# An Age Structured Population Model

## Written for the Envision It! Workshop Steven McKelveyMathematics and Computer ScienceSaint Olaf College

### The Model

A very common and useful tool for investigating future demographics is the age structured model. In this model populations are not tracked in their totality, but rather according to their age class. The first age class is made up of newborns. As time progresses these individuals either move up into subsequent (older) age classes or perish.

The key for the modeler is to determine how many individuals of a given age class die during each time interval, how many move into the next age class and how many newborns are created by the members of each age class. It is typical for the fecundity of each age class to differ.

Ideally, each age class would include individuals of identical ages. However, as we shall see, it is complicated and tedious to split a population into too many subdivisions. In human populations it is common to use ten years as a generation length, and to split up populations into decade long age classes.

The length of each age class, along with the time step used in the model, influences the value of the parameters in the model. If no deaths are occuring in an age class, it is traditional to send a certain fraction of an age class on to the next age class at the end of every time step. This fraction is the ratio of the time step to the length of the age class. For example, if a model uses 10 year long age classes and a time step of two year, 2/10=20% of each age class would be assumed to move into the next age class during each time step. For long age classes this can lead to some significant errors, but for reasonable age classes it is a good approximation.

In the STELLA model below a population with four age classes, each of length two years, is considered. Newborns are assumed to not give birth, but all other age classes contribute to the newborn population. Each age class is two years long. The time step in this model is a single year.

If deaths were not an issue, one half of each age class would "age" (move on to the next age class) every time step. However, each age class loses some of its individuals to death. Clearly, the dead individuals do not move on to the next age class.

To properly account for these deaths the number of individuals who would have moved on had they been alive, but did not survive must be subtracted from the individuals who do move on. In the case of our example, if Pn is the number of individuals in age class n, then 0.5*Pn would be the number of individuals we would send on to the next age class in the absence of death. If Dn represents the death rate (as a proportion) for age class n, then 0.5*Dn*Pn of these individuals would be dead, leaving 0.5*(1-Dn)*Pn individuals to move up into the next age class.

The birth rates in the model are given in units of newborns/individual in the indicated age class.

The STELLA diagram for the example model described above is: 