The King Laboratory
of Theoretical Ecology & Evolution
at the University of Michigan

Epidemiological consequences of imperfect vaccines for immunizing infections

F. Magpantay, M. Riolo, M. de Cellès, A. A. King, and P. Rohani
SIAM Journal on Applied Mathematics 74:  1810–1830, 2014.

The control of some childhood diseases has proved to be difficult even in countries that maintain high vaccination coverage. This may be due to the use of imperfect vaccines, and there has been much discussion on the different modes by which vaccines might fail. To understand the epidemiological implications of some of these different modes, we performed a systematic analysis of a model based on the standard susceptible-infectious-recovered equations with a vaccinated component that permits vaccine failure in degree (“leakiness”), take (“all-or-nothingness”), and duration (waning of vaccine-derived immunity). The model was first considered as a system of ordinary differential equations and then extended to a system of partial differential equations to accommodate age structure. We derived analytic expressions for the steady states of the system and the final age distributions in the case of homogenous contact rates. The stability of these equilibria are determined by a threshold parameter R_p, a function of the vaccine failure parameters and the coverage p. The value of p for which R_p=1 yields the critical vaccination ratio, a measure of herd immunity. Using this concept, we can compare vaccines that confer the same level of herd immunity to the population but may fail at the individual level in different ways. For any fixed R_p>1, the leaky model results in the highest prevalence of infection, while the all-or-nothing and waning models have the same steady state prevalence. The actual composition of a vaccine cannot be determined on the basis of steady state levels alone, but the distinctions can be made by looking at transient dynamics (such as after the onset of vaccination), the mean age of infection, the age distributions at steady state of the infected class, and the effect of age-specific contact rates.

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