Chapter 1 gives a brief introduction to modelling epidemics in both a deterministic and stochastic framework. Chapter 2 gives a brief introduction to the single-group Susceptible-Infected-Susceptible (SIS) model and we mention and establish all the necessary background theory we apply to the model with the aim of extending these theories and techniques to our own multi-group SIS model in order to obtain results concerning the short and long-term behaviour of the epidemic. In Chapter 3 we define a k-group SIS model which will allow us to examine the effects of heterogeneity in three different categories - the infectivity of infectious individuals, their mixing behaviour and an individual's susceptibility to the disease. In doing this, we describe the dynamics of the model including how to represent it in a deterministic framework, the quasi-stationary distribution and the time to extinction. We apply a branching process approximation to the model, viable for the early stages of a disease, using well established theory. In Chapter 4 we look specifically at the early stages of the epidemic based on the branching process approximation and produce numerical results on how the probability of disease emergence behaves as the basic reproduction number Ra increases. We contrast 2-group heterogeneous models against a homogeneous model, for epidemics assuming either an exponential or constant infectious period. We then analyse these results with an iterative and inductive proof showing that the emergence probability for a heterogeneous model will always be less than that for a homogeneous model, not just in the limit but at all stages of iterative convergence. Next we provide a proof which shows that for the non-separable general model this ordering exists for any given infectious period. We then go on to look at comparing two heterogeneous models to one another under various sets of parameters and use majorization theory as a tool for doing so. We use orderings referred to as ordinary majorization, p-majorization and pq-majorization to show that there is an inferred ordering of emergence probabilities when comparing multi-group he- terogeneous models to one another. In Chapter 5 we study the long-term behaviour of the stochastic multi-group SIS model. We begin by formulating conditions for the general model under which feasible equilibria exist and conditions where either the disease-free or endemic equilibria are stable. For a 2-group version, we calculate numerically the determi- nistic equilibrium values, stochastic means and quasi-stationary distributions for a range of Ra values. We use an Ornstein-Uhlenbeck process to approximate the quasi-stationary distribution and assess the accuracy of the approximation. We then calculate the expected time to extinction and use a coefficient of variation approximation as a proxy for this and discuss the suitability of such an approxi- mation to the exact results. These analyses are all carried out for models which exhibit heterogeneity in infectivity, mixing or susceptibility.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:576951 |
| Date | January 2011 |
| Creators | Pearce, Christopher |
| Publisher | University of Liverpool |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.002 seconds