Since social interactions have been shown to lead to symmetric clusters, we propose here that symmetries play a key role in epidemic modeling. Mathematical models on d-ary tree graphs were recently shown to be particularly effective for modeling epidemics in simple networks [Seibold & Callender, 2016]. To account for symmetric relations, we generalize this to a new type of networks modeled on d-cliqued tree graphs, which are obtained by adding edges to regular d-trees to form d-cliques. This setting gives a more realistic model for epidemic outbreaks originating, for example, within a family or classroom and which could reach a population by transmission via children in schools. Specifically, we quantify how an infection starting in a clique (e.g. family) can reach other cliques through the body of the graph (e.g. public places). Moreover, we propose and study the notion of a safe zone, a subset that has a negligible probability of infection.