Identification and control of SARS-CoV-2 epidemic model parameters

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.