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. 2019 Nov 1;10:5003. doi: 10.1038/s41467-019-12862-7

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© The Author(s) 2019

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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Fig. 2 — Schematic representation of the two-time-scales model. The model describes disease and introgression dynamics between two species. The figure shows the transition between two time steps at the evolutionary time scale, $t - 1$ and $t$ . a Transmission rates $β_{i j}$ ( $i = j$ , within-species transmission; $i \neq j$ , between-species transmission from species $i$ to species $j$ ) determine the average impact of pathogens $F_{i}$ at the ecological time scale, according to Eqs. (3)-(9). The pathogen package size $P_{i}$ and the average impact of each pathogen $F_{i}$ determine the disease burden $D_{i}$ (purple), according to Eqs. (1) and (2). The species respond to disease burden by adjusting contact rates $β_{i j}$ (green), according to Eqs. (10)–(13). Inter-species contact results in gene flow and adaptive introgression, reducing pathogen package sizes $P_{i}$ (orange), according to Eqs. (14) and (15). b Impact at the ecological time scale (dashed box in a) is modeled in an SIR epidemic framework, as the average impact of an epidemic. Individuals transition from state S (susceptible) to state I (infectious) from either within-species infections or between-species infections (Eqs. (3)–(8)). Individuals in state I transition to state R (recovered/removed) at rate $γ$ (Eqs. (3)–(8)). The impact of an epidemic $F_{i}$ is measured as the overall proportion of individuals in species $i$ infected throughout the run of the epidemic (9)