A useful relationship between epidemiology and queueing theory: The distribution of the number of infectives at the moment of the first detection

In this paper we establish a relation between the spread of infectious diseases and the dynamics of so called M/G/1M/G/1 queues with processor sharing. The relation between the spread of epidemics and branching processes, which is well known in epidemiology, and the relation between M/G/1M/G/1 queues and birth death processes, which is well known in queueing theory, will be combined to provide a framework in which results from queueing theory can be used in epidemiology and vice versa.In particular, we consider the number of infectious individuals in a standard SIR epidemic model at the moment of the first detection of the epidemic, where infectious individuals are detected at a constant per capita rate. We use a result from the literature on queueing processes to show that this number of infectious individuals is geometrically distributed.