Its not necessarily a case of strength quite, possibly the antibiotic chosen hasn't been able to penetrate all of the affected tissue, possibly it needed a longer course (second jab of the same) as it did improve after one treatment, obviously maybe the bacteria involved are resistant to the chosen drug - what have you.
All drugs struggle where there are abscesses and areas of dead tissue as they cannot get into it to kill the bacteria though some work better than others in this situation