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