Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1591–1617 | Cite as

Vaccine impact in homogeneous and age-structured models



A general model of an imperfect vaccine for a childhood disease is presented and the effects of different types of vaccine failure on transmission were investigated using models that consider both homogeneous and age-specific mixing. The models are extensions of the standard SEIR equations with an additional vaccinated component that allows for five different vaccine parameters: three types of vaccine failure in decreasing susceptibility to infection via failure in degree (“leakiness”), take (“all-or-nothingness”) and duration (waning of vaccine-derived immunity); one parameter reflecting the relative reduction in infectiousness of vaccinated individuals who get infected; and one parameter that reflects the relative reduction in reporting probability of vaccinated individuals due to a possible reduction in severity of symptoms. Only the first four parameters affect disease transmission (as measured by the basic reproduction number). The reduction in transmission due to vaccination is different for age-structured models than for homogeneous models. Notably, if the vaccine exhibits waning protection this could be larger for an age-structured model with high contact rates between young children who are still protected by the vaccine and lower contact rates between adults for whom protection might have already waned. Analytic expressions for age-specific “vaccine impacts” were also derived. The overall vaccine impact is bounded between the age-specific impact for the oldest age class and that of the youngest age class.


Age-structure Disease ecology Imperfect vaccines Mathematical modeling 

Mathematics Subject Classification




The author thanks M. Domenech de Cellès, A.A. King and P. Rohani for their helpful insights. This work is supported by the Natural Sciences and Engineering Research Council of Canada, the National Institutes of Health (Grant No. R01AI101155) and by MIDAS, National Institute of General Medical Sciences (Grant No. U54-GM111274).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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