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1999 | Buch

Introduction and Simple Examples Kapitel lesen Erstes Kapitel lesen

Compartmental Modeling with Networks

verfasst von: Gilbert G. Walter, Martha Contreras

Verlag: Birkhäuser Boston

Buchreihe : Modeling and Simulation in Science, Engineering and Technology

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Introduction and Simple Examples

1. Introduction and Simple Examples
Abstract
The purpose of a mathematical model is to explain or predict some phenomenon in the “real world.” This real world is the one in which measurements and observations are made. These, in turn, may be informal such as the observation that a traffic jam always forms on a certain corner, or they may consist of precise measurements of the outcome of a physics experiment. By themselves, any measurements are meaningless; they must be put into some context to give them sense. This context is a model. Sometimes the model is in form of a verbal or visual model, but often it is, in fact, the mathematical model in question and assumes the form of equations. These equations may then be solved to obtain desired predictions. Of course, there are many cases in which the equations cannot be solved, but are used instead to derive properties of the solutions.
Gilbert G. Walter, Martha Contreras

Structure of Models: Directed Graphs

Frontmatter
2. Digraphs and Graphs: Definitions and Examples
Abstract
In the Introduction, we referred to digraphs or directed graphs several times, but we did not define them. We saw that they are objects that look like those in Figure 2.1.
Gilbert G. Walter, Martha Contreras
3. A Little Simple Graph Theory
Abstract
In this chapter, we concentrate on some of the theoretical concepts and results in graph theory. These are usually intuitive and easy to understand, particularly in terms of the associated diagram. However, there are a lot of definitions to keep straight; furthermore there is no unanimity in the literature about these definitions, so other references may not be helpful. We present the proofs of theorems in some cases as well. These are more difficult than the other materials and can be omitted on first reading.
Gilbert G. Walter, Martha Contreras
4. Orientation of Graphs and Related Properties
Abstract
A graph can be converted into a digraph by assigning a direction to each of its edges, i.e., by giving it an orientation. In this chapter, we discuss desirable properties of such orientations. These include the Eulerian orientation of multigraphs and strongly connected orientations of graphs. Related concepts are those of vertex basis, i.e., a minimal set of vertices from which it is possible to reach all vertices, and spanning trees. Various ways of constructing the latter are introduced; in particular, the greedy algorithm, which leads to spanning tree which is optimal in some cases. If the original graph is a weighted graph, this is optimal in the sense of minimum cost.
Gilbert G. Walter, Martha Contreras
5. Tournaments
Abstract
The directed graphs in which every pair of vertices has exactly one arc joining them is called a tournament. They are used, e.g., in experiments involving paired comparisons, in round robin tournaments in which each player plays every other one, in studying pecking order in a barnyard or in an organization. Some natural questions that arise with these digraphs are: (i) Is there always a winner? (ii) Is there an ordering of the players determined by the tournament? (iii) If so, is it unique?
Gilbert G. Walter, Martha Contreras
6. Planar Graphs
Abstract
Although the diagrams of those graphs we have drawn have always been in a plane, sometimes the edges cross each other. If the graph is isomorphic to one where this doesn’t occur, it is said to be planar. Thus, the graphs in Figure 6.1 are planar, but the one in Figure 6.2 is not, but this is not so easy to see yet.
Gilbert G. Walter, Martha Contreras
7. Graphs and Matrices
Abstract
In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’s correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.
Gilbert G. Walter, Martha Contreras

Digraphs and Probabilities: Markov Chains

Frontmatter
8. Introduction to Markov Chains
Abstract
In Chapter 7, we saw how weighted digraphs and adjacency matrices are related. In this chapter, we consider particular types of weights and matrices that are used with Markov chains, and their associated special terms, which differ from those used previously.
Gilbert G. Walter, Martha Contreras
9. Classification of Markov Chains
Abstract
We assume that we have a Markov chain with transition matrix P and stochastic digraph D, as described in the last chapter. The digraph can be assumed to be weakly connected since, otherwise, the chain can be split into several noninteracting parts.
Gilbert G. Walter, Martha Contreras
10. Regular Markov Chains
Abstract
Let P be the transition matrix of a regular Markov Chain. Since by the definition of regular, there is a k such that any two vertices are joined by a path of length k, it follows that P k has all positive elements. Recall that the powers of the adjacency matrix of a nonweighted digraph, A k , count the number of paths from the vertex u i to u i . The same argument works for P k .
Gilbert G. Walter, Martha Contreras
11. Absorbing Markov Chains
Abstract
The prototypes of the absorbing chains are the Russian roulette and random walk chains:
Gilbert G. Walter, Martha Contreras
12. From Markov Chains to Compartmental Models
Abstract
For our Markov chain models, we have always assumed that the state vectors are probabilities. However, in some applications such as ecosystems, we are more interested in the quantities of, say, a nutrient in a particular state at a particular time rather than the probability that a molecule is in that state at that time. We shall see that the two concepts are equivalent.
Gilbert G. Walter, Martha Contreras

Compartmental Models: Applications

Frontmatter
13. Introduction to Compartmental Models
Abstract
In this chapter, we first present elements of the theory of compartmental models. We then present a few special cases and examples and examine the structure of the associated matrices.
Gilbert G. Walter, Martha Contreras
14. Models for the Spread of Epidemics
Abstract
In order to study the spread of epidemics, the population at risk is divided into compartments consisting of the number of persons with the disease (I, for infectives), those recovered from the disease and no longer susceptible (R), and those susceptible to infection (S). The population (N) is assumed to be relatively constant and that therefore N = S + I + R. The particular model used depends on the disease, but most can be analyzed using compartmental models.
Gilbert G. Walter, Martha Contreras
15. Three Traditional Examples as Compartmental Models
Abstract
The three models considered in this chapter are usually not treated as compartmental models. However, since they involve flows between compartments, they can be treated as such.
Gilbert G. Walter, Martha Contreras
16. Ecosystem Models
Abstract
Compartment al models and their extensions are natural for studying ecosystems since flows of energy and nutrients (nitrates, phosphates, carbon, etc.) drive the system.
Gilbert G. Walter, Martha Contreras
17. Fisheries Models
Abstract
Ecosystem models are generally used to try to understand and predict the behavior of the system. However, in the case of fisheries models, they are used for more than this. They are used to try to determine the optimum harvest and, for management of fisheries, to establish the allowable catch of the various species.
Gilbert G. Walter, Martha Contreras
18. Drug Kinetics
Abstract
Drug kinetics have already been briefly discussed in Chapter 13 and an example given. However, there are many other examples in the literature. The interested reader is referred to the books by Gibaldi and Perrier (1982), Godfrey (1983), or Jacquez (1996) and their references. In this section, we take up a few additional examples.
Gilbert G. Walter, Martha Contreras

Compartmental Models Theory

Frontmatter
19. Basic Properties of Linear Models
Abstract
As we saw in Chapter 13, the equation of a compartmental model with donor-controlled flow rates has the form
$$ \frac{{dx}} {{dt}} = Ax + f_i - E_0 x, $$
(19.1)
whereas that with recipient-controlled flow rates has the form
$$ \frac{{dx}} {{dt}} = Bx + E_i x - f_0 . $$
(19.2)
The matrices A and B are both singular since the sum of the columns is zero in both cases. The difference is that all diagonal elements in A are nonpositive while the off-diagonal elements are non-negative, whereas in B, the opposite is true. The matrices Eo and Ei are both diagonal matrices with non-negative elements.
Gilbert G. Walter, Martha Contreras
20. Structure and Dynamical Properties
Abstract
In the last chapter, we saw that compartimentai matrices had eigenvalues whose real part, if not zero, was negative. Hence, any solution to the associated differential equation was asymptotically stable. We also saw that positive initial values lead to non-negative solutions. However, there is still the possibility that a solution have a zero component in finite time. We also would like to determine structural conditions under which the asymptotic levels are nonzero; i.e. conditions for which x(t) > 0 as t → ∞. We also consider a method for simplifying the structure. Certain cases with a particular structure are shown to have associated dynamical behavior.
Gilbert G. Walter, Martha Contreras
21. Identifiability of a Compartmental System
Abstract
In most of the work in the previous chapters, we have assumed a knowledge of the flow rates and have tried to find properties of the solution to the system of differential equations. In this chapter, we attack the inverse problem: Given a solution, what are the properties of the flow rates?
Gilbert G. Walter, Martha Contreras
22. Parameter Estimation
Abstract
Identifiability, as discussed in the last chapter, is primarily a theoretical concept. It may be possible in theory to find the flow rates given the output but may be impossible in practice. There are a number of reasons for this. One is that the output is usually only known at discrete times and, then, only approximately. This must be used to estimate the true output or its Laplace transform. Another reason is that the equations involving the coefficients must be solved numerically since they are coupled nonlinear equations. Still another reason is that the exact structure of the model is seldom known so that the equations to be estimated may not be correct.
Gilbert G. Walter, Martha Contreras
23. Complexity and Stability
Abstract
A recurring problem in ecology is the relation between the complexity of an ecosystem and its stability. Most ecologists assumed the two concepts went together, i.e., greater complexity was associated with greater stability. However, May (1973) challenged this assumption and, indeed, showed that for Lotka—Volterra models of ecosystems, the opposite is sometimes true. However, he did not consider compartmental models and used only the number of nonzero flows as an indicator of complexity. For compartmental models, another approach, that we shall use, is possible.
Gilbert G. Walter, Martha Contreras
Backmatter
Metadaten
Titel
Compartmental Modeling with Networks
verfasst von
Gilbert G. Walter
Martha Contreras
Copyright-Jahr
1999
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-1590-5
Print ISBN
978-1-4612-7207-6
DOI
https://doi.org/10.1007/978-1-4612-1590-5