Risk Analysis and Genomic Epidemiology Unit, Istituto Zooprofilattico Sperimentale della Lombardia e dell'Emilia Romagna, Via dei Mercati 13, Parma 43126, Italy.
Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy.
Math Biosci. 2019 Sep;315:108232. doi: 10.1016/j.mbs.2019.108232. Epub 2019 Jul 19.
The total number of infections (epidemic size) and the time needed for the infection to go extinct (epidemic duration) represent two of the main indicators for the severity of infectious disease epidemics in human and livestock. However, few attempts have been made to address the problem of minimizing at the same time the epidemic size and duration from a theoretical point of view by using optimal control theory. Here, we investigate the multi-objective optimal control problem aiming to minimize, through either vaccination or isolation, a suitable combination of epidemic size and duration when both maximum control effort and total amount of resources available during the entire epidemic period are limited. Application of Pontryagin's Maximum Principle to a Susceptible-Infected-Removed epidemic model, shows that, when the resources are not sufficient to maintain the maximum control effort for the entire duration of the epidemic, the optimal vaccination control admits only bang-bang solutions with one or two switches, while the optimal isolation control admits only bang-bang solutions with one switch. We also find that, especially when the maximum control effort is low, there may exist a trade-off between the minimization of the two objectives. Consideration of this conflict among objectives can be crucial in successfully tackling real-world problems, where different stakeholders with potentially different objectives are involved. Finally, the particular case of the minimum time optimal control problem with limited resources is discussed.
感染总数（疫情规模）和感染灭绝所需的时间（疫情持续时间）是衡量人类和家畜传染病严重程度的两个主要指标。然而，从理论角度出发，利用最优控制理论同时最小化疫情规模和持续时间的尝试很少。在这里，我们研究了多目标最优控制问题，旨在通过接种疫苗或隔离来最小化，当最大控制努力和整个流行期间可用的总资源都受到限制时，适当组合的疫情规模和持续时间。将庞特里亚金极大值原理应用于易感染-感染-消除的流行模型，表明当资源不足以维持整个流行期间的最大控制努力时，最优接种控制仅允许有一个或两个开关的 bang-bang 解，而最优隔离控制仅允许有一个开关的 bang-bang 解。我们还发现，特别是当最大控制努力较低时，两个目标之间可能存在权衡。在成功解决涉及不同目标的不同利益相关者的现实世界问题时，考虑到目标之间的这种冲突至关重要。最后，讨论了资源有限的最小时间最优控制问题的特殊情况。
