how finishing a course of anti biotics kills all the bacteria?

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How does finishing the prescription ensure that all the bacteria is killed and that there are no surviving or remaining bacteria? Have always been told that finishing the prescription does this but never how

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Finishing a course finishes off stragglers who have slightly elevated immunity relative to the background level of immunity so they don’t survive to reproduce and pass on their genes, producing a new generation of slightly more resistant bacteria.

Suppose you have a population of bacteria of which 99.99% experience a certain antibiotic drug lethal at a certain dose `d` sustained for a certain amount of time `t`.

In a population with random variation, there will naturally be some that have a chance of being more resistant—meaning they can tolerate a sustained dose of `d` for `t` time and still survive. But maybe they’re not so strong and resistant that if they are exposed to a dose of `d` for `t * 1.05` time they will die. Then prescribing such a course will ensure those bugs that won the genetic lottery won’t survive to pass on their superior genes.

But if you stop short of the full course, they’ll slip through the cracks and go on to produce a new generation of bacteria with slightly stronger immunity. Repeat this enough times and you get superbugs that resist antibiotics dosages 1000x the original bacteria could survive.

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