At last... Jimmy Savile is having a good effect

It is being called the "Savile effect". I'm not sure that anyone would have predicted, when the DJ's criminal career began to be exposed just over a year ago, that the police inquiry would have such a dramatic impact. But rape crisis centres are reporting a surge in calls, with some charities recording increases of 40 per cent. At the same time, more women are going to the police: in London alone, there has been an increase of 29 per cent in rape reports in the past 12 months.

Of course, that still means that most rapes go unreported. Ten days ago, I went with the Mayor of London, Boris Johnson, to the Solace Women's Aid centre in north London. The centre helps 5,000 survivors of sexual and domestic violence each year but staff say that only 20 per cent of their clients report their experience to the police.

It is a chastening fact that, when Johnson was first elected, there was only one rape crisis centre in the country's capital city. He spent £1.4m to set up three more, but the Savile effect is so pronounced that he has had to find an extra £25,000 for the south London centre over the past year.

The breathtaking extent of Savile's crimes – he is believed to have committed sex offences against 450 people, including more than 30 rapes – has given many women the confidence to talk about their own traumatic experiences for the first time. The Savile cases, along with the conviction of the broadcaster Stuart Hall, has smashed the myth of sexual attacks as one-off crimes committed by "normal" men in a moment of madness. A slew of trials, including that of the prolific "black cab" rapist John Worboys, has shown that rapists tend to be serial criminals who select their victims carefully. It is also their downfall: juries may not believe a single victim but they are more likely to convict when several women describe similar experiences.

A landmark case which ended on Friday suggests that the criminal justice system is finally treating rape with the gravity it deserves. A serial rapist, Harbinder Khatkar, received a life sentence after he was found guilty of attacking six women in Derby in a single night. But the Khatkar case is notable for another reason; it is the first time a defendant has been retried in Derbyshire after a change in the double jeopardy law. Khatkar was acquitted of rape in 2011, just six weeks before he attacked three women in the street and another three in their own homes, but the Crown Prosecution Service successfully challenged the verdict.

In London, prosecutions for rape have gone up by almost a fifth in the past year. That is good news, even though police and rape crisis centres desperately need more resources. Jimmy Savile would be turning in his grave, I suspect, if he knew how public discussion of his crimes is helping to bring other rapists to justice.

Joan Smith is co-chair of the Mayor of London's Violence Against Women and Girls Panel;