Stability Analysis of a Fractional-Order Differential Equation System of a GBM-IS Interaction Depending on the Density

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APPLIED MATHEMATICS & INFORMATION SCIENCES, vol.8, no.3, pp.1021-1028, 2014 (SCI-Expanded) identifier identifier


In this paper, fractional-order is introduced into the interaction model between GBM and IS. GBM (Glioblastoma Multiforme) is a brain tumor, that has a monoclonal origin and produces after reaching a specific density another tumor with different growth rate and treatment susceptibilities. The IS cells are also divided into two populations, namely, the macrophages and the activated macrophages. Hence, this model show two conversions, the conversion from sensitive tumor cell to resistant tumor cell and the conversion from macrophages to active macrophages, and an interaction between the tumor cell and the macrophages. In this work, it is shown that the model possesses non-negative solutions. Furthermore, the stability, existence and uniqueness were studied. To investigate the conditions for an extinction of the tumor cells, Allee threshold is considered. Numerical simulations will give a detailed description of the behavior of the constructed models at the end of the paper.