Stochastic Epidemic Models and Their Statistical Analysis

by Andersson, Hakan; Britton, Tom

• ISBN13: 9780387950501
• ISBN10: 0387950508
• Format: Paperback
• Copyright: 2000-07-01
• Publisher: Springer Verlag

Summary

This book describes stochastic epidemic models and methods for statistically analyzing them. It is aimed at statisticians, biostatisticians, and biomathematicians.

Table of Contents

Preface	v
Part I: STOCHASTIC MODELLING	1	(84)
Introduction	3	(8)
Stochastic versus deterministic models	3	(1)
A simple epidemic model: The Reed-Frost model	4	(2)
Stochastic epidemics in large communities	6	(1)
History of epidemic modelling	7	(4)
Exercises	9	(2)
The standard SIR epidemic model	11	(8)
Definition of the model	11	(1)
The Sellke construction	12	(2)
The Markovian case	14	(1)
Exact results	15	(4)
Exercises	18	(1)
Coupling methods	19	(8)
First examples	19	(3)
Definition of coupling	22	(1)
Applications to epidemics	22	(5)
Exercises	26	(1)
The threshold limit theorem	27	(12)
The imbedded process	27	(1)
Preliminary convergence results	28	(2)
The case Mn/n → μ > 0 as n → ∞	30	(2)
The case Mn = m for all n	32	(2)
Duration of the Markovian SIR epidemic	34	(5)
Exercises	36	(3)
Density dependent jump Markov processes	39	(12)
An example: A simple birth and death process	39	(1)
The general model	40	(1)
The Law of Large Numbers	41	(2)
The Central Limit Theorem	43	(3)
Applications to epidemic models	46	(5)
Exercises	48	(3)
Multitype epidemics	51	(12)
The standard SIR multitype epidemic model	51	(2)
Large population limits	53	(2)
Household model	55	(1)
Comparing equal and varying susceptibility	56	(7)
Exercises	61	(2)
Epidemics and graphs	63	(10)
Random graph interpretation	64	(1)
Constant infectious period	65	(1)
Epidemics and social networks	66	(4)
The two-dimensional lattice	70	(3)
Exercises	72	(1)
Models for endemic diseases	73	(12)
The SIR model with demography	73	(4)
The SIS model	77	(8)
Exercises	83	(2)
PART II: ESTIMATION	85	(42)
Complete observation of the epidemic process	87	(12)
Martingales and log-likelihoods of counting processes	87	(4)
ML-estimation for the standard SIR epidemic	91	(8)
Exercises	94	(5)
Estimation in partially observed epidemics	99	(8)
Estimation based on martingale methods	99	(4)
Estimation based on the EM-algorithm	103	(4)
Exercises	105	(2)
Markov Chain Monte Carlo methods	107	(10)
Description of the techniques	107	(2)
Important examples	109	(2)
Practical implementation issues	111	(2)
Bayesian inference for epidemics	113	(4)
Exercises	114	(3)
Vaccination	117	(10)
Estimating vaccination policies based on one epidemic	117	(3)
Estimating vaccination policies for endemic diseases	120	(3)
Estimation of vaccine efficacy	123	(4)
Exercises	124	(3)
References	127	(8)
Subject index	135