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**Backward bifurcation of an epidemic model with infectious force in infected and immune period and treatment.**
*(English)*
Zbl 1278.92044

In this paper, an epidemic model, with an infectious force in the infected and immune periods and a treatment rate of infectious individuals, is proposed to understand the effect of the capacity for treatment on the disease spread. It is assumed that the treatment rate is proportional to the number of infective individuals below the capacity and is constant when the number of infective individuals is greater than the capacity. It is proved that the existence and stability of equilibria for the model is not only related to the basic reproduction number but also to the capacity for treatment of infective agents. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.

Reviewer: Xueyong Zhou (Xinyang)

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\textit{Y. Xue} and \textit{J. Wang}, Abstr. Appl. Anal. 2012, Article ID 647853, 14 p. (2012; Zbl 1278.92044)

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### References:

[1] | J. Zhang and Z. E. Ma, “Global analysis of the SEI epidemic model with constant inflows of different compartments,” Journal of Xi’an Jiaotong University, vol. 37, no. 6, pp. 653-656, 2003. |

[2] | S. L. Yuan, L. T. Han, and Z. E. Ma, “A kind of epidemic model having infectious force in both latent period and infected period,” Journal of Biomathematics, vol. 16, no. 4, pp. 392-398, 2001. · Zbl 1059.92509 |

[3] | G. H. Li and Z. Jin, “Global stability of an SEI epidemic model,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 925-931, 2004. · Zbl 1045.34025 |

[4] | G. H. Li and Z. Jin, “Global stability of an SEI epidemic model with general contact rate,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 997-1004, 2005. · Zbl 1062.92062 |

[5] | G. H. Li and Z. Jin, “Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period,” Chaos, Solitons and Fractals, vol. 25, no. 5, pp. 1177-1184, 2005. · Zbl 1065.92046 |

[6] | J. Arino, C. C. McCluskey, and P. van den Driessche, “Global results for an epidemic model with vaccination that exhibits backward bifurcation,” SIAM Journal on Applied Mathematics, vol. 64, no. 1, pp. 260-276, 2003. · Zbl 1034.92025 |

[7] | J. Dushoff, W. Huang, and C. Castillo-Chavez, “Backwards bifurcations and catastrophe in simple models of fatal diseases,” Journal of Mathematical Biology, vol. 36, no. 3, pp. 227-248, 1998. · Zbl 0917.92022 |

[8] | P. van den Driessche and J. Watmough, “A simple SIS epidemic model with a backward bifurcation,” Journal of Mathematical Biology, vol. 40, no. 6, pp. 525-540, 2000. · Zbl 0961.92029 |

[9] | K. P. Hadeler and P. van den Driessche, “Backward bifurcation in epidemic control,” Mathematical Biosciences, vol. 146, no. 1, pp. 15-35, 1997. · Zbl 0904.92031 |

[10] | Y. A. Kuznetsov and C. Piccardi, “Bifurcation analysis of periodic SEIR and SIR epidemic models,” Journal of Mathematical Biology, vol. 32, no. 2, pp. 109-121, 1994. · Zbl 0786.92022 |

[11] | S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135-163, 2003. · Zbl 1028.34046 |

[12] | X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433-443, 2008. · Zbl 1144.92038 |

[13] | Z. Hu, S. Liu, and H. Wang, “Backward bifurcation of an epidemic model with standard incidence rate and treatment rate,” Nonlinear Analysis. Real World Applications, vol. 9, no. 5, pp. 2302-2312, 2008. · Zbl 1156.34320 |

[14] | X.-Z. Li, W.-S. Li, and M. Ghosh, “Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 141-150, 2009. · Zbl 1159.92036 |

[15] | X. Zhang and X. Liu, “Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 565-575, 2009. · Zbl 1167.34338 |

[16] | W. Wang and S. Ruan, “Bifurcation in an epidemic model with constant removal rate of the infectives,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 775-793, 2004. · Zbl 1054.34071 |

[17] | W. Wang, “Backward bifurcation of an epidemic model with treatment,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 58-71, 2006. · Zbl 1093.92054 |

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