×

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.

MSC:

92D30 Epidemiology
45J05 Integro-ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.