Analytic calculation of finite-population reproductive numbers for direct- and vector-transmitted diseases with homogeneous mixing.

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.