Lower edges (in years) of the age groups: fine in infancy, where immunity and
maternal dynamics move fast, and coarse in adulthood. Group edges are pinned
at the conventional reporting boundaries – 0, 1, 2, 3, 5, 7, 10, 15, 20, 30,
40, 60 years – and the groups between two anchors are of equal width, with
the budget of groups spread across the anchors in proportion to the width in
log(1 + age). Each interval gets at least one group; the remainder is
shared out largest-remainder.
Arguments
- max_age
oldest age-group lower edge, in years (the absorbing top group). Must be a single finite number
>= 20: below that the grid has too few anchors left to resolve the ages over which immunity develops.- n_group
total number of age groups, the absorbing top group included. The default, 53, is one more than the 52 the old grid used, and is what the error figures above were measured at. Raising it refines the whole grid while keeping its shape, which is what a grid-convergence check wants.
Details
Log rather than linear width, because what the grid has to resolve is the
rise of immunity with age, and that is far closer to a function of log age
than of age. (Of log(1 + age) rather than log(age), which is infinite
over the first year. The offset is in years, so the allocation is not
scale-free.) An equal-width grid at the same cost is four times as far from
the converged solution as this one, so the grading matters more than the
count.
This replaced a grid of fixed monthly / quarterly / yearly / 5-yearly sections, halving the discretisation error at the same number of groups: the largest departure across age bands falls from 10.7% to 4.5% and the rms from 5.4% to 3.2%. The old grid over-resolved infancy (12 of its 52 groups in the first year, against 7 here) and under-resolved everything above 15.
Refining removes that error: the departure from the converged profile halves
with each doubling of n_group, first order. What refinement does not
remove is the mean-field approximation itself – one immunity value per
stratum, where the IBM holds a spread of infection histories at the same age
– so a persistent difference from the IBM is not evidence that the grid is
too coarse. validations/age-grid/run.R in fleetcheck measures both.
Band aggregation weights each age group by the exact fraction of its own width that falls inside the band, so a band edge landing inside a group (as one can, away from the anchors) apportions that group between the two bands instead of handing it whole to one of them. The same is true of intervention age targeting. Neither is snapped to the grid.
Examples
default_age_lower()
#> [1] 0.0000000 0.1428571 0.2857143 0.4285714 0.5714286 0.7142857
#> [7] 0.8571429 1.0000000 1.2000000 1.4000000 1.6000000 1.8000000
#> [13] 2.0000000 2.2500000 2.5000000 2.7500000 3.0000000 3.4000000
#> [19] 3.8000000 4.2000000 4.6000000 5.0000000 5.5000000 6.0000000
#> [25] 6.5000000 7.0000000 7.7500000 8.5000000 9.2500000 10.0000000
#> [31] 11.2500000 12.5000000 13.7500000 15.0000000 16.6666667 18.3333333
#> [37] 20.0000000 22.5000000 25.0000000 27.5000000 30.0000000 33.3333333
#> [43] 36.6666667 40.0000000 44.0000000 48.0000000 52.0000000 56.0000000
#> [49] 60.0000000 65.0000000 70.0000000 75.0000000 80.0000000
# coarser top of the grid
default_age_lower(max_age = 60)
#> [1] 0.00000 0.12500 0.25000 0.37500 0.50000 0.62500 0.75000 0.87500
#> [9] 1.00000 1.20000 1.40000 1.60000 1.80000 2.00000 2.25000 2.50000
#> [17] 2.75000 3.00000 3.40000 3.80000 4.20000 4.60000 5.00000 5.50000
#> [25] 6.00000 6.50000 7.00000 7.75000 8.50000 9.25000 10.00000 11.00000
#> [33] 12.00000 13.00000 14.00000 15.00000 16.66667 18.33333 20.00000 22.00000
#> [41] 24.00000 26.00000 28.00000 30.00000 32.50000 35.00000 37.50000 40.00000
#> [49] 44.00000 48.00000 52.00000 56.00000 60.00000
# twice the resolution everywhere, same shape
length(default_age_lower(n_group = 105))
#> [1] 105