A fractional diffusion model of CD8+T cells response to parasitic infection in the brain

Toxoplasma gondii (T. gondii) is a parasitic pathogen that causes serious brain diseases in fetuses and patients with immunodeficiency, particularly AIDS patients. In the field of immunology, a large number of studies have shown that effector CD8 + T cells respond to T. gondii infection in the brain tissue through controlling the proliferation of intracellular parasites and killing infected brain cells. These protective mechanisms do not occur without T cell movement and searching for infected cells, as a fundamental  feature of the immune system. Following infection with a pathogen in a tissue, in their search for infected cells, CD8 + T cells can perform different stochastic searches, including Levy and Brownian random walks. Statistical analysis of CD8 + T cells in response to infected brain cells could be described by a Levy random walk., In this work, by considering a Levy distribution for the displacements, we propose a space fractional-order diffusion equation for the T cell density in the infected brain tissue. Furthermore, we derive a mathematical model representing CD8 + T cell response to infected brain cells. By solving the model equations numerically, we perform a comparison between Levy and Brownian search strategies.