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. 2022 Jul 29;12:13039. doi: 10.1038/s41598-022-16532-5

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© The Author(s) 2022

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Optimal trajectories for minimum use of non-quarantine interventions. Optimal trajectories when the departments would like to minimise use of masks, social distancing and environmental disinfection. (a) and (b) The epidemic evolution among students and staff, respectively. (c) The optimal strategies for mask wearing ( $κ_{m}$ ), social distancing ( $κ_{d}$ ), and environmental disinfection ( $κ_{e}$ ). (d) The optimal strategies for mandatory quarantine on infected students ( $κ_{qy}$ ) and infected staff ( $κ_{qs}$ ).