Abstract
On account of the effect of limited treatment resources on the control of epidemic disease, a saturated removal rate is incorporated into Hethcote’s SIR epidemiological model (Hethcote, SIAM Rev. 42:599–653, 2000). Unlike the original model, the model has two endemic equilibria when R 0<1. Furthermore, under some conditions, both the disease free equilibrium and one of the two endemic equilibria are asymptotically stable, i.e., the model has bistable equilibria. Therefore, disease eradication not only depends on R 0 but also on the initial sizes of all sub-populations. By the Poincaré-Bendixson theorem, Poincaré index, center manifold theorem, Hopf bifurcation theorem and Lyapunov-Lasalle theorem, etc., the existence and asymptotical stability of the equilibria, the existence, stability and direction of Hopf bifurcation are discussed, respectively.
Article PDF
Similar content being viewed by others
References
Corbett, B.D., Moghadas, S.M., Gumel, A.B.: Subthreshold domain of bistable equilibria for a model of HIV epidemiology. Int. J. Math. Math. Sci. 2003, 3679–3698 (2003)
Cui, J., Mu, X., Wan, H.: Saturation recovery leads to multiple endemic equilibria and backward bifurcation. J. Theor. Biol. 254, 275–283 (2008)
Feng, Z., Thieme, H.R.: Recurrent outbreaks of childhood diseases revisited: the impact of isolation. Math. Biosci. 128, 93–130 (1995)
Gukenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1990)
Hale, J.K.: Ordinary Differential Equations, 2nd edn. Krieger, Melbourne (1980)
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)
Hyman, J.M., Li, J.: Modeling the effectiveness of isolation strategies in preventing STD epidemics. SIAM J. Appl. Math. 58, 912–925 (1998)
Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Englewood Cliffs (2002)
Kribs-Zaleta, C.M., Velasco-Hernsandez, J.X.: A simple vaccination model with multiple endemic states. Math. Biosci. 164, 183–201 (2000)
Matallana, L.G., Blanco, A.M., Bandoni, J.A.: Estimation of domains of attraction in epidemiological models with constant removal rates of infected individuals. J. Phys. Conf. Ser. 90, 1–7 (2007)
Miller, R.K., Michel, A.N.: Ordinary Differential Equations. Academic Press, San Diego (1982)
Moghadas, S.M.: Analysis of an epidemic model with bistable equilibria using the Poincaré index. Appl. Math. Comput. 149, 689–702 (2004)
Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1996)
Smith, H.L., Thieme, H.R.: Stable coexistence and bi-stability for competitive systems on ordered Banach spaces. J. Differ. Equ. 176, 195–222 (2001)
Wang, W., Ruan, S.: Bifurcations in an epidemic model with constant removal rate of the infectives. J. Math. Anal. Appl. 291, 775–793 (2004)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin (1990)
Wu, L., Feng, Z.: Homoclinic bifurcation in an SIQR model for childhood diseases. J. Differ. Equ. 168, 150–167 (2000)
Zhang, Z.F., Ding, T.R., Huang, W.Z., et al.: Qualitative Theory of Differential Equations. Science Press, Beijing (1985) (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Training Fund of Xi’an University of Science and Technology under the contract 200836 and the Dr. Start-up Fund of Xi’an University of Science and Technology.
Rights and permissions
About this article
Cite this article
Zhonghua, Z., Yaohong, S. Qualitative analysis of a SIR epidemic model with saturated treatment rate. J. Appl. Math. Comput. 34, 177–194 (2010). https://doi.org/10.1007/s12190-009-0315-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-009-0315-9