Journal of Mathematical Biology

, Volume 78, Issue 6, pp 1713–1725 | Cite as

Global stability properties of a class of renewal epidemic models

  • Michael T. MeehanEmail author
  • Daniel G. Cocks
  • Johannes Müller
  • Emma S. McBryde


We investigate the global dynamics of a general Kermack–McKendrick-type epidemic model formulated in terms of a system of renewal equations. Specifically, we consider a renewal model for which both the force of infection and the infected removal rates are arbitrary functions of the infection age, \(\tau \), and use the direct Lyapunov method to establish the global asymptotic stability of the equilibrium solutions. In particular, we show that the basic reproduction number, \(R_0\), represents a sharp threshold parameter such that for \(R_0\le 1\), the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when \(R_0 > 1\), i.e. when it exists.


Global stability Lyapunov Renewal Kermack–McKendrick 

Mathematics Subject Classification

92D30 34D23 34A34 37N25 



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Copyright information

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

Authors and Affiliations

  1. 1.Australian Institute of Tropical Health and MedicineJames Cook UniversityTownsvilleAustralia
  2. 2.Research School of Science and EngineeringAustralian National UniversityCanberraAustralia
  3. 3.Centre for Mathematical Sciences, Technische Universität München, and Institute of Computational Biology, German Research Center for Environmental HealthMünchenGermany

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