Compute evolution of a distribution over time

Question

We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the distribution of ages over time.

Denote the mass of people at or below age $a$ by $F(a,t)$

$$ F(a,t) = \int_0^{a} m(\tilde a,t) d\tilde a $$

Ultimately, I am after some Kolmogorov forward equation, that is, the solution for

$$ \partial_t F(a,t)$$

My approach Let $f(a, t)$ denote the density of people at age $a$ and point in time $t$. I will start with a discrete time approximation and let $\Delta$ go to zero. At each discrete point in time,

$$ f(a+\Delta, t+\Delta) = (1-P(a, t))f(a, t)$$

where $P(a, t)$ is the discrete time analogue of $d(a,t)$. As I'm going to let $\Delta\to 0$, I can approximate $1-P$ with $1-\Delta d)$:

$$ f(a+\Delta, t+\Delta) = (1- \Delta d(a,t))f(a, t)\\ \frac{f(a+\Delta, t+\Delta) -f(a,t)}{\Delta} = -d(a,t))f(a, t)\\ (\partial_t + \partial_a)f(a,t) = \lim_{\Delta\to 0}\frac{f(a+\Delta, t+\Delta) -f(a,t)}{\Delta} = -d(a,t))f(a, t)\\ $$

I can integrate both sides w.r.t. a and get

$$ \partial_t F(t, a) = - f(t, a) - \int q(t, a) f(t, a) da \\ \partial_t F(t, a) = - \partial_a F(t, a) - \int q(t, a) \partial_a F(t, a) da $$

I know that $\partial_a q(t, a) = q(t, a) (1-q(t, a))$. However, that doesn't really help me with solving the integral. Is there perhaps another angle to attack this problem? Or did I miss something?

Answer 1

Here's my best guess. I haven't checked to thoroughly if this is right, but maybe it will help.

Evolution of population density

I understand the model as follows. $f(a,t)$ is the density of people of age $a$ at time $t$. Suppose at time $t=0$, the density of the population is $f_0(a)$. To model the aging process as well as the mortality rate, the density $f$ must evolve over time so that it satisfies the condition that you derived. Thus, $m$ must satisfy the following partial differential equation $$ \frac{\partial f}{\partial t} + \frac{\partial f}{\partial a} + d(a, t) f(a,t) = 0 $$ and the initial condition $$ f(a,0) = f_0(a). $$ I believe that PDEs of this form can have relatively straightforward solutions depending on the functional form chosen for $d$. Here are some cases.

Case 1: Constant mortality rate, $d(a,t) = d_0$.

Suppose a constant mortality rate, $d(a,t) = d_0$. Then (using Mathematica),

DSolve[
 {D[f[a, t], t] + D[f[a, t], a] + d0 f[a, t] == 0, 
  f[a, 0] ==  f0[a]},
 f[a, t], {a, t}]

results in

{{f[a, t] -> E^(-a d0 + d0 (a - t)) f0[a - t]}}

So, as we can see, this gives a simple solution $$ f(a,t) = \exp\{-a d_0 + d_0 (a-t)\} \cdot f_0(a-t) $$ Case 2: Log Mortality, $d(a,t) = \log(a+1)$.

Here we have

DSolve[
 {D[f[a, t], t] + D[f[a, t], a] + Log[a+1] f[a, t] == 0, 
  f[a, 0] ==  f0[a]},
 f[a, t], {a, t}]

which results in

{{f[a, t] -> (1 + a)^(-1 - a) E^t (1 + a - t)^(1 + a - t) f0[a - t]}}

This also gives a simple solution $$ f(a,t) = (1+a)^{-a-1} e^t (1+a-t)^{1+a-t} \cdot f_0(a-t) $$

Case 3: The general case

For the case where we do not yet specify $d(a,t)$,

DSolve[
 {D[f[a, t], t] + D[f[a, t], a] + d[a, t] f[a, t] == 0, 
  f[a, 0] ==  f0[a]},
 f[a, t], {a, t}]

gives us (in TeXForm for clarity)

$$ \begin{align*} \left\{\left\{f(a,t)\to \\ \text{f0}(a-t) \cdot \\ \exp \left(\int_1^a -d(K[1],K[1]-a+t) \, dK[1]-\int_1^{a-t} -d(K[1],K[1]-a+t) \, dK[1]\right)\right\}\right\} \end{align*} $$