title: An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing.
abstract: The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.
authors.affiliation: Department of Mathematics, Purdue University, West Lafayette, IN, USA.
authors.forename: Zhilan
authors.initials: Z
authors.lastname: Feng
authors.affiliation: National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention, CDC, Atlanta, GA, USA.
authors.forename: Andrew N
authors.initials: AN
authors.lastname: Hill
authors.affiliation: National Center for Immunization and Respiratory Diseases, CDC, 1600 Clifton Road, NE, Mail Stop A-34, Atlanta, GA 30333, USA.
authors.forename: Philip J
authors.initials: PJ
authors.lastname: Smith
authors.affiliation: National Center for Immunization and Respiratory Diseases, CDC, 1600 Clifton Road, NE, Mail Stop A-34, Atlanta, GA 30333, USA.
authors.forename: John W
authors.initials: JW
authors.lastname: Glasser
mesh_terms: D050936:Basic Reproduction Number; D003141:Communicable Diseases / Q000453:epidemiology / Q000276:immunology / Q000635:transmission; D004196:Disease Outbreaks / Q000517:prevention & control*; D006801:Humans; D032521:Immunity, Herd; D017589:Immunization Programs; D008954:Models, Biological; D011157:Population Dynamics
keywords: Heterogeneity; Population-immunity threshold; Vaccine coverage