Skip to main content
SpringerLink
Log in

Global stability for SIRS epidemic models with general incidence rate and transfer from infectious to susceptible

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

We study a class of SIRS epidemic dynamical models with a general nonlinear incidence rate and transfer from infectious to susceptible. The incidence rate includes a wide range of monotonic, concave incidence rates and some non-monotonic or concave cases. We apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number \(R_0\le 1\), and the endemic equilibrium is globally asymptotically stable if \(R_0>1\), under some conditions imposed on the incidence function f(S, I).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Agrawal, A., Tenguria, A., Modi, G.: Stability analysis of an SIR epidemic model with specific nonliner incidence rate. Math. Theory Model. 6(1), 45–51 (2016)

    Google Scholar 

  2. Anderson, R.M., May, R.M.: Regulation and stability of host-parasite population interactions: I. Regulatory processes. J. Anim. Ecol. 47, 219–247 (1978)

    Article  Google Scholar 

  3. Bai, Y., Mu, X.: Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible. J. Appl. Anal. Comput. 8(2), 402–412 (2018)

    MathSciNet  Google Scholar 

  4. Buonomo, B., Rionero, S.: On the lyapunov stability for SIRS epidemic models with general nonlinear incidence rate. Appl. Math. Comput. 217(8), 4010–4016 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 35(2), 274–284 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Capasso, V., Serio, G.: A generalization of the Kermack–McKendrick deterministic epidemic model. Math. Biosci. 42(1), 43–61 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Capasso, V., Grosso, E., Serio, G.: I modelli matematici nella indagine epidemiologica. Applicazione all’epidemia di colera verificatasi in Bari nel 1973. Ann. Sclavo 19, 193–208 (1977)

    Google Scholar 

  8. Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361–404 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cui, J., Sun, Y., Zhu, H.: The impact of media on the control of infectious diseases. J. Dyn. Differ. Equ. 20(1), 31–53 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, H., Li, M.Y., Shuai, Z.: Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can. Appl. Math. Q. 14(3), 259–284 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Hethcote, H.W.: Qualitative analyses of communicable disease models. Math. Biosci. 28(3–4), 335–356 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A: Math., Phys. Eng. Sci. 115, 700–721 (1927)

    Article  MATH  Google Scholar 

  14. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68(3), 615–626 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Korobeinikov, A.: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69(6), 1871–1886 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Korobeinikov, A., Maini, P.K.: Non-linear incidence and stability of infectious disease models. Math. Med. Biol. 22(2), 113–128 (2005)

    Article  MATH  Google Scholar 

  17. Lahrouz, A., Omari, L., Kiouach, D.: Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model. Nonlinear Anal.: Model. Control 16(1), 59–76 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lahrouz, A., Omari, L., Kiouach, D., Belmaâti, A.: Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination. Appl. Math. Comput. 218(11), 6519–6525 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Li, T., Zhang, F., Liu, H., Chen, Y.: Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible. Appl. Math. Lett. 70, 52–57 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, W., Levin, S.A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23(2), 187–204 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mena-Lorca, J., Hethcote, H.W.: Dynamic models of infectious diseases as regulators of population sizes. J. Math. Biol. 30(7), 693–716 (1992)

    MathSciNet  MATH  Google Scholar 

  22. Muroya, Y., Enatsu, Y., Nakata, Y.: Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate. J. Math. Anal. Appl. 377(1), 1–14 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188(1), 135–163 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sun, C., Lin, Y., Tang, S.: Global stability for an special SEIR epidemic model with nonlinear incidence rates. Chaos, Solitons Fractals 33(1), 290–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tang, Q., Teng, Z., Abdurahman, X.: A new lyapunov function for SIRS epidemic models. Bull. Malays. Math. Sci. Soc. 40(1), 237–258 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1), 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vargas-De-León, C.: On the global stability of SIS, SIR and SIRS epidemic models with standard incidence. Chaos, Solitons Fractals 44(12), 1106–1110 (2011)

    Article  MATH  Google Scholar 

  28. Wang, L., Zhang, X., Liu, Z.: An SEIR epidemic model with relapse and general nonlinear incidence rate with application to media impact. Qual. Theory Dyn. Syst., 1–21 (2017)

  29. Wang, X., Liu, S.: An epidemic model with different distributed latencies and nonlinear incidence rate. Appl. Math. Comput. 241, 259–266 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Wei, C., Chen, L.: A delayed epidemic model with pulse vaccination. Discret. Dyn. Nat. Soc. 2008, 13 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, R., Ma, Z.: Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos, Solitons Fractals 41(5), 2319–2325 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou, X., Cui, J.: Global stability of the viral dynamics with Crowley-Martin functional response. Bull. Korean Math. Soc. 48(3), 555–574 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by Sistema Nacional de Investigadores (15284) and Conacyt-Becas.

Author information

Authors and Affiliations

  1. Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje 13615, 97119, Mérida, Yucatán, Mexico

    Eric J. Avila-Vales & Ángel G. Cervantes-Pérez

Authors
  1. Eric J. Avila-Vales
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ángel G. Cervantes-Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric J. Avila-Vales.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avila-Vales, E.J., Cervantes-Pérez, Á.G. Global stability for SIRS epidemic models with general incidence rate and transfer from infectious to susceptible. Bol. Soc. Mat. Mex. 25, 637–658 (2019). https://doi.org/10.1007/s40590-018-0211-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40590-018-0211-0

Mathematics Subject Classification

Search

Navigation