An expert in this fast-developing field is UK mathematician Keith Still. In 1989, he established Crowd Dynamics, a consultancy which has since advised on everything from worldscale engineering projects to the annual Glastonbury music festival. Over the last five years, the firm has been involved in a project to redesign one of the most densely crowded structures in the world - the Jamarat bridge at Mina near Mecca.

Every year during the four-day Hajj pilgrimage to Mecca, Muslims perform one of the most sacred rites at Mina, where three pillars representing the Devil are stoned in a symbolic rejection of evil. The act of stoning is obligatory to the successful fulfilment of the Hajj and, during the allocated five-hour period, 3 million pilgrims gather on the elevated walkway to perform the rite.

Over the years, this has led to severe overcrowding and it is estimated that during the rite the current two-tier structure holds 15-20 per cent more pilgrims than it is designed to. Inevitably, crushes occur. The worst accident in recent years took place in February 2004, when 249 pilgrims died at the eastern entrance.

Following the incident, Riyadh announced an ambitious programme to improve safety for pilgrims and prevent any repeat of the accident. The estimated $1,400 million Jamarat bridge replacement involves replacing the existing structure with a five-tier, 130-metre-long walkway, complete with associated approach ramps, lifts and escalators and cut-and-cover tunnels. The design has been prepared by Beirut-based Dar al-Handasah (Shair & Partners).

Crucial to the design is Crowd Dynamics' crowd control study. Combining mathematical modelling and historical data, the study highlights the deficiencies of the existing bridge structure and outlines the optimal size and shape of the proposed scheme.

Crowd Dynamics' report focuses on the three key stages of the stoning process - access, lapidation (the act of stoning) and egress. If the various parts of the crowd flow formula fall out of balance, the queue of people at either point can grow exponentially - triggering accidents such as the one in 2004. 'Any crowd in a confined space is like water in a bucket, being fed and drained by different pipes,' says Still. 'We have to stop the bucket overflowing.'

Perhaps uniquely in engineering, crowd modelling also requires an understanding of human psychology. 'On the Jamarat bridge project, I was trying to work out why normally calm people would push and shove so violently, and whether the religious significance of the experience raised hysteria levels,' he says.

Crowd Dynamics' principal tool for analysing the Jamarat crowd was Myriad, a software programme developed by Still. Myriad uses a variety of graphical representations to show where the structure is most strained. Although capable of analysing the movement of individuals, Still says Myriad assesses risk probability 'macroscopically', treating the crowd as a unit. 'One of the most important features of the software is its speed. From the input of a map or architectural plan to the detailed assessment of crowd risk takes five-10 minutes, where consultancies traditionally take weeks.'

Myriad is not without its rivals, however. An alternative approach to crowd modelling has been taken by Nicole Hoffman, associate at