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. 2018 Jun 25;8:9642. doi: 10.1038/s41598-018-27716-3

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© The Author(s) 2018

Open Access This 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 license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license 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 license, visit http://creativecommons.org/licenses/by/4.0/.

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Schematic diagram explaining the bacteria-bacteriophage interactions given by Models I and II. The system consists of four compartments: Susceptible bacteria (S), infected bacteria in lysogenic (I₁) and lytic (I₂) state and the free bacteriophages (P). Arrows show loses in some compartments or transitions between compartments. The function denoted by d_τ in the figure is $(K_{2} (T (t - τ)) S (t - τ) P (t - τ) -$ $λ_{1} (T (t - τ)) I_{1} (t - τ)) \exp (- \int_{t - τ}^{t} m (u (x)) dx)$ .