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**Models for transmission of disease with immigration of infectives.**
*(English)*
Zbl 0995.92041

Summary: Simple models for disease transmission that include immigration of infective individuals and variable population size are constructed and analyzed. A model with a general contact rate for a disease that confers no immunity admits a unique endemic equilibrium that is globally stable. A model with mass action incidence for a disease in which infectives either die or recover with permanent immunity has the same qualitative behavior.

This latter result is proved by reducing the system to an integro-differential equation. If mass action incidence is replaced by a general contact rate, then the same result is proved locally for a disease that causes fatalities. Threshold-like results are given, but in the presence of immigration of infectives there is no disease-free equilibrium. A considerable reduction of infectives is suggested by the incorporation of screening and quarantining of infectives in a model for HIV transmission in a prison system.

This latter result is proved by reducing the system to an integro-differential equation. If mass action incidence is replaced by a general contact rate, then the same result is proved locally for a disease that causes fatalities. Threshold-like results are given, but in the presence of immigration of infectives there is no disease-free equilibrium. A considerable reduction of infectives is suggested by the incorporation of screening and quarantining of infectives in a model for HIV transmission in a prison system.

### MSC:

92D30 | Epidemiology |

45J05 | Integro-ordinary differential equations |

34D23 | Global stability of solutions to ordinary differential equations |

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\textit{F. Brauer} and \textit{P. van den Driessche}, Math. Biosci. 171, No. 2, 143--154 (2001; Zbl 0995.92041)

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### References:

[1] | S. Busenberg, K.L. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Biomathematics, vol. 53, Springer, Berlin, 1993; S. Busenberg, K.L. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Biomathematics, vol. 53, Springer, Berlin, 1993 · Zbl 0837.92021 |

[2] | Gani, J.; Yakowitz, S.; Blount, M., The spread and quarantine of HIV infection in a prison system, SIAM J. Appl. Math., 57, 1510 (1997) · Zbl 0892.92021 |

[3] | H.W. Hethcote, Three basic epidemiological models, in: S.A. Levin, T.G. Hallam, L.J. Gross (Eds.), Applied Mathematical Ecology; Biomathematics, vol. 18, Springer, Berlin, 1989, p. 119; H.W. Hethcote, Three basic epidemiological models, in: S.A. Levin, T.G. Hallam, L.J. Gross (Eds.), Applied Mathematical Ecology; Biomathematics, vol. 18, Springer, Berlin, 1989, p. 119 |

[4] | Hale, J. K.; Koçak, H., Dynamics and Bifurcations (1991), Springer: Springer Berlin · Zbl 0745.58002 |

[5] | Bochner, S., Monotone Funktionen, Stieltjessche Integrale und Harmonische Analyse, Math. Ann., 108, 378 (1933) |

[6] | Bochner, S., Lectures on Fourier Integrals (1959), Princeton University: Princeton University Princeton, NJ · Zbl 0085.31802 |

[7] | Halanay, A., On the asymptotic behaviour of the solutions of an integro-differential equation, J. Math. Anal. Appl., 10, 319 (1965) · Zbl 0136.10202 |

[8] | Nohel, J. A.; Shea, D. F., Frequency domain methods for Volterra equations, Adv. Math., 22, 278 (1976) · Zbl 0349.45004 |

[9] | Gripenberg, G.; Londen, S.-O.; Staffans, O., Volterra Integral and Functional Equations (1990), Cambridge University: Cambridge University Cambridge · Zbl 0695.45002 |

[10] | Jeffries, C.; Klee, V.; van den Driessche, P., When is a matrix sign stable?, Can. J. Math., 29, 315 (1977) · Zbl 0383.15005 |

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