Stochastic Epidemic Models and Their Statistical Analysis

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.

Contenu

I: Stochastic Modelling.- 1. Introduction.- 1.1. Stochastic versus deterministic models.- 1.2. A simple epidemic model: The Reed-Frost model.- 1.3. Stochastic epidemics in large communities.- 1.4. History of epidemic modelling.- Exercises.- 2. The standard SIR epidemic model.- 2.1. Definition of the model.- 2.2. The Sellke construction.- 2.3. The Markovian case.- 2.4. Exact results.- Exercises.- 3. Coupling methods.- 3.1. First examples.- 3.2. Definition of coupling.- 3.3. Applications to epidemics.- Exercises.- 4. The threshold limit theorem.- 4.1. The imbedded process.- 4.2. Preliminary convergence results.- 4.3. The casemn/n??> 0 asn? ?.- 4.4. The casemn=mfor alln.- 4.5. Duration of the Markovian SIR epidemic.- Exercises.- 5. Density dependent jump Markov processes.- 5.1. An example: A simple birth and death process.- 5.2. The general model.- 5.3. The Law of Large Numbers.- 5.4. The Central Limit Theorem.- 5.5. Applications to epidemic models.- Exercises.- 6. Multitype epidemics.- 6.1. The standard SIR multitype epidemic model.- 6.2. Large population limits.- 6.3. Household model.- 6.4. Comparing equal and varying susceptibility.- Exercises.- 7. Epidemics and graphs.- 7.1. Random graph interpretation.- 7.2. Constant infectious period.- 7.3. Epidemics and social networks.- 7.4. The two-dimensional lattice.- Exercises.- 8. Models for endemic diseases.- 8.1. The SIR model with demography.- 8.2. The SIS model.- Exercises.- II: Estimation.- 9. Complete observation of the epidemic process.- 9.1. Martingales and log-likelihoods of counting processes.- 9.2. ML-estimation for the standard SIR epidemic.- Exercises.- 10. Estimation in partially observed epidemics.- 10.1. Estimation based on martingale methods.- 10.2. Estimation based on the EM-algorithm.- Exercises.- 11. Markov Chain Monte Carlo methods.- 11.1. Description of the techniques.- 11.2. Important examples.- 11.3. Practical implementation issues.- 11.4. Bayesian inference for epidemics.- Exercises.- 12. Vaccination.- 12.1. Estimating vaccination policies based on one epidemic.- 12.2. Estimating vaccination policies for endemic diseases.- 12.3. Estimation of vaccine efficacy.- Exercises.- References.