A worked example: what the constraint does to one destination

Portugal, 2019, five endemic origins. Reproduced from the model’s own inputs by LCD_mobility/worked_example_v1_v2.py, which reimplements distanceMatrix → getsMatrix → travelProbMat → travelRateMat → constrain_travel_matrix in Python.

Status of the numbers. The constrained side (A_obs, φ, the constrained pair totals) is exact: it comes from UN Tourism and arithmetic, and needs no kernel. The kernel side is a faithful reimplementation, not a bit-identical copy of a MATLAB run — the reconstructed 2019 panel holds 1,667 rows (170 countries + 1,497 NUTS3) against MATLAB’s 1,634, so kernel figures should be read to two significant figures, and the ratios rather than the last digits are the point.

1. The travel input

v1 carries one number per origin per year — TravProp, total outbound trips per head, with no destination dimension. Where those trips land is decided entirely by the radiation kernel.

v2 carries the same TravProp plus the observed bilateral total A_obs(O,J,y) from UN Tourism, and rescales the kernel’s pair total onto it.

origin	tProp (trips/head)	population	A_obs → Portugal	T_ker (kernel pair total)	φ = A_obs/T_ker
Brazil	0.0523	211,782,878	1,298,714	2,967.0	437.7
Thailand	0.1728	71,307,763	15,683	184.4	85.0
India	0.0144	1,383,112,050	73,123	16,494.2	4.4
Mexico	0.1805	125,085,311	47,926	1,245.9	38.5
Philippines	0.0719	110,380,804	15,832	165.4	95.7

The first column is identical in v1 and v2. Everything to the right of it exists only in v2.

Read the last column as the size of the error the submitted model was carrying. The kernel put 2,967 Brazilians into Portugal in 2019; UN Tourism counts 1.3 million. The kernel is not wrong by a percentage, it is wrong by a factor of 438 on this pair — and by only 4.4 on India, which is the more damaging fact, because it means the error is not a constant the incidence term could absorb. It is a reallocation between origins. The radiation model has no colonial history, no language, no diaspora; Brazil→Portugal is exactly the pair it cannot see.

2. The rate matrix

travelRateMat returns rates, not probabilities: r(a,b) = tp(b)·N(b) · p(a,b), where b is the column and the origin. Rates into Portugal’s NUTS3 regions, six largest by population:

NUTS3	region	v1 (kernel)	v2 (constrained)	within-country share
PT170	Área Metropolitana de Lisboa	812.0	355,410	0.2737
PT11A	Área Metropolitana do Porto	395.8	173,235	0.1334
PT150	Algarve	139.2	60,925	0.0469
PT16E	Região de Coimbra	124.0	54,282	0.0418
PT11C	Tâmega e Sousa	96.0	42,003	0.0323
PT119	Ave	94.9	41,540	0.0320

(Brazil column. The full five-origin table is worked_example_v1_v2_nuts3.csv.)

Every v2 entry is its v1 entry times the single scalar φ(BRA,PRT) = 437.7. The share column is therefore identical in v1 and v2, and sums to 1 in both. The constraint takes the allocation between countries away from the kernel and leaves the allocation within a country entirely to it.

Pair totals, as a check on the whole operation:

origin	v1 kernel total	v2 constrained total	A_obs
BRA	2,967.006	1,298,714.000	1,298,714
THA	184.433	15,683.000	15,683
IND	16,494.213	73,123.000	73,123
MEX	1,245.887	47,926.000	47,926
PHL	165.398	15,832.000	15,832

Binding to machine precision, by construction, on every observed pair.

3. The p matrix

travelProbMat — p(a,b) = (nᵢnⱼ)/((nᵢ+s)(nᵢ+nⱼ+s)) — is unchanged between v1 and v2. The constraint multiplies the rate matrix, never the probability kernel:

NUTS3	region	BRA	THA	IND	MEX	PHL
PT170	A. M. de Lisboa	7.333e-05	4.131e-06	2.268e-04	1.379e-05	5.773e-06
PT11A	A. M. do Porto	3.574e-05	2.514e-06	1.395e-04	8.529e-06	3.494e-06
PT150	Algarve	1.257e-05	6.469e-07	3.500e-05	2.106e-06	8.906e-07
PT16E	Região de Coimbra	1.120e-05	6.338e-07	3.547e-05	2.140e-06	8.809e-07
PT11C	Tâmega e Sousa	8.666e-06	6.093e-07	3.410e-05	2.067e-06	8.469e-07

This is the sentence to give a referee: the radiation model is retained in full and unmodified; what changed is that its country-pair totals are now anchored to observed bilateral arrivals rather than left to a kernel calibrated on U.S. county commuting.

4. What this means for the paper

Three points worth carrying into the Methods.

The constraint is one scalar per (origin, destination, year). It has no free parameters and nothing in it is fitted to the ECDC/CDC validation series.
It cannot change the sub-national pattern of any figure, because within-country shares are algebraically invariant under it. Every sub-national result in the paper is still the radiation model’s.
It changes the between-country composition by up to two orders of magnitude on individual pairs, and unequally across origins — which is why the resident channel moved from 3–5% of importations to the majority. That was not a bug; it was the kernel’s misallocation being removed.

Files

LCD_mobility/worked_example_v1_v2.py
LCD_mobility/worked_example_v1_v2_summary.csv — the φ table
LCD_mobility/worked_example_v1_v2_nuts3.csv — per-NUTS3, all five origins
LCD_mobility/worked_example_v1_v2_pmatrix.csv — the unchanged p matrix

Written 26 August 2026.