Vaccination and herd immunity thresholds in heterogeneous populations

Elamin H. Elbasha, Abba B. Gumel

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


It has been suggested, without rigorous mathematical analysis, that the classical vaccine-induced herd immunity threshold (HIT) assuming a homogeneous population can be substantially higher than the minimum HIT obtained when considering population heterogeneities. We investigated this claim by developing, and rigorously analyzing, a vaccination model that incorporates various forms of heterogeneity and compared it with a model that considers a homogeneous population. By employing a two-group vaccination model in heterogeneous populations, we theoretically established conditions under which heterogeneity leads to different HIT values, depending on the relative values of the contact rates for each group, the type of mixing between the groups, the relative vaccine efficacy, and the relative population size of each group. For example, under biased random mixing assumption and when vaccinating a given group results in disproportionate prevention of higher transmission per capita, we show that it is optimal to vaccinate that group before vaccinating the other groups. We also found situations, under biased assortative mixing assumption, where it is optimal to vaccinate more than one group. We show that regardless of the form of mixing between the groups, the HIT values assuming a heterogeneous population are always lower than the HIT values obtained from a corresponding model with a homogeneous population. Using realistic numerical examples and parametrization (e.g., assuming assortative mixing together with vaccine efficacy of 95% and the value of the basic reproduction number, R, of the model set at R= 2.5), we demonstrate that the HIT value generated from a model that considers population heterogeneity (e.g., biased assortative mixing) is significantly lower (40%) compared with a HIT value of 63% obtained if the model uses homogeneous population.

Original languageEnglish (US)
Article number73
JournalJournal Of Mathematical Biology
Issue number6-7
StatePublished - Dec 2021


  • Basic reproduction number
  • Herd immunity threshold
  • Heterogeneity
  • Homogeneous population
  • Mixing pattern
  • SVIR model

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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