Epidemic models often incorporate contact net- works along which the disease can be passed. This study incorporates a restarting-recentering evolutionary algorithm, previously developed to locate extremal epidemic networks, together with a new representation, the toggle-delete representa- tion, for evolvable networks. The goal is to locate networks that were likely to have produced a given epidemic behavior. This goal subsumes a new fitness function for driving selection in network evolution. Earlier representations used networks with a fixed sequence of contact numbers. The new representation can add and remove edges from the network, permitting a search that varies contact numbers within the network. A parameter setting study is performed on an epidemic profile obtained from an random network and then tested on a bimodal profile invented by the researchers. The algorithm succeeds in producing networks that cause epidemics run on them to mimic the specified epidemic profiles.