Contact reciprocity in a two-group epidemic ($N_H{c}_H{ϵ}_H=N_L{c}_L{ϵ}_L$)

Each undirected cross-group partnership is counted once from each group's contact total. Thus group size times per-person contact rate times cross-group fraction must agree. If one group occupies fraction $p$ while its cross-group per-person rate stays fixed, aggregate cross-group contact is order $p$ and the other group's cross-contact fraction is order $p$. Assigning both per-person rates independently generally breaks this balance.