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 \({\beta }_{ij}\) (\(i=j\), within-species transmission; \(i \, \ne \, j\), 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 \({\beta }_{ij}\) (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 \(\gamma\) (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)
