Analysis of a model of a virus that replicates selectively in tumor cells

We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. This process is modelled as a free boundary problem for a nonlinear system of hyperbolic-parabolic differential equations, where the free boundary is the surface of the tumor. The unknowns are the densities of uninfected cells, infected cells, necrotic cells and free virus particles, and the velocity of cells within the tumor as well as the free boundary r = R( t). The purpose of this paper is to establish a rigorous mathematical analysis of the model, and to explore the reduction of the tumor size that can be achieved by this therapy.