From maps to minutes

How does a map become an estimate of the time it takes to reach care? The unequal road to care uses modelled travel times to the nearest mapped health facility. These are calculated journeys, not trips someone has driven. This notebook follows that healthcare calculation. The girls' school analysis uses the same broad approach, but requires its own destination list and travel-time surface.

One approach is to calculate routes along a road network, from a village or another chosen starting point to nearby facilities. Another divides the country into a grid and estimates the time needed to move between neighbouring cells. The grid approach gives a travel time to every cell, including places away from mapped roads. Its detail depends on the size of the cells and the movement assumptions built into them.

For healthcare, the Malaria Atlas Project has already published that travel-time grid. Adaad combined it with boundaries and population estimates to calculate local and national averages. This notebook explains how the grid is made, then follows the steps Adaad carried out. It shows why the same map gives an average of about 204 minutes across cells and 22 minutes across people.

Two interactive exercises let you explore a grid and change where people live. A spreadsheet exercise reproduces the national healthcare figures from the tehsil table. By the end, you should be able to explain a travel-time surface, calculate a population-weighted mean, and say what a national average leaves out.

A map made of cells

A digital map can describe shapes or a grid. A tehsil boundary is a polygon, a road is a line, and a clinic is a point. These are vector data. A raster divides space into cells and stores a value in each one: a travel time, a population count, or a cost of movement. This analysis brings the shapes and the grid together.

The travel grid has cells 30 arc-seconds on a side, or one 120th of a degree. That is roughly 0.9 kilometres north to south; the width shrinks with latitude, from about 0.8 kilometres around Karachi to about 0.75 around Hunza. The clipped rectangle is 1,608 rows by 2,018 columns, with about 1.2 million valid cells inside the reporting boundaries.

A road narrower than a cell can influence its value. But the output still assigns one travel time to the whole cell. It cannot distinguish two homes within that cell, and a sub-minute result should not be read as a precisely measured trip. Resolution limits the questions a map can answer.

The friction surface

The first layer describes how costly movement is. The Malaria Atlas Project's friction surface stores minutes per metre, not minutes per cell. Multiplying that cost by a movement distance gives a time. A diagonal step between cells covers more ground than a horizontal one, so a routing calculation must account for distance as well as the cell values.

The researchers combined roads, land cover, terrain and other transport information. Their road inputs draw on OpenStreetMap and Google sources. Roads can permit faster movement; slopes and land cover can slow it. Water is handled through transport assumptions too, rather than treated as an impassable barrier everywhere. These choices turn geography into modelled movement costs. The friction-layer documentation gives the units and inputs.

The motorized scenario allows vehicle travel where the model permits it. The walking scenario removes that option. They are scenarios, not reliable upper and lower bounds on any household's journey. Waiting for transport, road damage, fares and seasonal conditions can all change the trip. When a map disagrees with local knowledge, inspect the destination list, grid, dates and movement assumptions before blaming the routing algorithm.

The shortest path

Imagine starting at the mapped clinics and working outward across the grid. At each step, visit the cell with the shortest travel time found so far among those still waiting to be visited. Then check whether travelling through it offers a quicker route to any neighbouring cell. For example, reaching a cell by a road may reveal a faster way to its neighbour than the route across rough ground found earlier. This is the idea behind Dijkstra's algorithm.

With movement costs of zero or more, a cell's time becomes final when it is visited in that order; the algorithm calls this cell ‘settled’. Finding a possible route to a cell is not the same as settling it. The calculation can be exact for the grid it is given while the grid remains an approximation of the world.

An exact answer to a model is still a modelled answer to a journey.

The Malaria Atlas Project has already published global travel-time surfaces to hospitals and clinics mapped in OpenStreetMap and Google Maps. For this issue, Adaad reused those surfaces; we did not run a new shortest-path calculation. The study and methods explain their construction, and the project download page supplies the layers.

“Nearest” here means the mapped facility reached at the lowest modelled travel cost. It does not mean a facility with the right service, an available clinician, medicine in stock, or an affordable fee. Nor is the estimate a guaranteed minimum for an actual journey. Omit a real facility and modelled times can rise or stay unchanged; include a closed or wrongly located one and they can fall. The destination list is part of the definition of access.

From cells to places

The surface answers questions about cells. To report on a tehsil, we assign cells to its boundary and summarise their values. This is called zonal statistics. Our build uses the cell centre to decide which polygon receives a cell, so small boundaries and coastlines are especially sensitive to the grid's placement.

The 2015 boundary frame contains 147 districts and 553 tehsils; later subdivisions remain within their older parent boundaries. Of those tehsils, 552 have valid cells and positive modelled population in this build. Manora Cantonment has no valid cells and its estimates are missing, not zero. The median among the 552 is 1,254 cells. Karachi Cantonment has seven, representing about 252,000 modelled people.

Publish the cell count because it reveals how coarse the spatial representation is. It is not a sample size or a guarantee of statistical reliability: neighbouring raster cells are related, and thousands of cells can share the same underlying model error.

Weighting by people

There are two different questions to ask of the same surface. What is the average travel time across cells? And what is the average across the people allocated to those cells? In the tehsil-covered grid, the answers are about 204 minutes and 22 minutes respectively. Each cell counts equally in the first calculation. Each modelled person counts equally in the second.

The first is a mean over cells, not strictly an area-weighted mean: these degree-based cells have different physical areas at different latitudes. It describes the grid's geography. The population-weighted mean is the relevant measure for a statement about average access among the represented population. Neither calculation describes the experience of every resident.

The same weighting rule lets us combine tehsils. Killa Saifullah's six tehsils range from 41.3 minutes in Kan Mehtarzai to 374.5 in Badini, about a nine-fold spread. Weighting each tehsil's mean by its population gives 112.3 minutes across the six. A district average hides these local differences.

In Shigar, the mean across cells is 1,086.8 minutes, against 431.3 when weighted by population. The difference is consistent with people being concentrated in more accessible parts of a mountainous tehsil. It does not independently prove where every household lives. WorldPop estimates the distribution of people using census information and covariates, including roads; its unconstrained method can allocate population across the landscape rather than only to identified buildings.

If a population model wrongly places people close to roads, it can pull weighted travel times down. If it places people in uninhabited remote cells, it can push them up. Sharing a road input with the travel model is a reason to check sensitivity, not evidence that the net error must go in one direction. Better weights require better information about settlement, not a presumption that weighting removes uncertainty.

For most tehsils, population weighting lowers the mean. But in 13 of the 552 valid rows it raises it. This is possible whenever the model places more people in cells that are less accessible than the tehsil's typical cell. The diagonal in Figure 3 marks agreement; points on either side answer different spatial questions.

A mean also hides the tail. The national population-weighted mean is 22.1 minutes, yet the table implies about 7.45 per cent of its represented population lives in cells more than 60 modelled motorized minutes from a mapped facility: roughly 16.36 million people on the 2020 population base. Publish a threshold share alongside the mean. A small average can coexist with a large group facing long journeys.

The recipe, and what it needs

For an existing healthcare surface, the practical sequence is to align the travel and population grids, assign valid cells to boundaries, then calculate summaries with a common mask. A mask is simply the set of cells admitted to a calculation. Grid alignment, missing values, boundary rules and resampling all need explicit choices. The published CSV lets you practise the final step without installing mapping software.

For new destinations, start earlier. Define exactly what belongs in the list: for example, government girls' secondary schools operating on a stated date. Record coordinates, source, service type and verification status. Check duplicates, misplaced points and coverage against a second source or a local sample. An accessibility map for that list then requires a new least-cost surface before the same boundary and population summaries can be calculated.

An advanced assignment is to choose one district, audit a destination list, and compare results after removing unverified locations or changing travel assumptions. Report which places change most, the population affected, and the evidence needed to decide which scenario is credible. Runtime depends on the grid, implementation and hardware; the important deliverable is an auditable calculation, not a promise that a national map takes a few minutes.

The choices continue through the computation: which neighbours connect, how water is treated, which cells are missing, and how population is moved between grids. A useful map makes those choices visible. A useful sentence then says whose access is being summarised, to what kind of destination, under which assumptions, and on what date.