The aim of this study is to model the transmission of COVID-19 and investigate the impact of some control strategies on its spread. We propose an extension of the classical SEIR model, which takes into account the age structure and uses fractional-order derivatives to have a more realistic model. For each age group $j$ the population is divided into seven classes namely susceptible $S^j,$ exposed $E^j,$ infected with high risk $I_h^{{}_j},$ infected with low risk $I_l^{{}_j},$ hospitalized $H^j,$ recovered with and without psychological complications $R_1^j$ and $R_2^j,$ respectively. In our model, we incorporate three control variables which represent: awareness campaigns, diagnosis and psychological follow-up. The purpose of our control strategies is protecting susceptible individuals from being infected, minimizing the number of infected individuals with high and low risk within a given age group $j,$ as well as reducing the number of recovered individuals with psychological complications. Pontryagin’s maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. Numerical simulations performed using Matlab, are provided to show the effectiveness of three control strategies and the effect of the order of fractional derivative on the efficiency of these control strategies. Using a cost-effectiveness analysis method, our results show that combining awareness with diagnosis is the most effective strategy. To the best of our knowledge, this work is the first that propose a framework on the control of COVID-19 transmission based on a multi-age model with Caputo time-fractional derivative.

本研究的目的是模拟 COVID-19 的传播，并调查一些控制策略对其传播的影响。我们提出了经典 SEIR 模型的扩展，它考虑了年龄结构并使用分数阶导数来获得更真实的模型。针对每个年龄段$j$人口分为七个等级，即易感人群${\mathrm{小号}}^j,$裸露${乙}^j,$高危感染${我}_H^{{}_j},$低风险感染${我}_{升}^{{}_j},$住院$H^j,$康复后有无心理并发症$R_{\mathrm{1个}}^j$和$R_{\mathrm{2个}}^j,$分别。在我们的模型中，我们纳入了三个控制变量，分别代表：宣传活动、诊断和心理随访。我们控制策略的目的是保护易感人群免受感染，最大限度地减少给定年龄组中高风险和低风险感染者的数量$j,$以及减少患有心理并发症的康复者人数。Pontryagin 的最大值原理用于表征最优控制，最优系统通过迭代方法求解。提供了使用 Matlab 执行的数值模拟，以显示三种控制策略的有效性以及分数阶导数对这些控制策略效率的影响。使用成本效益分析方法，我们的结果表明，将意识与诊断相结合是最有效的策略。据我们所知，这项工作是第一个提出基于具有 Caputo 时间分数阶导数的多年龄模型的 COVID-19 传输控制框架的工作。