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. 2017 Apr 26;17:308. doi: 10.1186/s12879-017-2399-4

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© The Author(s). 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

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Fig. 3 — a-b SIRS model describing the transmission of infection in a population (S: susceptible, I: infectious, R: immune, V vaccinated). a Standard SIRS model with protective vaccination. b Modified SIRS model with vaccinees who can become infected. Parameters: per capita birth and death rate μ, contact rate β, recovery rate γ, vaccination rate ϕ, loss rate of naturally acquired immunity ρ, loss rate of vaccination-derived immunity τ, population size N. The full model description is given in the Additional file 1