Cockroach populations are rapidly evolving to become “almost impossible” to kill with chemicals alone, a team of US scientists said.

When cockroaches survive an insecticide treatment, they and their offspring quickly become “essentially immune” to it, a Purdue University study published in Scientific Reports found.

But crucially, they also develop immunity to a range of other insecticides, even if they were never exposed to them - something the scientists call “cross-resistance”.

“We didn’t have a clue that something like that could happen this fast,” Michael Scharf, professor of entomology - the study of insects and their relationship to humans - said in a statement.

“We would see resistance increase four- or six-fold in just one generation.”

Female cockroaches can produce up to 50 offspring every three months, passing their immunity on to them. So even if just a fraction of a cockroach population survives an insecticide treatment and becomes cross-resistant, a decimated population could skyrocket again.

“Cockroaches developing resistance to multiple classes of insecticides at once will make controlling these pests almost impossible with chemicals alone,” said Prof Scharf.

For the study, three different treatments were tested on cockroaches in Indiana and Illinois over a six-month period.

When they rotated three different insecticides over six months, researchers managed to contain cockroach populations - but not to reduce them.

When they tried two insecticides, the results got even worse, as populations boomed.

They also tried a single-insecticide treatment, with mixed results: if the cockroaches had no resistance to it at all, the researchers wiped them out. But if there was even just 10 per cent resistance, populations rose quickly.

The scientists said in the statement insecticides should still be an important component in controlling cockroach populations.

However, they said that when the insecticide alone could not completely eliminate cockroaches, the most effective way to deal with infestations was to combine pesticides with other methods - including traps, improved sanitation and vacuums.

Cockroaches are pests that live exclusively in human habitats, with a preference for moist areas such as sewers, pipes and bathrooms.

They have been found to produce asthma-triggering allergens, carry pathogens including salmonella and E coli and six types of parasitic worms.