The innate immune system plays a vital role in the control of infection, but many aspects of its behaviour are not fully understood. In this dissertation, I use mathematical modelling approaches to elucidate two linked processes which are key to the innate immune response to Salmonella infection.

First, I consider the inflammasome, a large protein complex which forms in cells following infection. The inflammasome is responsible for coordinating cell death and the release of cytokines, which promote further activation of the immune system. Despite its importance, there are many conflicting accounts of this process in the existing literature, and there is no accepted single conceptual model of inflammasome formation.

In the first chapters of this dissertation, I present a suite of deterministic ordinary differential equation models and discrete stochastic models which highlight different elements of the inflammasome formation process. By comparing the results from these models with an existing dataset describing the innate immune response to infection in macrophages, I construct a cohesive conceptual model of inflammasome formation. In particular, I propose a novel `branching' mechanism of protein recruitment to the inflammasome complex. I also show that variation in inflammasome formation times can arise from differences in the initial abundance of NLR oligomers present in the cell prior to inflammasome formation.

The remainder of this dissertation focuses on the formation of clusters of infected cells (or `lesions') in tissues following Salmonella infection. Inflammasome formation on the cellular level is a crucial part of this larger-scale process; however, the precise downstream effects of inflammasome-coordinated signalling and cell death in this context remain unclear.

In the later chapters of this thesis, I outline a spatial partial differential equation model of lesion formation in liver tissue following Salmonella infection, with a particular focus on motility of infected cells and multiple forms of cell death. Through use of Turing instability analysis and simulated solutions of the lesion model, I show that the spatial structure of lesions can arise simply through the interactions of infected and uninfected phagocytes, bacteria and a chemokine. I also demonstrate the importance of decreased motility of infected cells, and a balance of different forms of cell death with influx of uninfected immune cells, for the successful control of infection.