Threshold condition R
= 1 for Europe and Mexico. Threshold condition R∗ = 1 as a function of ε and α for Europe (bottom curves) and Mexico (top curves): comparison of the no-immunity case with the case of pre-existing immunity. Here we consider: R0 = 1.2 in Europe and R0 = 1.4 in Mexico. All travellers are adults (r = 0). The two lines red and blue correspond to pre-existing immunity and no-immunity. Global epidemic invasion region is above each critical curve. The patterned grey area refers to the region of parameter values that do not satisfy the consistency relation.