# American Institute of Mathematical Sciences

June  2016, 21(4): 1101-1117. doi: 10.3934/dcdsb.2016.21.1101

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

 1 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged,, Hungary 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  March 2016

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.
Citation: Attila Dénes, Gergely Röst. Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
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