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

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.

^* Corresponding author: afshin.farhadi@uclouvain.be; afshinfarhadi19@gmail.com

Received: 3 January 2021
Accepted: 16 January 2022

Abstract

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.