We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton-Jacobi-Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin's maximum principle and directly gives open-loop controls, via the solution of an optimality system of ordinary differential equations. This method, however, may not converge to the optimal solution. We propose a combination of the two methods in order to obtain high-quality and reliable solutions. Several simulations are presented and discussed.

翻译：我们提出并比较了两种应用于流行病学SEIR模型的最优控制方法，这些方法能够制定控制流行病传播的策略。第一种方法利用动态规划，将问题的值函数表征为偏微分方程（即Hamilton-Jacobi-Bellman方程）的解，并推导出反馈形式的最优策略。第二种方法基于Pontryagin最大值原理，通过求解由常微分方程构成的最优性系统，直接给出开环控制。然而，该方法可能无法收敛到最优解。为此，我们提出将这两种方法相结合，以获得高质量且可靠的解。文中还展示并讨论了多项仿真结果。