Mathematical modelling of cytokine-mediated immune response and autoimmunity

One of the major outstanding challenges in immunology is the development of a comprehensive, quantitative and accurate approach to understanding the causes and dynamics of immune responses. The immune system normally protects the body against infections, but at the same time it is possible that it can fail to distinguish the host’s own cells from the cells affected by the infection, which can lead to autoimmune disease. The question of what releases the auto-pathogenic potential of T lymphocytes is at the heart of understanding autoimmune disease. Among various possible causes of autoimmune disease, an important role is played by infections that can result in a breakdown of immune tolerance, primarily through the mechanism of molecular mimicry, where the introduction of pathogenic peptides that structurally resemble self-peptides, derived from infection, may induce T lymphocytes to proliferate and leave them with the ability to respond to self, as well as foreign antigens. Deterministic and stochastic models have been extensively used in the past to study the dynamics of immune responses and analyse a possible onset of autoimmunity. The main focus of this thesis is the development and analysis of mathematical models of immune response to infection, as well as the onset and progress of autoimmunity. Particular emphasis is made on developing new mathematical approaches for elucidating the roles played by various cytokines in the immune dynamics. In the first part of the thesis I develop a mathematical model for dynamics of immune response to hepatitis B. This model explicitly includes contributions from innate and adaptive immune responses, as well as from cytokines. Analysis of the model identifies parameter regimes where the model exhibits clearance of infection, maintenance of a chronic infection, or periodic oscillations. Effects of nucleoside analogues and interferon treatments are analysed, and the critical drug efficiency is determined. The second part of the thesis investigates the dynamics of immune response to a general viral infection and a possible onset of autoimmunity, which account for regulatory T cells, T cells with different activation thresholds, and cytokines. Feasibility and stability analyses of different steady states yield boundaries of stability and bifurcations in terms of system parameters. This model exhibits bi-stability and shows different regimes of normal clearance of viral infection, chronic infection, or autoimmune behaviour. Therefore, it can provide significant new insights into autoimmune dynamics. To investigate the role of stochasticity in immune dynamics, I developed a stochastic version of the model, and the major result is that adding stochasticity can lead to the emergence of sustained oscillations around deterministically stable steady states, thus providing a possible explanation for experimentally observed variations in the progression of autoimmune disease. I also have investigated stochastic dynamics in the regime of bi-stability and computed the magnitude of these fluctuations. I have also analysed the effects of different time delays, as well as the inhibiting effect of regulatory T cells on secretion of interleukin-2 on autoimmune dynamics. To this end, I have performed a systematic analysis of stability of all steady states of the corresponding model both analytically, and numerically. The identification of basins of attraction of different steady states and periodic solutions indicates that time delays can change the shape of these basins of attraction, and the new results show better qualitative agreement with the experimental observations. My thesis culminates with the last part, where I explore stochastic effects in a time-delayed model for autoimmunity. The major achievement in this part of the thesis is the development of a new methodology for deriving an Itô stochastic delay differential equation (SDDE) from delay discrete stochastic models, as well as showing the equivalency of previously proposed methods. Using this equivalence, I derived a simpler SDDE model to perform numerical simulations. I have used a linear noise approximation (LNA) to determine the magnitude of stochastic fluctuations around deterministic steady states, and to obtain insights into how the coherence of stochastic oscillations around deterministically stable steady states depends on system parameters.