Global stability properties of a class of renewal epidemic models

J Math Biol. 2019 May;78(6):1713-1725. doi: 10.1007/s00285-018-01324-1. Epub 2019 Feb 9.


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, [Formula: see text], 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, [Formula: see text], represents a sharp threshold parameter such that for [Formula: see text], the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when [Formula: see text], i.e. when it exists.

Keywords: Global stability; Kermack–McKendrick; Lyapunov; Renewal.

MeSH terms

  • Basic Reproduction Number*
  • Communicable Diseases / epidemiology*
  • Communicable Diseases / transmission
  • Computer Simulation
  • Epidemics / prevention & control*
  • Epidemics / statistics & numerical data
  • Humans
  • Models, Biological*
  • Time Factors