Quantum Markov processes are widely used models of the dynamics open quantum systems, a fundamental topic in theoretical and mathematical physics with important applications in experimental realisations of quantum systems such as ultracold atomic gases and new quantum information technologies such as quantum metrology and quantum control. In this thesis we present a mathematical framework which effectively characterises dynamical phase transitions in quantum Markov processes, using the theory of large deviations, by combining insights developed in non-equilibrium dynamics with techniques from quantum information and probability. We provide a natural decomposition for quantum Markov chains into phases, paving the way for the rigorous treatment of critical features of such systems such as phase transitions and phase purification. A full characterisation of dynamical phase transitions beyond properties of the steady state is described in terms of a dynamical perspective through critical behaviour of the quantum jump trajectories. We extend a fundamental result from large deviations for classical Markov chains, the Sanov theorem, to a quantum setting; we prove this Sanov theorem for the output of quantum Markov chains, a result which could be extended to a quantum Donsker-Varadhan theory. We perform an in-depth analysis of the atom maser, an infinite-dimensional quantum Markov process exhibiting various types of critical behaviour: for certain parameters it exhibits strong intermittency in the atom detection counts, and has a bistable stationary state. We show that the atom detection counts satisfy a large deviations principle, and therefore we deal with a phase cross-over rather than a genuine phase transition, although the latter occurs in the limit of infinite pumping rate. As a corollary, we obtain the Central Limit Theorem for the counting process.

The process of soft tissue injury repair at the cellular lew I can be decomposed into three phases: acute inflammation including coagulation, proliferation and remodelling. While the later phases are well understood the early phase is less so. We produce a series of new mathematical models for the early phases coagulation and inflammation. The models produced are relevant not only to soft tissue injury repair but also to the many disease states in which coagulation and inflammation play a role. The coagulation cascade and the subsequent formation of the enzyme thrombin are central to the creation of blood clots. By focusing on a subset of reactions that occur within the coagulation cascade, we develop a model that exhibits a rich asymptotic structure. Using singular perturbation theory we produce a sequence of simpler time-dependent model which enable us to elucidate the physical mechanisms that underlie the cascade and the formation of thrombin. There is considerable interest in identifying new therapeutic targets within the coagulation cascade, as current drugs for treating pathological coagulation (thrombosis) target multiple factors and cause the unwelcome side effect of excessive bleeding. Factor XI is thought to be a potential therapeutic target, as it is implicated in pathological coagulation but not in haemostasis (the stopping of bleeding), but its mechanism of activation is controversial. By extending our previous model of the coagulation cascade to include the whole cascade (albeit in a simplistic way) we use numerical methods to simulate experimental data of the coagulation cascade under normal as well as specific-factor-deficient conditions. We then provide simulations supporting the hypothesis that thrombin activates factor XI. The interest in inflammation is now increasing due to it being implicated in such diverse conditions as Alzmeimer's disease, cancer and heart disease. Inflammation can either resolve or settle into a self-perpetuating condition which in the context of soft tissue repair is termed chronic inflammation. Inflammation has traditionally been thought gradualIy to subside but new biological interest centres on the anti-inflammatory processes (relating to macrophages) that are thought to promote resolution and the pro-inflammatory role that neutrophils can provide by causing damage to healthy tissue. We develop a new ordinary differential equation model of the inflammatory process that accounts for populations of neutrophils and macrophages. We use numerical techniques and bifurcation theory to characterise and elucidate the physiological mechanisms that are dominant during the inflammatory phase and the roles they play in the healing process. There is therapeutic interest in modifying the rate of neutrophil apoptosis but we find that increased apoptosis is dependent on macrophage removal to be anti-inflammatory. We develop a simplified version of the model of inflammation reducing a system of nine ordinary equations to six while retaining the physical processes of neutrophil apoptosis and macrophage driven anti-inflammatory mechanisms. The simplified model reproduces the key outcomes that we relate to resolution or chronic inflammation. We then present preliminary work on the inclusion of the spatial effects of chemotaxis and diffusion.

The severity of the outbreak of an infectious disease is highly dependent upon the structure of the population through which it spreads. This thesis considers the stochastic SIR (susceptible → infective → removed) household epidemic model, in which individuals mix with other individuals in their household at a far higher rate than with any other member of the population. This model gives a more realistic view of dynamics for the transmission of many diseases than the traditional model, in which all individuals in a population mix homogeneously, but retains mathematical tractability, allowing us to draw inferences from disease data. This thesis considers inference from epidemics using data which has been acquired after an outbreak has finished and whilst it is still in its early, `emerging' phase. An asymptotically unbiased method for estimating within household infectious contact rate(s) from emerging epidemic data is developed as well as hypothesis testing based on final size epidemic data. Finally, we investigate the use of both emerging and final size epidemic data to estimate the vaccination coverage required to prevent a large scale epidemic from occurring. Throughout the thesis we also consider the exact form of the households epidemic model which should be used. Specifically, we consider models in which the level of infectious contact between two individuals in the same household varies according to the size of their household.