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 toT. gondiiinfection 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 ofT. 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.
Highlights
After the discovery of the Toxoplasma gondii (T. gondii) parasite in 1907, it has been extensively studied in various fields, such as morphology, immunology, and identification of diseases caused by T. gondii parasite [47, 80]
Observation of the Levy random walk (RW) pattern performed by CD8+ T cells in their search for the T. gondii infected brain cells has led to more studies on such a pattern of the movement in the body [10, 43]
Our goal is to investigate the efficacy of a Levy search pattern performed by effector CD8+ T cells in the T. gondii-infected brain tissue
Summary
After the discovery of the Toxoplasma gondii (T. gondii) parasite in 1907, it has been extensively studied in various fields, such as morphology, immunology, and identification of diseases caused by T. gondii parasite [47, 80]. Mathematical models have been widely applied to study the dynamics of host immune responses to tumour cells and viral infections, like hepatitis, human immunodeficiency virus (HIV), and influenza in the form of an ordinary differential system of equations with integer or fractional orders (see for instance [19, 41]). Observation of the Levy RW pattern performed by CD8+ T cells in their search for the T. gondii infected brain cells has led to more studies on such a pattern of the movement in the body [10, 43] Following such studies, it would be essential to derive appropriate mathematical models based on the Levy RW that can shed some light on differences with the classical Brownian motion, which has been observed in the immune system. Our model is used to estimate the efficiency of effector CD8+ T cells for a Levy RW versus a Brownian motion in Section 5, and in the last section, we finish the paper with a review of the results and some suggestions for future work
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