Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates
We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic rep...
| Autores: | , , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2012 |
| País: | España |
| Institución: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/409 |
| Acceso en línea: | http://hdl.handle.net/20.500.11824/409 |
| Access Level: | acceso abierto |
| Palabra clave: | Global asymptotic stability Hopf bifurcation Lyapunov functional Nonlinear incidence rate SIR epidemic model |
| Sumario: | We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R 0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R 0 ≤ 1. Second we show that the model is permanent if and only if R 0 > 1. Moreover, using a threshold parameter R 0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1< R0≤R 0 and it loses stability as the length of the delay increases past a critical value for 1<R 0< R0. Our result is an extension of the stability results in [J.-J. Wang, J.-Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal. RWA 11 (2009) 2390-2402]. |
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