This thesis develops probabilistic models for the spread of infectious diseases in which individuals experience a period of transient immunity after recovering from infection. Quasi-stationary distributions (QSDs) and limiting conditional distributions (LCDs) are used to describe the temporary equilibrium that is reached between an initial exponential growth phase and the epidemic dying out. This thesis includes results characterising QSDs corresponding to existing birth-death processes and epidemic models and to new processes such as the Evolving Strain SIRS model which we define to describe the progression of a disease undergoing antigenic drift, such as seasonal influenza. Existence and uniqueness results are proven for specific LCDs. Results regarding marginals of special cases of these processes are proven, including the preservation of x-invariance for Q as discussed in Pollett [1988].

Many of the models considered in this thesis are multidimensional, which makes explicit calculation of QSDs extremely challenging. To combat this, specialised techniques for simulating QSDs are developed to illustrate and explore these distributions. These novel methods, involving variants on SMC samplers, are shown to facilitate the simulation of QSDs for discrete-valued stochastic processes, particularly reducible processes. A formal proof of convergence of the SMC sampler is provided for some simple examples.

The simulation methods are then used to characterise the properties of QSDs and LCDs related to endemic epidemic models with evolving strains under an equivalence relation. These QSDs are used to define a reproduction number similar to Ro when the process starts from quasi-stationarity. The epidemic models with evolving strains are shown to have the standard SIR and SIRS epidemic models arising as limiting processes as evolution at each infection becomes certain.