Mathematical modelling is an important tool in helping make projections of infectious disease dynamics. In this thesis, we address epidemiological modelling questions at both the population and within-host scales. By focussing on problems relating to immunity and within-host immunological dynamics, we aim to establish how the accuracy of model projections can be improved.

First, we consider the spread of SARS-CoV-2, using a stochastic model to determine the probability that infected individuals introduced into a population with low case prevalence initiate an outbreak. We consider Israel and the Isle of Man as case studies, two regions that have had few cases and that are quickly vaccinating their populations. Our results suggest that when travel restrictions are relaxed, in order to prevent local transmission, it will still be necessary to implement surveillance of incoming passengers to identify infected individuals.

We then consider forecasting influenza epidemics. Previous exposure to influenza viruses confers cross-immunity against future infections with related strains, however this effect is not always included in influenza forecasting models. We use case notification data from Japan from the 2009 H1N1 influenza pandemic to estimate parameter values and then consider a hypothetical future epidemic. Our results show that a more epidemiologically realistic model, in which cross-immunity is represented explicitly, can generate more accurate projections during outbreaks than if cross-immunity is not included.

We next model influenza virus dynamics at the within-host scale. We consider a hierarchy of three ordinary differential equation models which describe the kinetics of influenza infection; a complex model which explicitly accounts for an immune response to the virus, and two simpler models in which the infection is limited by the availability of target-cells. We show that the simplest target-cell limited model can reproduce synthetic data generated from the complex immune response model, and furthermore that the parameters of the target-cell limited model vary in a systematic manner as the parameters in the complex model used to generate the data are changed. Finally, we extend these within-host models to describe the severity of symptoms experienced by a host. We show that a simple model can be used to make real-time forecasts of the severity of influenza symptoms and be used to classify patients based on estimated model parameter values.