Traveling-wave solution for McKendrick age-structure model with finite lifespan

Question

The model was described many times before, so I keep the details concise. We have $$ \frac{\partial}{\partial t} \rho(t,a) + \frac{\partial}{\partial a} \rho(t,a) = -\delta(a) \rho(t,a) $$ where $\rho$ is population (number) density and $\delta$ is instantaneous mortality. Moreover, $$ \Pi(a) = \exp \left( - \int_{0}^{a} \delta (\alpha) d\alpha \right) $$ is the probability of surviving to age $a$. According to Cushing (1998) one can redefine $$ \psi(t,a) = \frac{\rho(t,a)}{\Pi(a)} $$ such that the McKendrick PDE above simplifies to $$ \frac{\partial}{\partial t} \psi(t,a) + \frac{\partial}{\partial a} \psi(t,a) = 0 $$ leading to a traveling-wave solution $\psi(t,a) = \varphi(t-a)$.

What I'm wondering is how this kind of simplification is possible when there is a maximum age $a_{M} < \infty$? Because then $\Pi(a_{M}) = 0$, leaving $\psi(t,a)$ undefined. Cushing doesn't mention anything in that regard, so am I missing something trivial?

Answer 1

$\Pi$ cannot have any zeros if it is defined as

$$\Pi(a)=\exp\left(-\int_0^a \delta(a)da\right)$$

since the exponential function has no zeros. That being said, in real life, there is presumably an age after which the probability of living is zero. If you want to include that possibility in your model, then $\delta(a)$ will not be well-defined, since it is given by

$$\delta(a)=\Pi'(a)/\Pi(a)$$

In this case, the original McKendrick equation cannot be posed.

Answer 2

There is absolutely no issue in writing a model in which the instantaneous mortality rate contains a singularity at $a=a_M$. Dividing by $\Pi(a)$ still works for $a<a_M$ and the resulting equation should be used only in the half strip $(0,\infty)\times(0,a_M)$. One can explicitly solve the model, and the general solution is given by

$$\rho(t,a)= \exp\left({-\int_0^a\delta(a')da'}\right)\varphi(t-a)$$

You can check this solves the PDE for any function $\varphi$. Now one can use appropriate BC's - usually by specifying the functions $\rho(0,a), \rho(t,0)$ - to find a solution in the half strip that vanishes precisely at $a=a_M$. This is hardly a surprising feature, it just says that the population density is zero at the maximum possible age. For a specific example, choose say $\delta(a)=1/(a_M-a)$ in which case it is easy to see that the general solution has a zero exactly at the maximum age for sufficiently regular $\varphi$:

$$\rho(t,a)=\left(1-\frac{a}{a_M}\right)\varphi(t-a)$$

Thus it can be seen that simple poles in the instantaneous mortality naturally implement a $C^0$ profile for the population density with a boundary (with the reasonable physical assumption that $\rho=0, a\geq a_M$). Of course, certain singular boundary conditions can in principle produce spurious behavior at the boundary, but these should be ruled out when one considers only the space of physically sensible ones.