Skip to main content

Advertisement

Log in

Global dynamics of treatment models with time delay

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

The problem of the asymptomatic dynamics of a treatment model with time delay is considered, subject to two incidence functions, namely standard incidence and Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that for each of the two incidence functions, the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally-asymptotically stable for a special case. Finally, for each of the two incidence functions, the disease burden decreases with increasing time delay (incubation period). In summary the results in this article is similar to those established in Safi and Gumel (Nonlinear Anal Ser B Real World Appl 12:215–235, 2011) (i.e., treatment models considered here have the same dynamics of quarantine-isolation models in Safi and Gumel (Nonlinear Anal Ser B Real World Appl 12:215–235, 2011).

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.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson RM, May RM (1982) Population biology of infectious diseases. Springer, Berlin

    Book  Google Scholar 

  • Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University, London

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Cooke KL, van den Driessche P (1996) Analysis of an SEIRS epidemic model with two delays. J Math Biol 35:240–260

    Article  MATH  MathSciNet  Google Scholar 

  • Hale J (1977) Theory of functional differential equations. Springer, Heidelberg

    MATH  Google Scholar 

  • Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653

    Article  MATH  MathSciNet  Google Scholar 

  • Hou J, Teng Z (2009) Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates. Math Comput Simul 79:3038–3054

    Article  MATH  MathSciNet  Google Scholar 

  • Kribs-Zaleta C, Velasco-Hernandez J (2000) A simple vaccination model with multiple endemic states. Math Biosci 164:183–201

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Mukandavire Z, Chiyaka C, Garira W, Musuka G (2009) Mathematical analysis of a sex-structured HIV/AIDS model with a discrete time delay. Nonlinear Anal 71:1082–1093

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Safi MA, Gumel AB (2011) Effect of incidence function on the dynamics of quarantine/isolation model with time delay. Nonlinear Anal Ser B Real World Appl 12:215–235

    Article  MATH  MathSciNet  Google Scholar 

  • Sharomi O et al (2007) Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math Biosci 210:436–463

    Article  MATH  MathSciNet  Google Scholar 

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

  • Xu R, Ma Z (2009) Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal Real World Appl 10:3175–3189

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mohammad A. Safi.

Additional information

Communicated by Maria do Rosario de Pinho.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Safi, M.A. Global dynamics of treatment models with time delay. Comp. Appl. Math. 34, 325–341 (2015). https://doi.org/10.1007/s40314-014-0119-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-014-0119-x

Keywords

Mathematics Subject Classification (2010)

Navigation