Let's say we use a blockchain to store these user data in a decentralized way. Now, how will blocks get created (new blocks added) in the chain? How are you planning to get a consensus? Who decides?
However all the design I came through included a state-run centralized database to contact people who have been in contact with someone having the covid, and generally gets access to the info of which covid-positive person you have been in contact with... How would design a system including more privacy ?
My feeling is you don't necessarily need to store anything on the blockchain. If you are more than Xminutes at less than 2 meter of someone, you get a unique address from this person. Then you just use the unique addresses of all the people to send them the info that you have the covid (after validation by a doctor?). That could use existing blockchain to do these simple transaction, with existing coin (eg. using OP-RETURN) or by creating a specific token. Monero would be interesting for the privacy side. Less private, but I like Komodo too.