With the current pandemic situation around the world, it comes to attention that if contact between people is tracked (via mobile devices - GPS), people could take much safer precautions by knowing if they were in contact with an infected. One of the biggest challenges in generating such a ledger is privacy. Blockchain and other security measures could provide a viable solution. But, I am raising this question to the research community to derive a mechanism where we could guarantee privacy to achieve 100% public involvement to create an active database.
https://www.eff.org/deeplinks/2020/04/challenge-proximity-apps-covid-19-contact-tracing
https://apps.who.int/iris/bitstream/10665/185258/1/WHO_EVD_Guidance_Contact_15.1_eng.pdf?ua=1
https://www.wired.com/story/covid-19-contact-tracing-apps-cryptography/
Generally, privacy fall under four major umbrellas; Identity, Data, Location, and Usage privacy. To make it simple, some applications (such as your proposed idea) require location and identity information, hence, I guess you can't provide 100% privacy in such a scenario.
On the other hand, Blockchain deployment is typically implemented for storing data that should never be altered/modified or deleted. In your suggestion, I think people who had the disease and recovered after a while, shall be deleted from a database that consists of people who are infected.
Following the mentioned insights, I believe:
1- people who contribute their private information should intentionally agree to exposing some of their private data, and
2- Blockchain may not be helpful; data stored can not be altered, Blockchain requires higher energy consumption levels, Blockchain consumes higher RTT rates.
You should have a look at the Pan-European Privacy-Preserving Proximity Tracing (PEPP-PT) initiative at https://www.pepp-pt.org/ which tries to address this problem.
Blockchain looks like a rather questionable tool for this kind of application, since one of its core features is that data are stored permanently and cannot be deleted or modified - more or less the opposite of what you want in this type of tracing system. The information whether I have been in contact with a certain person should only be stored for about two or three weeks, after that it is no longer relevant and should be deleted.