An SIRS Pulse Vaccination Model with Nonlinear Incidence Rate and Time Delay

Ardak Kashkynbayev, Meruyert Yeleussinova, Shirali Kadyrov

Research output: Contribution to journalArticlepeer-review


In this paper, we investigate the effectiveness of pulse vaccination as a control strategy for a time-delayed SIRS epidemic model with varying population size. The dynamics of the infectious disease are closely tied to the basic reproduction number, denoted as R0 . Traditional epidemic models evaluate R0 using the next generation matrix, but this approach is unsuitable for non-autonomous systems. As our study focuses on pulse vaccination strategies, our system naturally falls into the non-autonomous category. To address this, we adopt a general approach that derives R0 in terms of spectral radii of Poincaré maps. Furthermore, we demonstrate the existence of an infectious-free periodic solution and establish its global attractiveness for R0 < 1 while highlighting the persistence of the infectious disease for R0 > 1. Lastly, we conduct a comprehensive sensitivity analysis for R0 under the framework of the Holling type II functional response.

Original languageEnglish
Pages (from-to)133-148
Number of pages16
JournalLetters in Biomathematics
Issue number1
Publication statusPublished - 2023


  • epidemic models
  • global attractiveness
  • pulse vaccination system (PVS)
  • spectral radius
  • the Poincaré map

ASJC Scopus subject areas

  • Statistics and Probability
  • Biochemistry, Genetics and Molecular Biology (miscellaneous)
  • Applied Mathematics


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