Global stability for SIR and SIRS models with nonlinear incidence and removal terms  via Dulac functions

Attila  Dénes; Gergely Röst

doi:10.3934/dcdsb.2016.21.1101

Article Contents

2016, Volume 21, Issue 4: 1101-1117. Doi: 10.3934/dcdsb.2016.21.1101

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Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions

Attila Dénes^1, and
Gergely Röst^2,

1.
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged,
2.
Analysis and Stochastics Research Group, Hungarian Academy of Sciences, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1.

Received: November 2014

Revised: September 2015

Published: June 2016

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Abstract

We prove the global asymptotic stability of the disease-free and the endemic equilibrium for general SIR and SIRS models with nonlinear incidence. Instead of the popular Volterra-type Lyapunov functions, we use the method of Dulac functions, which allows us to extend the previous global stability results to a wider class of SIR and SIRS systems, including nonlinear (density-dependent) removal terms as well. We show that this method is useful in cases that cannot be covered by Lyapunov functions, such as bistable situations. We completely describe the global attractor even in the scenario of a backward bifurcation, when multiple endemic equilibria coexist.

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Mathematics Subject Classification: Primary: 37C70, 92D30; Secondary: 34C23, 34D23.

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