Bulletin of Mathematical Biology

, Volume 73, Issue 12, pp 3047–3070

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


    • Department of Mathematics and StatisticsUniversity of Maine
  • Isaac J. Michaud
    • Department of Mathematics and StatisticsUniversity of Maine
  • Hamilton Hoxie Ackerman
    • Department of Mathematics and StatisticsBoston University
    • Biogen Idec, 14 Cambridge Center
    • Department of StatisticsUniversity of California, Berkeley
  • Shannon Reed Iosevich
    • Department of MathematicsUniversity of Missouri
    • Warner School of EducationUniversity of Rochester
  • Andre Robinson
    • Department of MathematicsMedgar Evers College
Original Article

DOI: 10.1007/s11538-011-9660-6

Cite this article as:
Hiebeler, D.E., Michaud, I.J., Ackerman, H.H. et al. Bull Math Biol (2011) 73: 3047. doi:10.1007/s11538-011-9660-6


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 modelsVaccinationSpatial clustering

Copyright information

© Society for Mathematical Biology 2011