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, also checking first and second order necessary optimality conditions for the corresponding numerical solutions.

翻译：我们提出并比较了两种应用于流行病学SEIR模型的不同最优控制方法，这些方法使我们能够获得控制疫情传播的若干策略。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——哈密顿-雅可比-贝尔曼方程的解，并推导出反馈形式下的最优策略。第二种方法基于庞特里亚金最大值原理，通过求解常微分方程组的最优性系统，直接给出开环控制。然而，该方法可能无法收敛到最优解。我们提出将两种方法相结合，以获得高质量且可靠的解。文中展示并讨论了若干数值模拟结果，同时检验了相应数值解的一阶和二阶必要最优性条件。