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Table of contents
1 Multitype branching processes
1.1 Continuous-time multitype Bienaym´e-Galton-Watson process
1.1.1 Preliminaries: generating functions and infinitesimal generator
1.1.2 Moments and the Perron-Frobenius Theorem
1.1.3 Extinction probability
1.1.4 Some basic examples
1.2 Multitype Feller diffusion process
1.2.1 Definitions and preliminaries
1.2.2 Associated martingale problem and SDE
1.2.3 Extinction of the process
2 Branching processes conditioned on non-extinction
2.1 Historical introduction: various ways of conditioning
2.2 Multitype branching processes forced to extinction
2.2.1 BGWc process forced to extinction
2.2.2 Feller diffusion process forced to extinction
2.3 Yaglom-type limits
2.3.1 Yaglom-type limits for the multitype BGWc process
2.3.2 Yaglom-type limits for the multitype Feller diffusion process
2.4 Q-process
2.4.1 Q-process associated with the multitype BGWc process
2.4.2 Q-process associated with the multitype Feller diffusion process
3 Commutativity results
3.1 Commutativity of the long-time limits
3.1.1 Long-time limits of the conditioned BGWc process
3.1.2 Long-time limits of the Feller diffusion process
3.2 Commutativity between rescaling and conditioning
3.2.1 Rescaled BGWc process
3.2.2 Scaling limit of the BGWc process conditioned on extinction
3.2.3 Scaling limit of the Yaglom-type distributions
3.2.4 Scaling limit of the Q-process
3.2.5 Scaling limit of the time asymptotic of the Q-process
3.3 Overview of the commutativity results
4 Risk analysis for vanishing branching populations
4.1 Stochastic branching model
4.2 Estimation of the unknown parameter
4.2.1 A CLSE with asymptotic properties, as |X0| → ∞
4.2.2 A CLSE with asymptotic properties, as n → ∞
4.2.3 An explicit estimator with asymptotic properties, as n → ∞
4.2.4 Comparison of the estimators and illustration of the asymptotic
4.3 Study of the very late extinction case
4.3.1 Q-process associated with the model
4.3.2 CLSE for the Q-process
5 BSE epidemic in Great-Britain
5.1 The epidemic model
5.1.1 Description
5.1.2 Theoretical results
5.1.3 The geometrical case
5.1.4 Origin of the model
5.2 Prediction of the disease spread
5.2.1 Estimation of the infection parameter
5.2.2 Prediction of the incidences of cases and infected cattle
5.2.3 Prediction of the extinction time
5.2.4 Prediction of the total size of the epidemic
5.3 Prediction of the disease spread in case of a very late extinction
5.3.1 Estimation of the infection parameter
5.3.2 Prediction of the disease spread
