| Citation: |
Li Jin, Yunxian Dai, Yu Xiao, Yiping Lin. RANK-ONE CHAOS IN A DELAYED SIR EPIDEMIC MODEL WITH NONLINEAR INCIDENCE AND TREATMENT RATES[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1779-1801. doi: 10.11948/20200190
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RANK-ONE CHAOS IN A DELAYED SIR EPIDEMIC MODEL WITH NONLINEAR INCIDENCE AND TREATMENT RATES
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Abstract
The rank one chaos in a SIR model with two time-delays is studied in this paper. By using center manifold theorem, normal form theory and Hassard's method, the existence, direction and stability of Hopf bifurcation are discussed. Based on the rank-one chaos theory for delayed differential equations, the conditions for the existence of rank-one strange attractor in disturbed system are obtained. Finally, numerical simulations are given to verify the theoretical analysis results.
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References
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Li Jin, Yunxian Dai, Yu Xiao, Yiping Lin. RANK-ONE CHAOS IN A DELAYED SIR EPIDEMIC MODEL WITH NONLINEAR INCIDENCE AND TREATMENT RATES[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1779-1801. doi: 10.11948/20200190
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Figure 1.
$ E $ is locally asymptotically stable with$ \varepsilon = 0 $ ,$ \tau_{1} = \tau_{2} = 0 $ -
Figure 2.
$ E $ is asymptotically stable with$ \varepsilon = 0 $ ,$ \tau_{1} = 0 $ ,$ \tau_{2} = 5 $ . -
Figure 3. A Hopf bifurcation occurs from
$ E $ with$ \varepsilon = 0 $ ,$ \tau_{1} = 0,\tau_{2} = 7.14 $ . -
Figure 4.
$ E $ is asymptotically stable with$ \varepsilon = 0 $ ,$ \tau = 0.9 $ . -
Figure 5. A Hopf bifurcation occurs from
$ E $ with$ \varepsilon = 0 $ ,$ \tau = 1.04 $ . -
Figure 6.
$ E $ is asymptotically stable with$ \varepsilon = 0 $ ,$ \tau_{1} = 0.2 $ ,$ \tau_{2} = 5 $ . -
Figure 7. Stable periodic solutions bifurcate from
$ E $ with$ \varepsilon = 0 $ ,$ \tau_{1} = 0.56 $ ,$ \tau_{2} = 5 $ . -
Figure 8.
$ E $ is unstable, and a rank-one strange attractor occurs with$ \tau_{1} = 0.56 $ ,$ \tau_{2} = 5 $ ,$ \varepsilon = 0.1 $ ,$ T = 2000 $ . -
Figure 9.
$ E $ is unstable, and a rank-one strange attractor occurs with$ \tau_{1} = 0.56 $ ,$ \tau_{2} = 5 $ ,$ \varepsilon = 0.1 $ ,$ T = 4000 $ . -
Figure 10. Largest Lyapunov exponent
$ \lambda $ versus$ \varepsilon $ with$ \tau_{1} = 0.56 $ ,$ \tau_{2} = 5 $ ,$ T = 2000 $ ,$ \varepsilon $ varying from$ 0 $ to$ 1 $ . -
Figure 11. Largest Lyapunov exponent
$ \lambda $ versus$ \varepsilon $ with$ \tau_{1} = 0.56 $ ,$ \tau_{2} = 5 $ ,$ T = 4000 $ ,$ \varepsilon $ varying from$ 0 $ to$ 1 $ . -
Figure 12. Largest Lyapunov exponent
$ \lambda $ versus$ T $ with$ \tau_{1} = 0.56 $ ,$ \tau_{2} = 5 $ ,$ \varepsilon = 0.1 $ ,$ T $ varying from$ 1000 $ to$ 6000 $