Spreading processes in post-epidemic environments

We analyze infection spreading processes in a system where only fraction $p$ of individuals can be affected by disease, while remaining $1-p$ are immune. Such picture emerge as natural consequence previously terminated epidemic process or arise formerly vaccinated population. To this end, we apply the synchronous cellular automata algorithm studying stationary states and spatial patterning SI, SIS SIR models on square lattice with active sites. A concept "safety patterns" susceptible agents surrounded immune naturally arises proposed system, which plays an important role course under consideration. Detailed analysis distribution such patterns is given, turn determine infected state $I^*(p)$. Estimates for threshold values basic reproduction number $R_0^c$ function obtained well. In particular, our results allow to predict optimal individuals, needed advance order get maximal unaffected given curing rate.