Chapter 2

Let $p$ denote the population density, and let it depend on time $t$ and age $r$. The number of people of ages between $r$ and $r+dr$ at a certain time $t$ is given by $p(t,r)dr$. Define the mortality rate $\mu (t,r)$ in the following way: $\mu (t,r)drdt$ is the fraction of people in the age class $[r,r+dr]$ who die in the time interval $[t,t+dt]$. Based on the infinitesimal equality $$p(t+dt,r+dt)dr-p(t,r)dr=-\mu pdrdt.$$ Show that $p$ satisfies the following partial differential equation $$\frac{\partial p}{\partial r}+\frac{\partial p}{\partial t}=-\mu p$$ Let the initial age distribution be given as $$p(0,r)=p_0(r),0\le r\le 1$$ and the birth rate function as the boundary condition $$p(t,0)=u(t),t\ge 0.$$ Here it assumed that the age $r$ is scaled in such a way that nobody reaches an age $r>1$. One can consider $u(t)$ as the input to the system and as output $y(t)$ for instance the number of people in the working age, say between the ages $a$ and $b$, \(0