Martha Contreras

Structure and Dynamical Properties

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.

Classification of Markov Chains

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.

Three Traditional Examples as Compartmental Models

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.

Introduction and Simple Examples

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.

Regular Markov Chains

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 .

Introduction to Compartmental Models

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.

A Little Simple Graph Theory

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.

Digraphs and Graphs: Definitions and Examples

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.

Models for the Spread of Epidemics

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.

Graphs and Matrices

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.