2.5: Discrete Age-Structured Populations

In a discrete model, population censuses occur at discrete times and individuals are assigned to age classes, spanning a range of ages. For model simplicity, we assume that the time between censuses is equal to the age span of all age classes.

Table 2.2: Definitions needed in an age-structured, discrete-time population model

$u_{i, n}$	number of females in age class $i$ at census $n$
$s_{i}$	fraction of females surviving from age class $i-1$ to $i$
$m_{i}$	expected number of female offspring from a female in age class $i$
$l_{i}=s_{1} \cdots s_{i}$	fraction of females surviving from birth to age class $i$
$f_{i}=m_{i} l_{i}$	basic reproductive ratio
$\mathcal{R}_{0}=\sum_{i} f_{i}$

An example is a country that censuses its population every five years, and assigns individuals to age classes spanning five years (e.g., 0-4 years old, 5-9 years old, etc.). Although country censuses commonly count both females and males separately, we will only count females and ignore males.

There are several new definitions in this Section and I place these in Table $2.2$ for easy reference. We define $u_{i, n}$ to be the number of females in age class $i$ at census $n$. We assume that $i=1$ represents the first age class and $i=\omega$ the last. No female survives past the last age class. We also assume that the first census takes place when $n=1$. We define $s_{i}$ as the fraction of females that survive from age class $i-1$ to age class $i$ (with $s_{1}$ the fraction of newborns that survive to their first census), and define $m_{i}$ as the expected number of female births per female in age class $i$.

We construct difference equations for $\left\{u_{i n+1}\right\}$ in terms of $\left\{u_{i n}\right\}$. First, newborns at census $n+1$ were born between census $n$ and $n+1$ to different aged females, with differing fertilities. Also, only a faction of these newborns survive to their first census. Second, only a fraction of females in age class $i$ that were counted in census $n$ survive to be counted in age class $i+1$ in census $n+1$. Putting these two ideas together with the appropriately defined parameters, the difference equations for $\left\{u_{i, n+1}\right\}$ are determined to be

$$\begin{aligned} u_{1, n+1} &=s_{1}\left(m_{1} u_{1, n}+m_{2} u_{2, n}+\cdots+m_{\omega} u_{\omega, n}\right) \\[4pt] u_{2, n+1} &=s_{2} u_{1, n} \\[4pt] u_{3, n+1} &=s_{3} u_{2, n} \\[4pt] \vdots & \\[4pt] u_{\omega, n+1} &=s_{\omega} u_{\omega-1, n} \end{aligned} \nonumber$$

which can be rewritten as the matrix equation

$$\left(\begin{array}{c} u_{1, n+1} \\[4pt] u_{2, n+1} \\[4pt] u_{3, n+1} \\[4pt] \vdots \\[4pt] u_{\omega, n+1} \end{array}\right)=\left(\begin{array}{ccccc} s_{1} m_{1} & s_{1} m_{2} & \ldots & s_{1} m_{\omega-1} & s_{1} m_{\omega} \\[4pt] s_{2} & 0 & \ldots & 0 & 0 \\[4pt] 0 & s_{3} & \ldots & 0 & 0 \\[4pt] \vdots & \vdots & \vdots & \vdots & \vdots \\[4pt] 0 & 0 & \ldots & s_{\omega} & 0 \end{array}\right)\left(\begin{array}{c} u_{1, n} \\[4pt] u_{2, n} \\[4pt] u_{3, n} \\[4pt] \vdots \\[4pt] u_{\omega, n} \end{array}\right) ; \nonumber$$

or in compact vector form as

$$\mathbf{u}_{n+1}=\mathrm{L} \mathbf{u}_{n} \nonumber$$

where $L$ is called the Leslie Matrix.

This system of linear equations can be solved by determining the eigenvalues and associated eigenvectors of the Leslie Matrix. One can solve directly the characteristic equation, $\operatorname{det}(\mathrm{L}-\lambda \mathrm{I})=0$, or reduce the system of first-order difference equations (2.5.2) to a single high-order equation for the number of females in the first age class. Following the latter approach, and beginning with the second row of $(2.5.2)$, we have

$$\begin{aligned} u_{2, n+1} &=s_{2} u_{1, n} \\[4pt] u_{3, n+1} &=s_{3} u_{2, n} \\[4pt] &=s_{3} s_{2} u_{1, n-1} \\[4pt] & \vdots \\[4pt] u_{\omega, n+1} &=s_{\omega} u_{\omega-1, n} \\[4pt] &=s_{\omega} s_{\omega-1} u_{\omega-2, n-1} \\[4pt] & \vdots \\[4pt] &=s_{\omega} s_{\omega-1} \cdots s_{2} u_{1, n-\omega+2} \end{aligned} \nonumber$$

If we define $l_{i}=s_{1} s_{2} \cdots s_{i}$ to be the fraction of females that survive from birth to age class $i$, and $f_{i}=m_{i} l_{i}$ to be the number of female offspring expected from a newborn female upon reaching age class $i$ (taking into account that she may not survive to age class $i$ ), then the first row of (2.5.2) becomes

$$u_{1, n+1}=f_{1} u_{1, n}+f_{2} u_{1, n-1}+f_{3} u_{1, n-2}+\cdots+f_{\omega} u_{1, n-\omega+1} . \nonumber$$

Here, we have made the simplifying assumption that $n \geq \omega$ so that all the females counted in the $n+1$ census were born after the first census.

The high-order linear difference equation (2.5.3) may be solved using the ansatz $u_{1, n}=\lambda^{n} .$ Direct substitution and division by $\lambda^{n+1}$ results in the discrete EulerLotka equation

$$\sum_{j=1}^{\omega} f_{j} \lambda^{-j}=1 \nonumber$$

which may have both real and complex-conjugate roots.

Once an eigenvalue $\lambda$ is determined from $(2.5.4)$, the corresponding eigenvector $\mathbf{v}$ can be computed using the Leslie matrix. We have

$$\left(\begin{array}{ccccc} s_{1} m_{1}-\lambda & s_{1} m_{2} & \ldots & s_{1} m_{\omega-1} & s_{1} m_{\omega} \\[4pt] s_{2} & -\lambda & \ldots & 0 & 0 \\[4pt] 0 & s_{3} & \cdots & 0 & 0 \\[4pt] \vdots & \vdots & \vdots & \vdots & \vdots \\[4pt] 0 & 0 & \cdots & s_{\omega} & -\lambda \end{array}\right)\left(\begin{array}{c} v_{1} \\[4pt] v_{2} \\[4pt] v_{3} \\[4pt] \vdots \\[4pt] v_{\omega} \end{array}\right)=\left(\begin{array}{c} 0 \\[4pt] 0 \\[4pt] 0 \\[4pt] \vdots \\[4pt] 0 \end{array}\right) \nonumber$$

Taking $v_{\omega}=l_{\omega} / \lambda^{\omega}$, and beginning with the last row and working backwards, we have:

$$\begin{aligned} v_{\omega-1} &=l_{\omega-1} / \lambda^{\omega-1} \\[4pt] v_{\omega-2} &=l_{\omega-2} / \lambda^{\omega-2} \\[4pt] \vdots & \\[4pt] v_{1} &=l_{1} / \lambda \end{aligned} \nonumber$$

so that

$$v_{i}=l_{i} / \lambda^{i}, \quad \text { for } i=1,2, \ldots, \omega \nonumber$$

We can obtain an interesting implication of this result by forming the ratio of two consecutive age classes. If $\lambda$ is the dominant eigenvalue (and is real and positive, as is the case for human populations), then asymptotically,

$$\begin{aligned} u_{i+1, n} / u_{i, n} & \sim v_{i+1} / v_{i} \\[4pt] &=s_{i+1} / \lambda \end{aligned} \nonumber$$

With the survival fractions $\left\{s_{i}\right\}$ fixed, increasing $\lambda$ implies a decreasing ratio: a faster growing population has relatively more younger people than a slower growing population. In fact, we are now living through a time when developed countries, particularly Japan and those in Western Europe, as well as Hong Kong and Singapore, have substantially lowered their population growth rates and are increasing the average age of their citizens.

If we want to simply determine if a population grows or decays, we can calculate the basic reproduction ratio $\mathcal{R}_{0}$, defined as the net expectation of female offspring to a newborn female. Stasis is obtained if the female only replaces herself before dying. If $\mathcal{R}_{0}>1$, then the population grows, and if $\mathcal{R}_{0}<1$ then the population decays. $\mathcal{R}_{0}$ is equal to the number of female offspring expected from a newborn when she is in age class $i$, summed over all age classes, or

$$\mathcal{R}_{0}=\sum_{i=1}^{\omega} f_{i} \nonumber$$

For a population with approximately equal numbers of males and females, $\mathcal{R}_{0}=1$ means a newborn female must produce on average two children over her lifetime. News stories in the western press frequently state that for zero population growth, women need to have $2.1$ children. The term women used in these stories presumably means women of child-bearing age. Since girls who die young have no children, the statistic of $2.1$ children implies that $0.1 / 2.1$, or about $5 \%$ of children die before reaching adulthood.

A useful application of the mathematical model developed in this Section is to predict the future age structure within various countries. This can be important for economic planning-for instance, determining the tax revenues that can pay for the rising costs of health care as a population ages. For accurate predictions on the future age-structure of a given country, immigration and migration must also be modeled. An interesting website to browse is at

http://www.census.gov/ipc/www/idb.

This website, created by the US census bureau, provides access to the International Data Base (IDB), a computerized source of demographic and socioeconomic statistics for 227 countries and areas of the world. In class, we will look at and discuss the dynamic output of some of the population pyramids, including those for Hong Kong and China.