World health officials fear the disease could wreak havoc in Afghanistan, which is one of the world's poorest countries and has little healthcare for many inhabitants.
Government officials had by yesterday recorded 665 cases and 21 deaths, but a lack of testing means the true tally is expected to be far higher.
Large numbers of infected Afghans are feared to have streamed into Afghanistan from neighbouring Iran during recent weeks.
Hamdullah Nomani, a former Taliban minister and now deputy director of the insurgents' health commission, said that the movement's rural anti-coronavirus campaign was "in full swing".
He said: "In areas under our influence, those returned from Iran must be quarantined at home for 15 days. They are banned from going to worship places, the market, or any other social places. If anyone gets Covid-19 symptoms it's compulsory to visit a nearby hospital."
The movement is keen to demonstrate to Afghans, and the international community, that it could mount a responsible administration as well as a ruthlessly determined insurgency, analysts believe.
Yet it also remains unclear how much action the militants are willing or able to take against the virus and how much is propaganda.
Residents of Taliban-controlled areas gave a mixed picture.
Haji Shah Baluch, 50, a shopkeeper in the southern province of Helmand, said life was unaffected and locals had only heard of the virus from the media. The Taliban had done nothing, he said.
In the northern Takhar province, 43-year-old Nabi Khan said the Taliban had visited, but provided little support. "We are so poor we even could not afford handwash, forget about a mask."
