The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay.

*(English)*Zbl 1208.34127Summary: The problem of the asymptotic dynamics of a quarantine/isolation model with time delay, subject to two incidence functions, namely standard incidence and the Holling type II (saturated) incidence function, is considered. Rigorous qualitative analysis of the model shows that it exhibits essentially the same (equilibrium) dynamics regardless of which of the two incidence functions is used. In particular, 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 the Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally asymptotically stable for a special case. The permanence of the disease is also established for the model with the Holling type II incidence function. Finally, numerical simulations of the model with standard incidence show that the disease burden decreases with increasing time delay (incubation period).

##### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

92D30 | Epidemiology |

34K20 | Stability theory of functional-differential equations |

34K25 | Asymptotic theory of functional-differential equations |

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\textit{M. A. Safi} and \textit{A. B. Gumel}, Nonlinear Anal., Real World Appl. 12, No. 1, 215--235 (2011; Zbl 1208.34127)

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##### References:

[1] | Chowell, G.; Hengartner, N.W.; Castillo-Chavez, C.; Fenimore, P.W.; Hyman, J.M., The basic reproductive number of ebola and the effects of public health measures: the cases of congo and uganda, Journal of theoretical biology, 1, 119-126, (2004) |

[2] | Hethcote, H.W.; Ma, Zhien; Liao, Shengbing, Effects of quarantine in six endemic models for infectious diseases, Mathematical biosciences, 180, 141-160, (2002) · Zbl 1019.92030 |

[3] | Lipsitch, M., Transmission dynamics and control of severe acute respiratory syndrome, Science, 300, 1966-1970, (2003) |

[4] | Lloyd-Smith, J.O.; Galvani, A.P.; Getz, W.M., Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Proceedings of the royal society of London, series B, 170, 1979-1989, (2003) |

[5] | McLeod, R.G.; Brewster, J.F.; Gumel, A.B.; Slonowsky, D.A., Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs, Mathematical biosciences and engineering, 3, 527-544, (2006) · Zbl 1092.92042 |

[6] | Riley, S., Transmission dynamics of etiological agent of SARS in Hong Kong: the impact of public health interventions, Science, 300, 1961-1966, (2003) |

[7] | Wang, W.; Ruan, S., Simulating the SARS outbreak in Beijing with limited data, Journal of theoretical biology, 227, 369-379, (2004) |

[8] | Webb, G.F.; Blaser, M.J.; Zhu, H.; Ardal, S.; Wu, J., Critical role of nosocomial transmission in the Toronto SARS outbreak, Mathematical biosciences and engineering, 1, 1-13, (2004) · Zbl 1060.92054 |

[9] | Yan, X.; Zou, Y., Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Mathematical and computer modelling, 47, 235-245, (2008) · Zbl 1134.92033 |

[10] | Safi, M.A.; Gumel, A.B., Global asymptotic dynamics of a model for quarantine and isolation, Discrete and continuous dynamical systems. series B, 14, 209-231, (2010) · Zbl 1193.92075 |

[11] | Hethcote, H.W., The mathematics of infectious diseases, SIAM review, 42, 599-653, (2000) · Zbl 0993.92033 |

[12] | Sharomi, O., Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Mathematical biosciences, 210, 436-463, (2007) · Zbl 1134.92026 |

[13] | Anderson, R.M.; May, R.M., Population biology of infectious diseases, (1982), Springer-Verlag Berlin, Heidelberg, New York |

[14] | R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University, London, New York, 1991. |

[15] | Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Mathematical biosciences, 42, 43-61, (1978) · Zbl 0398.92026 |

[16] | Hou, J.; Teng, Z., Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates, Mathematics and computers in simulation, 79, 3038-3054, (2009) · Zbl 1166.92034 |

[17] | Liu, W.; Levin, S.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of mathematical biology, 23, 187-204, (1986) · Zbl 0582.92023 |

[18] | Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of differential equations, 188, 135-163, (2003) · Zbl 1028.34046 |

[19] | Cooke, K.L.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, Journal of mathematical biology, 35, 240-260, (1996) · Zbl 0865.92019 |

[20] | Donnelly, C., Epidemiological determinants of spread of a causal agent of severe acute respiratory syndrome in Hong Kong, Lancet, 361, 1761-1766, (2003) |

[21] | Gumel, A.B., Modelling strategies for controlling SARS outbreaks, Proceedings of the royal society of London, series B, 271, 2223-2232, (2004) |

[22] | Leung, G., The epidemiology of severe acute respiratory syndrome in the 2003 Hong Kong epidemic: an analysis of all 1755 patients, Annals of internal medicine, 9, 662-673, (2004) |

[23] | Wikipedia, Incubation Period. www.en.wikipedia.org (accessed May 2010). |

[24] | Chowell, G.; Castillo-Chavez, C.; Fenimore, P.W.; Kribs-Zaleta, C.M.; Arriola, L.; Hyman, J.M., Model parameters and outbreak control for SARS, Eid, 10, 1258-1263, (2004) |

[25] | Kribs-Zaleta, C.; Velasco-Hernandez, J., A simple vaccination model with multiple endemic states, Mathematical biosciences, 164, 183-201, (2000) · Zbl 0954.92023 |

[26] | Hale, J., Theory of functional differential equations, (1977), Springer-Verlag Heidelberg |

[27] | Mukandavire, Z.; Chiyaka, C.; Garira, W.; Musuka, G., Mathematical analysis of a sex-structured HIV/AIDS model with a discrete time delay, Nonlinear analysis, 71, 1082-1093, (2009) · Zbl 1178.34103 |

[28] | Smith, H.L.; Waltman, P., The theory of the chemostat, (1995), Cambridge University Press · Zbl 0860.92031 |

[29] | Xu, R.; Ma, Z., Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear analysis: real world applications, 10, 3175-3189, (2009) · Zbl 1183.34131 |

[30] | Gao, Shujing; Chenc, Lansun; Teng, Zhidong, Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear analysis: real world applications, 9, 599-607, (2008) · Zbl 1144.34390 |

[31] | Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Applied mathematics letters, 17, 1141-1145, (2004) · Zbl 1071.34082 |

[32] | Wang, W., Global behavior of an SEIRS epidemic model with time delays, Applied mathematics letters, 15, 423-428, (2002) · Zbl 1015.92033 |

[33] | Zhang, T.; Teng, Z., Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model, Chaos, solitons and fractals, 39, 2411-2425, (2009) · Zbl 1197.34090 |

[34] | Xu, R.; Ma, Z., Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, solitons and fractals, 41, 2319-2325, (2009) · Zbl 1198.34098 |

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