Article Contents
Article Contents

# Dynamics of a stochastic HIV/AIDS model with treatment under regime switching

• * Corresponding author: Daqing Jiang

This work is supported by the National Natural Science Foundation of China under Grant No. 11871473 and Natural Science Foundation of Shandong Province under Grant No. ZR2019MA010

• This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.

Mathematics Subject Classification: Primary: 34E10, 34F05; Secondary: 37A50.

 Citation:

• Figure 1.  The solution of subsystem with state 1. (Color figure online)

Figure 2.  The solution of subsystem with state 2. (Color figure online)

Figure 3.  The pictures (a), (b) and (c) are the solution of system (3). The picture (d) is the corresponding Markov chain with $\pi = (\frac{3}{5},\frac{2}{5})$. (Color figure online)

Figure 4.  The solution of subsystem with state 1. (Color figure online)

Figure 5.  The solution of subsystem with state 2. (Color figure online)

Figure 6.  The pictures (a), (b) and (c) are the solution of system (3). The picture (d) is the corresponding Markov chain with $\pi = (\frac{3}{5},\frac{2}{5})$. (Color figure online)

Figure 7.  The diagrams track the variation trends of $S(t)$ and $I_{k}(t),k = 1,2$ with different transmission rate $\lambda(m),m = 1,2$. (Color figure online)

Figure 8.  The diagrams track the variation trends of $I_{k}(t)$ and $T_{k}(t),k = 1,2$ with different treatment rate $\tau$. (Color figure online)

Table 1.  List of the biological parameters

 Parameter Definition Value Source $\rho_{k}$ Transition rates per year from stage $k$ to stage $k+1$ for untreated individuals $\rho_{1}=1/0.271, \rho_{2}=1/8.31$ [7,2,13] $\gamma_{k}$ Transition rates per year from stage $k$ to stage $k+1$ for treated individuals $\gamma_{1}=1/8.21, \gamma_{2}=1/54$ [2,46] $\tau$ Rate per year of moving from the untreated to the treated population range 0-100$\%$ [20] $\phi$ Rate of moving from the treated back to the untreated population range 0-100$\%$ [20] $\epsilon$ Infectivity of individuals under treatment around 0.01 [10] $h_{k}$ Infectivity of untreated individuals in stage $k$ of infection per year around 2.76 for $h_{1}$, around 0.106 for $h_{2}$ [13]
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