Bulletin of Mathematical Biology

, Volume 73, Issue 12, pp 3047–3070 | Cite as

Multigeneration Reproduction Ratios and the Effects of Clustered Unvaccinated Individuals on Epidemic Outbreak

  • David E. Hiebeler
  • Isaac J. Michaud
  • Hamilton Hoxie Ackerman
  • Shannon Reed Iosevich
  • Andre Robinson
Original Article


An SIR epidemiological community-structured model is constructed to investigate the effects of clustered distributions of unvaccinated individuals and the distribution of the primary case relative to vaccination levels. The communities here represent groups such as neighborhoods within a city or cities within a region. The model contains two levels of mixing, where individuals make more intra-group than inter-group contacts. Stochastic simulations and analytical results are utilized to explore the model. An extension of the effective reproduction ratio that incorporates more spatial information by predicting the average number of tertiary infections caused by a single infected individual is introduced to characterize the system. Using these methods, we show that both the vaccination coverage and the variation in vaccination levels among communities affect the likelihood and severity of epidemics. The location of the primary infectious case and the degree of mixing between communities are also important factors in determining the dynamics of outbreaks. In some cases, increasing the efficacy of a vaccine can in fact increase the effective reproduction ratio in early generations, due to the effects of population structure on the likely initial location of an infection.


Epidemiological models Vaccination Spatial clustering 


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Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  • David E. Hiebeler
    • 1
  • Isaac J. Michaud
    • 1
  • Hamilton Hoxie Ackerman
    • 2
    • 3
    • 6
  • Shannon Reed Iosevich
    • 4
    • 7
  • Andre Robinson
    • 5
  1. 1.Department of Mathematics and StatisticsUniversity of MaineOronoUSA
  2. 2.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  3. 3.Biogen Idec, 14 Cambridge CenterCambridgeUSA
  4. 4.Department of MathematicsUniversity of MissouriColumbiaUSA
  5. 5.Department of MathematicsMedgar Evers CollegeBrooklynUSA
  6. 6.Department of StatisticsUniversity of California, BerkeleyBerkeleyUSA
  7. 7.Warner School of EducationUniversity of RochesterRochesterUSA

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