Bulletin of Mathematical Biology

, Volume 71, Issue 1, pp 75–83 | Cite as

Global Properties of SIR and SEIR Epidemic Models with Multiple Parallel Infectious Stages

  • Andrei Korobeinikov
Original Article


We consider global properties of compartment SIR and SEIR models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number R 0, this state can be either endemic (R 0>1), or infection-free (R 0≤1).


Infectious disease Mass-action Endemic equilibrium state Global stability Direct Lyapunov method Lyapunov function 


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  1. Barbashin, E.A., 1970. Introduction to the Theory of Stability. Wolters-Noordhoff, Groningen. zbMATHGoogle Scholar
  2. Georgescu, P., Hsieh, Y.-H., 2006. Global stability for a virus dynamics model with nonlinaer incidence of infection and removal. SIAM J. Appl. Math. 67(2), 337–353. CrossRefMathSciNetGoogle Scholar
  3. Guo, H., Li, M.Y., 2006. Global dynamics of a staged progression model for infectious diseases. Math. Biosci. Eng. 3(3), 513–525. zbMATHMathSciNetGoogle Scholar
  4. Korobeinikov, A., 2004a. Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. J. IMA 21(2), 75–83. zbMATHCrossRefGoogle Scholar
  5. Korobeinikov, A., 2004b. Global properties of basic virus dynamics models. Bull. Math. Biol. 66(4), 879–883. CrossRefMathSciNetGoogle Scholar
  6. Korobeinikov, A., 2006. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68(3), 615–626. CrossRefMathSciNetGoogle Scholar
  7. Korobeinikov, A., 2007. Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871–1886. zbMATHCrossRefMathSciNetGoogle Scholar
  8. Korobeinikov, A., 2008. Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate. Math. Med. Biol. J. IMA, to appear. Google Scholar
  9. Korobeinikov, A., Maini, P.K., 2004. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. Eng. 1(1), 57–60. zbMATHMathSciNetGoogle Scholar
  10. Korobeinikov, A., Maini, P.K., 2005. Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. J. IMA 22, 113–128. zbMATHCrossRefGoogle Scholar
  11. Korobeinikov, A., Petrovskii, S.V., 2008. Toward a general theory of ecosystem stability: plankton-nutrient interaction as a paradigm. In: Hosking, R.J., Venturino, E. (Eds.), Aspects of Mathematical Modelling, pp. 27–40. Birkhäuser, Basel. CrossRefGoogle Scholar
  12. Korobeinikov, A., Wake, G.C., 2002. Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models. Appl. Math. Lett. 15(8), 955–961. zbMATHCrossRefMathSciNetGoogle Scholar
  13. La Salle, J., Lefschetz, S., 1961. Stability by Liapunov’s Direct Method. Academic, New York. zbMATHGoogle Scholar
  14. Okuonghae, D., Korobeinikov, A., 2006. Dynamics of tuberculosis: the effect of direct observation therapy strategy (DOTS) in Nageria. Math. Model. Nat. Phenom. Epidemiol. 2(1), 99–111. MathSciNetGoogle Scholar
  15. van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48. zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  1. 1.MACSI, Department of Mathematics and StatisticsUniversity of LimerickLimerickIreland

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