the dependent variable is a continuous variable (total number of infarcts per person)
But then the dependent variable is not a continuous variable. It is a discreet variable, with values like 0, 1, 2, 3, 4,.....
But maybe you mean ratio scale. So the four infarcts are twice as many as two infartcs. But for an interval variable like temperature four degrees of Celsius in not twice as much as two degrees of Celsius.
For the variable number of infarcts it would better to specify a Poisson distribution with a log link (and skip the talk about odds ratios, which just seems to confuse).