Optimal control of an epidemiological Covid-19 model with state constraint

The outbreak of the Covid-19 pandemic has forced governments to impose restrictions on the individual liberty of people. Such containment measures have considerably reduced the number of infections but have also caused substantial damage. In this context the following main issue arises: which policy is the best to contain fatalities and economic losses complying with the intensive care units capacity? This issue is investigated through the study of an optimal control problem based on a SEAIRD epidemic model referring to Covid-19. A state constraint is imposed on the number of infected individuals in order to maintain the infectious level under the health-facilities capacity threshold. The challenge is to find a control function that minimizes the total cost which represents a trade-off between economic losses and human deaths. After showing the existence of an optimal solution, the necessary optimality conditions provided by Pontryagin Minimum Principle are derived. Numerical solutions are obtained by discretizing the optimal control problem and applying nonlinear optimization methods. Various scenarios with different initial conditions representing different degrees of infection are studied and the solutions are compared.The COVID-19 control problem treated here may also serve as a prototypical example for solving an epidemiological control model with state constraints.