Cross-scale dynamics and the evolutionary emergence of infectious diseases

Sebastian J. Schreiber; Ruian Ke; Claude Loverdo; Miran Park; Priyanna Ahsan; James O. Lloyd-Smith

The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. T 0 min{K, (N − 1) exp(r w t)}dt is almost independent from N . Therefore, most of the dependence of b w N/(N − 1) T 0 min{K, (N − 1) exp(r w t)}f (t, N )dt with N stems from the dependence of f (t, N ) with N . Since a → a/(1 + a) is an increasing function bounded above by 1 for positive a, the expression exp(τ + st)/(N − 1)/(1 + exp(τ + st)/(N − 1)) decreases when N increases. As N → (a/(1 + a)) N −1 is a decreasing function of N ≥ 1 for a > 0, we get that the probability of emergence decreases at least exponentially with the bottleneck size, as claimed in the main text. Fig Appendix-1 illustrates that these approximations work especially when s is sufficiently negative.