Workshop on Branching Processes and Their Applications (Lecture Notes in Statistics, 197)

✍ Scribed by Miguel González (editor), Inés M. Puerto (editor), Rodrigo Martínez (editor), Manuel Molina (editor), Manuel Mota (editor), Alfonso Ramos (editor)

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 303
• Category: Library

✦ Synopsis

One of the charms of mathematics is the contrast between its generality and its applicability to concrete, even everyday, problems. Branching processes are typical in this. Their niche of mathematics is the abstract pattern of reproduction, sets of individuals changing size and composition through their members reproducing; in other words, what Plato might have called the pure idea behind demography, population biology, cell kinetics, molecular replication, or nuclear ?ssion, had he known these scienti?c ?elds. Even in the performance of algorithms for sorting and classi?cation there is an inkling of the same pattern. In special cases, general properties of the abstract ideal then interact with the physical or biological or whatever properties at hand. But the population, or bran- ing, pattern is strong; it tends to dominate, and here lies the reason for the extreme usefulness of branching processes in diverse applications. Branching is a clean and beautiful mathematical pattern, with an intellectually challenging intrinsic structure, and it pervades the phenomena it underlies.

✦ Table of Contents

Foreword
Preface
Contents
Contributors
Part I Population Growth Models in Random and Varying Environments
1 A refinement of limit theorems for the critical branching processes in random environment
Vladimir Vatutin
1.1 Introduction and main results
1.2 Branching in conditioned environment
1.3 Proof of Theorems 1.1 and 1.2
References
2 Branching processes in stationary random environment: The extinction problem revisited
Gerold Alsmeyer
2.1 Introduction
2.2 Classical results revisited
2.3 Main result and a counterexample
2.4 Some useful facts from Palm-duality theory
2.5 Proofs
References
3 Environmental versus demographic stochasticity in population growth
Carlos A. Braumann
3.1 Introduction
3.2 Density-independent models and their local behavior
3.3 Density-independent models and extinction
3.4 Density-dependent models for environmental stochasticity
3.5 Conclusions
References
4 Stationary distributions of the alternating branching processes
Penka Mayster
4.1 Introduction
4.2 Alternating branching process
4.3 Alternating branching process with explicit immigration
4.4 Reproduction by n cycles
4.5 Criticality
4.6 Stationary distribution in random environment
4.7 Unconditional probability generating functions
4.8 Feed-back control
References
Part II Special Branching Processes
5 Approximations in population-dependent branching processes
Fima C. Klebaner
5.1 Introduction and a motivating example
5.2 A Representation of the process and its re-scaled version
5.2.1 Re-scaled process: Dynamics plus small noise
5.2.2 Dynamics without noise in binary splitting
5.3 Time to extinction
5.4 The size of the population after a long time providedit has survived
5.5 Case of small initial population
5.5.1 Probability of becoming large and time for it to happen
5.6 Behaviour before extinction
References
6 Extension of the problem of extinction on Galton--Watson family trees
George P. Yanev
6.1 Introduction
6.2 Critical phenomenon
6.3 Distribution of the number of complete and disjoint subtrees, rooted at the ancestor
6.4 Ratio of expected values of Zns provided infinite subtrees exist
6.5 Geometric offspring distribution
6.6 Poisson offspring distribution
6.7 One-or-many offspring distribution
6.8 Concluding remarks
References
7 Limit theorems for critical randomly indexed branching processes
Kosto V. Mitov, Georgi K. Mitov and Nikolay M. Yanev
7.1 Introduction
7.2 A conditional limit theorem for random time change
7.3 Renewal processes
7.4 BGW branching processes starting with random number of particles
7.5 Limit theorems for the process Y(t)
7.6 Concluding remarks
References
8 Renewal measure density for distributions with regularly varying tails of order (0,1/2]
Valentin Topchii
8.1 Introduction
8.2 Effects of attraction to a stable law
8.3 Asymptotics of renewal function density
References
Part III Limit Theorems and Statistics
9 Approximation of a sum of martingale differences generated by a bootstrap branching process
Ibrahim Rahimov
9.1 Introduction
9.2 Main theorems
9.3 Array of processes
References
10 Critical branching processes with immigration
Márton Ispány and Gyula Pap
10.1 Introduction
10.2 Branching and autoregressive processes
10.3 Functional limit theorems
10.4 Nearly critical branching processes with immigration
10.5 Conditional least squares estimators
References
11 Weighted conditional least squares estimation in controlled multitype branching processes
Miguel González and Inés M. del Puerto
11.1 Introduction
11.2 Probability model
11.3 Weighted conditional least squares estimator of the offspring mean matrix
References
Part IV Applications in Cell Kinetics and Genetics
12 Branching processes in cell proliferation kinetics
Nikolay M. Yanev
12.1 Introduction
12.2 Distributions of discrete marks over a proliferatingcell populations
12.3 Distributions of continuous labels in branching populationsof cells
12.4 Age and residual lifetime distributions for branching processes
12.5 Branching processes with immigration as models of leukemia cell kinetics
12.6 Age-dependent branching populations with randomly chosen paths of evolution
12.7 Multitype branching populations with a large numberof ancestors
12.8 Concluding remarks
References
13 Griffiths--Pakes branching process as a model for evolution of Alu elements
Marek Kimmel and Matthias Mathaes
13.1 Introduction
13.2 Alu repeat sequences
13.2.1 Background on Alus
13.2.2 Alu sequence data used in this study
13.3 Discrete branching process of Griffiths and Pakes with infinite allele mutations
13.3.1 Linear fractional offspring distribution
13.4 Fitting results
13.5 Discussion
References
14 Parametric inference for Y-linked gene branching models: Expectation-maximization method
Miguel González, Cristina Gutiérrez and Rodrigo Martínez
14.1 Introduction
14.2 The probability model
14.3 The estimation problem: The expectation-maximization method
14.3.1 Determining the distribution ofFRrN|(FMN,,R,r)
14.3.2 The expectation-maximization method
14.4 Simulation study
References
Part V Applications in Epidemiology
15 Applications of branching processes to the final size of SIR epidemics
Frank Ball and Peter Neal
15.1 Introduction
15.2 Early stages of epidemic
15.3 Final outcome of Reed--Frost epidemic
15.3.1 Preliminaries
15.3.1.1 Susceptibility sets
15.3.1.2 Mean and variance of final size
15.3.2 Many initial infectives
15.3.2.1 Limiting mean final size
15.3.2.2 Limiting variance final size
15.3.3 Few initial infectives
15.3.4 Central limit theorem
References
16 A branching process approach for the propagation of the Bovine Spongiform Encephalopathy in Great-Britain
Christine Jacob, Laurence Maillard-Teyssier, Jean-Baptiste Denis and Caroline Bidot
16.1 Introduction
16.2 Initial branching model
16.3 Limit process as N0
16.4 Behavior of the BGW limit process
16.4.1 Extinction probability
16.4.2 Extinction time distribution
16.4.3 Size of the epidemic
16.5 Estimation
16.5.1 Observations
16.5.2 Model and parameters
16.5.3 Prior distributions
16.5.4 Algorithm and software
16.5.5 Main results
16.5.5.1 Parameters estimation
16.5.5.2 Prediction of the epidemic
16.6 Conclusion
References
17 Time to extinction of infectious diseases through age-dependent branching models
Miguel González, Rodrigo Martínez and Maroussia Slavtchova-Bojkova
17.1 Introduction
17.2 Model of epidemic spread
17.3 The epidemic's time to extinction
17.4 Determining vaccination policies
17.4.1 Vaccination based on the mean value of the time to extinction
17.4.2 Analyzing the control measures for avian influenza in Vietnam
17.5 Concluding remarks
17.6 Proofs
References
18 Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus
Dominik Heinzmann
18.1 Introduction
18.2 Prevalence-based interaction model
18.3 Approximating branching processes
18.4 Coupling
18.5 Time to extinction
18.6 Numerical illustration
References
Part VI Two-Sex Branching Models
19 Bisexual branching processes with immigration depending on the number of females and males
Shixia Ma and Yongsheng Xing
19.1 Introduction
19.2 The bisexual process with immigration
19.3 The asymptotic growth rate
19.4 Limit behavior for the supercritical case
References
20 Two-sex branching process literature
Manuel Molina
20.1 Introduction
20.2 The Daley's two-sex branching process
20.3 Discrete time two-sex branching processes
20.3.1 Processes with immigration
20.3.2 Processes in varying or in random environments
20.3.3 Processes depending on the number of couples in the population
20.3.4 Processes with control on the number of progenitor couples
20.3.5 Others classes of two-sex processes
20.4 Continuous time two-sex branching processes
20.5 Applications
20.5.1 Application in the field of the Epidemiology
20.5.2 Applications in the field of the Genetics
20.5.3 Applications in population dynamics
20.6 Some suggestions for research
References
Index