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Space-time fractional diffusion equations in chemotaxis and immunology

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MMVW04 - Modelling non-Markov Movement Processes

In the presence of sparse attractants, the movement of both cells and large organisms has been shown to be governed by long distance runs, according to an approximate L´evy distribution. In this talk we clarify theform of biologically relevant PDE descriptions for such movements. Motivated by experiments we consider a microscopic velocity-jump model in which an individual performs occasional long jumps according to an approximate L´evy distribution. We derive the appropriate kinetic-transport equation, where the collision term describes the nonlocal motion. Under a perturbation argument and an appropriate parabolic scaling in space and time, we derive fractional Patlak-Keller-Segel equations [2]. Further, we consider the implications of such biological diffusion in the context of T cell movement in the brain [3]. In response to Toxoplasma gondii infections, these cells have been shown to exhibit a mixed L´evy walk in which an additional resting time occurs between directional changes. The resulting equation generates a time-fractional term and allows us to formally interpret the results of [1]. Consequently, we shed light on the extent to which L´evy flight behaviour impacts on the average time taken for T cells to locate the sparsely distributed infected targets. We use the PDE description to present numerical results. This work is in collaboration with H. Gimperlein, K. Painter and J. Stocek. References1 T. Harris, et al. Generalized L´evy walks and the role of chemokines in migration of effector CD8 + T cells, Nature 486, 545–548, 2012.[2] G. Estrada, H. Gimperlein, K.J. Painter. Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion, SIAP , 78(2): 1155-1177, 2018.[3] G. Estrada, H. Gimperlein, K.J. Painter, J. Stocek. Space-time fractional diffusion in immune cell models with delay, M3AS , 29, 65-88, 2019.  

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