We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition, every edge is deleted at constant rate, a mechanism which extends previous partial duplication models. In this model, we obtain results on the degree distribution, which shows a phase transition such that either -- if $p$ is small enough -- the frequency of isolated vertices converges to 1, or there is a positive fraction of vertices with unbounded degree. We derive results on the degrees of the initial vertices as well as on the sub-graph of non-isolated vertices. In particular, we obtain expressions for the number of star-like subgraphs and cliques.