Re: If there is no infection at all then 1% of the population will still test positive.
This gets into understanding how medical tests work (IAAD, and this is a bit of a niche) and which tests you should use.
If there is a very low infection rate, and you are looking to 'rule in' you need a test that is very specific (is only true in disease) otherwise you suffer from too many false positives.
Equally, once the infection is fairly widespread, it is generally safe to assume that anyone who looks like a zombie, is a zombie.
We generally use medical tests with a 1% 'error' rate because in practice that is usually good enough.
To look at this problem in a better way, Likelihood Ratios are the way to go.