In this paper, we discuss the numerical representation of the linearly implicit Euler method for an age-structured HIV infection model with a general transmission mechanism. We first define the basic reproduction number of the continuous model, and present the stability results of the equilibriums. For the numerical process, we establish the solvability of the system and the non-negativity and convergence of numerical solutions. In the analysis of the long-term dynamical behavior, this paper mainly focus on the existence of the infection equilibrium determined by the numerical reproduction number . To overcome the difficulty caused by the complexity of epidemic transmission mechanisms, the 1-order convergence analysis of numerical basic reproduction numbers is implemented by using the properties of the fundamental solution matrix. By a comparison principle, we show that the disease-free equilibrium is globally asymptotically stable if . Moreover, for , a unique numerical endemic equilibrium exists, which converges to the exact one, is locally asymptotically stable. Hence, numerical processes visually represent the dynamic properties of nonlinear age-structured HIV models. Finally, some numerical experiments demonstrate the verification and the efficiency of our results.

中文翻译：

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具有一般传播机制的年龄结构HIV模型的数值分析

在本文中，我们讨论了具有一般传播机制的年龄结构 HIV 感染模型的线性隐式欧拉方法的数值表示。我们首先定义连续模型的基本再生数，并给出平衡点的稳定性结果。对于数值过程，我们建立了系统的可解性以及数值解的非负性和收敛性。在对长期动态行为的分析中，本文主要关注由数值再生数决定的感染平衡的存在性。为了克服流行病传播机制复杂性带来的困难，利用基本解矩阵的性质，实现了数值基本再生数的一阶收敛分析。通过比较原理，我们证明如果 ，则无病平衡是全局渐近稳定的。此外，对于 ，存在一种独特的数值地方性平衡，该平衡收敛于精确的平衡，并且是局部渐近稳定的。因此，数值过程直观地代表了非线性年龄结构 HIV 模型的动态特性。最后，一些数值实验证明了我们的结果的验证和效率。