Of course, we can’t have a 50 percent rule for humans. Because who decides? These are vulnerable people, and while the world is full of dedicated, self-sacrificing caregivers, it also contains far too many people who stand to gain from death (through inheritance) or from continuing life (in the form of Social Security checks or cheap housing).

FOR Byron, the fateful day came both slowly and suddenly. Over his last year, he’d had a number of health problems, and we had intended to take a palliative approach: doing only those treatments that lessened his suffering and avoiding tests and stressful vet visits at all costs. But then his paw hurt, so we took him in. He didn’t respond to the first antibiotic so sedation and a biopsy were required. Next, he developed bloody, watery stool. We talked the vet into prescribing standard antibiotics without a visit, and he improved. But a few months later, he was short of breath. He needed an X-ray to determine that it was pneumonia, then oxygen and more antibiotics. He stayed overnight. A few months after that, pus dripped from his red, swollen eyes — conjunctivitis. Each time he had to go to the vet, he shook, panted, climbed up our bodies, and tugged on his leash, his tiny body straining for the door.

Suddenly, I fully understood something I observed at work all the time — how it was possible to love a frail relative, prioritize his comfort and well-being, and yet repeatedly find oneself doing things that felt awful to everyone.

Finally we made an appointment with a hospice vet. When I returned home from work that night, the humans of our family were cradling Byron and looking sad. He ran to me, wiggling his tail. He hadn’t eaten all day. I thawed some chicken, and he gobbled it down. Someone said, “You can’t kill him.”

Then he followed me to the bathroom and vomited the chicken onto the floor at my feet. He stood, tail down, facing the wall.

That night, the vet gave him the injection, and Byron died in our arms.

Since then, I have often wondered whether we waited too long. We counted the time he spent sleeping as contentment, tipping the scale above the 50 percent mark. But I know that in elderly humans, sleep is more often a sign of chronic exhaustion, depression and avoidance of pain. In dealing with the guilt brought on by our mixed feelings — we love him; he’s ruining our lives — I realize we may have overcompensated to his detriment.

With dying humans, similar situations arise every day: hospital stays that fix the acute problem and worsen the chronic ones; emergency department visits that yield diagnoses but require weeks of recovery from the waiting and testing; surgeries that are themselves minor but provoke major confusion, complications and hated nursing home stays. On the other hand, there are the relatively simple problems that might be addressed by a doctor if only seeing one didn’t require an ambulance for transportation, or time off work by an adult child, or more taxi fare than remains in the Social Security check, or more effort than seems worth the while.