Workshop on Branching Processes and Their Applications (eBook)

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