Bulletin of Mathematical Biology

, Volume 69, Issue 6, pp 1871–1886 | Cite as

Global Properties of Infectious Disease Models with Nonlinear Incidence

  • Andrei KorobeinikovEmail author
Original Article


We consider global properties for the classical SIR, SIRS and SEIR models of infectious diseases, including the models with the vertical transmission, assuming that the horizontal transmission is governed by an unspecified function f(S,I). We construct Lyapunov functions which enable us to find biologically realistic conditions sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state. This state can be either endemic, or infection-free, depending on the value of the basic reproduction number.


Direct Lyapunov method Lyapunov function Endemic equilibrium state Global stability Nonlinear incidence 

Mathematics Subject Classification (2000)

92D30 34D20 


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Laboratory of Nonlinear Science and Computation, Research Institute for Electronic ScienceHokkaido UniversitySapporoJapan

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