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Applied mathematics and nonlinear sciences in the war on cancer

may benefit from the use of mathematical ideas [ 27 , 29 – 31 ]. The first textbooks treating mathematical oncology as a unified subject are appearing [ 22 , 32 ] suggesting a maturing of the field. Furthermore, a search in the Web of Science database provides more than 6000 results for “mathematical model” and “cancer” (including statistical models); 537 results for “cancer” and “differential equation” and 912 on “cancer” and “mathematics”. Thus, on the basis of such a large number of results, one would expect applied mathematics and nonlinear sciences to

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Nonlinear waves in a simple model of high-grade glioma

Abstract

We present an analysis of a mathematical model describing the key features of the most frequent and aggressive type of primary brain tumor: glioblastoma. The model captures the salient physiopathological characteristics of this type of tumor: invasion of the normal brain tissue, cell proliferation and the formation of a necrotic core. Our study, based on phase space analysis, geometric perturbation theory, exact solutions and numerical simulations, proves the existence of bright solitary waves in the tumor coupled with kink and anti-kink fronts for the normal tissue and the necrotic core. Finally, we study the linear stability of the solutions to calculate the time of tumor recurrence.

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A Continuous–Time Markov Chain Modeling Cancer–Immune System Interactions

References 1. A. R. A. Anderson and P. K. Maini, Special issue: Mathematical oncology, Bull. Math. Biol., vol. 280, pp. 945{953, 2018. 2. P. M. Altrock, L. L. Liu, and F. Michor, The mathematics of cancer: integrating quantitative models, Nat. Rev. Cancer, vol. 15, pp. 730{745, 2015. 3. M. P. Little, Cancer models, genomic instability and somatic cellular darwinian evolution, Biology Direct, vol. 5, pp. 1{19, 2010. 4. A. Konstorum, A. T. Vella, A. J. Adler, and R. C. Laubenbacher, Addressing

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