In early February, my sister posted a video in our family’s WhatsApp group.It was a seven-minute CNN report on Malawi, a country in East Africa that is one of the world’s poorest. Coronavirus vaccines were nowhere to be found in Malawi, the report said, because richer countries were hogging the supplies. The video focused on Queen Elizabeth Central Hospital in Blantyre, Malawi’s second-largest city, showing the strain the facility was under as it battled the virus. The hospital’s workers were tending to infected patients but had little prospect of getting vaccinated soon.My sister Pui-Ying, a pediatrician, was one of those unprotected workers.“Sorry to see this,” I texted back, lamely.Unwritten was a question: Would Pui-Ying, a frontline doctor who would have been vaccinated against Covid-19 months ago if she worked in the United States, be the last of my family to be inoculated?For more than a year, the pandemic has divided families around the world, with relatives unable to travel to see one another without fearing for their health, waiting out quarantines and navigating red tape. But even as vaccines become available, the hopes for reunions have been tempered by the higgledy-piggledy nature of the rollouts.Many families are at the mercy of a global game of vaccine geopolitics, as wealthier countries struggle to balance immunizing their own citizens against providing supplies to others. Some nations have no doses. Where there are supplies, there are also vaccine misinformation, technological hurdles and personal doubts about the efficacy and safety of vaccines.My family, spread across three continents, has been a microcosm of these forces. Our parents live in Hong Kong; Pui-Ying in Blantyre; and another sister, Pui Ling, and I in the San Francisco Bay Area. We are separated by many time zones and thousands of miles.My parents have worried about the safety of the vaccines, their anxiety fueled by a combination of media reports and their political views and health histories. Pui Ling and I have grumbled about our chaotic local vaccine situations, knowing that we would soon get our chance — even as Pui-Ying, who works in a two-room Covid ward for children, wondered if she would get any shot at all.Dr. Kate O’Brien, director of immunizations and vaccines at the World Health Organization, said that what my family faced was the result of a “very discoordinated and incoherent set of approaches, where every country was making decisions for their own context.” Many families — including her own — were grappling with the same inequities, she said.Once countries procure vaccines, they generally give priority access to the highest-risk groups, including health care workers and seniors. If that were the case with my family, Pui-Ying would be at the front of the line, along with Dad and Mom, who are in their mid-70s. Pui Ling, who works at a foundation, and I, a New York Times editor, would be at the back of the queue.That was not how it turned out.Hong KongAs Britain and the United States began administering vaccines in December, Mom and Dad surprised us. Maybe, they said during a WhatsApp video call, they wouldn’t get vaccinated. What if the shots were unsafe?I was shocked. The coronavirus had made it impossible for us to see one another in person. Hong Kong, a densely packed city of 7.5 million, had instituted strict travel and quarantine rules. In the United States, the pandemic was out of control. Vaccines held the key to freeing us from a year of stifling restrictions.And it wasn’t like my parents to fall for anti-vaccine conspiracy theories. What was I missing?They explained that their concerns were rooted in Mom’s medical history. She had once had a severe allergic reaction to medication (something that they hadn’t deemed important enough to tell us at the time). And they were closely following Chinese and Western media reports about people suffering health problems after getting vaccinated, even if it was unclear whether those were coincidences or actual reactions to the shots.I called Benjamin Cowling, the head of epidemiology and biostatistics at Hong Kong University. Vaccine hesitancy was widespread in Hong Kong, he said. In a January survey, 50 percent of respondents said they intended to get vaccinated, compared with 89 percent in China and 75 percent in the United States.The low numbers in part reflected Hong Kong’s success at containing the virus, Dr. Cowling said. Roughly one-tenth of 1 percent of the city’s population had received a Covid diagnosis. With the risk of infection seemingly small, people were in no hurry to be inoculated.Pui-Wing Tam’s parents in Hong Kong. They initially had concerns about getting vaccinated.Lam Yik Fei for The New York TimesThat didn’t stop us from urging Mom and Dad to get the vaccines. Pui-Ying, who carries the voice of authority as a doctor, told them that she might be able to visit Hong Kong this summer and would prefer them to be alive.Our parents’ thinking evolved. It helped when an 80-year-old uncle in Tallahassee, Fla., was vaccinated in January. He was the first in our extended family to be inoculated, and the news sparked much excitement on a WhatsApp thread among my aunts, uncles and parents.“It would be safer to get it” than not to, Dad concluded.Then came another curveball. Mom and Dad announced that they were willing to get only vaccines made in China. At least four Chinese biopharmaceutical companies, such as Sinovac and Sinopharm, had developed Covid vaccines, joining a field that also included Britain and Sweden’s AstraZeneca, Russia’s Gamaleya Research Institute and America’s Johnson & Johnson, Moderna and Pfizer, the last of which had teamed up with the German company BioNTech.The wrinkle was that Sinovac’s two-shot vaccine had an efficacy rate of about 50 percent, according to clinical trials. That was substantially lower than the European and American vaccines, in particular the shots from Pfizer-BioNTech and Moderna. Those prevented about 90 percent of infections, according to the Centers for Disease Control and Prevention.Yet Dad was firm. Years of a deteriorating relationship between China and the United States, including a trade war, had made him skeptical of American superpower. A proponent of a strong, united China, he was proud of the country’s rise in recent decades. “I’m patriotic,” he said about the vaccine choice.China and other countries have nurtured such vaccine patriotism. Last month, Beijing promised expedited visa processing to foreigners inoculated with Chinese-made vaccines. Britain has also wrapped the Union Jack around the AstraZeneca vaccine, which was developed with researchers at Oxford University, said Claire Wardle, the U.S. director of First Draft, a nonprofit focused on global misinformation.My sisters and I were simply relieved that Dad and Mom would take a vaccine. Get whatever you can, we told them, because any vaccine was better than none.Ultimately, the matter of which one they could get was dictated not by nationalism but by supply. In late February, Hong Kong got its first vaccine shipments: one million doses of Sinovac. (Hong Kong would later receive 585,000 doses of the BioNTech vaccine via a Chinese company, Fosun.)On Feb. 22, Mom texted that she and Dad had booked a March 11 appointment to get their first shots, followed by second doses in April. A day later, she reported that Dad hadn’t pressed the button to confirm the appointment on the online booking system and had lost the slots.The next week, they texted again: They had walked to a private clinic that was dispensing Sinovac shots. After a short wait, they received the vaccine. On April 2, they told us that they had gotten their second dose of Sinovac and were feeling fine. Mom groused that even though they had an appointment, they “still need to wait for half an hour.”Our responses were more enthusiastic.“Great news,” I wrote.“Yay!” Pui-Ying texted, followed by celebratory emojis.“Congrats!” Pui Ling said.Blantyre, MalawiPui-Ying had moved with her family to Malawi in 2016 to work as a doctor and conduct clinical research on children’s health. Resources at the Queen Elizabeth Central Hospital, where she works, were limited. When Madonna’s charity helped finance the construction of a new children’s wing at the hospital, which opened in 2017, it was big news.Staffing was tight even before the coronavirus, Pui-Ying said. When the pandemic came, the hospital decided on a one-week-on, one-week-off routine to reduce staff exposure to Covid-19 while ensuring that enough medical professionals would be working at all times. Masks, gloves and other protective equipment were scarce.In pediatrics, Pui-Ying and her colleagues set up a “respiratory zone” for children with Covid-19. It was essentially a two-room ward, with about a dozen beds in the main room. The second room, which was an isolation unit, had space for four children.For a while, Malawi kept the coronavirus under control. But in December, the country was crushed by a second wave of Covid-19, which may have been supercharged by a South African variant.Staffing was tight and protective equipment scarce at Pui-Ying’s hospital, she said.Thoko Chikondi for The New York TimesPositivity rates for Covid-19 soared at one point to 40 percent, said Dr. Queen Dube, who was head of pediatrics at Queen Elizabeth Central Hospital and recently was appointed chief of health services at Malawi’s Ministry of Health. (By comparison, the peak positivity rate in the United States was around 22 percent last April, according to Johns Hopkins University.)The number of adult Covid patients at Pui-Ying’s hospital tripled to 106; it had to open two additional wards, Dr. Dube said.“I don’t remember in my career such a ravaging disease,” Dr. Dube said. “We lost colleagues. We lost close friends.”Fortunately, Pui-Ying only glimpsed the worst. The isolation room for children never had more than three patients, she said.Britain, the United States and other wealthy countries began ramping up campaigns to immunize their populations — something that was possible because they had spent billions of dollars last year placing advance orders for hundreds of millions of doses. Malawi did not have a single dose of a vaccine. The CNN report about the country’s lack of vaccines, which aired on Feb. 8, was forwarded throughout the medical community there.“I don’t think we knew what to expect with vaccines,” Pui-Ying said after she posted the video in our family WhatsApp group. “We just knew we wouldn’t be able to afford them.”In early March, after an application from Malawi’s government, the vaccine-sharing initiative Covax sent 360,000 doses of the AstraZeneca vaccine. When the shipment arrived at the airport in Malawi’s capital, Lilongwe, health workers there were photographed flashing celebratory V-signs.The bad news was that the shipment covered less than 2 percent of the population.Dr. Dube said she hoped for 960,000 more doses soon. The goal was to be able to vaccinate 60 percent of the country by the end of next year. By contrast, the United States is inoculating more than three million people a day, and all adults who want a vaccine will be able to get one by this summer.In the meantime, what doses there were had been earmarked for high-risk groups. Pui-Ying, who was eligible, said she was elated. Based on what she had read in the media, she hoped she would get a shot within days.San FranciscoAs Pui-Ying waited to hear about a vaccine, the situation in California brightened.The rollouts of the Moderna and Pfizer-BioNTech vaccines, which the U.S. Food and Drug Administration authorized for emergency use in December, were rocky. Federal, state and local officials had underestimated the challenges of a mass-vaccination campaign. Supplies were tight, yet thawed vaccines sometimes had to be discarded.Slowly, though, the situation improved. In February, as segments of the general population became eligible for vaccines, ballparks and conference centers — such as the Coliseum stadium in Oakland and the giant Moscone Center in San Francisco — were transformed into efficient vaccination hubs. Most important, supplies were becoming abundant.Pui-Wing, right, and Pui Ling Tam in Emeryville, Calif. By April, more than 18.4 million doses had been given in the state.Cayce Clifford for The New York TimesAs of April 1, more than 18.4 million vaccine doses had been administered in California, up from 3.5 million two months earlier. Gov. Gavin Newsom declared that everyone 16 and older would be eligible for the vaccines on April 15.Even with the occasional hiccup, California’s situation mirrored the country’s, said Dr. Bob Wachter, chairman of the department of medicine at the University of California, San Francisco. He called it a “true triumph of science and policy.” The United States has commitments from manufacturers for enough doses to cover 400 million people, about 70 million more than its total population.Even so, Covid-19 continues to spread. In early March, my husband’s aunt and uncle, both in their 70s and living in Queens, N.Y., died from Covid-19. The disease has killed more than 550,000 Americans.As more vaccines became available in California, colleagues and friends urged one another to make appointments or seek leftovers, sending spreadsheets of various inoculation sites. I signed up for a slot.On the evening of March 10, I got one of the last Pfizer-BioNTech shots of the day at a Walgreens. A bored pharmacist injected me with the vaccine in a screened-off area of the dimly lit drugstore. It was strangely anticlimactic. But after a year of lockdowns, it was also a great relief.I texted the good news to Mom and Dad, using many exclamation marks. They were pleased and immediately quizzed me on whether I felt any side effects. (I didn’t, apart from a slightly sore shoulder.) I called Pui Ling and prodded her to try to get a shot. She said she would wait, knowing her turn would come before long.Pui-Ying receiving the first shot of the AstraZeneca vaccine last month in Malawi.via Pui-Ying TamA few days later, Mom forwarded a photo to our family WhatsApp group. It was of Pui-Ying, mask on and with a sleeve of her T-shirt pushed up. She was getting an AstraZeneca shot outside Queen Elizabeth Central Hospital. Malawi had started vaccinating people on March 11, when a live broadcast showed top officials being immunized. Pui-Ying got hers five days later.I had gotten the vaccine six days before my sister, the frontline doctor.In a phone call, I mentioned to Pui-Ying that it looked like she was smiling underneath her mask when she got the vaccine. “I was!” she said.I asked when her second dose was.“May,” she said.I got mine on Wednesday.