the spread of an infectious disease in a population of moving individuals is studied. The local rule consists of two subrules. The first one, applied synchronously, models infection and recovery. It is a probabilistic cellular automaton rule. The second, applied sequentially, describes the motion of the individuals. The model contains three parameters, the probabilities p i to get infected and p r to recover, and the average number of tentative moves per individual m. Depending upon the values of these parameters, in the infinite-time limit, the system is either in the disease-free state or in the endemic state. It goes from one state to the other through a transcritical bifurcation similar to a second-order phase transition characterized by a non-negative order parameter, whose role is played, in this model, by the stationary density of infected individuals. The (p i ,p r ) phase diagram and the critical behaviour of the stationary density of infectives in the neighbourhood of the phase transition, are studied as a function of m. According to whether the individuals perform short- or long-range moves, it is found that the parameters characterizing the transition have a qualitatively different behaviour as m varies. When m is very large, the correlations created by the application of the subrule modelling infection and recovery are destroyed, and, as expected, the behaviour of the system is then correctly predicted by a mean-field-type approximation which assumes a homogeneous mixing of the individuals. When m is not large, this assumption is no longer correct.