Table Of ContentMDd.llng and Simulation In Se/.ne., Englnllrlng and T.ehnology
S"IIB Editor
Nicola Bellomo
Politecnico Torino
Italy
Advisory Edltor/al Board
K.J. Bathe P.Degond
Massachusetts Institute of Technology Universit6 P. Sabatier Toulouse 3
USA France
W. Kliemann P. Le Tallec
Iowa State University INRIA
USA France
S. Nikitin K.R. Rajagopal
Arizona State University Texas A&M University
USA USA
V. Protopopescu Y. Sone
CSMD Kyoto University
Oak Ridge National Laboratory Japan
USA
E.S. Subuhi
Istanbul Technical University
Turkey
Gilbert G. Walter
Martha Contreras
Compartmental Modeling
with Networks
Springer Science+Business Media, LLC
Gilbert G. Walter Martha Contreras
Department of Mathematical Sciences Department of Biometry
University of Wisconsin-Milwaukee Cornell University
P.O. Box 413 434 Warren Hall
Milwaukee, WI 53201-0413 Ithaca, NY 14853-7801
USA USA
Library of Congress Cataloging-in-PubIication Data
Walter, Gilbert G.
Compartmental modeling with networks / Gilbert Walter, Martha
Contreras.
p. cm. - (Modeling and simulat ion in science, engineering and
technology)
Includes bibliographical references and index.
ISBN 978-1-4612-7207-6 ISBN 978-1-4612-1590-5 (eBook)
DOI 10.1007/978-1-4612-1590-5
1. Mathematical models. 2. Computer simulation. 1. Contreras,
Martha. II. Title. III. Series.
TA342.W35 1999
511'.8--dc21 98-44616
CIP
AMS Subject Classifications: 05, 60, 92
Printed on acid-free paper.
© 1999 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 1999
Softcover reprint of the hardcover lst edition 1999
AII rights reserved. This work may not be translated or copied in whole or in part without
the written permission of the publisher, Springer Science+Business Media, LLC
except for brief excerpts in connection
with reviews or scholarly analysis. Use in connection with any form of information storage
and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publicat ion, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
byanyone.
ISBN 978-1-4612-7207-6
Formatted from the authors' TeX files.
9 8 7 6 5 4 321
Contents
Preface ix
List of Figures xviii
1 Introduction and Simple Examples 1
1.1 Mathematical Models 1
1.2 Examples of Models ........ 2
Part I. Structure of Models: Directed Graphs 9
2 Digraphs and Graphs: Definitions and Examples 11
2.1 Definitions. 11
2.2 Examples 12
2.3 Problems . 15
3 A Little Simple Graph Theory 17
3.1 Isomorphic Graphs and Digraphs 17
3.2 Connected Graphs and Digraphs 19
4 Orientation of Graphs and Related Properties 25
4.1 Vertex Basis. . . . . . 25
4.2 Multigraphs...... 27
4.3 Orientation of Graphs 31
4.4 Spanning Trees . . . . 33
4.5 Minimum Connector Problem 37
5 Tournaments 41
5.1 Definitions and Basic Results 41
5.2 Transitive Tournaments 44
6 Planar Graphs 47
6.1 Bipartite Graphs . . . . . . . . . . . . . . . . . . 48
6.2 A Necessary Condition for a Graph to Be Planar 49
v
VI Contents
7 Graphs and Matrices 53
7.1 Adjacency and Reachability Matrices. 53
7.2 Eigenvalues of Adjacency Matrices 56
7.3 Using Maple with Graphs ...... . 59
Part II. Digraphs and Probabilities: Markov Chains 63
8 Introduction to Markov Chains 65
8.1 Relation to Digraphs ..... . 65
8.2 More Definitions and Examples 66
9 Classification of Markov Chains 71
9.1 Definitions............ 71
9.2 Condensation of a Stochastic Digraph 76
10 Regular Markov Chains 81
10.1 Theory of Regular Chains 81
10.2 Fixed-Point Probability Vector 84
10.3 Influence Digraph ..... 86
11 Absorbing Markov Chains 89
11.1 An Example. . . . . . . . . . . . . . . . . . . . . . . . .. 89
11.2 Some General Results .................... 92
11.3 Population Genetics: An Example of an Absorbing Chain 95
11.4 Small Group Decision Making: An Absorbing Markov Chain 97
12 From Markov Chains to Compartmental Models 101
12.1 Comparison of Quantities 101
12.2 Two Examples ................ . 104
Part III. Compartmental Models: Applications 109
13 Introduction to Compartmental Models 111
13.1 Basic Concepts ......... . 111
13.2 One-Compartment Applications. 114
13.2.1 Linear Case ... 115
13.2.2 Logistic Growth 116
13.3 Other Examples ... . 118
13.4 Matrix Forms ..... . 120
13.5 Two-Compartment Models 121
14 Models for the Spread of Epidemics 125
14.1 The SIR Model ...... . 125
14.2 Other Models of Epidemics ..... 128
Contents
Vll
15 Three Traditional Examples as Compartmental Models 131
15.1 Predator-Prey or Host-Parasite Equations. 131
15.2 The Leslie Matrix ........ . 135
15.3 Leontief Input-Output Analysis. 137
16 Ecosystem Models 141
16.1 Dissolved Oxygen. 141
16.2 Forest Ecosystem 143
16.3 Food Webs . . 146
17 Fisheries Models 149
17.1 Logistic Equation Approach 150
17.2 Nonequilibrium Yield ... . 152
17.3 A Multispecies Model .. . 156
17.4 An Ecosystem Fisheries Model 159
18 Drug Kinetics 163
18.1 Bilirubin Metabolism (Simple) 163
18.2 Bilirubin Metabolism (Complex) 165
18.3 Lead Kinetics . 166
18.4 HIV Dynamics . . . . . . . . . . 167
Part IV. Compartmental Models Theory 173
19 Basic Properties of Linear Models 175
19.1 Compartmental Matrices. 176
19.2 Eigenvalues ... . 178
19.3 Analytic Solution .... . 180
20 Structure and Dynamical Properties 183
20.1 Positivity of Solutions .... 183
20.2 Condensation of the Digraph 185
20.3 Eigenvalues and Structure 189
20.4 Some Special Cases ..... . 192
21 Identifiability of a Compartmental System 197
21.1 General Input and Output ......... . 197
21.2 An Example of an Identifiable System . . . 199
21.3 Another Example Which Is Not Identifiable 200
21.4 Another Approach: Using Laplace Transforms. 201
21.5 The General Case. . . . . . . . . . . . . . . . 203
21.6 Size Identifiability of Compartmental Models 204
21.6.1 Example ....... . 206
21.6.2 Some More Examples 207
21.6.3 Main Result ..... . 209
viii Contents
22 Parameter Estimation 211
22.1 Estimation Problem 211
22.2 Statistical Estimation 215
23 Complexity and Stability 218
23.1 Complexity ..... 218
23.2 The Shannon Index. 221
23.3 Other Indices . . . . 223
23.4 Relation to Stability 226
A Mathematical Prerequisites 228
Appendix 228
A.l Matrix Operations .......... . 228
A.2 Finding Eigenvalues and Eigenvectors 233
A.3 Systems of Differential Equations 235
A.4 Matrices with Maple . . . . . . . . . . 238
Bibliography 242
Index 247
Preface
The subject of mathematical modeling has expanded considerably in the
past twenty years. This is in part due to the appearance of the text by
Kemeny and Snell, "Mathematical Models in the Social Sciences," as well
as the one by Maki and Thompson, "Mathematical Models and Applica
tions." Courses in the subject became a widespread if not standard part of
the undergraduate mathematics curriculum. These courses included var
ious mathematical topics such as Markov chains, differential equations,
linear programming, optimization, and probability. However, if our own
experience is any guide, they failed to teach mathematical modeling; that
is, few students who completed the course were able to carry out the mod
eling paradigm in all but the simplest cases. They could be taught to
solve differential equations or find the equilibrium distribution of a regular
Markov chain, but could not, in general, make the transition from "real
world" statements to their mathematical formulation. The reason is that
this process is very difficult, much more difficult than doing the mathemat
ical analysis. After all, that is exactly what engineers spend a great deal
of time learning to do. But they concentrate on very specific problems and
rely on previous formulations of similar problems. It is unreasonable to
expect students to learn to convert a large variety of real-world problems
to mathematical statements, but this is what these courses require.
Fortunately, there is a large class of problems for which the transition step
is not as difficult. These are problems for which the appropriate model is a
flow model. They are used when there is a flow of something such as a fluid
or money or energy between the components of a system. They are widely
used in biomedicine for tracer experiments, but have applications in other
areas of biology such as the study of ecosystems, as well as in input-output
analysis in economics, arms races, and the study of epidemics. What's more
is that the transition step from the problem statement to the mathematical
formulation is very intuitive and easily learned. Most of our students chose
such models for their course projects even though they formerly constituted
a small portion of the course.
These flow models are usually called compartmental models to distinguish
IX
x Preface
them from diffusion models. To construct them, a system is divided up
into homogeneous compartments and the flows of material between the
various compartments and to and from the outside are traced. This leads
immediately to a directed graph, which can be used as a simple model and
may be adequate for an initial analysis of the system. If information on
flow rates is known, the model becomes a system of differential or difference
equations. These can be solved by traditional means, but because of the
special nature of these models, properties of the solutions can be inferred
without knowing the solutions explicitly.
In this work, we shall concentrate on these models and their applica
tions. This will involve consideration of some properties of digraphs (di
rected graphs), their relations to the systems of differential equations, and
their conversion to Markov chains. The applications will be to ecosystem
models, to fluid transfer, to competition models, to predator-prey systems,
to fisheries management, to regular and absorbing Markov chain models,
to Leontief input-output models, to Leslie matrices, to tracer experiments,
to epidemic models, and to network flows in engineering.
The book will be organized in four parts as follows:
The first part will be devoted to the theory of digraphs; it will be at a
level accessible to most students who have had any college mathematics
course.
The second part deals with Markov chains. Although the terminology
involves concepts from probability theory, the mathematics is pure matrix
theory. It requires only the material found in a first course in matrix theory
or finite mathematics.
The third part will introduce the differential equations associated with
compartmental models and develop their theory to some extent. This part
will require a knowledge of calculus and matrices, but not necessarily dif
ferential equations.
The final part will go into the theory of compartmental models somewhat
more deeply. The relations between the dynamics of the solution and the
structure of the model will be studied. The required level of sophistication
is higher but still requires only calculus and matrix theory.
Although these compartmental models are not designed to be computer
models, they lend themselves very well to computer simulation and approx
imation. The code for implementation of an approximation scheme comes
directly out of the differential equation formats.
Previous versions of this book have been used for a course in mathe
matical modeling at the University of Wisconsin-Milwaukee for about a
dozen years. This is a third-year course requiring a semester of calculus
and a semester of matrix theory. The students have come from a number
of different disciplines: about one-half from mathematics and others from
biology, economics, sociology, and even architecture.
Typically, about half the material in this book was covered in a semester
