We consider real-time timely tracking of infection status (e.g., covid-19) of individuals in a population. In this work, a health care provider wants to detect infected people as well as people who recovered from the disease as quickly as possible. In order to measure the timeliness of the tracking process, we use the long-term average difference between the actual infection status of the people and their real-time estimate by the health care provider based on the most recent test results. We first find an analytical expression for this average difference for given test rates, and given infection and recovery rates of people. Next, we propose an alternating minimization based algorithm to minimize this average difference. We observe that if the total test rate is limited, instead of testing all members of the population equally, only a portion of the population is tested based on their infection and recovery rates. We also observe that increasing the total test rate helps track the infection status better. In addition, an increased population size increases diversity of people with different infection and recovery rates, which may be exploited to spend testing capacity more efficiently, thereby improving the system performance. Finally, depending on the health care provider's preferences, test rate allocation can be altered to detect either the infected people or the recovered people more quickly.