Journal of Mathematical Biology

, Volume 78, Issue 5, pp 1553–1579 | Cite as

A periodic SEIRS epidemic model with a time-dependent latent period

  • Fuxiang LiEmail author
  • Xiao-Qiang Zhao


Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. In order to investigate the impact of time-dependent delays on disease control, we propose an SEIRS epidemic model with a periodic latent period. We introduce the basic reproduction ratio \(R_0\) for this model and establish a threshold type result on its global dynamics in terms of \(R_0\). More precisely, we show that the disease-free periodic solution is globally attractive if \(R_0<1\); while the system admits a positive periodic solution and the disease is uniformly persistent if \(R_0>1\). Numerical simulations are also carried out to illustrate the analytic results. In addition, we find that the use of the temporal average of the periodic delay may underestimate or overestimate the real value of \(R_0\).


Periodic SEIRS model Time-dependent latent period Basic reproduction ratio Periodic solution Uniform persistence 

Mathematics Subject Classification

34K13 37N25 92D30 



This work was supported in part by the China Scholarship Council (201506460020) and the Natural Science and Engineering Research Council of Canada. We are grateful to two referees for their valuable comments and suggestions which led to an improvement of our original manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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