title: A useful relationship between epidemiology and queueing theory: the distribution of the number of infectives at the moment of the first detection.
abstract: In this paper we establish a relation between the spread of infectious diseases and the dynamics of so called M/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/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.
authors.affiliation: Julius Center for Health Research & Primary Care, University Medical Center Utrecht, Heidelberglaan 100, P.O. Box 85500, 3508 GA Utrecht, The Netherlands.
authors.forename: Pieter
authors.initials: P
authors.lastname: Trapman
authors.forename: Martinus Christoffel Jozef
authors.initials: MC
authors.lastname: Bootsma
mesh_terms: D000465:Algorithms; D000818:Animals; D003140:Communicable Disease Control; D003141:Communicable Diseases / Q000453:epidemiology* / Q000662:veterinary; D003198:Computer Simulation; D003428:Cross Infection / Q000453:epidemiology; D004812:Epidemiologic Methods; D006801:Humans; D008390:Markov Chains; D055624:Methicillin-Resistant Staphylococcus aureus; D008954:Models, Biological; D040242:Risk Reduction Behavior; D013203:Staphylococcal Infections / Q000453:epidemiology; D013269:Stochastic Processes; D013598:Systems Theory