| Citation: |
Songnan Liu, Xiaojie Xu, Zhangyi Dong. LONG-TIME BEHAVIOR OF STOCHASTIC STAGED PROGRESSION EPIDEMIC MODEL WITH HYBRID SWITCHING FOR THE TRANSMISSION OF HIV[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 125-152. doi: 10.11948/20210085
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LONG-TIME BEHAVIOR OF STOCHASTIC STAGED PROGRESSION EPIDEMIC MODEL WITH HYBRID SWITCHING FOR THE TRANSMISSION OF HIV
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Abstract
In this paper, one stochastic hybrid switching SP (staged progression) model for the transmission of HIV is proposed and investigated. The system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent. Furthermore, sufficient conditions for extinction of disease are established. At last, some examples and simulations are provided to illustrate our results.
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References
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Songnan Liu, Xiaojie Xu, Zhangyi Dong. LONG-TIME BEHAVIOR OF STOCHASTIC STAGED PROGRESSION EPIDEMIC MODEL WITH HYBRID SWITCHING FOR THE TRANSMISSION OF HIV[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 125-152. doi: 10.11948/20210085
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Figure 1. In the SP model every infected individual goes through the same series of stages. This model can account for a short early highly infectious stage equivalent to the acute phase of infection, a middle period of low infectiousness, and a late chronic stage with higher infectiousness, where
$ \lambda = r\sum_{i = 1}^n\beta_iI_i(t)S(t) $ . -
Figure 2.
$ ( S(t) , I_1(t), I_2(t), I_3(t), I_4(t)) $ has ergodic property. The pictures on the right are the density functions of system (1.1) for$ \sigma_1 = 0.3 $ ,$ \sigma_2 = 0.2 $ ,$ \sigma_3 = 0.1 $ ,$ \sigma_4 = 0.2 $ and$ \sigma_5 = 0.2 $ . We employ the Milstein's Higher Order Method with initial value$ ( S(0) , I_1(0), I_2(0), I_3(0), I_4(0)) = (0.3, 0.7, 0.3) $ . -
Figure 3.
$ I_1(t)) $ $ I_2(t) $ ,$ I_3(t) $ and$ I_4(t) $ will tends to zero exponentially with probability one. The pictures on the right are the density functions of system (1.1) for$ \sigma_1 = 0.8 $ ,$ \sigma_2 = 0.9 $ ,$ \sigma_3 = 0.9 $ ,$ \sigma_4 = 0.9 $ and$ \sigma_5 = 0.9 $ . We use the Milstein's Higher Order Method with initial value$ ( S(0) , I_1(0), I_2(0), I_3(0), I_4(0)) = (0.7, 0.3, 0.3, 0.3, 0.3) $ . -
Figure 4.
$ ( S(t) , I_1(t), I_2(t), I_3(t), I_4(t)) $ is positive recurrence. The pictures on the left are Markovian chain. The pictures on the right are the density functions of system (1.4) for$ l\in\mathcal{M} = \{1, 2\} $ . We employ the Milstein's Higher Order Method with initial value$ ( S(0) , I_1(0), I_2(0), I_3(0), I_4(0)) = (0.7, 0.3, 0.3, 0.3, 0.3) $ . -
Figure 5. Computer simulation of a single path of
$ ( S(t) , I_1(t), I_2(t), I_3(t), I_4(t)) $ for the SDE model (1.4) with intial condition$ (0.7, 0.3, 0.3, 0.3, 0.3) $ . -
Figure 6. Computer simulation of 1000 pathes of
$ ( S(t) , I_1(t), I_2(t), I_3(t), I_4(t)) $ for the SDE model (1.4). - Figure 7. This picture is the 95% confidence intervals from 1000 stochastic simulations.
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Figure 8.
$ I_1(t) $ ,$ I_2(t) $ ,$ I_3(t) $ and$ I_4(t) $ will tends to zero exponentially with probability one. The pictures on the left are Markovian chain. The pictures on the right are the density functions of system (1.4) for$ l\in\mathcal{M} = \{1, 2\} $ . We employ the Milstein's Higher Order Method with initial value$ ( S(0) , I_1(0), I_2(0), I_3(0), I_4(0)) = (0.7, 0.3, 0.3, 0.3, 0.3) $ .