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

Afshin Farhadi¹ ; Emmanuel Hanert^1, 2

¹ Earth and Life Institute (ELI), Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium.
² Institute of Mechanics, Material and Civil Engineering (IMMC), Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium.

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 3.

Résumé

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 can 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 Lévy and Brownian random walks. Statistical analysis of CD8+ T cell movement in the brain of T. gondii-infected mouse has determined that the search strategy of CD8+ T cells in response to infected brain cells could be described by a Lévy random walk. In this work, by considering a Lévy 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 Lévy and Brownian search strategies. we demonstrate that the Lévy search pattern enables CD8+ T cells to spread over the whole brain tissue and hence they can rapidly destroy infected cells distributed throughout the brain tissue. However, with the Brownian motion assumption, CD8+ T cells travel through the brain tissue more slowly, leading to a slower decline of the infected cells faraway from the source of T cells. Our results show that a Lévy search pattern aids CD8+ T cells in accelerating the elimination of infected cells distributed broadly within the brain tissue. We suggest that a Lévy search strategy could be the result of natural evolution, as CD8+ T cells learn to enhance the immune system efficiency against pathogens.

Détail

DOI : 10.1051/mmnp/2022003

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