Skip to main content
Log in

Persistence in seasonally forced epidemiological models

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this paper we address the persistence of a class of seasonally forced epidemiological models. We use an abstract theorem about persistence by Fonda. Five different examples of application are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Aronsson G, Kellogg RB (1978) On a differential equation arising from compartmental analysis. Math Biosci 38: 113–122

    Article  MathSciNet  MATH  Google Scholar 

  • Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53: 421–436

    Article  MathSciNet  MATH  Google Scholar 

  • Bacaër N (2007) Approximation of the basic reproduction number R 0 for vector-borne diseases with a periodic vector population. Bull Math Biol 69: 1067–1091

    Article  MathSciNet  MATH  Google Scholar 

  • Bacaër N, Ouifki R (2007) Growth rate and basic reproduction number for population models with a simple periodic factor. Math Biosci 210: 647–658

    Article  MathSciNet  MATH  Google Scholar 

  • Bacaër N, Ouifki R, Pretorius C, Wood R, Williams B (2008) Modeling the joint epidemics of TB and HIV in a South African township. J Math Biol 57: 557–593

    Article  MathSciNet  MATH  Google Scholar 

  • Bacaër N, Ait Dads EH (2011) Genealogy with seasonality, the basic reproduction number, and the influenza pandemic. J Math Biol 62: 741–762

    Article  MathSciNet  MATH  Google Scholar 

  • Cooke K, Kaplan JL (1976) A periodicity threshold theorem for epidemics and population growth. Math Biosci 31: 87–104

    Article  MathSciNet  MATH  Google Scholar 

  • Fonda A (1988) Uniformly persistent semidynamical systems. Proc Am Math Soc 104: 111–116

    Article  MathSciNet  MATH  Google Scholar 

  • Freedman HI, Ruan S, Tang M (1994) Uniform persistence and flows near a closed positively invariant set. J Dyn Differ Equ 6: 583–600

    Article  MathSciNet  MATH  Google Scholar 

  • Garay B, Hofbauer J (2003) Robust permanence for ecological differential equations, minimax, and discretizations. SIAM J Math Anal 34: 1007–1039

    Article  MathSciNet  MATH  Google Scholar 

  • Gedeon T, Bodelón C, Kuenzi A (2010) Hantavirus transmission in sylvan and peridomestic environments. Bull Math Biol 72: 541–564

    Article  MathSciNet  MATH  Google Scholar 

  • Hethcote HW (1973) Asymptotic behavior in a deterministic epidemic model. Bull Math Biol 35: 607–614

    MATH  Google Scholar 

  • Hethcote H (1994) A thousand and one epidemic models. In: Levin S (eds) Frontiers in mathematical biology. Springer, Berlin, pp 504–515

    Google Scholar 

  • Hirsch M (1985) Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere. SIAM J Math Anal 16: 423–439

    Article  MathSciNet  MATH  Google Scholar 

  • Hirsch M, Morris W, Smith HL, Zhao X (2001) Chain transitivity, attractivity, and strong repellors for semidynamical systems. J Dyn Differ Equ 13: 107–131

    Article  MATH  Google Scholar 

  • Hofbauer J, Schreiber S (2010) Robust permanence for interacting structured populations. J Differ Equ 248: 1955–1971

    Article  MathSciNet  MATH  Google Scholar 

  • Liu L, Zhao X, Zhou Y (2010) A tuberculosis model with seasonality. Bull Math Biol 72: 931–952

    Article  MathSciNet  MATH  Google Scholar 

  • Lou Y, Zhao X (2010) A climate-based malaria transmission model with structured vector population. SIAM J Appl Math 70: 2023–2044

    Article  MathSciNet  MATH  Google Scholar 

  • Margheri A, Rebelo C (2003) Some examples of persistence in epidemiological models. J Math Biol 46: 564–570

    Article  MathSciNet  MATH  Google Scholar 

  • Nakata Y, Kuniya T (2010) Global dynamics of a class of SEIRS epidemic models in a periodic environment. J Math Anal Appl 363: 230–237

    Article  MathSciNet  MATH  Google Scholar 

  • Nussbaum RD (1977) Periodic solutions of some integral equations from the theory of epidemics. In: Lakshmikantham V (eds) Nonlinear systems and applications. Academic Press, New York, pp 235–257

    Google Scholar 

  • Nussbaum RD (1978) A periodicity threshold theorem for some nonlinear integral equations. SIAM J Math Anal 9: 356–376

    Article  MathSciNet  MATH  Google Scholar 

  • Salceanu P, Smith H (2010) Persistence in a discrete-time, stage-structured epidemic model. J Differ Equ Appl 16: 73–103

    Article  MathSciNet  MATH  Google Scholar 

  • Schreiber S (2000) Criteria for C r robust permanence. J Differ Equ 162: 400–426

    Article  MathSciNet  MATH  Google Scholar 

  • Smith HL (1977) On periodic solutions of a delay integral equation modelling epidemics. J Math Biol 4: 69–80

    Article  MathSciNet  MATH  Google Scholar 

  • Smith HL (1983) Subharmonic bifurcation in an S-I-R epidemic model. J Math Biol 17: 163–177

    Article  MathSciNet  MATH  Google Scholar 

  • Smith HL (1983) Multiple stable subharmonics for a periodic epidemic model. J Math Biol 17: 179–190

    Article  MathSciNet  MATH  Google Scholar 

  • Smith HL, Thieme HR (2011) Dynamical systems and population persistence. AMS, Providence

    MATH  Google Scholar 

  • Smith HL, Waltman P (1995) The theory of the chemostat. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Thieme HR (2000) Uniform persistence and permanence for non-autonomous semiflows in population biology. Math Biosci 166: 173–201

    Article  MathSciNet  MATH  Google Scholar 

  • Thieme HR (2003) Mathematics in population biology. Princeton University Press, New Jersey

    MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Wang W, Zhao X (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20: 699–717

    Article  MATH  Google Scholar 

  • Zhang F, Zhao X (2007) A periodic epidemic model in a patchy environment. J Math Anal Appl 325: 496–516

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carlota Rebelo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rebelo, C., Margheri, A. & Bacaër, N. Persistence in seasonally forced epidemiological models. J. Math. Biol. 64, 933–949 (2012). https://doi.org/10.1007/s00285-011-0440-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-011-0440-6

Keywords

Mathematics Subject Classification (2000)

Navigation