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Stochastic Epidemic Models and Their Statistical Analysis

verfasst von: Håkan Andersson, Tom Britton

Verlag: Springer New York

Buchreihe : Lecture Notes in Statistics

Enthalten in: Professional Book Archive

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The present lecture notes describe stochastic epidemic models and methods for their statistical analysis. Our aim is to present ideas for such models, and methods for their analysis; along the way we make practical use of several probabilistic and statistical techniques. This will be done without focusing on any specific disease, and instead rigorously analyzing rather simple models. The reader of these lecture notes could thus have a two-fold purpose in mind: to learn about epidemic models and their statistical analysis, and/or to learn and apply techniques in probability and statistics. The lecture notes require an early graduate level knowledge of probability and They introduce several techniques which might be new to students, but our statistics. intention is to present these keeping the technical level at a minlmum. Techniques that are explained and applied in the lecture notes are, for example: coupling, diffusion approximation, random graphs, likelihood theory for counting processes, martingales, the EM-algorithm and MCMC methods. The aim is to introduce and apply these techniques, thus hopefully motivating their further theoretical treatment. A few sections, mainly in Chapter 5, assume some knowledge of weak convergence; we hope that readers not familiar with this theory can understand the these parts at a heuristic level. The text is divided into two distinct but related parts: modelling and estimation.

Inhaltsverzeichnis

Frontmatter

Stochastic Modelling

Frontmatter

1. Introduction

Abstract

These lecture notes focus on stochastic models and their statistical analysis. Deterministic epidemic models have perhaps received more attention in the literature (see also Section 1.4 about the history of epidemic models). F’or example, the monograph by Anderson and May (1991), probably the most cited reference in the recent literature on epidemic models, treats almost exclusively deterministic models.

Håkan Andersson, Tom Britton

2. The standard SIR epidemic model

Abstract

In this chapter we present a simple model for the spread of an infectious disease. Several simplifying assumptions are made. In particular, the population is assumed to be closed, homogeneous and homogeneously mixing. Also, the effects of latent periods, change in behaviour, time varying infectivity and temporary or partial immunity are not taken into account. In later chapters we shall indicate ways of handling some of these complicating features of a real-life epidemic.

Håkan Andersson, Tom Britton

3. Coupling methods

Abstract

Let us assume that we are interested in comparing two or more random elements with each other. It is sometimes possible to construct versions of these random elements on the same probability space, in such a way that the comparison suddenly becomes easy (indeed, often trivial) to carry out. This procedure is calledcouplingthe term referring to the fact that the random elements so constructed are often highly dependent. The coupling method has found many important applications in various fields of probability theory, including Markov processes, renewal processes and Poisson approximation. The book by Lindvall (1992) provides a nice introduction to the subject.

Håkan Andersson, Tom Britton

4. The threshold limit theorem

Abstract

We will now explore in greater detail the large population limit of the final size distribution for the standard SIR epidemic model E_n,m(λI).We have seen (Section 3.3) that, if the population of susceptibles is large and we introduce a small number of initial infectives, the number of infectious individuals behaves like a branching process in the beginning. If the basic reproduction number R_o= λι is less than or equal to 1, a small outbreak will occur. On the other hand, if R_oexceeds 1, then there is a positive probability that the approximating branching process explodes; this implies, of course, that the branching process approximation will break down after some time. Then it is reasonable to expect that the final epidemic size will satisfy a law of large numbers. This indicates that the asymptotic distribution of the final size actually consists of two parts, one close to zero arid the other concentrated around some deterministic value. In this chapter we sketch the derivation of these results, using the Sellke construction and the beautiful imbedding representation of Scalia-Tomba (1985, 1990). A fluctuation result for the final size, giver a large outbreak, will also be given. In the final section we combine earlier results and indicate the proof of a theorem due to Barbour (1975) on the duration of the (Markovian) standard SIR epidemic.

Håkan Andersson, Tom Britton

5. Density dependent jump Markov processes

Abstract

In the present section we shall approximate certain jump Markov processes as a parameter n, interpreted as the population size, becomes large. The results will be presented in a form general enough for our purposes. More general results, as well as other extensions, may be found in Chapter 11 of Ethier and Kurtz (1986), which has served as our main source. With the aim to explain the intuition behind the theory we start with a simple example, a birth and death process with constant birth rate (`immigration’) and constant individual death rate. The results in Sections 5.3 and 5.4 are applied to this example, thus giving explicit solutions. In Section 5.5 we apply the results to the Markovian version of the epidemic model described in Section 2.3. It is shown that this process converges weakly to a certain Gaussian process but in this case it is not possible to obtain explicit solutions for the deterministic limit and the covariance function. It is worth mentioning that the techniques presented in this chapter may be applied to a wide range of problems such as more general epidemic models and models for chemical reactions and population genetics, as well as other population processes.

Håkan Andersson, Tom Britton

6. Multitype epidemics

Abstract

The epidemic model studied so farE _n,m(λ, I), assumes that the population is homogeneous (with regard to the disease) and that individuals mix uniformly. In real life epidemics, this is rarely the case. For example, children are usually more susceptible to influenza, and sometimes individuals with a previous history of the disease have acquired some partial immunity. For STDs (sexually transmitted diseases), some individuals have higher infectivity in that they are more promiscuous (varying infectivity is often the case in other transmittable diseases as well). It may also be that the infectious periods of different individuals are not identically distributed; however, the assumption of independence seems reasonable in most cases. These heterogeneities can be characterised asindividual.A second group of heterogeneities is caused by the social structure in the population. The model E_n,m(λ,I) assumes that an individual has contact with each individual atequalrate (=λ/n), so that there is uniform mixing. In real life, the presence of social structures, such as households, friendly (including work) relations, and geographical structures, violates this assumption.

Håkan Andersson, Tom Britton

7. Epidemics and graphs

Abstract

Random graphs provide us with a useful tool in understanding the structure of stochastic epidemic models. By representing the individuals in a population by vertices and transmission links by arrows between these vertices, we obtain a graph that contains information on many important characteristics, such as the final epidemic size, the basic reproduction number and the probability of a large outbreak. The connection between SIR epidemics and random graphs was observed by Ludwig (1974), and has then been more fully exploited by von Bahr and Martin-Löf (1980), Ball and Barbour (1990) and Barbour and Mollison (1990). In the first two sections of this chapter the random graph interpretation of the standard SIR epidemic model will be given; we also show that the special case of aconstantinfectious period yields a particularly nice class of random graphs.

Håkan Andersson, Tom Britton

8. Models for endemic diseases

Abstract

All the epidemic models encountered so far have assumed aclosedpopulation, i.e. births, deaths, immigration and emigration of individuals are not considered. However, when modelling the spread of a disease with a very long infectious period or a disease in a very large population, dynamic changes in the population itself cannot be ignored. Indeed, in a large community the susceptible population might be augmented fast enough for the epidemic to be maintained for a long time without introducing new infectious individuals into the community; our common childhood diseases are typical examples. Such a disease is calledendemic.

Håkan Andersson, Tom Britton

Estimation

Frontmatter

9. Complete observation of the epidemic process

Abstract

In this chapter we assume that the standard SIR epidemic process E_n,m (λ, I) is observed completely. By complete observation is meant that the infection times τ _i and removal times ρ_i (hence also the length of the infectious period I_i=ρi-τi for formulas to be consistent. From the data it is obviously possible to deduce how many individuals are susceptible, infectious (and removed) for each time t implying that (X,Y)= {(X)(t),Y(t);t≥0} as well as the final size Z=n-X(∞) are observed. Based on the observed data we want to draw inferences on the transmission parameter λ and the distribution of the infectious periodI.We do this by means of Maximum Likelihood (ML) theory. First, we have to make the meaning of ‘likelihood’clear for such epidemic processes. Hence, in Section 9.1 we derive the log-likelihood for a vector of counting processes, and also state some properties of martingales which will turn out useful in the statistical analysis.

Håkan Andersson, Tom Britton

10. Estimation in partially observed epidemics

Abstract

In the previous chapter, statistical analysis was based on what we called complete observation of the epidemic process; both the times of infection and removal (recovery) for all infected individuals were observed. In real life such detailed data is rarely available. In the present chapter we study estimation procedures for less detailed partial data. The likelihood for partial data is usually cumbersome to work with, being a sum or integral over all complete data sets resulting in the observed partial data. Other estimation techniques can turn out to be much simpler. Below we present two general techniques which have been successful in several epidemic applications: martingale methods and the EM-algorithm.

Håkan Andersson, Tom Britton

11. Markov Chain Monte Carlo

Abstract

Markov Chain Monte Carlo (MCMC) is a computer-intensive statistical tool that has received considerable attention over the past few years. Using MCMC theory, it is often quite simple to write efficient algorithms for sampling from extremely complicated target distributions; thus, it is not difficult to understand why these techniques have found important applications in a vast number of different areas. Although the literature on MCMC methods is growing rapidly, the excellent book by Gilks, Richardson and Spiegelhalter (1996) provides a good starting point for the interested reader.

Håkan Andersson, Tom Britton

12. Vaccination

Abstract

Perhaps the most important practical reason for the statistical analysis of epidemics lies in its application to vaccination policy. In the present chapter we focus on this topic. More precisely we consider the following question: who and how many individuals should be vaccinated to prevent future epidemic outbreaks? In Section 12.1 we study this question for the standard SIR epidemic, and in Section 12.2 for endemic diseases, focusing on estimates and neglecting uncertainties. The chapter concludes with a section devoted to the estimation of the vaccine efficacy, which measures the reduction in susceptibility resulting from vaccination. In Sections 12.1 and 12.3 we make use of the multitype epidemic presented in Chapter 6, where the types consist of vaccinated and unvaccinated individuals.

Håkan Andersson, Tom Britton

Backmatter

Titel: Stochastic Epidemic Models and Their Statistical Analysis
verfasst von: Håkan Andersson
Tom Britton
Verlag: Springer New York
Electronic ISBN: 978-1-4612-1158-7
Print ISBN: 978-0-387-95050-1
DOI: https://doi.org/10.1007/978-1-4612-1158-7

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Stochastic Modelling

Frontmatter

1. Introduction

2. The standard SIR epidemic model

3. Coupling methods

4. The threshold limit theorem

5. Density dependent jump Markov processes

6. Multitype epidemics

7. Epidemics and graphs

8. Models for endemic diseases

Estimation

Frontmatter

9. Complete observation of the epidemic process

10. Estimation in partially observed epidemics

11. Markov Chain Monte Carlo

12. Vaccination

Backmatter