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 *p*_{i} at times
.
Use
and *p*_{i}[*t*_{n}] 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
*T*_{ij} 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 *T*_{ij}.

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

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 *t*_{n+1} for the people in region *j* at time *t*_{n}.