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Type: Theses
Title: Hybrid methodology for Markovian epidemic models
Author: Rebuli, Nicolas Peter
Issue Date: 2018
School/Discipline: School of Mathematical Sciences
Abstract: In this thesis, we introduce a hybrid discrete-continuous approach suitable for analysing a wide range of epidemiological models, and an approach for improving parameter estimation from data describing the early stages of an outbreak. We restrict our attention to epidemiological models with continuous-time Markov chain (CTMC) dynamics, a ubiquitous framework also commonly used for modelling telecommunication networks, chemical reactions and evolutionary genetics. We introduce our methodology in the framework of the well-known Susceptible–Infectious–Removed (SIR) model, one of the simplest approaches for describing the spread of an infectious disease. We later extend it to a variant of the Susceptible–Exposed–Infectious– Removed (SEIR) model, a generalisation of the SIR CTMC that is more realistic for modelling the initial stage of many outbreaks. Compartmental CTMC models are attractive due to their stochastic individual-to-individual representation of disease transmission. This feature is particularly important when only a small number of infectious individuals are present, during which stage the probability of epidemic fade out is considerable. Unfortunately, the simple SIR CTMC has a state space of order N², where N is the size of the population being modelled, and hence computational limits are quickly reached as N increases. There are a number of approaches towards dealing with this issue, most of which are founded on the principal of restricting one’s attention to the dynamics of the CTMC on a subset of its state space. However, two highly-efficient approaches published in 1970 and 1971 provide a promising alternative to these approaches. The fluid limit [Kurtz, 1970] and diffusion limit [Kurtz, 1971] are large-population approximations of a particular class of CTMC models which approximate the evolution of the underlying CTMC by a deterministic trajectory and a Gaussian diffusion process, respectively. These large-population approximations are governed by a compact system of ordinary differential equations and are suitably accurate so long as the underlying population is sufficiently large. Unfortunately, they become inaccurate if the population of at least one compartment of the underlying CTMC is close to an absorbing boundary, such as during the initial stages of an outbreak. It follows that a natural approach to approximating a CTMC model of a large population is to adopt a hybrid framework, whereby CTMC dynamics are utilised during the initial stages of the outbreak and a suitable large-population approximation is utilised otherwise. In the framework of the SIR CTMC, we present a hybrid fluid model and a hybrid diffusion model which utilise CTMC dynamics while the number of infectious individuals is low and otherwise utilises the fluid limit and the diffusion limit, respectively. We illustrate the utility of our hybrid methodology in computing two key quantities, the distribution of the duration of the outbreak and the distribution of the final size of the outbreak. We demonstrate that the hybrid fluid model provides a suitable approximation of the distribution of the duration of the outbreak and the hybrid diffusion model provides a suitable approximation of the distribution of the final size of the outbreak. In addition, we demonstrate that our hybrid methodology provides a substantial advantage in computational-efficiency over the original SIR CTMC and is superior in accuracy to similar hybrid large-population approaches when considering mid-sized populations. During the initial stages of an outbreak, calibrating a model describing the spread of the disease to the observed data is fundamental to understanding and potentially controlling the disease. A key factor considered by public health officials in planning their response to an outbreak is the transmission potential of the disease, a factor which is informed by estimates of the basic reproductive number, R₀, defined as the average number of secondary cases resulting from a single infectious case in a naive population. However, it is often the case that estimates of R₀ based on data from the initial stages of an outbreak are positively biased. This bias may be the result of various features such as the geography and demography of the outbreak. However, a consideration which is often overlooked is that the outbreak was not detected until such a time as it had established a considerable chain of transmissions, therefore effectively overcoming initial fade out. This is an important feature because the probability of initial fade out is often considerable, making the event that the outbreak becomes established somewhat unlikely. A straightforward way of accounting for this is to condition the model on a particular event, which models the disease overcoming initial fade out. In the framework of both the SIR CTMC and the SEIR CTMC we present a conditioned approach to estimating R₀ from data on the initial stages of an outbreak. For the SIR CTMC, we demonstrate that in certain circumstances, conditioning the model on effectively overcoming initial fade out reduces bias in estimates of R₀ by 0.3 on average, compared to the original CTMC model. Noting that the conditioned model utilises CTMC dynamics throughout, we demonstrate the flexibility of our hybrid methodology by presenting a conditioned hybrid diffusion approach for estimating R₀. We demonstrate that our conditioned hybrid diffusion approach still provides estimates of R₀ which exhibit less bias than under an unconditioned hybrid diffusion model, and that the diffusion methodology enables us to consider larger outbreaks then would have been computationally-feasible in the original conditioned CTMC framework. We demonstrate the flexibility of our conditioned hybrid approach by applying it to a variant of the SEIR CTMC and using it to estimate R₀ from a range of real outbreaks. In so doing, we utilise a truncation rule to ensure the initial CTMC dynamics are computationally-feasible.
Advisor: Bean, Nigel Geoffrey
Ross, Joshua
Dissertation Note: Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2018
Keywords: epidemiology
Markov chain
fluid approximation
diffusion approximation
basic reproductive number
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