Epidemic models commonly consider the spread of disease through successful contacts between susceptible and infectious individuals. The SIR-epidemic model considers that recovered individuals are permanently immune, while the SIS model considers recovered individuals to be immediately resusceptible.

In the first part of the thesis, we study the case of temporary immunity in an SIR-based model for a single strain with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay-differential equations. We find conditions for which the endemic steady-state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.

In the second part, we consider the coupled-system dynamics of multi-strain diseases such as influenza, meningitis, dengue virus and malaria. When the probability of reinfection to a second strain is weak, the global coupling between the strains leads to a set of endemic steady states. We show that the equilibria are globally attracting but that stochastic noise causes solutions to drift near endemic states. We then investigate the oscillatory epidemics that result from seasonal forcing and temporary immunity.

In the final part, we consider spatial dynamics of the SIR model with temporary immunity over a discrete domain. The SIR model studied consists of a set of spatially coupled ordinary differential equations. We study the case of temporary immunity which results in a delay coupling between the removed and susceptible groups in the populations on each node. Numerical simulations are used to investigate the stability of the endemic state and examine oscillatory epidemics that can occur with temporary immunity.