Actuarial studies · 2025

Specialist Projections

How many specialists a place actually needs.

Workforce planning usually starts from headcount and works outward. This starts from the population and works in: how many people, of what ages, carrying what hazard, with what capacity to seek care — and how many practitioners that implies. The morbidity term is the hard part, because morbidity by age is not published at the granularity the model needs, so the central death rate stands in for it and the approximation is stated rather than hidden.

Status: Research
Domain: Demand modelling · Health workforce
Role: Model design, derivation
Year: 2025

Stack

Actuarial modelling
Life tables
Census geography
Python
R
LaTeX

An actuarial demand model that projects required specialist supply for an area from its population age structure, a mortality-derived morbidity proxy, and income elasticity.

Demand, derived rather than assumed

The model resolves an area to a required number of practitioners.

Population is decomposed into age bands, each band contributes a baseline visit rate plus a hazard-scaled increment, the total is adjusted for income, and the result is divided by the caseload one practitioner can absorb. Every term is separately observable, which means every term can be argued with — the point of writing it this way rather than fitting a single regression to historical headcount.

Mortality standing in for morbidity

The quantity the model actually wants is age-specific morbidity, and it is not published at the resolution required.

The central death rate is used instead, corrected for bracket width and adjusted by how far the band midpoint sits from a population-weighted reference age. The two co-move strongly with age, which is what makes the substitution defensible — and it would fail for a specialty whose caseload is dominated by conditions people live with rather than die from. That limitation is part of the model, not a caveat appended to it.

Income as a multiplicative adjustment

Ability to seek care varies with income, and the adjustment is deliberately multiplicative around unity rather than additive.

An area at the reference income neither inflates nor deflates the estimate, and the elasticity term behaves identically at both ends of the income range. An additive term would have made the correction asymmetric for no defensible reason.

A stress parameter instead of a point estimate

The age-gradient adjustment carries an explicit stress parameter, so the model produces a fan rather than a number.

Sensitivity is a first-class output: the useful answer to a workforce question is rarely a single figure, it is the range the figure sits in and which assumption moves it most.

The mathematics

What the model
actually computes.

Equations and every symbol defined. Fitted parameters — shares, tapers, elasticities, reconciliation bands — stay in the configuration they belong to.

Required specialist supply

Demand for a specialty in an area is population, weighted by age structure, scaled by a morbidity proxy, adjusted for income, and divided by what one practitioner can absorb.

D_req = P [ Σ ki=0 p_i ( v₀ + v_k · μ_i,eff ) ] · [ 1 + ε (I − 1) ] C

P: Population of the catchment
i: Age band
p_i: Proportion of the population in band i
v₀: Baseline visits per year
v_k: Additional visits per unit of hazard
μ_i,eff: Effective hazard for the band — see below
I: Income index for the area
ε: Income elasticity of demand
C: Patient visits one practitioner can absorb

The income term is a multiplicative adjustment around unity, so an area at the reference income neither inflates nor deflates the estimate — a deliberate choice, because an additive term would make the model behave differently at the two ends of the income range for no defensible reason.

Central death rate as a morbidity proxy

Morbidity by age is not published at the granularity this needs, but mortality is. The central death rate stands in for it, bracket-width corrected and adjusted for how far the band sits from the population-weighted reference age.

μ_i,eff = d_x P_x · ω_i · [ 1 + δ x̄_k − x̄_ref x̄_ref ]

d_x: Annual deaths in the band
P_x: Mid-period population of the band
ω_i: Width of the age bracket
x̄_k: Midpoint age, x_k + ω_k/2
x̄_ref: Population-weighted reference age
δ: Stress parameter

Using mortality as a stand-in for morbidity is an approximation and is stated as one. It holds because the two co-move strongly with age; it would fail for a specialty whose caseload is dominated by conditions people live with rather than die from, and that limitation belongs on the page rather than in a footnote.

Skills exercised

What the build
actually demanded.

Against the corpus · 15 systems

Stack breadth 6
Design decisions 4

This system Corpus median

Actuarial

Central death rate and life-table construction
Age-banded exposure and bracket-width correction
Morbidity proxying from mortality, with stated limits
Elasticity-adjusted demand under an income index

Modelling

Population-weighted reference ages
Stress parameters for sensitivity fans
Capacity conversion to required practitioner counts
Catchment-level aggregation over census geography

What it establishes

Every term observable

Each factor can be argued with separately
Mortality as proxy

Stated as an approximation, with its failure mode
Multiplicative income term

Symmetric around the reference
Fan, not a point

Stress parameter makes sensitivity an output