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Constant mortality with age-localized fertility

Example

Take constant mortality $μ (a) = 1$ , initial density $n_{0} = 1_{[1, 2]}$ , and triangular fertility

$β (a) = \max {1 - 2 ∣ a - 3 / 2 ∣, 0} .$

The fertility is localized to ages $[1, 2]$ . Use cutoff $A = 4$ and horizon $T = 1 / 4$ . Compute the initial birth rate, exhibit the corner incompatibility, and take one positive finite-volume step on a unit age grid with $Δ t = 1 / 4$ .

Verification

Given: The data above, the characteristic renewal formula and the upwind mass update.

[F1] Linear age-structured renewal transport problem: the inflow is $B (t) = \int_{0}^{\infty} β (a) n (t, a) d a$ and the initial datum is specified as an $L^{1}$ function; this example separately uses its right-hand trace at age zero.

[F2] Characteristic formula and Volterra solution for age renewal transport: the old cohort contributes $G (t) = \int_{0}^{\infty} β (x + t) S (x, t) n_{0} (x) d x$ to the Volterra birth equation.

[F3] Finite-volume upwind scheme for age renewal transport: $M_{j}^{k + 1} = M_{j}^{k} - Δ t (F_{j}^{k} - F_{j - 1}^{k}) - Δ t μ_{j} M_{j}^{k}$ with $F_{0} = B_{h}$ and $F_{j} = M_{j} / Δ a$ .

[F4] Positivity, L1 growth and data stability of the renewal finite-volume step: if $Δ t / Δ a + Δ t μ_{\infty} \leq 1$ , the updated masses remain nonnegative.

1.1F1algebra

The triangle has base length $1$ and height $1$ , so $B (0) = \int_{1}^{2} β (a) d a = 1 / 2$ . The chosen initial density has right-hand trace $n_{0} (0 +) = 0$ , which differs from $B (0)$ ; hence smooth renewal-corner compatibility fails. The example uses the characteristic/weak formulation rather than assuming that compatibility.

1.2F2algebra

Here $S (x, t) = e^{- t}$ , so [F2] gives $G (t) = e^{- t} \int_{1}^{2} β (x + t) d x$ and $B (t) = G (t) + \int_{0}^{t} β (a) e^{- a} B (t - a) d a$ . The first formula records the initially present age cohort; the second is the renewal feedback term.

1.3F1constructalgebra

Use cells $[0, 1], [1, 2], [2, 3], [3, 4]$ , each of width $1$ , and initial masses $(0, 1, 0, 0)$ . Four-panel composite midpoint quadrature on the second cell samples fertility values $1 / 4, 3 / 4, 3 / 4, 1 / 4$ , so $B_{h}^{0} = (1 / 4) (1 / 4 + 3 / 4 + 3 / 4 + 1 / 4) = 1 / 2$ .

2.1F3F4step 1.3algebra∎

With $λ = Δ t / Δ a = 1 / 4$ and $μ_{j} Δ t = 1 / 4$ , the CFL sum is $1 / 2$ . The fluxes are $F_{0} = 1 / 2, F_{1} = 0, F_{2} = 1, F_{3} = 0, F_{4} = 0$ , so the first updated mass vector is $(1 / 8, 1 / 2, 1 / 4, 0)$ . Every entry is nonnegative, in agreement with [F4].

Depends on

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Sources

Benoît Perthame, Introduction to Structured Equations in Biology, full lecture notes (standard reference, not scraped)
MATH 575 lecture notes, §9 Finite difference methods for the transport equation (standard reference, not scraped)
MATH 373 lecture notes, §8 Approximation of Integrals (standard reference, not scraped)