# Analytic Calculation of Finite-Population Reproductive Numbers for Direct- and Vector-Transmitted Diseases with Homogeneous Mixing

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Bulletin of Mathematical Biology

, Volume 76, Issue 5, pp 1143–1154

First Online: 23 April 2014Received: 25 May 2013Accepted: 13 March 2014DOI: 10.1007-s11538-014-9950-x

Cite this article as: Keegan, L. & Dushoff, J. Bull Math Biol 2014 76: 1143. doi:10.1007-s11538-014-9950-x

Abstract

The basic reproductive number, \\mathcal {R} {0}\, provides a foundation for evaluating how various factors affect the incidence of infectious diseases. Recently, it has been suggested that, particularly for vector-transmitted diseases, \\mathcal {R} {0}\ should be modified to account for the effects of finite host population within a single disease transmission generation. Here, we use a transmission factor approach to calculate such -finite-population reproductive numbers,- under the assumption of homogeneous mixing, for both vector-borne and directly transmitted diseases. In the case of vector-borne diseases, we estimate finitepopulation reproductive numbers for both host-to-host and vector-to-vector generations, assuming that the vector population is effectively infinite. We find simple, interpretable formulas for all three of these quantities. In the direct case, we find that finite-population reproductive numbers diverge from \\mathcal {R} {0}\ before \\mathcal {R} {0}\ reaches half of the population size. In the vector-transmitted case, we find that the host-to-host number diverges at even lower values of \\mathcal {R} {0}\, while the vector-to-vector number diverges very little over realistic parameter ranges.

KeywordsBasic reproductive number Transmission factors Malaria Modeling Vector-borne diseases  Download fulltext PDF

Autor: Lindsay Keegan - Jonathan Dushoff