Statistical analysis of a stochastic automata model for the spread of disease among mobile individuals

We present techniques that allow for the statistical identification of the infection front and for the microscopic control of macroscopic statistics in a simple stochastic lattice automata model for the spread of an infectious disease through a mobile host population. The individual based model consists of susceptible and infected individuals that are free to move about a regular lattice. These individuals interact with each other when located at the same node of the lattice, and susceptible individuals become infected with a probability of infection that is dependent on the number of infected individuals present. By using statistics from the healthy population alone, we present a method by which the spread of an infection in the model can be located spatially, even in a low-density population. A parameter which governs the local mobility rules of the model is shown to be functionally related to the non-dimensional statistical values of skewness and flatness for various macroscopic quantities. We show formal convergence to reaction-diffusion equations from the lattice Boltzmann equations of the model via a Hilbert expansion. The validity of both the lattice Boltzmann equations and the reaction-diffusion equations is shown in a low-density population regime.