For antibiotic resistant bacteria that is handled by two rounds of antibiotics, why are the two prescribed sequentially rather than at the same time?
Based on the very specific pair of antibiotics you have asked about, I will assume that your hypothetical senario is regarding Mycobacterium genitalium infection.
The US Centers for Disease Control and prevention recommends the following as a potential treatment regimen:
Doxycycline 100 mg orally 2 times/day for 7 days followed by moxifloxacin 400 mg orally once daily for 7 days
The CDC goes on to note:
As part of this approach, doxycycline is provided as initial empiric therapy, which reduces the organism load and facilitates organism clearance
A paper by Read and colleagues (2019. PMCID 6355821) concurs:
Because several studies have shown that M. genitalium infections with lower loads were more likely to be cured, it may be that doxycycline lowered bacterial load, rendering M. genitalium more susceptible to a subsequent macrolide [11, 13, 20].
Thus, the two treatments are consecutive because the first (doxycycline) decreases the number of bacteria which can be better handled by the body during the second treatment.
I have not been able to find any specific trials of using the two drugs together concurrently, but online drug interaction checkers do not indicate any conflicts between doxycycline and moxifloxacin.