Lifesaving maths…you know something, John Snow

Many Londoners may walk past the John Snow pub in Soho and assume it is the work of a passionate Game of Thrones fan, but it’s actually a memorial to a physician who developed an ingenious mathematical device to prove the waterborne transmission of cholera.

By September 1854, a cholera outbreak had decimated Soho, killing 10% of the population.  Doctors believed that the disease was spread by “bad air” emanating from the stinking open sewers.

John Snow’s theory and how he proved it

For all its progress, nineteenth century London was a filthy city.  No sewerage system meant that waste was thrown into cesspits or open sewers that regularly contaminated drinking water supplies. Doctors blamed the regular outbreaks of disease in the city on the stench, believing the miasma theory of disease, which held that disease was spread through “bad air”.

But the physician John Snow had his own ideas and realised that data would prove his theory that the disease spread through water.

John Snow’s map. Each bar represents a death at an address. The curve
marks points at equal distance from the Broad Street pump and another pump.

Mapping the data

Snow created a map showing the geographic spread of deaths in the outbreak.  Each bar on the graph represents a death at that address, showing as many as 18 people dying in particular households.  This representation of the data shows that most of the deaths were tightly clustered around the water pump at 40 Broad Street (now Broadwick Street) in Soho.  Snow’s research had led him to believe that the Broad Street pump was the source of the disease, and the data backed up that theory. But how could he show that it was most likely that this particular pump was the source when there were other pumps nearby?

His next step was to represent the time it took to travel to the Broad Street pump on his map and to calculate who was most likely to use each water pump in the area. Snow drew a curve on the map that marked the points where the Broad Street pump was at equal walking distance from neighbouring water pumps. The Broad Street pump was the closest source of water for those who lived inside this curve. Almost all the deaths marked on the map lay inside this curve and anecdotal evidence explained the few cases that did not.

The Voronoi diagram

This mathematical device is called a Voronoi diagram. Suppose you have a number of sites (such as the water pumps in Snow’s maps) spread out over an area you can map. The dots on a Voronoi diagram represent these sites and the points on the edges on the diagram are exactly those points that are equidistant between two (or more if you are on a corner of a region) sites. The edges divide the diagram up into regions, or cells that enclose all the points that are closest to the site in that particular region. Voronoi diagrams are widely used to study spatial relationships, for example to study competition between plant species and to model economic markets.

A Voronoi diagram. Image by David Austin

This convincing mathematical analysis of the cholera outbreak in Soho convinced the authorities that Snow’s theory that the disease was transmitted through water was correct. The handle to the Broad Street pump was removed and the outbreak died away, though Snow himself said that by that time the disease was already on the wane as people had already fled the area.

Either way, Snow’s mathematical evidence that cholera was waterborne is one of the founding moments of epidemiology and the use of mathematics to understand disease, one of the greatest advances in medicine that has saved millions of lives.

Further reading:
Uncovering the Cause of Cholera by Rachel Thomas

Pamela Hellig
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