Martínez / Molina / Mota Workshop on Branching Processes and Their Applications

1. Auflage 2010
ISBN: 978-3-642-11156-3
Verlag: Springer
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 296 Seiten

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.

Autoren/Hrsg.

Martínez, Rodrigo

Molina, Manuel

Mota, Manuel

R, Alfonso

González, Miguel

Ramos, Alfonso

Inhaltsverzeichnis

1;Foreword;6
2;Preface;8
3;Contents;12
4;Contributors;18
5;Part I Population Growth Models in Random and Varying Environments;21
5.1;1 A refinement of limit theorems for the critical branching processes in random environment;22
5.2;Vladimir Vatutin;22
5.2.1;1.1 Introduction and main results;22
5.2.2;1.2 Branching in conditioned environment;27
5.2.3;1.3 Proof of Theorems 1.1 and 1.2;36
5.2.4;References;37
5.3;2 Branching processes in stationary random environment: The extinction problem revisited;39
5.4;Gerold Alsmeyer;39
5.4.1;2.1 Introduction;39
5.4.2;2.2 Classical results revisited;42
5.4.3;2.3 Main result and a counterexample;43
5.4.4;2.4 Some useful facts from Palm-duality theory;46
5.4.5;2.5 Proofs;47
5.4.6;References;54
5.5;3 Environmental versus demographic stochasticity in population growth;55
5.6;Carlos A. Braumann;55
5.6.1;3.1 Introduction;55
5.6.2;3.2 Density-independent models and their local behavior;57
5.6.3;3.3 Density-independent models and extinction;62
5.6.4;3.4 Density-dependent models for environmental stochasticity;63
5.6.5;3.5 Conclusions;67
5.6.6;References;70
5.7;4 Stationary distributions of the alternating branching processes;71
5.8;Penka Mayster;71
5.8.1;4.1 Introduction;71
5.8.2;4.2 Alternating branching process;73
5.8.3;4.3 Alternating branching process with explicit immigration;74
5.8.4;4.4 Reproduction by n cycles;76
5.8.5;4.5 Criticality;77
5.8.6;4.6 Stationary distribution in random environment;80
5.8.7;4.7 Unconditional probability generating functions;81
5.8.8;4.8 Feed-back control;82
5.8.9;References;84
6;Part II Special Branching Processes;86
6.1;5 Approximations in population-dependent branching processes;87
6.2;Fima C. Klebaner;87
6.2.1;5.1 Introduction and a motivating example;87
6.2.2;5.2 A Representation of the process and its re-scaled version;89
6.2.2.1;5.2.1 Re-scaled process: Dynamics plus small noise;90
6.2.2.2;5.2.2 Dynamics without noise in binary splitting;90
6.2.3;5.3 Time to extinction;91
6.2.4;5.4 The size of the population after a long time providedit has survived;91
6.2.5;5.5 Case of small initial population;92
6.2.5.1;5.5.1 Probability of becoming large and time for it to happen;93
6.2.6;5.6 Behaviour before extinction;93
6.2.7;References;94
6.3;6 Extension of the problem of extinction on Galton--Watson family trees;95
6.4;George P. Yanev;95
6.4.1;6.1 Introduction;95
6.4.2;6.2 Critical phenomenon;96
6.4.3;6.3 Distribution of the number of complete and disjoint subtrees, rooted at the ancestor;99
6.4.4;6.4 Ratio of expected values of Zns provided infinite subtrees exist;100
6.4.5;6.6 Poisson offspring distribution;105
6.4.6;6.7 One-or-many offspring distribution;107
6.4.7;6.8 Concluding remarks;109
6.4.8;References;109
6.5;7 Limit theorems for critical randomly indexed branching processes;111
6.6;Kosto V. Mitov, Georgi K. Mitov and Nikolay M. Yanev;111
6.6.1;7.1 Introduction;111
6.6.2;7.2 A conditional limit theorem for random time change;113
6.6.3;7.3 Renewal processes;116
6.6.4;7.4 BGW branching processes starting with random number of particles;119
6.6.5;7.5 Limit theorems for the process Y(t);121
6.6.6;7.6 Concluding remarks;123
6.6.7;References;124
6.7;8 Renewal measure density for distributions with regularly varying tails of order (0,1/2];125
6.8;Valentin Topchii;125
6.8.1;8.1 Introduction;125
6.8.2;8.2 Effects of attraction to a stable law;127
6.8.3;8.3 Asymptotics of renewal function density;130
6.8.4;References;134
7;Part III Limit Theorems and Statistics;135
7.1;9 Approximation of a sum of martingale differences generated by a bootstrap branching process;136
7.2;Ibrahim Rahimov;136
7.2.1;9.1 Introduction;136
7.2.2;9.2 Main theorems;138
7.2.3;9.3 Array of processes;142
7.2.4;References;148
7.3;10 Critical branching processes with immigration;149
7.4;Márton Ispány and Gyula Pap;149
7.4.1;10.1 Introduction;149
7.4.2;10.2 Branching and autoregressive processes;150
7.4.3;10.3 Functional limit theorems;152
7.4.4;10.4 Nearly critical branching processes with immigration;154
7.4.5;10.5 Conditional least squares estimators;156
7.4.6;References;159
7.5;11 Weighted conditional least squares estimation in controlled multitype branching processes;161
7.6;Miguel González and Inés M. del Puerto;161
7.6.1;11.1 Introduction;161
7.6.2;11.2 Probability model;162
7.6.3;11.3 Weighted conditional least squares estimator of the offspring mean matrix;164
7.6.4;References;169
8;Part IV Applications in Cell Kinetics and Genetics;170
8.1;12 Branching processes in cell proliferation kinetics;171
8.2;Nikolay M. Yanev;171
8.2.1;12.1 Introduction;171
8.2.2;12.2 Distributions of discrete marks over a proliferatingcell populations;173
8.2.3;12.3 Distributions of continuous labels in branching populationsof cells;174
8.2.4;12.4 Age and residual lifetime distributions for branching processes;176
8.2.5;12.5 Branching processes with immigration as models of leukemia cell kinetics;180
8.2.6;12.6 Age-dependent branching populations with randomly chosen paths of evolution;183
8.2.7;12.7 Multitype branching populations with a large numberof ancestors;185
8.2.8;12.8 Concluding remarks;189
8.2.9;References;189
8.3;13 Griffiths--Pakes branching process as a model for evolution of Alu elements;191
8.4;Marek Kimmel and Matthias Mathaes;191
8.4.1;13.1 Introduction;191
8.4.2;13.2 Alu repeat sequences;192
8.4.2.1;13.2.1 Background on Alus;192
8.4.2.2;13.2.2 Alu sequence data used in this study;192
8.4.3;13.3 Discrete branching process of Griffiths and Pakes with infinite allele mutations;193
8.4.3.1;13.3.1 Linear fractional offspring distribution;196
8.4.4;13.4 Fitting results;198
8.4.5;13.5 Discussion;199
8.4.6;References;201
8.5;14 Parametric inference for Y-linked gene branching models: Expectation-maximization method;202
8.6;Miguel González, Cristina Gutiérrez and Rodrigo Martínez;202
8.6.1;14.1 Introduction;202
8.6.2;14.2 The probability model;203
8.6.3;14.3 The estimation problem: The expectation-maximization method;206
8.6.3.1;14.3.1 Determining the distribution ofFRrN|(FMN,,R,r);208
8.6.3.2;14.3.2 The expectation-maximization method;210
8.6.4;14.4 Simulation study;211
8.6.5;References;215
9;Part V Applications in Epidemiology;216
9.1;15 Applications of branching processes to the final size of SIR epidemics;217
9.2;Frank Ball and Peter Neal;217
9.2.1;15.1 Introduction;218
9.2.2;15.2 Early stages of epidemic;221
9.2.3;15.3 Final outcome of Reed--Frost epidemic;223
9.2.3.1;15.3.1 Preliminaries;223
9.2.3.1.1;15.3.1.1 Susceptibility sets;223
9.2.3.1.2;15.3.1.2 Mean and variance of final size;223
9.2.3.2;15.3.2 Many initial infectives;226
9.2.3.2.1;15.3.2.1 Limiting mean final size;226
9.2.3.2.2;15.3.2.2 Limiting variance final size;227
9.2.3.3;15.3.3 Few initial infectives;229
9.2.3.4;15.3.4 Central limit theorem;231
9.2.4;References;232
9.3;16 A branching process approach for the propagation of the Bovine Spongiform Encephalopathy in Great-Britain;234
9.4;Christine Jacob, Laurence Maillard-Teyssier, Jean-Baptiste Denis and Caroline Bidot;234
9.4.1;16.1 Introduction;234
9.4.2;16.2 Initial branching model;235
9.4.3;16.3 Limit process as N0;238
9.4.4;16.4 Behavior of the BGW limit process;242
9.4.4.1;16.4.1 Extinction probability;243
9.4.4.2;16.4.2 Extinction time distribution;244
9.4.4.3;16.4.3 Size of the epidemic;244
9.4.5;16.5 Estimation;244
9.4.5.1;16.5.1 Observations;245
9.4.5.2;16.5.2 Model and parameters;245
9.4.5.3;16.5.3 Prior distributions;246
9.4.5.4;16.5.4 Algorithm and software;246
9.4.5.5;16.5.5 Main results;247
9.4.5.5.1;16.5.5.1 Parameters estimation;247
9.4.5.5.2;16.5.5.2 Prediction of the epidemic;247
9.4.6;16.6 Conclusion;248
9.4.7;References;249
9.5;17 Time to extinction of infectious diseases through age-dependent branching models;250
9.6;Miguel González, Rodrigo Martínez and Maroussia Slavtchova-Bojkova;250
9.6.1;17.1 Introduction;250
9.6.2;17.2 Model of epidemic spread;252
9.6.3;17.3 The epidemic's time to extinction;253
9.6.4;17.4 Determining vaccination policies;255
9.6.4.1;17.4.1 Vaccination based on the mean value of the time to extinction;256
9.6.4.2;17.4.2 Analyzing the control measures for avian influenza in Vietnam;257
9.6.5;17.5 Concluding remarks;260
9.6.6;17.6 Proofs;260
9.6.7;References;265
9.7;18 Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus;266
9.8;Dominik Heinzmann;266
9.8.1;18.1 Introduction;266
9.8.2;18.2 Prevalence-based interaction model;267
9.8.3;18.3 Approximating branching processes;268
9.8.4;18.4 Coupling;269
9.8.5;18.5 Time to extinction;271
9.8.6;18.6 Numerical illustration;272
9.8.7;References;274
10;Part VI Two-Sex Branching Models;276
10.1;19 Bisexual branching processes with immigration depending on the number of females and males;277
10.2;Shixia Ma and Yongsheng Xing;277
10.2.1;19.1 Introduction;277
10.2.2;19.2 The bisexual process with immigration;278
10.2.3;19.3 The asymptotic growth rate;279
10.2.4;19.4 Limit behavior for the supercritical case;281
10.2.5;References;284
10.3;20 Two-sex branching process literature;286
10.4;Manuel Molina;286
10.4.1;20.1 Introduction;286
10.4.2;20.2 The Daley's two-sex branching process;287
10.4.3;20.3 Discrete time two-sex branching processes;291
10.4.3.1;20.3.1 Processes with immigration;291
10.4.3.2;20.3.2 Processes in varying or in random environments;292
10.4.3.3;20.3.3 Processes depending on the number of couples in the population;292
10.4.3.4;20.3.4 Processes with control on the number of progenitor couples;294
10.4.3.5;20.3.5 Others classes of two-sex processes;294
10.4.4;20.4 Continuous time two-sex branching processes;294
10.4.5;20.5 Applications;295
10.4.5.1;20.5.1 Application in the field of the Epidemiology;296
10.4.5.2;20.5.2 Applications in the field of the Genetics;296
10.4.5.3;20.5.3 Applications in population dynamics;297
10.4.6;20.6 Some suggestions for research;297
10.4.7;References;298
11;Index;302