Sampling a changing distribution over time I'm interested in learning more about sampling a changing population over time. For example, these statistics from the University of Washington coronavirus testing show a 7% positive rate day after day. Some news outlets are interpreting this as a positive: "One encouraging sign is that in Washington State, which had an early outbreak, the number of positive tests appears stable."
http://depts.washington.edu/labmed/covid19/
https://www.nytimes.com/2020/03/20/opinion/sunday/coronavirus-outcomes.html
Can someone suggest a branch of statistics or papers for further study that would help me assess the veracity of such claims? My intuitive reasoning, as described below, shows why I am questioning the optimistic interpretation, but I would like to learn a more formalized framework.

For instance, let's say that the population proportion that tests positive for an attribute is 7%, and that proportion is stable. If I took a random sample each day, I should be getting 7% consistently, subject to some variance.
However, let's say the population proportion is subject to change over time. Let's also add the assumption that I am trying to prioritize sampling people (without replacement) that I believe are most likely to be positive. Eventually I will sample the whole population. Under these new assumptions, if the population proportion were stable, you would expect my initial sample proportions to overshoot, and my final sample proportions to under shoot as I ran out of likely positives. That would imply that if my sample proportion was constant and did NOT drift down over time, then my population proportion MUST be increasing over time. And if my sample proportion was increasing over time, then my population proportion must be increasing by a heck of a lot over time.
These are just qualitative conclusions I can come to. Is there a branch of statistics I can learn about to help me make concrete conclusions about a changing population proportion given sample proportions over time? I am sure that as you vary the sample size across time, the conclusions also change. Any framework to help understand the tweaks and their effects would be great.
Thank you.
 A: OK, the service area is likely many hundreds of thousands for the U of Washington and for certain patients covers a five state area. The detection rate for a tertiary/quaternary referral center reflects the patient selection process, same criteria for referring patients in distress might lead to same detection rate. However, more precisely one cannot make inferences on that basis because the criteria for patient selection for testing is an uncontrolled variable.
To explain this a bit more, a primary or community hospital would refer patients to a larger regional hospital which refers patients to a specialized tertiary care hospital, and for certain problems, tertiary care hospitals would send patients to a quaternary care facility. Even more complicated, a local physician in the community, might refer a patient directly to a quaternary care facility, or the patient might walk in to the emergency room off of the street.  So, there are so many confounders that any correlation  you might make risks being spurious. BTW, the raw data you linked to was not organized for peer review, I review for 17 medical journals, and I am telling you as simply as I can that without controlling for the referral pattern, there is only limited use for the raw data.
