Non-homogeneous stochastic birth and death processes with applications to epidemic outbreak data

Abstract

The subject of this thesis is the non-homogeneous birth-death process with some of its special cases and its use in modeling epidemic data. This model describes changes in the size of a population. New population members can appear with a rate, called the birth rate or the reproductive power, and members can leave the population with a rate called the death rate. These rates are taken to be non-homogeneous in time. The Lagrange transform is used to derive the probability distribution function of the non-homogeneous birth-death process. Because of the non-homogeneous rates the reproduction and death times are described with general survival functions. A parametric choice for these distributions is discussed in Chapter 2 and in Chapter 4 and three types of models are examined: a proportional rate model, an accelerated failure time model and a model combining both. The non-homogeneous birth-death model gives rise to the net reproduction ratio. This net reproduction ratio is used as tool to study the avian influenza outbreak in the Netherlands in 2003. In Chapter 3, a non-homogeneous martingale model is derived from the non-homogeneous birth-death process, by taking the reproductive power equal to the death rate, to model volatility in a stochastic time series of counts with a constant mean. A model is obtained for which the expected value at a certain time point equals what has been observed at a previous time point. The variance, however, can depend on time. It is shown that the net reproduction ratio can be used as a additional tool. These models and procedures are illustrated with MRSA (Methicillin-Resistant Staphylococcus aureus) prevalence data registered since 2001 from three Acute Trusts of hospitals from the National Health Service in Great Britain. In Chapter 4, two special cases of the non-homogeneous birth-death model are considered: the non-homogeneous death process and the non-homogeneous birth process. In both of these models the end of the epidemic is incorporated through a modification of the survival function with a final size parameter, as is done in long-term survival models. These models are applied to three outbreaks: The Dutch classical swine fever outbreak from 1997-1998, the foot and mouth disease outbreak in Great Britain from 2001 and the Dutch avian influenza (H7N7) outbreak from 2003. In Chapter 5 the non-homogeneous birth model is applied to the transmission of avian influenza (H5N1) among poultry in Thailand. A proportional rate regression model was formulated. In Chapter 6 methods of estimating the reproductive power and the survival function with communicable events are discussed. Using a non-homogeneous birth process, one can estimate the reproductive power and then the survival function and their standard errors directly from the data using the (log-)likelihood, instead of using a parametric form as is done in the other chapters. Methods are developed to compare empirical estimates in two independent groups by means of the (log) reproduction power ratio. These methods are applied to the Dutch avian influenza (H7N7) outbreak from 2003 and to data from avian influenza (H5N1) outbreaks among poultry in Thailand.