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## Markov Chain Models

Suppose a population consists of k disjoint sub-populations labeled by the integers . Let pi be the fraction of the population in sub-population i. Form the vector called the state vector of the system:

The sum of the population fractions is assumed to be 1.

The state vector will depend on time t. In a discrete model, we might measure the population fractions pi at times . Use and pi[tn] respectively to denote values at the end of the n'th time period.

In a Markov chain model one assumes that there are transition probabilities Tij representing the proportion of the population in sub-population j during time period n which enter sub-population i for time period n+1. These transition probabilities are independent of time. Let T be the k x k matrix whose i,j entry is Tij.

We also assume that the entire population in time period n+1 comes by this transition process from the population during time period n. Thus

This is just the formula for the i'th entry of a matrix product of T and the n'th state vector . Thus in matrix form, .

More generally, we can keep track of the change in population over r time periods by computing the r'th power of T. Specifically .

The sum of the entries in each column (e.g. the j'th) of the transition matrix T should be 1 because the entries of that column reflect all the possible destinations at time tn+1 for the people in region j at time tn.

Next: Analyzing the Population Changes Up: Due in Recitation on Previous: A Population Problem
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2002-08-21