Replication-competent viruses have shown considerable promise in overcoming the inefficient gene transduction experienced by traditional gene therapy approaches to cancer treatment. The viruses infect tumor cells and replicate inside them, eventually causing lysis. Virus particles released during lysis are then able to infect other tumor cells, and, in this way, continuous rounds of infection and lysis allow the virus to spread throughout the tumor. Motivated by this novel cancer treatment, we formulate and analyse a system of partial differential equations that is essentially a radially-symmetric epidemic model embedded in a Stefan problem. We compare three, alternative virus-injection strategies: a fixed fraction of cells pre-infected with the virus are introduced throughout the entire tumor volume, within the tumor core, or within the tumor rim. For all three injection methods, simple and accurate conditions that predict whether the virus will control the tumor are derived.