Comment by MatthiasWandel

Like so many clever puzzle answers, this doesn't take into account the effect of measurement error. The tests apparently have a false negative rate of 20% as it is. Now start mixing samples together, causing more dilution of the positive samples, and the false negative rate is bound to go up further. Now you have to do studies to figure out what the false negative rate is going to be, and put that into your model as well. This could get impractical fast.