An optimal control problem for virus propagation and economic loss

We introduce a new optimal control problem aiming to suggest possible lockdown strategies in case of pandemic virus proliferation. Starting from the (nonlinear) logistic equation, the model takes into account the drawback of the lockdown strategy by coupling it with a second logistic equation evaluating the health effects deriving from economic losses. The control parameter is the lockdown strength and the optimal control is sought in order to minimize the percentage of the overall affected population, counting both the directly affected humans and the humans affected by the economic loss. Our results show that, in some cases, the optimal control is a total lockdown strategy while, in other cases, the lockdown should be milder.