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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
Article
  • 286 Downloads

Abstract

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.

Keywords

Global stability Lyapunov Renewal Kermack–McKendrick 

Mathematics Subject Classification

92D30 34D23 34A34 37N25 

Notes

References

  1. Bichara D, Iggidr A, Sallet G (2013) Global analysis of multi-strains SIS, SIR and MSIR epidemic models. J Appl Math Comput 44(1):273–292MathSciNetzbMATHGoogle Scholar
  2. Breda D, Diekmann O, De Graaf W, Pugliese A, Vermiglio R (2012) On the formulation of epidemic models (an appraisal of Kermack and McKendrick). J Biol Dyn 6(sup2):103–117MathSciNetCrossRefGoogle Scholar
  3. Chen Y, Zou S, Yang J (2016) Global analysis of an SIR epidemic model with infection age and saturated incidence. Nonlinear Anal Real World Appl 30:16–31MathSciNetCrossRefzbMATHGoogle Scholar
  4. Diekmann O (1977) Limiting behaviour in an epidemic model. Nonlinear Anal Theory Methods Appl 1(5):459–470MathSciNetCrossRefzbMATHGoogle Scholar
  5. Diekmann O, Gyllenberg M (2012) Equations with infinite delay: blending the abstract and the concrete. J Differ Equ 252(2):819–851MathSciNetCrossRefzbMATHGoogle Scholar
  6. Diekmann O, Getto P, Gyllenberg M (2008) Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J Math Anal 39(4):1023–1069MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fan M, Li MY, Wang K (2001) Global stability of an SEIS epidemic model with recruitment and a varying total population size. Math Biosci 170(2):199–208MathSciNetCrossRefzbMATHGoogle Scholar
  8. Huang G, Takeuchi Y (2011) Global analysis on delay epidemiological dynamic models with nonlinear incidence. J Math Biol 63(1):125–139MathSciNetCrossRefzbMATHGoogle Scholar
  9. Jensen JLWV (1906) Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math 30(1):175–193MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond Math Phys Eng Sci 115(772):700–721CrossRefzbMATHGoogle Scholar
  11. Korobeinikov A (2004) Global properties of basic virus dynamics models. Bull Math Biol 66(4):879–883MathSciNetCrossRefzbMATHGoogle Scholar
  12. Korobeinikov A (2008) Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages. Bull Math Biol 71(1):75–83MathSciNetCrossRefzbMATHGoogle Scholar
  13. Korobeinikov A, Wake G (2002) Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl Math Lett 15(8):955–960MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li MY, Muldowney JS (1995) Global stability for the SEIR model in epidemiology. Math Biosci 125(2):155–164MathSciNetCrossRefzbMATHGoogle Scholar
  15. Li MY, Graef JR, Wang L, Karsai J (1999) Global dynamics of a SEIR model with varying total population size. Math Biosci 160(2):191–213MathSciNetCrossRefzbMATHGoogle Scholar
  16. Magal P, McCluskey C, Webb G (2010) Lyapunov functional and global asymptotic stability for an infection-age model. Appl Anal 89(7):1109–1140MathSciNetCrossRefzbMATHGoogle Scholar
  17. Martcheva M, Li XZ (2013) Competitive exclusion in an infection-age structured model with environmental transmission. J Math Anal Appl 408(1):225–246MathSciNetCrossRefzbMATHGoogle Scholar
  18. McCluskey CC (2008) Global stability for a class of mass action systems allowing for latency in tuberculosis. J Math Anal Appl 338(1):518–535MathSciNetCrossRefzbMATHGoogle Scholar
  19. McCluskey CC (2009) Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Math Biosci Eng 6(3):603–610MathSciNetCrossRefzbMATHGoogle Scholar
  20. McCluskey CC (2010a) Complete global stability for an SIR epidemic model with delay: distributed or discrete. Nonlinear Anal Real World Appl 11(1):55–59MathSciNetCrossRefzbMATHGoogle Scholar
  21. McCluskey CC (2010b) Global stability for an SIR epidemic model with delay and nonlinear incidence. Nonlinear Anal Real World Appl 11(4):3106–3109MathSciNetCrossRefzbMATHGoogle Scholar
  22. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. Lecture notes in biomathematics, vol. 68. Springer, BerlinGoogle Scholar
  23. Mikusiński J, Ryll-Nardzewski C (1951) Sur le produit de composition. Studia Mathematica 12:51–57MathSciNetCrossRefzbMATHGoogle Scholar
  24. Müller J, Kuttler C (2015) Methods and models in mathematical biology. Springer, BerlinGoogle Scholar
  25. O’Regan SM, Kelly TC, Korobeinikov A, O’Callaghan MJ, Pokrovskii AV (2010) Lyapunov functions for SIR and SIRS epidemic models. Appl Math Lett 23(4):446–448MathSciNetCrossRefzbMATHGoogle Scholar
  26. Smith H (2010) An introduction to delay differential equations with applications to the life sciences. Springer, BerlinGoogle Scholar
  27. Soufiane B, Touaoula TM (2016) Global analysis of an infection age model with a class of nonlinear incidence rates. J Math Anal Appl 434(2):1211–1239MathSciNetCrossRefzbMATHGoogle Scholar
  28. Thieme HR (1991) Stability change of the endemic equilibrium in age-structured models for the spread of S-I–R type infectious diseases. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 139–158zbMATHGoogle Scholar
  29. Thieme HR (2011) Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators. J Differ Equ 250(9):3772–3801MathSciNetCrossRefzbMATHGoogle Scholar
  30. Wang X, Liu S (2012) Global properties of a delayed SIR epidemic model with multiple parallel infectious stages. Math Biosci Eng 9(3):685–695MathSciNetCrossRefzbMATHGoogle Scholar

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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