Age-cohort quadrature along exact characteristics

Input a finite horizon $T>0$, a time step $\mathrm{\Delta }t>0$, an age cutoff $A>0$, bounded nonnegative rates $\mu ,\beta$ on $[0,A]$, and a finite positive initial quadrature $(a_i,m_i^0{)}_{i=1}^I$ supported in $[0,A)$ for the initial age measure on that interval. Set $N=\lfloor T\mathrm{/}\mathrm{\Delta }t\rfloor$; the procedure returns levels at $t_k=k\mathrm{\Delta }t$ for $0\le k\le N$, with final time $N\mathrm{\Delta }t\le T$. One option, when a nonnegative real-valued initial density $n_0$ is evaluable at every point of $[0,A]$, is to choose any finite partition $0=b_0<\cdots <b_I=A$ and apply Composite midpoint rule on a uniform grid with $n=1$ separately on each cell $[b_{i-1},b_i]$: set $\mathrm{\Delta }{a}_i=b_i-b_{i-1}$, $a_i=(b_{i-1}+b_i)\mathrm{/}2$, and $m_i^0=Q_M(n_0;b_{i-1},b_i,1)=\mathrm{\Delta }{a}_i{n}_0(a_i)$ for $1\le i\le I$. The cell widths need not be equal. A cohort at age $a_i^k$ and mass $m_i^k$ follows the exact age characteristic

$a_i^{k+1}=a_i^k+\mathrm{\Delta }t$

and updates its survival mass by forward Euler,

$m_i^{k+1}=(1-\mathrm{\Delta }t\mu (a_i^k))m_i^k\mathrm{.}$

If $\mathrm{\Delta }t\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}>1$, return status invalid-step with no updates. Otherwise initialize the birth and outflow records as empty and set ${\nu }_h^0={\sum }_i{m}_i^0{\delta }_{a_i^0}$. For each $k=0,\dots ,N-1$, first calculate $B_h^k={\sum }_i\beta (a_i^k)m_i^k$. Advance every existing cohort by $a_i^{k+1}=a_i^k+\mathrm{\Delta }t$ and update its survival mass by $m_i^{k+1}=(1-\mathrm{\Delta }t\mu (a_i^k))m_i^k$. If an updated age is at least $A$, record that updated mass as discrete outflow and discard the cohort; otherwise retain it. Then insert a newborn cohort of age $0$ and mass $\mathrm{\Delta }t{B}_h^k$ at time $t_{k+1}$. Record $B_h^k$ and the step's outflow, and form ${\nu }_h^{k+1}$ from the retained cohorts and this newborn. A failed rate evaluation returns failure with no partial update. The procedure returns the finite sequence $({\nu }_h^k{)}_{k=0}^N$ and its birth/outflow records. The bounded-Lipschitz quadrature defect of the initial measure and the Euler survival error are separate inputs to any consistency claim.