  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.
