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Mathematical analysis for an age-structured SIRS epidemic model

1 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan
2 Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501 Japan

Special Issues: Mathematical Modeling to Solve the Problems in Life Sciences

In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number R0 to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on R0 and the critical coverage of immunization based on the reinfection threshold.
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Keywords SIRS epidemic; basic reproduction number; age structure; forward bifurcation; persistence; compact attractor

Citation: Kento Okuwa, Hisashi Inaba, Toshikazu Kuniya. Mathematical analysis for an age-structured SIRS epidemic model. Mathematical Biosciences and Engineering, 2019, 16(5): 6071-6102. doi: 10.3934/mbe.2019304


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