Some of the tepid response may come down to apathy. A nationwide study released in February found that one in five Indian people were likely to have already had Covid-19. Surveys in cities show even higher prevalence rates. The disease is just one among many that people in India worry about, joining tuberculosis, dengue fever and avian flu. Many people are struggling to recover from the huge financial hit of India’s lockdown last year and can’t afford to take time off work to stand in line for a shot.

“These are hand-to-mouth people. Bread, butter depends on their daily work. They can’t sit back and relax and wait for the wave to go,” said Kiran Dighavkar, the assistant commissioner of the Mumbai ward that includes Dharavi. “They can’t afford quarantine, so the only option is to vaccinate these people as early as possible.”

Health experts are prodding Mr. Modi to do more, including making the vaccine available to more people. Older adults, health-care and frontline workers and some people with medical conditions are currently eligible for shots.

“I would try to put the injection in the arm of every Indian that is 18 years and above, and I would do it now,” said Dr. N.K. Ganguly, the president of a medical research institute in New Delhi.

Persuading the 800,000 residents of Dharavi, Asia’s largest slum, to get vaccinated is seen as critical. Residents travel for work to every corner of the city of 20 million. Officials are reintroducing what earlier in the pandemic they called the Dharavi model: If the disease can be contained there, transmission can be curbed citywide and even further afield.

It won’t be easy, even though just three miles away, a jumbo vaccination center is administering about 15,000 shots a day, free of charge.

Day and night, Dharavi is teeming with life. People overflow from thin, corrugated metal houses, stacked on top of each other like matchboxes, onto crowded, mostly unpaved lanes strung with loose electrical wire. Animals skitter between parked motorcycles and piles of debris. Shops, tanneries and factories are squeezed next to houses of worship and community toilets.

“We have been OK all this while,” Abdul Razad Rakim, a 61-year-old diabetic, said from a foldout chair in front of the tiny apartment he shares with his wife, Shamim. “Why do we have to go?”

A short walk away, Janabai Shinde, a former janitor for the city health department, was squatting on her front step, rising every few minutes to spit red tobacco juice into a drain.

“I take walks in this lane. I sit here for fresh air. I have not stepped out much since the lockdown,” Ms. Shinde said. Her son, who works for the city, has already registered her for a turn at a vaccination center. She said she hoped her neighbors would join her.

“It’s for our good,” she said.

The Mumbai government has enlisted aid groups to set up help desks in Dharavi, where residents can ask questions and complete online registration to make an appointment for a free shot.

Plans are underway to set up a vaccination center within the confines of the slum, and to reopen an institutional quarantine center with thousands of beds, according to Mr. Dighavkar, the assistant commissioner.

Last week, as Maharashtra recorded its highest new case numbers since September, the chief executive of a disaster relief group delivered a pep talk at Gold Filled Heights, an apartment complex largely occupied by members of the Jain religious group, who run many of the jewelry businesses in Dharavi.

“We can’t let the virus spread again,” said the chief executive, Shantilal Muttha. “If it spreads in Dharavi, it becomes a threat for the entire Mumbai and Maharashtra.”

Jyoti Shelar contributed reporting.

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