  By means of the asynchronous cellular automata algorithm we study stationary
states and spatial patterning in an $SIS$ model, in which the individuals' are
attached to the vertices of a graph and their mobility is mimicked by varying
the neighbourhood size $q$. The versions with fixed $q$ and those taken at
random at each step and for each individual are studied. Numerical data on the
local behaviour of the model are mapped onto the solution of its zero
dimensional version, corresponding to the limit $q\to +\infty$ and equivalent
to the logistic growth model. This allows for deducing an explicit form of the
dependence of the fraction of infected individuals on the curing rate $\gamma$.
A detailed analysis of the appearance of spatial patterns of infected
individuals in the stationary state is performed.
