The competitive dynamics between tumor cells, a replication-competent virus and an immune response

Replication-competent viruses have been used as an alternative therapeutic approach for cancer treatment. However, new clinical data revealed an innate immune response to virus that may mitigate the effects of treatment. Recently, Wein, Wu and Kirn have established a model which describes the interaction between tumor cells, a replication-competent virus and an immune response (Cancer Research 63 (2003):1317-1324). The purpose of this paper is to extend their model from the viewpoints of mathematics and biology and then prove global existence and uniqueness of solution to this new model, to study the dynamics of this novel therapy for cancers, and to explore a explicit threshold of the intensity of the immune response for controlling the tumor. We also study a time-delayed version of the model. We analytically prove that there exists a critical value tau0 of the time-delay tau such that the system has a periodic solution if tau > tau0. Numerical simulations are given to verify the analytical results. Furthermore, we numerically study the spatio-temporal dynamics of the model. The effects of the diffusivity of the immune response on the tumor growth are also discussed.