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E-grāmata: Stochastic Models In The Life Sciences And Their Methods Of Analysis

(Univ Of California, Irvine, Usa)
  • Formāts: 476 pages
  • Izdošanas datums: 29-Aug-2019
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813274624
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Stochastic Models In The Life Sciences And Their Methods Of AnalysisPalielināt
  • Formāts: 476 pages
  • Izdošanas datums: 29-Aug-2019
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813274624
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' the volume is impressively accessible. The result is a book that is valuable and approachable for biologists at all levels, including those interested in deepening their skills in mathematical modeling and those who seek an overview to aid them in communicating with collaborators in mathematics and statistics. The former group of readers may especially appreciate the first chapter, an introduction to key concepts in probability, and the set of ten assignments provided as an appendix.'CHOICEBiological processes are evolutionary in nature and often evolve in a noisy environment or in the presence of uncertainty. Such evolving phenomena are necessarily modeled mathematically by stochastic differential/difference equations (SDE), which have been recognized as essential for a true understanding of many biological phenomena. Yet, there is a dearth of teaching material in this area for interested students and researchers, notwithstanding the addition of some recent texts on stochastic modelling in the life sciences. The reason may well be the demanding mathematical pre-requisites needed to 'solve' SDE.A principal goal of this volume is to provide a working knowledge of SDE based on the premise that familiarity with the basic elements of a stochastic calculus for random processes is unavoidable. Through some SDE models of familiar biological phenomena, we show how stochastic methods developed for other areas of science and engineering are also useful in the life sciences. In the process, the volume introduces to biologists a collection of analytical and computational methods for research and applications in this emerging area of life science. The additions broaden the available tools for SDE models for biologists that have been limited by and large to stochastic simulations.
Preface xiii
Part 1 Discrete Stage Markov Chains
1(76)
Chapter 1 Discrete Sample Space Probability
3(8)
1.1 Terminology
3(2)
1.2 Intuitive Probability for Finite Sample Space
5(2)
1.3 The Binomial Distribution
7(1)
1.4 Conditional Probability
8(3)
Chapter 2 Discrete Stage Regular Markov Chains
11(28)
2.1 Introduction
11(1)
2.2 A Simple Mouse Experiment
12(2)
2.3 Transition Matrix for a DMC
14(6)
2.4 Regular Markov Chains
20(5)
2.5 DNA Mutation
25(4)
2.6 Linear Difference Equations
29(7)
2.7 Appendix A --- Proof of a Basic Existence Theorem for Regular MC
36(3)
Chapter 3 Discrete Stage Absorbing Markov Chains
39(18)
3.1 Introduction
39(1)
3.2 Gambler's Ruin
40(7)
3.3 Expected Transient Stops to an Absorbing State
47(1)
3.4 Birth and Death Processes
48(4)
3.5 A Simplified Infectious Disease Problem
52(5)
Chapter 4 Discrete Stage Nonlinear Markov Processes
57(20)
4.1 Mendelian Genetics and Difference Equation
57(2)
4.2 Hardy-Weinberg Stability Theorem
59(3)
4.3 Selective Breeding I
62(2)
4.4 Gene Frequencies
64(1)
4.5 Selective Breeding II
64(4)
4.6 Mutation
68(1)
4.7 A Nonlinear Infectious Disease Model
69(1)
4.8 Single Nonlinear Difference Equations
70(7)
Part 2 Continuous Time Markov Chains
77(84)
Chapter 5 Continuous Time Birth and Death Type Processes
79(32)
5.1 The Poisson Process
80(4)
5.2 Pure Birth and Pure Death Processes
84(7)
5.3 Simple Birth and Death Models
91(1)
5.4 Other Birth and Death Processes
92(1)
5.5 Surname Survival
93(6)
5.6 Chapman--Kolmogorov Equation
99(2)
5.7 Appendix --- The Method of Characteristics
101(10)
Chapter 6 Spread of Chlamydia and Stochastic Optimization
111(22)
6.1 Stochastic Models for the Development of C. Trachomatis
111(3)
6.2 A Birth and Death Process Model
114(5)
6.3 Uncertain Host Death
119(6)
6.4 Uniform Density on a Finite Interval
125(5)
6.5 Theory and Experimental Results
130(3)
Chapter 7 Random Walk, Diffusion and Heat Conduction
133(28)
7.1 One-Dimensional Random Walk
133(5)
7.2 Diffusion on a Bounded Domain
138(7)
7.3 Fourier Series
145(8)
7.4 Sturm--Liouville Problems
153(3)
7.5 The Rayleigh Quotient
156(5)
Part 3 Continuous State Random Variables
161(88)
Chapter 8 Continuous Sample Space Probability
163(20)
8.1 Random Variables
163(5)
8.2 Multivariate Random Variables
168(4)
8.3 Mean Square Convergence
172(6)
8.4 Chebyshev Inequality and Sample Size
178(3)
8.5 Characteristic Functions and Central Limit Theorem
181(2)
Chapter 9 Transformations and Stochastic ODE
183(44)
9.1 Introduction
183(2)
9.2 Functions of a Random Variable
185(10)
9.3 A Function of Two Random Variables
195(6)
9.4 Several Functions of Several Random Variables
201(6)
9.5 Applications to Stochastic ODE
207(8)
9.6 Liouville Equation for Random Initial Data
215(7)
9.7 ODE with Random Coefficients
222(5)
Chapter 10 Continuous Stochastic Processes
227(22)
10.1 Random Variables with Continuous Indexing
227(4)
10.2 Moments and Characteristic Functions
231(1)
10.3 Stationary Stochastic Processes
232(3)
10.4 Random Walk and the Wiener Process
235(3)
10.5 Mean Square Continuity
238(1)
10.6 Mean Square Differentiation
239(2)
10.7 Mean Square Integration
241(4)
10.8 Additional Tools in Mean Square Calculus
245(1)
10.9 White Noise
246(3)
Part 4 Stochastic Ordinary Differential Equations
249(70)
Chapter 11 Linear ODE with Random Forcing
251(28)
11.1 Existence and Uniqueness of a Mean Square Solution
251(5)
11.2 Integrate and Fire Models of Motoneuron
256(7)
11.3 The Scalar Linear Problem
263(2)
11.4 White Noise Excitation
265(2)
11.5 Correlated Forcing
267(2)
11.6 Linear Vector IVP with Random Forcing
269(2)
11.7 Time-Invariant Systems
271(4)
11.8 Storage Reduction for Separable PDE
275(1)
11.9 The Nonautonomous Case
276(3)
Chapter 12 General Nonlinear ODE Systems
279(40)
12.1 Nonlinear Stochastic ODE Models
279(4)
12.2 Existence and Uniqueness
283(2)
12.3 Kinetic Equation for a Stochastic Process
285(5)
12.4 Langevin's Equation
290(8)
12.5 Diffusion Processes
298(3)
12.6 The Ito Type SDE
301(8)
12.7 Stochastic Stability
309(10)
Part 5 Stochastic Partial Differential Equations
319(54)
Chapter 13 Linear PDE with Random Forcing
321(32)
13.1 Cable Model Neuron with Random Forcing
321(10)
13.2 The Method of Spatial Correlations
331(5)
13.3 Mean and Correlation
336(1)
13.4 Temporal White Noise Excitation
337(5)
13.5 Linear Stochastic PDE Systems
342(4)
13.6 Cable Model Neuron with O-U Input Current
346(7)
Chapter 14 Dpp Gradient in the Wing Imaginal Disc of Drosophila
353(20)
14.1 An Extracellular Morphogen Gradient Model
353(5)
14.2 Transient States
358(3)
14.3 Noisy Morphogen Synthesis Rate
361(8)
14.4 Effects of Noisy Environment on System Properties
369(4)
Part 6 First Exit Time Statistics
373(56)
Chapter 15 First Exit Time
375(26)
15.1 Threshold for Action Potential in Nerve Axon
375(1)
15.2 The Moments of First Exit Time
376(3)
15.3 Input Variability and Neuronal Firing
379(9)
15.4 Stochastic HIV-1 Models
388(5)
15.5 First Exit Time with a Moving Threshold
393(8)
Chapter 16 The Hodgkin-Huxley Model Neuron
401(14)
16.1 The Hodgkin-Huxley Model
401(3)
16.2 The Fitzhugh-Nagumo Model
404(4)
16.3 A Model for the Fast Variables
408(7)
Chapter 17 Numerical Simulations
415(14)
17.1 Stochastic Simulations
415(6)
17.2 Simpler Approaches for Less Information
421(1)
17.3 Genetic Instability and Carcinogenesis
422(7)
Appendix
429(16)
A.1 Assignment I
429(1)
A.2 Assignment II
430(2)
A.3 Assignment III
432(2)
A.4 Assignment IV
434(1)
A.5 Assignment V
435(2)
A.6 Assignment VI
437(1)
A.7 Assignment VII
438(1)
A.8 Assignment VIII
439(1)
A.9 Assignment IX
440(2)
A.10 Assignment X
442(3)
Bibliography 445(4)
Index 449
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