# American Institute of Mathematical Sciences

February  2019, 24(2): 783-800. doi: 10.3934/dcdsb.2018207

## Global dynamics of a latent HIV infection model with general incidence function and multiple delays

 1 School of Science and Technology, Zhejiang International Studies University, Hangzhou 310023, China 2 Faculty of Engineering, Kyushu University, Fukuoka 819-0395, Japan 3 College of Science and Engineering, Aoyama Gakuin University, Kanagawa 252-5258, Japan

* Corresponding author: yangyu@zisu.edu.cn

Received  August 2017 Revised  January 2018 Published  June 2018

Fund Project: The first author was partially supported by National Natural Science Foundation of China (No. 11501519) and the third author was partially supported by the Japan Society Promotion of Science) through the "Grant-in-Aid 26400211".

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

Citation: Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207
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##### References:
The solution converges to the infection-free equilibrium $E_0(10^6, 0, 0, 0)$ when $\mathcal{R}_0 = 0.1781<1$. The parameter values are taken from Data1 in Table 1
The solution converges to the infected equilibrium $E^*(9.0262\times10^5, 68.5219,944.7647, 8.1662\times10^4)$ when $\mathcal{R}_0 = 2.0126>1$. The parameter values are taken from Data2 in Table 1
The function $\mathcal{R}_0(\tau_3)$ with respect to $\tau_3$
Effect of different values of $\tau_3$ on the dynamics of system (12)
The solution converges to $E_0(10^6, 0, 0, 0)$ when $\tau_3$ becomes large. Here we choose $\tau_3 = 80$. The parameter values are taken from Data3 in Table 1
List of parameters
 Parameters Data1 Data2 Data3 Source $s$ (cells ml$^{-1}$ day$^{-1}$) 10$^{4}$ 10$^{4}$ 10$^{4}$ [4] $d_T$ (day$^{-1}$) 0.01 0.01 0.01 [30] $\beta$ (ml virion$^{-1}$ day$^{-1}$) 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ [43] $\alpha_1$ 0.00001 0.00001 0.00001 Assumed $k$ 1.5$\times10^{-6}$ 0.001 0.001 [4,2] $\delta_1$ (day$^{-1}$) 0.05 0.05 0.05 [2] $\delta_L$ (day$^{-1}$) 0.004 0.004 0.004 [4] $\delta_2$ (day$^{-1}$) 0.01 0.01 0.01 [28] $\alpha$ (day$^{-1}$) 0.01 0.01 0.01 [47] $\delta$ (day$^{-1}$) 0.7 1 1 [4,22] $N$ (virions cell$^{-1}$) 100 2000 2000 [4,47] $c$ (day$^{-1}$) 13 23 23 [4,45] $\tau_1$ 0.3 0.3 0.3 [2] $\tau_2$ 0.6 0.6 0.6 [2] $\tau_3$ 0.6 0.6 Assumed [41]
 Parameters Data1 Data2 Data3 Source $s$ (cells ml$^{-1}$ day$^{-1}$) 10$^{4}$ 10$^{4}$ 10$^{4}$ [4] $d_T$ (day$^{-1}$) 0.01 0.01 0.01 [30] $\beta$ (ml virion$^{-1}$ day$^{-1}$) 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ [43] $\alpha_1$ 0.00001 0.00001 0.00001 Assumed $k$ 1.5$\times10^{-6}$ 0.001 0.001 [4,2] $\delta_1$ (day$^{-1}$) 0.05 0.05 0.05 [2] $\delta_L$ (day$^{-1}$) 0.004 0.004 0.004 [4] $\delta_2$ (day$^{-1}$) 0.01 0.01 0.01 [28] $\alpha$ (day$^{-1}$) 0.01 0.01 0.01 [47] $\delta$ (day$^{-1}$) 0.7 1 1 [4,22] $N$ (virions cell$^{-1}$) 100 2000 2000 [4,47] $c$ (day$^{-1}$) 13 23 23 [4,45] $\tau_1$ 0.3 0.3 0.3 [2] $\tau_2$ 0.6 0.6 0.6 [2] $\tau_3$ 0.6 0.6 Assumed [41]
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