Geodesic
Global stability of a vaccination model with immigration
Electronic journal of differential equations, Tome 2015 (2015)
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.
Classification : 34K20, 92D30
Keywords: global stability, Lyapunov function, epidemiology, immigration 
@article{EJDE_2015__2015__a54,
     author = {Henshaw,  Sarah and McCluskey,  C.Connell},
     title = {Global stability of a vaccination model with immigration},
     journal = {Electronic journal of differential equations},
     year = {2015},
     volume = {2015},
     zbl = {1318.34065},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a54/}
}
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Henshaw,  Sarah; McCluskey,  C.Connell. Global stability of a vaccination model with immigration. Electronic journal of differential equations, Tome 2015 (2015). http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a54/