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In this paper, we study the immune response in a fractional order model for HIV dynamics, for distinct disease transmission rates and saturated cytotoxic T-lymphocyte (CTL) response. Our goal is twofold: (i) to analyze the role of the order of the fractional derivative, α, on the efficacy of the immune response, (ii) to examine the immune response for distinct transmission functions, in the presence of saturated CTL response. We compute the reproduction number of the model and state the stability of the disease-free equilibrium. We discuss the results of the model from an epidemiological point of view. [ABSTRACT FROM AUTHOR]

This paper proposes a quantitative model of the reaction-diffusion type to examine the distribution of interferon-α (IFNα) in a lymph node (LN). The numerical treatment of the model is based on using an original unstructured mesh generation software Ani3D and nonlinear finite volume method for diffusion equations. The study results in suggestion that due to the variations in hydraulic conductivity of various zones of the secondary lymphoid organs the spatial stationary distribution of IFNα is essentially heterogeneous across the organs. Highly protected domains such as sinuses, conduits, co-exist with the regions in which where the stationary concentration of IFNα is lower by about 100-fold. This is the first study where the spatial distribution of soluble immune factors in secondary lymphoid organs is modelled for a realistic three-dimensional geometry. [ABSTRACT FROM PUBLISHER]

This paper is devoted to the study of global existence of periodic solutions of a delayed tumor-immune competition model. Also some numerical simulations are given to illustrate the theoretical results [ABSTRACT FROM PUBLISHER]

In this paper several models in virus dynamics with and without immune response are discussed concerning asymptotic behaviour. The case of immobile cells but diffusing viruses and T-cells is included. It is shown that, depending on the value of the basic reproductive number R0 of the virus, the corresponding equilibrium is globally asymptotically stable. If R0 < 1 then the virus-free equilibrium has this property, and in case R0 > 1 there is a unique disease equilibrium which takes over this property.. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the dynamical behavior of a stochastic viral infection model which is formulated by four nonlinear stochastic differential equations to describe the interactions between virus, host cells, and immune response represented by cytotoxic T lymphocytes (CTL) cells. The infection transmission process is modeled by Hattaf-Yousfi incidence function which includes many special cases existing in the literature. The positivity of solutions is investigated. In addition, the extinction of the infection is established in terms of the basic reproduction number R0. Moreover, sufficient conditions for the fluctuation of solutions around the two infection equilibria are obtained. Numerical simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The mathematical model of the human immunodeficiency virus (HIV) pathogenesis with the adaptive immune response, two saturated rates and therapy is presented and studied in this work. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies; the two saturated rates describe the viral infection and the CTL proliferation. Two kinds of treatments are incorporated into the model; the objective of the first one is to reduce the number of infected cells, while the aim of the second one is to block the free viruses. The positivity and boundedness of solutions are established. The local stability of the disease free steady state and the infection steady states are studied. Numerical simulations are performed to show the behavior of solutions and the effectiveness of the incorporated therapy in controlling the HIV replication which can improves significantly the patient's life quality. [ABSTRACT FROM AUTHOR]

Many tumours undergo periods in which they apparently do not grow but remain at a roughly constant size for extended periods. This is termed tumour dormancy. The mechanisms responsible for dormancy include failure to develop an internal blood supply, individual tumour cells exiting the cell cycle and a balance between the tumour and the immune response to it. Tumour dormancy is of considerable importance in the natural history of cancer. In many cancers, and in particular in breast cancer, recurrence can occur many years after surgery to remove the primary tumour, following a long period of occult disease. Mathematical modelling suggested that continuous growth of tumours was incompatible with data of the times of recurrence in breast cancer, suggesting that tumour dormancy was a common phenomenon. Modelling has also been applied to understanding the mechanisms responsible for dormancy, how they can be manipulated and the implications for cancer therapy. Here, the literature on mathematical modelling of tumour dormancy is reviewed. In conclusion, promising future directions for research are discussed. [ABSTRACT FROM AUTHOR]