Electron. J. Diff. Equ., Vol. 2015 (2015), No. 92, pp. 1-10.

Global stability of a vaccination model with immigration

Sarah Henshaw, C. Connell McCluskey

Abstract:
We study an SVIR model of disease transmission with immigration into all four classes. Vaccinated individuals may only receive partial immunity to the disease, giving a leaky vaccine. The incidence function permits a nonlinear response to the number of infectives, so that mass action and saturating incidence are included as special cases. Because of the immigration of infected individuals, there is no disease-free equilibrium and hence no basic reproduction number. We use the Brouwer Fixed Point Theorem to show that an endemic equilibrium exists and the Poincare-Hopf Theorem to show that it is unique. We show the equilibrium is globally asymptotically stable by using a Lyapunov function.

Submitted February 12, 2015. Published April 10, 2015.
Math Subject Classifications: 34K20, 92D30.
Key Words: Global stability; Lyapunov function; epidemiology; immigration.

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Sarah Henshaw
Department of Mathematics
Wilfrid Laurier University
Waterloo, Ontario, Canada
email: hens3420@mylaurier.ca
C. Connell McCluskey
Department of Mathematics
Wilfrid Laurier University
Waterloo, Ontario, Canada
email: ccmcc8@gmail.com

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