论文标题
SARS-COV-2流行病模型参数的识别和控制
Identification and control of SARS-CoV-2 epidemic model parameters
论文作者
论文摘要
我们提出了一个数学模型，其中有五个用于SARS-COV-2传输的隔间：易感$ S $，未发现的无症状$ a $，未发现的感染有症状的$ i $，确认的阳性和孤立的$ l $，并恢复了$ r $，我们的目标是双重目标。首先，我们提出并解决了一个反问题，主要集中在识别$ a_ {0} $和$ i_ {0} $的$ a_ {0} $的无症状和有症状的个体的$ t_ {0} $，通过在两个成功的时间和恢复的时间$ t_ {0} $确定，$ t_感染性症状个体在单位时间内感染的平均人数的速率站立。然后，我们提出了一个控制问题，旨在通过改进级别的$ a $ a $ a $ a $ a $ a $ a $ a $ $ l $ $ l $ $ $ a $的行动来控制感染类。这些目标被称为最小化问题，第二个目标包括状态限制，该问题由最佳控制技术处理。证明了最佳控制器的存在，并确定了最佳最佳条件的第一阶条件。对于第二个问题，它们是通过在适当定义的近似问题的最佳条件下传递到极限来推断出来的。在这种情况下，双重系统是奇异的，在度量空间中具有一个组成部分。在获得生命免疫力时，对系统进行的渐近稳定性的讨论表明，该疾病的渐近灭绝，并确定了良好的繁殖数。
We propose a mathematical model with five compartments for the SARS-CoV-2 transmission: susceptible $S$, undetected infected asymptomatic $A$, undetected infected symptomatic $I$, confirmed positive and isolated $L$, and recovered $ R$, for which we have a twofold objective. First, we formulate and solve an inverse problem focusing mainly on the identification of the values $A_{0}$ and $ I_{0}$ of the undetected asymptomatic and symptomatic individuals, at a time $t_{0}$, by available measurements of the isolated and recovered individuals at two succeeding times, $ t_{0}$ and $ T>t_{0}.$ Simultaneously, we identify the rate standing for the average number of individuals infected in unit time by an infective symptomatic individual. Then, we propose a control problem aiming at controlling the infected classes by improving the actions in view of isolating as much as possible the populations $ A$ and $ I$ in the class $ L$. These objectives are formulated as minimization problems, the second one including a state constraint, which are treated by an optimal control technique. The existence of optimal controllers is proved and the first order necessary conditions of optimality are determined. For the second problem, they are deduced by passing to the limit in the conditions of optimality calculated for an appropriately defined approximating problem. In this case, the dual system is singular and has a component in the space of measures. The discussion of the asymptotic stability of the system done for the case when life immunity is gained reveals an asymptotic extinction of the disease, with a well determined reproduction number.
