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Threshold dynamics in an SEIRS model with latency and temporary immunity

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Abstract

A disease transmission model of SEIRS type with distributed delays in latent and temporary immune periods is discussed. With general/particular probability distributions in both of these periods, we address the threshold property of the basic reproduction number \(R_0\) and the dynamical properties of the disease-free/endemic equilibrium points present in the model. More specifically, we 1. show the dependence of \(R_0\) on the probability distribution in the latent period and the independence of \(R_0\) from the distribution of the temporary immunity, 2. prove that the disease free equilibrium is always globally asymptotically stable when \(R_0<1\), and 3. according to the choice of probability functions in the latent and temporary immune periods, establish that the disease always persists when \(R_0>1\) and an endemic equilibrium exists with different stability properties. In particular, the endemic steady state is at least locally asymptotically stable if the probability distribution in the temporary immunity is a decreasing exponential function when the duration of the latency stage is fixed or exponentially decreasing. It may become oscillatory under certain conditions when there exists a constant delay in the temporary immunity period. Numerical simulations are given to verify the theoretical predictions.

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Acknowledgments

Thanks to Dr. X-Q. Zhao for valuable discussions and comments. We are grateful to the anonymous referees for helpful suggestions which led to an improvement of our original manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL,  A1C 5S7, Canada

    Yuan Yuan

  2. Départment de mathématiques et de statistique & Centre de recherches mathématiques, Université de Montréal, Montréal, QC,  H3C 3J7, Canada

    Jacques Bélair

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  1. Yuan Yuan
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  2. Jacques Bélair
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Correspondence to Yuan Yuan.

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Supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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Yuan, Y., Bélair, J. Threshold dynamics in an SEIRS model with latency and temporary immunity. J. Math. Biol. 69, 875–904 (2014). https://doi.org/10.1007/s00285-013-0720-4

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  • DOI: https://doi.org/10.1007/s00285-013-0720-4

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