Applications of occupancy urn models to epidemiology

Pages: 509 - 520, Volume 6, Issue 3, July 2009

doi:10.3934/mbe.2009.6.509       Abstract        Full Text (265.6K)

Carlos M. Hernández-Suárez - Facultad de Ciencias, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico (email)
Oliver Mendoza-Cano - Instituto de Innovación y Desarrollo Tecnológico, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico (email)

Abstract: This paper shows how occupancy urn models can be used to derive useful results in epidemiology. First we show how simple epidemic models can be re-interpreted in terms of occupancy problems. We use this reformulation to derive an expression for the expected epidemic size, that is, the total number of infected at the end of an outbreak. We also use this approach to derive point and interval estimates of the Basic Reproduction Ratio, $R_{0}$. We show that this construction does not require that the underlying SIR model be a homogeneous Poisson process, leading to a geometric distribution for the number of contacts before removal, but instead it supports a general distribution. The urn model construction is easy to handle and represents a rich field for further exploitation.

Keywords:  Urn models; $R_{0}$; basic reproductive ratio; stochastic epidemic models.
Mathematics Subject Classification:  Primary: 62P10; Secondary: 92B05.

Received: February 2008;      Accepted: October 2008;      Published: June 2009.