Compartmental Modeling with Networks

1. Front Matter

   Pages i-xviii

   PDF

2. Introduction and Simple Examples

   1. Introduction and Simple Examples

      • Gilbert G. Walter, Martha Contreras

      Pages 1-8

3. Structure of Models: Directed Graphs

   1. Front Matter

      Pages 9-9

      PDF

   2. Digraphs and Graphs: Definitions and Examples

      • Gilbert G. Walter, Martha Contreras

      Pages 11-16

   3. A Little Simple Graph Theory

      • Gilbert G. Walter, Martha Contreras

      Pages 17-24

   4. Orientation of Graphs and Related Properties

      • Gilbert G. Walter, Martha Contreras

      Pages 25-40

   5. Tournaments

      • Gilbert G. Walter, Martha Contreras

      Pages 41-46

   6. Planar Graphs

      • Gilbert G. Walter, Martha Contreras

      Pages 47-51

   7. Graphs and Matrices

      • Gilbert G. Walter, Martha Contreras

      Pages 53-61

4. Digraphs and Probabilities: Markov Chains

   1. Front Matter

      Pages 63-63

      PDF

   2. Introduction to Markov Chains

      • Gilbert G. Walter, Martha Contreras

      Pages 65-69

   3. Classification of Markov Chains

      • Gilbert G. Walter, Martha Contreras

      Pages 71-80

   4. Regular Markov Chains

      • Gilbert G. Walter, Martha Contreras

      Pages 81-87

   5. Absorbing Markov Chains

      • Gilbert G. Walter, Martha Contreras

      Pages 89-99

   6. From Markov Chains to Compartmental Models

      • Gilbert G. Walter, Martha Contreras

      Pages 101-108

5. Compartmental Models: Applications

   1. Front Matter

      Pages 109-109

      PDF

   2. Introduction to Compartmental Models

      • Gilbert G. Walter, Martha Contreras

      Pages 111-123

   3. Models for the Spread of Epidemics

      • Gilbert G. Walter, Martha Contreras

      Pages 125-129

   4. Three Traditional Examples as Compartmental Models

      • Gilbert G. Walter, Martha Contreras

      Pages 131-139

   5. Ecosystem Models

      • Gilbert G. Walter, Martha Contreras

      Pages 141-148

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.

• Applied math
• Maple
• Markov
• mathematical modeling
• model
• modeling

• Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, USA

  Gilbert G. Walter

• Department of Biometry, Cornell University, Ithaca, USA

  Martha Contreras

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