An interesting if highly technical paper called “The Dynamics of Crowd Disasters: An Empirical Study” by Dirk Helbing, Anders Johansson, and Habib Zein Al-Abideen derives mathematical modelling of crowd behaviourm, based on analysis of videos of the crowd disaster in Mecca during the Hajj on January 12, 2006.
The authors conclude that:
“…At occupancies of about 7 persons per square meter the crowd becomes almost a fluid mass. Shock waves can be propagated through the mass, sufficent to … propel them distances of 3 meters or more. … People may be literally lifted out of their shoes, and have clothing torn off … Turbulent waves are experienced in dozens of crowd- intensive events each year all over the world. Therefore, it is necessary to understand why, where and when potentially critical situations occur. Viewing real-time video recordings is not very suited to identify critical crowd conditions: While the average density rarely exceeds values of 6 persons per square meter, the local densities can vary considerably due to dynamical patterns in the crowd. Moreover, evaluating the local densities is not enough to identify the critical times and locations precisely, which also applies to an analysis of the velocity field. The decisive quantity is rather the variance of speeds, multiplied by the density, i.e. what we call the “pressure”. It allows one to identify critical locations and times. There are even advance warning signs of critical crowd conditions: The crowd accident on January 12, 2006 started about 10 minutes after “turbulent” crowd motion set in, i.e. after the “pressure” exceeded a value of 0.02/s2 . Moreover, it occured more than 30 minutes after the average flow dropped below a critical threshold , which can be identified by watching out for stop-and-go waves in accelerated surveillance videos . Such advance warning signs of critical crowd conditions can be evaluated on-line by an automated video analysis system. In many cases, this can help one to gain time for corrective measures such as flow control, pressure relief strategies, or the separation of crowds into blocks to stop the propagation of shockwaves ..”