Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals.Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is alway...
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AIMS Press
2014-02-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.785 |
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| author | Yoichi Enatsu Yukihiko Nakata |
| author_facet | Yoichi Enatsu Yukihiko Nakata |
| author_sort | Yoichi Enatsu |
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| description | We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals.Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable.We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results. |
| format | Article |
| id | doaj-art-cc7b3df790a54e18983ed140e9dc9e9d |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2014-02-01 |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-cc7b3df790a54e18983ed140e9dc9e9d2025-01-24T02:28:18ZengAIMS PressMathematical Biosciences and Engineering1551-00182014-02-0111478580510.3934/mbe.2014.11.785Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rateYoichi Enatsu0Yukihiko Nakata1Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals.Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable.We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results.https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.785nonlinear incidence ratehopf bifurcationlinearized stabilitydelay differential equation.epidemic model |
| spellingShingle | Yoichi Enatsu Yukihiko Nakata Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate Mathematical Biosciences and Engineering nonlinear incidence rate hopf bifurcation linearized stability delay differential equation. epidemic model |
| title | Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate |
| title_full | Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate |
| title_fullStr | Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate |
| title_full_unstemmed | Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate |
| title_short | Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate |
| title_sort | stability and bifurcation analysis of epidemic models with saturated incidence rates an application to a nonmonotone incidence rate |
| topic | nonlinear incidence rate hopf bifurcation linearized stability delay differential equation. epidemic model |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.785 |
| work_keys_str_mv | AT yoichienatsu stabilityandbifurcationanalysisofepidemicmodelswithsaturatedincidenceratesanapplicationtoanonmonotoneincidencerate AT yukihikonakata stabilityandbifurcationanalysisofepidemicmodelswithsaturatedincidenceratesanapplicationtoanonmonotoneincidencerate |