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The competitive dynamics between tumor cells, a replication-competent virus and an immune response. (English) Zbl 1066.92035
Summary: 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, L. M. Wein, J. T. Wu and D. H. Kirn [Cancer Research 63, 1317–1324 (2003)] have established a model which describes the interaction between tumor cells, a replication-competent virus and an immune response . The purpose of this paper is to extend their model from the view points of mathematics and biology and then prove global existence and uniqueness of solutions to this new model, to study the dynamics of this novel therapy for cancers, and to explore an 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 \(\tau_{0}\) of the time-delay \(\tau\) such that the system has a periodic solution if \(\tau > \tau_{0}\). 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.

92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C05 Biophysics
35R35 Free boundary problems for PDEs
Full Text: DOI
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