Hepatitis C is a severe disease and a prime cause for liver transplantation. Up to 3% of the world’s population are chronically infected with its causative agent, the Hepatitis C virus (HCV). This capacity to establish long (decades) lasting persistent infection sets HCV apart from other plus-strand RNA viruses that typically cause acute, self-limiting infections. In order to understand the molecular processes governing HCV RNA replication, we developed the first detailed mathematical model of the initial dynamic phase of intracellular HCV RNA replication. By model analysis, we recognized diverse but crucial roles for the membraneous replication compartment of HCV in regulating replication by balancing translation versus replication and thus effectively limiting RNA amplification. We further predict the existence of an essential limiting host factor (or function) required for establishing active RNA replication and thereby determining cellular permissiveness for HCV. Our model also proved valuable to understand and predict the effects of pharmacological inhibitors of HCV and might be a solid basis for the development of similar models for other plus-strand RNA viruses. In currently ongoing work, we extend the model by the innate antiviral response. This will eventually allow us to study and ultimately interfere with the intricate balance between viral exploitation of host cellular resources and intrinsic defence mechanisms of the cell, which is the basis of virus persistence.