In this paper, we incorporate the Beddington–DeAngelis incidence rate to a continuous-time HIV infection model with cure rate and a full logistic proliferation rate of \(CD4^+\) T cells in both uninfected and infected cells. Equilibria and their local stability analysis are discussed. It is shown that the HIV-free equilibrium point is locally asymptotically stable if \({\mathcal {R}}_0<1\) and unstable if \({\mathcal {R}}_0\ge 1\), where \({\mathcal {R}}_0\) is the basic reproduction number. Whereas, the HIV equilibrium point is locally asymptotically stable if \({\mathcal {R}}_0>1\). A nonstandard finite difference method is applied to the continuous model to obtain its discrete counterpart. The scheme applied preserves the main features of the continuous model such as positivity, boundedness of the solutions, equilibria and their local stability. Moreover, an optimal control strategy is applied to the discrete-time model in order to reduce the number of infected cells as well as the number of free HIV particles. Numerical simulations are performed to verify the theoretical analysis obtained.

在本文中，我们将Beddington–DeAngelis的发生率纳入连续时间的HIV感染模型，该模型具有治愈率和未感染和感染细胞中CD（CD4 ^ +） T细胞的完全逻辑增殖率。讨论了平衡及其局部稳定性分析。结果表明，如果\（{\ mathcal {R}} _ 0 <1 \）则无HIV平衡点在局部渐近稳定，而如果\（{\ mathcal {R}} _ 0 \ ge 1 \）则不稳定，其中\ （{\ mathcal {R}} _ 0 \）是基本复制编号。而如果\（{\ mathcal {R}} _ 0> 1 \），则HIV平衡点在局部渐近稳定。。将非标准有限差分方法应用于连续模型，以获得其离散对应物。所应用的方案保留了连续模型的主要特征，例如正性，解的有界性，平衡性及其局部稳定性。此外，将最佳控制策略应用于离散时间模型，以减少感染细胞的数量以及游离HIV颗粒的数量。进行数值模拟以验证获得的理论分析。