Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth
We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any τ, we show that the disease-free equilibrium is globally asymptotically stable; when R0<1, the disease will die out. Directly afterwards, we pro...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/916130 |
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| _version_ | 1832564372976697344 |
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| author | Yakui Xue Tiantian Li |
| author_facet | Yakui Xue Tiantian Li |
| author_sort | Yakui Xue |
| collection | DOAJ |
| description | We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any τ, we show that the disease-free equilibrium is globally asymptotically stable; when R0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any τ=0; when R0>1, the disease will persist. However, for any τ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions. |
| format | Article |
| id | doaj-art-f596e98393094b1bbff5ccc563f858b0 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-f596e98393094b1bbff5ccc563f858b02025-02-03T01:11:19ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/916130916130Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic GrowthYakui Xue0Tiantian Li1Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, ChinaDepartment of Mathematics, North University of China, Taiyuan, Shanxi 030051, ChinaWe study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any τ, we show that the disease-free equilibrium is globally asymptotically stable; when R0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any τ=0; when R0>1, the disease will persist. However, for any τ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.http://dx.doi.org/10.1155/2013/916130 |
| spellingShingle | Yakui Xue Tiantian Li Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth Abstract and Applied Analysis |
| title | Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth |
| title_full | Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth |
| title_fullStr | Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth |
| title_full_unstemmed | Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth |
| title_short | Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth |
| title_sort | stability and hopf bifurcation for a delayed sir epidemic model with logistic growth |
| url | http://dx.doi.org/10.1155/2013/916130 |
| work_keys_str_mv | AT yakuixue stabilityandhopfbifurcationforadelayedsirepidemicmodelwithlogisticgrowth AT tiantianli stabilityandhopfbifurcationforadelayedsirepidemicmodelwithlogisticgrowth |