Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia.
Math Biosci Eng. 2023 Jan 11;20(3):5298-5315. doi: 10.3934/mbe.2023245.
In this paper, we analyse a dynamical system taking into account the asymptomatic infection and we consider optimal control strategies based on a regular network. We obtain basic mathematical results for the model without control. We compute the basic reproduction number (R) by using the method of the next generation matrix then we analyse the local stability and global stability of the equilibria (disease-free equilibrium (DFE) and endemic equilibrium (EE)). We prove that DFE is LAS (locally asymptotically stable) when R<1 and it is unstable when R>1. Further, the existence, the uniqueness and the stability of EE is carried out. We deduce that when R>1, EE exists and is unique and it is LAS. By using generalized Bendixson-Dulac theorem, we prove that DFE is GAS (globally asymptotically stable) if R<1 and that the unique endemic equilibrium is globally asymptotically stable when R>1. Later, by using Pontryagin's maximum principle, we propose several reasonable optimal control strategies to the control and the prevention of the disease. We mathematically formulate these strategies. The unique optimal solution was expressed using adjoint variables. A particular numerical scheme was applied to solve the control problem. Finally, several numerical simulations that validate the obtained results were presented.
在本文中，我们分析了一个考虑无症状感染的动力系统，并考虑了基于正则网络的最优控制策略。我们为没有控制的模型获得了基本的数学结果。我们使用下一代矩阵方法计算基本繁殖数 (R)，然后分析平衡点（无病平衡点 (DFE) 和地方病平衡点 (EE)）的局部稳定性和全局稳定性。我们证明当 R<1 时 DFE 是 LAS（局部渐近稳定），当 R>1 时它是不稳定的。此外，还进行了 EE 的存在性、唯一性和稳定性分析。我们推断出当 R>1 时，EE 存在且唯一，且是 LAS。通过使用广义 Bendixson-Dulac 定理，我们证明了当 R<1 时 DFE 是 GAS（全局渐近稳定），并且当 R>1 时唯一的地方病平衡点是全局渐近稳定的。后来，我们通过使用庞特里亚金最大值原理，针对疾病的控制和预防提出了几种合理的最优控制策略。我们对这些策略进行了数学公式化。使用伴随变量表达了唯一最优解。应用了一种特殊的数值方案来解决控制问题。最后，提出了几个验证所得结果的数值模拟。
