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E-raamat: Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics

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Basic Approach to Age-Structured Population Dynamics: Models, Methods and NumericsSuurem pilt
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This book provides an introduction to age-structured population modeling which emphasizes the connection between mathematical theory and underlying biological assumptions.

Through the rigorous development of the linear theory and the nonlinear theory alongside numerics, the authors explore classical equations that describe the dynamics of certain ecological systems. Modeling aspects are discussed to show how relevant problems in the fields of demography, ecology and epidemiology can be formulated and treated within the theory. In particular, the book presents extensions of age-structured modeling to the spread of diseases and epidemics while also addressing the issue of regularity of solutions, the asymptotic behavior of solutions, and numerical approximation. With sections on transmission models, non-autonomous models and global dynamics, this book fills a gap in the literature on theoretical population dynamics.

The Basic Approach to Age-Structured Population Dynamics will appeal to graduate students and researchers in mathematical biology, epidemiology and demography who are interested in the systematic presentation of relevant models and mathematical methods.
1 Why Age Structure? An Introduction
1(48)
1.1 Human Demography: A Classic
2(23)
1.1.1 Demographic Age Structure
7(2)
1.1.2 The Death Process
9(5)
1.1.3 Fertility
14(3)
1.1.4 Migration
17(3)
1.1.5 Modeling Trends and Habitat Changes
20(2)
1.1.6 The Basic Elements of a Description
22(3)
1.2 Ecology
25(7)
1.2.1 Life Tables
26(2)
1.2.2 Juvenile-Adult Interaction
28(3)
1.2.3 Modeling Nonlinear Vital Rates
31(1)
1.3 Epidemics
32(17)
1.3.1 Essential Unstructured Modeling
35(2)
1.3.2 The Single Epidemic Outbreak
37(2)
1.3.3 Disease Endemicity
39(2)
1.3.4 The Age of the Disease: The Internal Clock
41(2)
1.3.5 Chronological Age: Demography and Epidemics
43(3)
References
46(3)
2 The Basic Linear Theory
49(40)
2.1 The Lotka--McKendrick Equation
50(2)
2.2 The Renewal Equation
52(3)
2.3 Existence of a Solution
55(4)
2.4 Regularity
59(3)
2.5 The Asymptotic Behavior
62(5)
2.6 The Age Profile
67(6)
2.7 The Open Population
73(4)
2.8 Infinite Maximum Age
77(2)
2.9 The Leslie Matrix
79(3)
2.10 Eigenvalues, Eigenvectors and the Characteristic Equation
82(1)
2.11 Comments and References
83(6)
References
85(4)
3 Numerical Methods for the Linear Model
89(34)
3.1 The Methodology of Characteristics
90(2)
3.2 Euler--Riemann Methods
92(3)
3.3 Convergence of ER Methods
95(4)
3.4 Higher-Order Methods
99(7)
3.5 Unbounded Mortality Rates
106(4)
3.6 Approximation of R and α*
110(4)
3.7 Numerical Simulations
114(7)
3.8 Comments and References
121(2)
References
122(1)
4 The Time-Dependent Case
123(18)
4.1 Extension of the Lotka-McKendrick Model
124(2)
4.2 The Case of Converging Rates
126(5)
4.3 Periodic Rates
131(2)
4.4 Strong and Weak Ergodicity
133(3)
4.5 Real-Life Data and Numerical Simulations
136(3)
4.6 Comments and References
139(2)
References
140(1)
5 Nonlinear Models
141(32)
5.1 A General Nonlinear Model
142(4)
5.2 The Solution to the Problem
146(4)
5.3 The Equilibria of the Model
150(2)
5.4 Modeling Logistic Growth
152(3)
5.5 Juvenile-Adult Dynamics
155(5)
5.6 Multiple Equilibria in Juvenile-Adult Dynamics
160(2)
5.7 The Allee Effect
162(3)
5.8 A Model for Cannibalism
165(4)
5.9 Comments and References
169(4)
References
171(2)
6 Stability of Equilibria
173(28)
6.1 The Basic Paradigm of Stability
174(5)
6.2 Some Results on the Characteristic Equation
179(6)
6.3 Back to the Logistic Model
185(5)
6.4 Adult-Juvenile Competition
190(7)
6.5 Backward Bifurcation
197(1)
6.6 Comments and References
198(3)
References
199(2)
7 Numerical Methods for the Nonlinear Model
201(18)
7.1 Finite Differences on Characteristics
203(4)
7.2 Analytic Representation of the Solution
207(4)
7.3 Methods for Hyperbolic Equations
211(3)
7.4 Methods Based on Integral Equations
214(1)
7.5 Comments and References
215(4)
References
216(3)
8 Global Behavior
219(22)
8.1 A General Approach to a Class of Models
220(4)
8.2 A Class of Logistic Models
224(5)
8.3 Separable Models
229(5)
8.4 The Case a† = +∞
234(5)
8.5 Comments and References
239(2)
References
239(2)
9 Class-Age Structure for Epidemics
241(36)
9.1 The Classical Kermack--McKendrick Model
242(2)
9.2 Reduction of the System
244(2)
9.3 Epidemic Outbreak and Extinction
246(4)
9.4 The SIS Model: Endemic States and Stability
250(4)
9.5 Modeling Variable Populations
254(2)
9.6 The Basic SIR Model
256(5)
9.7 Endemic States for the SIR Model
261(4)
9.8 A Model for the HIV/AIDS Epidemic
265(5)
9.9 A Model for Tuberculosis
270(4)
9.10 Comments and References
274(3)
References
275(2)
10 Epidemics and Demography
277(44)
10.1 SIR and SIS Age-Structured Models
278(5)
10.2 Analysis of the SIS Model
283(11)
10.3 Asymptotic Stability for the SIS Model
294(6)
10.4 Results for the SIR Model
300(9)
10.5 Stability of Steady States for the SIR Model
309(7)
10.6 Comments and References
316(5)
References
317(4)
A The Laplace Transform
321(14)
A.1 Definitions
321(2)
A.2 Basic Properties
323(1)
A.3 The Inversion Formula
324(2)
A.4 Asymptotic Behavior of the Transform
326(4)
A.5 The Behavior of the Original Function
330(2)
A.6 The Discrete Context and the Z-Transform
332(3)
References
333(2)
B Integral Equations Theory
335(10)
B.1 The Linear Theory
335(3)
B.2 The Characteristic Equation
338(4)
B.3 Some Basic Tools
342(3)
References
343(2)
List of Symbols 345(2)
Index 347
Mimmo Iannelli obtained his Laurea degree at the Istituto per le Applicazioni del Calcolo, Italy, in 1968. He was Full Professor at the University of Rome from 1976 to 1978. He then moved to the University of Trento, where he currently holds the title of Senior Professor. His research focuses on evolution equations and the analysis of mathematical methods in population theory. He has authored various publications on evolution equations and mathematical models of ecology and epidemiology, including three books.

Fabio Milner obtained the degree of Licenciado en Ciencias Matemáticas in Argentina in 1976 before moving to the USA in 1978. He obtained his PhD in Numerical Analysis in 1983 from the University of Chicago and joined the faculty of Purdue University immediately thereafter. He left Purdue in 2008 as Full Professor when he joined Arizona State University as Director of Mathematics for STEM Education. He shifted the focus of his research to mathematical biology after meeting Mimmo Iannelli in 1997. Milner is the author of numerous scientific publications, including 14 articles and one book that were co-authored by Mimmo Iannelli.
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