We consider a mathematical model with five compartments relevant to depict the feature of a certain type of epidemic transmission. We aim to identify some system parameters by means of a minimization problem for a functional involving available measurements for observable compartments, which we treat by an optimal control technique with a state constraint imposed by realistic considerations. The proof of the maximum principle is done by passing to the limit in the conditions of optimality for an appropriate approximating problem. The proof of the estimates for the dual approximating system requires a more challenging treatment since the trajectories are not absolutely continuous, but only with bounded variation. These allow to pass to the limit to obtain the conditions of optimality for the primal problem and lead to a singular dual backward system with a generalized solution in the sense of measure. As far as we know, this approach developed here for the identification of parameters in an epidemic model considering a state constraint related to the actions undertaken for the disease containment was not addressed in the literature and represents a novel issue in this paper.

我们考虑一个具有五个隔间的数学模型来描述某种类型的流行病传播的特征。我们的目标是通过涉及可观察隔间可用测量的函数的最小化问题来识别一些系统参数，我们通过最优控制技术处理这些问题，并根据现实考虑施加状态约束。最大原理的证明是通过在适当的近似问题的最优条件下传递到极限来完成的。对偶逼近系统的估计证明需要更具挑战性的处理，因为轨迹不是绝对连续的，而是只有有限的变化。这些允许传递到极限以获得原始问题的最优条件，并导致奇异的对偶后向系统在测度意义上具有广义解。据我们所知，这里开发的这种方法用于识别流行病模型中的参数，考虑与为遏制疾病采取的行动相关的状态约束，在文献中没有涉及，并且在本文中代表了一个新问题。