Mathematically speaking it looks that if we split the categories $C_i$ into subcategories with zero (or small) cross transmission edges then this suppress the exponential growth at some scale. (And this is also the case for other geometric limitation on the graph of possible transmission, e.g if there are no edges for people who arge geographically remote). In technical terms one need to make the transmission graph (and every large subgraph) a very "bad" expander.
Gil Kalai
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