India has become the third country to record more than 1m coronavirus infections, following the US and Brazil, as it reported 34,956 new cases in the past 24 hours, taking the national total to 1,003,832.

New peaks continue to appear around the world, including an alarming rise in the Brittany region of France.

Amid evidence that the disease was taking hold in poorer, rural areas of India with less effective public healthcare, the latest tally prompted renewed concerns about the country’s ability to cope with rising infections.

The figures have been released after a week in which authorities in India were forced to impose new lockdowns, including fresh restrictions on 128 million people in the state of Bihar, which came into force on Thursday.

The continuing and escalating outbreaks, on top of record cases in the United States – which passed 75,000 daily cases in the last count – have dampened hopes that the pandemic is anywhere close to being brought under control, even as researchers race to find a viable vaccine.

Three states in India – Maharashtra, Delhi and Tamil Nadu – account for more than half of the total cases in the country so far. Ashish Jha, the director of the Harvard Global Health Institute, warned that the majority of cases in the country were still being missed.

The continuing rise has forced authorities to reinstate lockdowns in some cities and states.

In Bangalore, the government ordered a week-long lockdown that began on Tuesday evening after a rapid rise in cases.

Dr Anant Bhan, a global health researcher, said India was likely to experience a series of peaks as the virus spread in rural areas. He said the capital, New Delhi, and the financial capital, Mumbai, had already recorded surges, while infections have now shifted to smaller cities.

India’s response to the virus was initially sluggish, but on 24 March the prime minister, Narendra Modi, imposed a three-week nationwide lockdown of its 1.3-billion population.