The use of networks to model the spread of epidemics through structured populations is widespread. However, epidemics on networks lead to intractable exact systems with the need to coarse grain and focus on some average quantities. Often, the underlying stochastic processes are Markovian and so are the resulting mean-field models constructed as systems of ordinary differential equations (ODEs). However, the lack of memory (or memorylessness) does not accurately describe real disease dynamics. For instance, many epidemiological studies have shown that the true distribution of the infectious period is rather centred around its mean, whereas the memoryless assumption imposes an exponential distribution on the infectious period. Assumptions such as these greatly affect the predicted course of an epidemic and can lead to inaccurate predictions about disease spread. Such limitations of existing approaches to modelling epidemics on networks motivated my efforts to develop non-Markovian models which would be better suited to capture essential realistic features of disease dynamics. In the first part of my thesis I developed a pairwise, multi-stage SIR (susceptible-infected-recovered) model. Each infectious node goes through some K 2 N infectious stages, which for K > 1 means that the infectious period is gamma-distributed. Analysis of the model provided analytic expressions for the epidemic threshold and the expected final epidemic size. Using available epidemiological data on the infectious periods of various diseases, I demonstrated the importance of considering the shape of the infectious period distribution. The second part of the thesis expanded the framework of non-Markovian dynamics to networks with heterogeneous degree distributions with non-negligible levels of clustering. These properties are ubiquitous in many real-world networks and make model development and analysis much more challenging. To this end, I have derived and analysed a compact pairwise model with the number of equations being independent of the range of node degrees, and investigated the effects of clustering on epidemic dynamics. My thesis culminated with the third part where I explored the relationships between several different modelling methodologies, and derived an original non-Markovian Edge-Based Compartmental Model (EBCM) which allows both transmission and recovery to be arbitrary independent stochastic processes. The major result is a rigorous mathematical proof that the message passing (MP) model and the EBCM are equivalent, and thus, the EBCM is statistically exact on the ensemble of configuration model networks. From this consideration I derived a generalised pairwise-like model which I then used to build a model hierarchy, and to show that, given corresponding parameters and initial conditions, these models are identical to MP model or EBCM. In the final part of my thesis I considered the important problem of coupling epidemic dynamics with changes in network structure in response to the perceived risk of the epidemic. This was framed as a susceptible-infected-susceptible (SIS) model on an adaptive network, where susceptible nodes can disconnect from infected neighbours and, after some fixed time delay, connect to a random susceptible node that they are not yet connected to. This model assumes that nodes have perfect information on the state of all other nodes. Robust oscillations were found in a significant region of the parameter space, including an enclosed region known as an 'endemic bubble'. The major contribution of this work was to show that oscillations can occur in a wide region of the parameter space, this is in stark contrast with most previous research where oscillations were limited to a very narrow region of the parameter space. Any mathematical model is a simplification of reality where assumptions must be made. The models presented here show the importance of interrogating these assumptions to ensure that they are as realistic as possible while still being amenable to analysis.

The purpose of this PhD thesis is to study mesoscopic and macroscopic fluctuations in Interacting Particle Systems. The thesis is split into two main parts. In the first part, we consider a system of Ising spins interacting via Kac potential evolving with Glauber dynamics and study the macroscopic motion of an one-dimensional interface under forced displacement as the result of large scale fluctuations. In the second part, we consider a diffusive system modelled by a Simple Symmetric Exclusion Process (SSEP) which is driven out of equilibrium by the action of current reservoirs at the boundary and study the non-equilibrium fluctuations around the hydrodynamic limit for the SSEP with current reservoirs. We give a brief summary of the first part. In recent years, there has been significant effort to derive deterministic models describing two-phase materials and their dynamical properties. In this context, we investigate the law that governs the power needed to force a motion of a one dimensional macroscopic interface between two different phases of a given ferromagnetic sample with a prescribed speed V at low temperature. We show that given the mesoscopic deterministic non-local evolution equation for the magnetisation (a non local version of the Allen-Cahn equation), we consider a stochastic Ising spin system with Glauber dynamics and Kac interaction (the underlying microscopic stochastic process) whose mesoscopic scaling limit (intermediate scale between microscale and macroscale) is the given PDE, and we calculate the corresponding large deviations functional which would provide the action functional. We obtain that by deriving upper and lower bounds of the large deviation cost functional. Concepts from statistical mechanics such as contours, free energy, local equilibrium allow a better understanding of the structure of the cost functional. Then we characterise the limiting behaviour of the action functional under a parabolic rescaling, by proving that for small values of the ratio between the distance and the time, the interface moves with a constant speed, while for larger values the occurrence of nucleations is the preferred way to make the transition. This led to a production of two published papers [12] and [14] with my supervisor D. Tsagkarogiannis and N. Dirr. In the second part we study the non-equilibrium fluctuations of a system modelled by SSEP with current reservoirs around its hydrodynamic limit. In particular, we prove that, in the limit, the appropriately scaled fluctuation field is given by a Generalised Ornstein- Uhlenbeck process. For the characterisation of the limiting fluctuation field we implement the Holley-Stroock theory. This is not straightforward due to the boundary terms coming from the nature of the model. Hence, by following a martingale approach (martingale decomposition) and the derivation of the equation of the variance for this model combined with “good” enough correlation estimates (the so-called v-estimates), we reduce the problem to a form whose Holley-Stroock result in [45] is now applicable. This is work in progress jointly with my supervisor and P. Gonçalves, [13].