| Citation: |
Yongchang Wei, Jinxiang Zhan, Jinhai Guo. ASYMPTOTIC BEHAVIORS OF A HEROIN EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE INFLUENCED BY STOCHASTIC PERTURBATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 1060-1077. doi: 10.11948/20230323
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ASYMPTOTIC BEHAVIORS OF A HEROIN EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE INFLUENCED BY STOCHASTIC PERTURBATIONS
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Abstract
In this paper, the dynamical behaviors of a stochastic heroin epidemic model with Lévy noises is investigated. First, we prove that this system has a unique global positive solution. Second, we derive the conditions of persistence in the mean and asymptotic stability in mean square using the Lyapunov and inequalities technique, and establish a criterion for positive recurrence. The results show that asymptotic behaviors are closely related to the Lévy measure. Finally, numerical simulations are used to illustrate the theoretical results.
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Keywords:
- Heroin epidemic model /
- persistence in the mean /
- positive recurrence /
- Lévy noises
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References
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Yongchang Wei, Jinxiang Zhan, Jinhai Guo. ASYMPTOTIC BEHAVIORS OF A HEROIN EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE INFLUENCED BY STOCHASTIC PERTURBATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 1060-1077. doi: 10.11948/20230323
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Figure 1.
$ \gamma=0.7,\; H(z)=0.55,\; \nu(Z)=0.65,\Delta=0.01 $ $ T=100 $ $ S(t) $ $ U_1(t) $ $ U_2(t) $ $ T=10 $ -
Figure 2.
The simulation of system (1.3) with
$ \alpha=0.065,\; \beta=0.3,\; \sigma=0.05,\; d=0.01,\; \delta=0.02,\; \gamma=0.15,\; H(z)=0.8,\; \nu(Z)=0.4,\; T=1000 $ $ \Delta=0.001 $ -
Figure 3.
The simulation of system (1.3) with
$ \alpha=0.065,\; \beta=0.3,\; \sigma=0.05,\; d=0.01,\; \delta=0.02,\; \gamma=0.15,\; H(z)=0.8,\; \nu(Z)=0.4, T=100 $ $ \Delta=0.001 $ $ S(t) $ $ U_1(t) $ $ U_2(t) $ $ S(t) $ $ U_1(t) $ $ U_2(t) $