for a proportionate social system. On the top R∗ as a function of η. Panel (A) shows the case r = 0, α = 0.4 and R0 = 1.2. Panel (B) shows the case r = 1, α = 0.4 and R0 = 1.08. The continuous curves represent the value as computed numerically, while the dashed curves represent the approximate solutions for η → 0 and η → 1. On the bottom threshold condition R∗ = 1 in the α, η plane as obtained numerically for different values for R0. Panels C and D consider the cases r = 0 and r = 1 respectively. For all the panels μ = 0.5, and the mobility network is characterised by γ = 2.3 and w0 = 0.05. The coloured regions are the one for which the invasion condition R∗ > 1 is satisfied. In panel D we also report the η range of values [ηc,min(α),ηc,max(α)] for which invasion is obtained for a given value α.