Analysis of a Delayed SIR Model with Nonlinear Incidence Rate
An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold value ℜ0 determining whether the disease dies out is found. The r...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2008/636153 |
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| author | Jin-Zhu Zhang Zhen Jin Quan-Xing Liu Zhi-Yu Zhang |
| author_facet | Jin-Zhu Zhang Zhen Jin Quan-Xing Liu Zhi-Yu Zhang |
| author_sort | Jin-Zhu Zhang |
| collection | DOAJ |
| description | An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is
of saturated form with the susceptible. The threshold value ℜ0 determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold value ℜ0 and time delay (i.e., incubation time length). If ℜ0 is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one there will be an endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns. |
| format | Article |
| id | doaj-art-6991727c62784f9dbbd2dbc8b485b503 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-6991727c62784f9dbbd2dbc8b485b5032025-02-03T01:21:45ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2008-01-01200810.1155/2008/636153636153Analysis of a Delayed SIR Model with Nonlinear Incidence RateJin-Zhu Zhang0Zhen Jin1Quan-Xing Liu2Zhi-Yu Zhang3School of Mechantronic Engineering, North University of China, Taiyuan 030051, ChinaDepartment of Mathematics, North University of China, Taiyuan 030051, ChinaDepartment of Mathematics, North University of China, Taiyuan 030051, ChinaDepartment of Basic Science, Taiyuan Institute of Technology, Taiyuan 030008, ChinaAn SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold value ℜ0 determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold value ℜ0 and time delay (i.e., incubation time length). If ℜ0 is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one there will be an endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.http://dx.doi.org/10.1155/2008/636153 |
| spellingShingle | Jin-Zhu Zhang Zhen Jin Quan-Xing Liu Zhi-Yu Zhang Analysis of a Delayed SIR Model with Nonlinear Incidence Rate Discrete Dynamics in Nature and Society |
| title | Analysis of a Delayed SIR Model with Nonlinear Incidence Rate |
| title_full | Analysis of a Delayed SIR Model with Nonlinear Incidence Rate |
| title_fullStr | Analysis of a Delayed SIR Model with Nonlinear Incidence Rate |
| title_full_unstemmed | Analysis of a Delayed SIR Model with Nonlinear Incidence Rate |
| title_short | Analysis of a Delayed SIR Model with Nonlinear Incidence Rate |
| title_sort | analysis of a delayed sir model with nonlinear incidence rate |
| url | http://dx.doi.org/10.1155/2008/636153 |
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