Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions
Attila Dénes Gergely Röst
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.
keywords: Dulac functions. global stability SIR and SIRS models
Global stability of a multistrain SIS model with superinfection
Attila Dénes Yoshiaki Muroya Gergely Röst

In this paper, we study the global stability of a multistrain SIS model with superinfection. We present an iterative procedure to calculate a sequence of reproduction numbers, and we prove that it completely determines the global dynamics of the system. We show that for any number of strains with different infectivities, the stable coexistence of any subset of the strains is possible, and we completely characterize all scenarios. As an example, we apply our method to a three-strain model.

keywords: Epidemic model multistrain model SIS dynamics asymptotically autonomous systems global stability superinfection

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