| Citation: |
Zhen Cao, Lin-Fei Nie. DYNAMICS OF A STOCHASTIC VECTOR-HOST EPIDEMIC MODEL WITH AGE-DEPENDENT OF VACCINATION AND DISEASE RELAPSE[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1274-1303. doi: 10.11948/20220099
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DYNAMICS OF A STOCHASTIC VECTOR-HOST EPIDEMIC MODEL WITH AGE-DEPENDENT OF VACCINATION AND DISEASE RELAPSE
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Abstract
Due to the ubiquitous stochastic interference in nature, the uncertainty of the disease relapse and the duration of immunity, we present a stochastic vector-host epidemic model with age-dependent of vaccination and disease relapse, where two general incidences are also introduced to depict the transmission of virus between vectors and hosts. By constructing a suitable Lyapunov function, the existence and uniqueness of the global positive solution of our model are proved. Further, the stochastic extinction of disease, the existence of stationary distribution are also discussed. Moreover, the stochastic extinction of disease and the existence of stationary distribution for special incidence are obtained as an application, where the general incidence degenerates into the billinear incidence. Finally, numerical simulations are given to illuminate the main results, which also suggest that the behaviors of vectors and the self-protection of hosts are the key factors to eliminate the disease relative to the quantity of vector population during the transmission of vector-host infectious diseases.
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References
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Zhen Cao, Lin-Fei Nie. DYNAMICS OF A STOCHASTIC VECTOR-HOST EPIDEMIC MODEL WITH AGE-DEPENDENT OF VACCINATION AND DISEASE RELAPSE[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1274-1303. doi: 10.11948/20220099
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Figure 1.
The stochastic extinction of disease and asymptotical stability of the disease-free steady state for model (6.1) with
$ \Lambda_h=1 $ $ \mu_{v}=5000 $ $ \psi_h=0.375 $ $ \mu_v=0.12 $ $ \beta_1=2.875\times10^{-6} $ $ \beta_2=4.275\times10^{-6} $ $ \sigma_1=1.0\times10^{-5} $ $ \sigma_2=9.0\times10^{-6} $ $ \sigma_3=3.2\times10^{-6} $ $ \sigma_4=2.3\times10^{-6} $ $ \widetilde{\mathcal{R}}_0^*\approx 0.9360<1 $ $ \sigma_1^2-\beta_1/S_h^0\approx-1.0781\times10^{-6}<0 $ $ \sigma_2^2-\beta_2/S_v^0\approx-2.16\times10^{-11}<0 $ -
Figure 2.
The persistence of disease of model (6.1) with
$ \Lambda_h=10 $ $ \psi_h=0.175 $ $ \mu_v=1/30 $ $ \beta_1=2.875\times10^{-4} $ $ \beta_2=4.275\times10^{-4} $ $ \sigma_3=8.678\times10^{-38} $ $ \sigma_4=8.678\times10^{-3} $ $ \mathcal{R}_*^s\approx 14.6748>1 $ $ (\mu_{h}\wedge\mu_{v})-(\sigma_{3}^{2}\vee\sigma_{4}^{2})/2\approx 3.9791 \times10^{-7}>0 $ $ I_v(t) $ $ I_v(t) $ $ I_h(t) $ $ I_v(t) $ -
Figure 3.
The effect of strength of
$ \sigma_{i}^2(i=1, 2, 3, 4) $ $ \Lambda_{h}=10 $ $ \Lambda_{v}=5000 $ $ \psi=0.175 $ $ k=1/7 $ $ \mu_h=1/(72\times 365) $ $ \beta_1=\beta_2=2.875\times 10^{-5} $ $ \mu_v=1/20 $ $ \sigma_{3}=\sigma_{4}=2.5\times 10^{-6} $ $ \sigma_{1}=\sigma_{2}=1.2\times 10^{-6} $ $ 1.2\times 10^{-5} $ $ 5.2\times 10^{-4} $ $ \sigma_{1}=\sigma_{1}=2.5\times 10^{-6} $ $ \sigma_{3}=\sigma_{4}=8.2\times 10^{-4} $ $ 8.2\times 10^{-2} $ $ 8.2\times 10^{-1} $ -
Figure 4.
The effect of strength of
$ \sigma_{i}^2(i=1, 2, 3, 4) $ $ \Lambda_{h}=10 $ $ \Lambda_{v}=5000 $ $ \psi=0.175 $ $ k=1/7 $ $ \mu_h=1/(72\times 365) $ $ \beta_1=\beta_2=2.875\times 10^{-5} $ $ \mu_v=1/20 $ $ (i) $ $ \sigma_{1}=\sigma_{2}=2.5\times 10^{-6} $ $ \sigma_{3}=\sigma_{4}=1.2\times 10^{-5} $ $ (ii) $ $ \sigma_{1}=\sigma_{2}=2.5\times 10^{-5} $ $ \sigma_{3}=\sigma_{4}=1.2\times 10^{-4} $ $ (iii) $ $ \sigma_{1}=\sigma_{2}=5.2\times 10^{-5} $ $ \sigma_{3}=\sigma_{4}=1.2\times 10^{-2} $