Nothing puts your home insurance premium up like having been burgled in the past – because it means you’re more likely to be burgled again. Stanford researcher Nancy Rodríguez, with colleagues Henri Berestycki (who is first author, for the record) and Lenya Ryzhik, has developed a travelling waves model to explain this phenomenon – and, more importantly, how to stop it.

Crime, according to past research, tends to cluster in particular neighbourhoods – and even individual houses. Once a crime epicentre has been established, criminal activity tends to spread out in a wave pattern, gradually engulfing larger and larger areas.

Travelling waves

The idea of travelling waves came to the fore in the 1950s – Russian scientist Boris Beloüsov discovered an oscillating reaction (warning, may hurt eyes) and reputedly submitted an article on it to a journal, only for it to be rejected as thermodynamically impossible. Beloüsov did what any of us would have done and threw his toys out of the pram, completely withdrawing from science.

Luckily, the discovery was picked up by Zhabotinsky, who persevered with it, or else we’d have neither that brilliant technicolour nor a good model for how crime spreads.

Reaction-diffusion-advection

Rodríguez’s model, a reaction-diffusion-advection system, is made of three elements. Anyone care to guess what they are? Well done. A reaction term, roughly meaning that once crime happens somewhere it tends to carry on happening there; a diffusion term, which damps down crime, and an advection term, which tends to spread crime around. More specifically:

\[ s_t = \Delta s – s + s_0(x) + (\alpha(x,t))u(x,y), \]
where $s(x,t)$ is the propensity towards crime, $(\alpha(x,y))$ is the total payoff (ill-gotten gains minus costs) of committing crime, $s_0$ is the base propensity and $u(x,y)$ is the moving average of crime.

One of the model’s innovations is to allow for the community attitude towards crime to vary. When everyone is perfectly happy to commit profitable crime ($s_0 > 0$ and $\alpha > 0$), it turns out (to everyone’s surprise) that crime tends to spread. When the population is neutral, things aren’t much better – when $s_0 = 0$ and crime is profitable enough, a hot-spot can still form; when the local populace is ready with the perfectly-legal pitchforks ($s_0 < 0$), though, you have two distinct spatial equilibria: you still get occasional waves of crime – but they’re mixed with waves of no crime at all.

Defined: the gap

Moreover, if you can surround the crime zone with a big enough ‘gap’ where no crime takes place – for example, by increasing police presence there, or bulldozing large sections of the city ((I don’t think Dr Rodríguez is advocating this, although it would probably work.)) – you can prevent the wave spreading. It’s a similar idea to containing forest fires: if you surround a fire with fireproof stuff, eventually it burns out.

Rodríguez says that simply increasing police numbers isn’t the solution to stopping crime: “It’s not enough to crack down on crime without cracking down on the attitude of the community.”

Further reading

• Stanford News summary
• Dr Rodríguez’s slides for a talk on the topic
• Link to submitted paper.

Thanks to George Stagg for bringing this to our attention.