NEW DELHI—As coronavirus cases rose rapidly across India in April, a 70-year-old woman died at her home in the eastern state of Bihar. A rapid antigen test for Covid-19 had been positive, and a lung scan had indicated viral pneumonia and the “possibility of Covid infection.”

But Shila Singh’s death hasn’t been counted among India’s Covid-19 toll.

India has officially recorded more than 390,000 coronavirus deaths, but families who have lost loved ones, health experts and statisticians say that vastly undercounts the true toll. Families like Mrs. Singh’s have been left struggling to get compensation that some states have set up for Covid-19 victims.

India’s undercount has also left a huge gap in the world’s understanding of the impact of the Delta variant, which health experts believe helped drive one of the world’s worst Covid-19 surges in April and May. India was the first to detect the highly infectious variant, which has hopscotched around the world. It is fueling a surge in the U.K., and is expected to become the dominant variant in the U.S.

An accurate count of Covid-19 infections and deaths is “a very important part of understanding how big a threat new variants are,” said Christopher Murray, director of the University of Washington’s Institute for Health Metrics and Evaluation.

Health workers transport a body to a crematorium in New Delhi.

Health workers transport a body to a crematorium in New Delhi.

Photo: Danish Siddiqui/Reuters

The institute’s modeling suggests the true Covid-19 death toll in India exceeds 1.1 million, close to three times the reported number. Dr. Murray estimates the scale of the undercount is similar to those in some countries in Latin America or Africa. He said the institute estimates India has detected only about 3% to 5% of all infections due to insufficient testing.

Murad Banaji, a mathematician at the Middlesex University in London who has been tracking the pandemic in India, estimates the country’s real death toll could be around five times the reported figure, based partly on mortality and serosurvey data, which shows the presence of antibodies in the population.

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The undercounting of infections and deaths is a problem world-wide, even in countries with widespread testing. The World Health Organization said last month that the global Covid-19 death toll could be two or three times the official number. The problem, however, is particularly acute in the developing world, where access to healthcare and coronavirus testing is often more limited.

Health experts point to a number of issues that have led to widespread undercounting in India. One is that hospitals were turning patients away in April and May as the rapid rise in cases overwhelmed the healthcare system. That left many people to die at home without being tested for Covid-19.

At the time Mrs. Singh tested positive, her family was under tremendous strain from the disease, according to her son, Neeraj Shekhar. He himself had tested positive five days earlier, and Mrs. Singh’s husband and two teenage grandchildren tested positive the same day that she did.

Shila Singh

Shila Singh

Photo: Neeraj Shekhar

To qualify for its Covid-19 compensation payment of 400,000 rupees, equivalent to about $5,400, the state requires a report from a certified lab, which at the time were taking days to process. The family got a test strip from the lab indicating that Mrs. Singh was positive and rushed to a doctor.

The physician, Dr. N.P. Narain, said he ordered a CT scan and told the family to take Mrs. Singh to the hospital if her condition worsened. “At that time they probably didn’t think they would need all this proper documentation,” he said.

With multiple members of the family sick, Mrs. Singh’s condition deteriorated rapidly and she died five days after testing positive. The family could now use the state compensation to pay the nearly $1,300 in medical bills they racked up caring for family members with Covid-19, said Mr. Shekhar.

Health experts say many Covid-19 deaths have gone uncounted among India’s vast population of rural poor, who have little access to healthcare or Covid-19 testing. In the village of Sirondhan in the northern state of Uttar Pradesh, for example, Vijay Pal Singh said his 38-year-old wife, Mithilesh Devi, fell ill last month with a fever and struggled to breathe. Mr. Singh said he took her to the village clinic and several hospitals in the district, but none had available beds or testing kits.

India's Covid-19 crisis has resulted in record numbers of cases and deaths. WSJ breaks down the chain of events that led to the fastest-growing wave of infection since the pandemic started, and what it means for the world. Photo: Samuel Rajkumar/Reuters The Wall Street Journal Interactive Edition

“She died at home, gasping for oxygen,” Mr. Singh said. Her death wasn’t included in the official Covid-19 tally, he added.

At least 30 people died in the village in late April and early May, many suffering from Covid-like symptoms, according to villagers and social workers. A further 47 died in two neighboring villages.

Villager Dharamvir Singh said one or two people died almost every day during the worst weeks of the second wave. He says he believes he and four members of his family were infected. No one was tested.

“The official numbers, at least for our village and a few others close to us, are totally wrong,” Mr. Singh said.

A senior official of the Uttar Pradesh government said there has been no underreporting of Covid-19 deaths, and that the state doesn’t plan to revise or update its official death tally.

Mr. Banaji, the mathematician, says the central government has tended to praise states with low death counts and castigate those with higher counts as incompetent. “This narrative of success and failure centered on fatality numbers is very dangerous,” he said.

Prime Minister Narendra Modi’s office didn’t respond to requests for comment. India’s health ministry said the government has regularly emphasized the need for robust systems to monitor and report cases and deaths in all districts.

Health workers tested a woman for Covid-19 at Jamsoti Village, Uttar Pradesh on June 8.

Health workers tested a woman for Covid-19 at Jamsoti Village, Uttar Pradesh on June 8.

Photo: Rajesh Kumar Singh/Associated Press

Some states and cities in India have revised their death tolls after criticism by politicians and judges, but experts say the adjustments fall far short. In Maharashtra, home to the financial capital of Mumbai, a weekslong effort has added more than 8,000 deaths to the official tally. The capital city of New Delhi also revised its infection and death counts, increasing them by more than 44,000 and 700 respectively, in the midst of the second wave in April.

A recent investigation in the state of Bihar found nearly 4,000 more Covid-19 deaths, raising its toll by over 72%. They included deaths at home, as well as in private hospitals and from health complications related to Covid, said Dr. Sunil Kumar, the Bihar state secretary for the Indian Medical Association.

To have a death counted, families have to produce test results from a certified lab or confirmation of infection from the healthcare system—but often, like Mrs. Singh’s family, they don’t have the required paperwork. Dr. Kumar said he has treated patients who died after suffering from common Covid-19 symptoms, but because their families couldn’t produce a positive test the deaths weren’t included in the recent count.

“The figures could be higher based on what we are seeing,” he said

—Krishna Pokharel contributed to this article

Write to Shan Li at shan.li@wsj.com, Suryatapa Bhattacharya at Suryatapa.Bhattacharya@wsj.com and Vibhuti Agarwal at vibhuti.agarwal@wsj.com