How should we react to yet another needless disaster on our train system? You’ll note I say “yet another" because apart from newsworthy tragedies like the Elphinstone Station stampede and others, about nine people die on the Mumbai suburban rail network every day. Not so newsworthy, sadly. Still, how should we react?

It’s likely most of us who have travelled on those trains thought “That could have been me": because we’ve walked along—or been propelled along—overbridges like at Elphinstone Road. Most of us probably also were angry at railway and other authorities for ignoring multiple warnings about the bridge and, worse, for focusing on frivolities like changing the names of stations—Elphinstone Road included—rather than commuter safety. (Did you know the railway minister was actually in the city that day for the function to rename Elphinstone Road to Prabhadevi?)

And some of us also checked what mathematicians have had to say about situations like the tragedy on that bridge. Here, I’d like to offer you a flavour of that.

Actually, mathematicians have studied collective behaviours for a long time. Ants, birds, humans—how do they behave when they gather in groups? How does that differ from each creature’s individual behaviour? Does the collective behaviour emerge from all those individual behaviours? Or is it a completely separate phenomenon? We’ve all heard of “mob behaviour". But what causes it? And in particular, how do crowds like at Elphinstone Road behave? Can we learn lessons that might prevent future tragedies? How can we use mathematical models and computer software to learn about the way crowds of humans interact with our geographical surroundings?

In 2014 Marie-Therese Wolfram, a mathematician at the University of Warwick, made a presentation titled “On the Mathematical Modeling and Simulation of Crowd Motion". To simulate the flow of pedestrians, she used the “Hughes model" (proposed by an Australian professor of civil engineering, Roger Hughes). 

Consider the assumptions Wolfram made to properly simulate crowd motion:

The speed at which a pedestrian moves depends on the density of the surrounding pedestrian flow

That is, when you’re in the middle of a tightly-packed crowd that is itself in motion, you can’t really choose your own speed, whether slower or faster than others around you. You’re really just swept along by them. (Conversely, if the crowd is relatively sparse, you can walk faster or slower than the others). 

Pedestrians have a common sense of the task

Meaning, everyone in the crowd is focused on getting to the destination—an exit from a stadium, or a platform at the end of the overbridge. But what’s more, there’s no advantage if you move to another position in the crowd that’s about the same distance from the goal. For the person in that position thinks the task ahead of her is as about as difficult as you think yours is.

Pedestrians try to reduce their travel time, but want to avoid high densities

Seems logical, right? You want to get to the exit as soon as possible, but you don’t want to be trapped in that crowd you see developing before you. Everybody else wants just the same.

Expressing those assumptions via mathematical equations, Wolfram built a model for pedestrian flow. She then applied it to a “fast exit scenario", in which a group wants to quickly exit an enclosure that they think has just one exit. The mathematics involved is beyond what I can explain here, but it suggested to her some intriguing avenues for more investigation:

What if pedestrians have only “local vision"? That is, only some of them are tall enough to see over the crowd and decide where the high densities are. The rest can only see their immediate surroundings. What does such a limitation—obviously, a better reflection of reality—do to crowd behaviour?

l Suppose you had one or a few “leaders" in the crowd, who make efforts to move in a different direction, aiming to ease the congestion. Can that work, and if so, how many such leaders are necessary?

Mathematicians have worked on these ideas. With two colleagues, Wolfram herself studied an improved Hughes model, incorporating local vision. They found scenarios in which part of the crowd would wait for congestion to ease at the exit, or even turn around and look for another exit. Whether these are rational behaviours depends on what the reality is. But it at least suggests that a crowd pressing towards an exit does not have to end up with some of its members dead.

Last year, a team headed by Massimo Fornasier at the Technical University of Munich reported the results of their modelling of group behaviour.

One experiment was with two groups of about 40 people each, both given the same task to work on. But unknown to the subjects, one of the groups contained two “incognito informed agents". These two moved “very determinedly" in a specific direction. The result? That group chose that direction too. That is, these two determined individuals were able to influence the whole group. Fornasier and his colleagues showed that this would happen with larger groups too. For every 100 individuals in a crowd, they estimated that just two or three such agents would be able to control the overall behaviour. 

Of course it would not have been possible to insert a few such individuals into the crowd on the Elphinstone bridge. But it raises the question: what if a few people already in the crowd on the bridge—3-4 friends perhaps—had suddenly started moving in the opposite direction, maybe shouting loudly to make their intention clear? No doubt they would have found it hard at first, but perhaps just the impression of leadership and purpose would have convinced people to follow them. Enough people, maybe, to ease the pressure on the staircase where 23 were killed. I’m not sure it would have worked, and perhaps the pressure would have simply moved to the other end of the bridge. But in the face of impending disaster, this might have been at least worth trying. 

Paul Torrens, professor of computer science at New York University, models crowd behaviour in “extraordinary situations" that would be impossible to test in real life— like a crowd trying to escape a building collapse, or fleeing from a burning car before it explodes. His starting point is the individual, expanding on this idea of local vision. Even if an individual is caught in a crowd, she has her own intelligence and emotions and respect for authority, her own ability to look around and plan, the tendency to converse with those around her. As he remarks on his website: “From the perspective of the crowd as an aggregate, it may appear that the crowd mass is moving in unison. However, in reality, crowd participants are likely pursuing their own individual behaviour, but have access to similar information in the immediate surroundings."

If each “crowd participant" acts like this, applies this information to the situation that’s unfolding around her—as any of us would do—and if everybody else in the crowd does the same, what kind of collective behaviour will result? 

Because Torrens gives each individual a certain level of human characteristics and reasoning power, there’s a certain sense of reality to his models. Therefore, there are some interesting lessons from them. For example, he can simulate a crowd caught in a confined area and starting to run because someone shouts “Fire!" From the simulation, we can learn where bottlenecks form and why; what effect the number, size and location of exits has on the crowd; and more. He found, for example, that if there’s a column in front of a gate, but off-centre, it tends to impede, slow and ultimately decongest a crowd that is pressing towards that gate. Perhaps that’s not a reasonable thing to happen if the crowd is trying to escape a fire and time is critical. Then again, decongesting a bottleneck at an exit would likely speed up the flow of people through the exit anyway.

One surprising result of such crowd analysis is the realization that “stampede"—in the sense of a mindless mob rushing headlong and trampling anyone who stumbles—is probably the wrong word for these crises, precisely because individuals retain their ability to think and reason. Torrens himself says “The idea of the hysterical mass is a myth." Keith Still, a widely consulted crowd science professor at Manchester Metropolitan University, put it this way: “People don’t die because they panic. They panic because they are dying."

Put that simply and starkly, it’s hard not to wonder. Simply by its ability to decongest crowds, an off-centre column might have saved lives at Elphinstone Road. Simple measures. Why not try them?

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance. His Twitter handle is @DeathEndsFun

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