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

Stationary states and spatial patterning in an $SIS$ epidemiology model with implicit mobility