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).
Similar content being viewed by others
References
Anderson RM, May RM (1982) Population biology of infectious diseases. Springer, Berlin
Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University, London
Capasso V, Serio G (1978) A generalization of the Kermack–Mckendrick deterministic epidemic model. Math Biosci 42:43–61
Cooke KL, van den Driessche P (1996) Analysis of an SEIRS epidemic model with two delays. J Math Biol 35:240–260
Hale J (1977) Theory of functional differential equations. Springer, Heidelberg
Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653
Hou J, Teng Z (2009) Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates. Math Comput Simul 79:3038–3054
Kribs-Zaleta C, Velasco-Hernandez J (2000) A simple vaccination model with multiple endemic states. Math Biosci 164:183–201
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
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
Ruan S, Wang W (2003) Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Differ Equ 188:135–163
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
Sharomi O et al (2007) Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math Biosci 210:436–463
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
Xu R, Ma Z (2009) Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solitons Fractals 41:2319–2325
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Maria do Rosario de Pinho.
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-014-0119-x