KöMaL 2021 A804

There is a city with $n$ citizens. The city wants to buy sceptervirus tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following: $\bullet$ Virus positive: there is at least one currently infected person among the people whose samples were analyzed, and none of them were healed from an earlier infection. $\bullet$ Antibody positive: there is at least one person who was healed from an earlier infection among the people whose samples were analyzed, and non of them are infected now. $\bullet$ Neutral: either all of the people whose samples were analyzed are not infected, or there is at least one currently infected person and one person who was healed from an earlier infection. (Viruses and antibodies in samples completely neutralize each other.) What is the smallest number of tests to buy if we would like to know if the sceptervirus is present now or it has been present earlier? (The samples are taken from the people at the same time. The people are either infected now, have been infected earlier, or haven't contacted the virus yet.)