You might be interested in this exposure notification app https://www.covidwatch.org/. It's a collaboration between Stanford University and the University of Waterloo and it's pretty good. It will soon be more widely available.
The question is not the Software. That can be easily written. The question is how you would like to map the available information, schematically and/or logically. And how you want to collect the data.
The rest is easy.
If the data is sufficient and the desired answer can be projected from the available data, an AI can be created that can answer the necessary question.
[bounty: $100 for recommending a tool that I use for more than two weeks]
I would love (and happily pay for) a piece of software that I could tell (a) a bunch of recent interactions between people ("Alice, masked, spent 3 hours, indoors, distanced, with Bob, unmasked, on Nov 1"), and (b) a bunch of evidence about their health over time ("Dolores tested negative on Nov 3"), and make queries about how likely various people are to have COVID.
I'd want it to be capable of complicated inferences like "Alice met with Bob yesterday; Bob met with Charlie 4d ago; Charlie separately met with Dolores that same day. Dolores just tested negative; given that, Alice is now less likely to be incubating COVID."
Ideally, it would take into account things like contagiousness-over-time profiles, and incubation periods, and asymptomatic cases, and how all those things differ between people, and tests' false negative rates -- but I realize that's a lot to ask.
Some non-solutions: