Fig. 2

Two epidemic transitions. Simulations of the epidemic recovered cluster R as a function of \(\beta\) for different values of \(\zeta\) on a log-linear graph with \(K = 4\) and \(k_{inter} = 10^{-3}\). The epidemic recovered cluster is measured once no infected nodes remain. Two distinct epidemic transitions are observed. The first (lower) transition at \(\beta ^{ER}_c = 1/ K\) (black dashed line) occurs when a small outbreak spread in a city but not in the entire country. The second (higher) transition at \(\beta ^{2D}_c\) when a global epidemic spread in the whole country is obtained from Eq. (3) and is denoted by black \(\times\). The inset shows the derivative of \(\log (R)\) with respect to \(\beta\) for different values of \(\zeta\). Two maxima appear corresponding to the two epidemic thresholds for each \(\zeta\) shown with \(\times\) in the main figure. As \(\zeta\) increase \(\beta ^{2D}_c\) decreases and for \(\zeta \rightarrow L\) the two maximums collide. Here \(N = L \times L \sim 10^8\) (\(L = 9960\) for \(\zeta = 60\) and \(L = 10^4\) for the other \(\zeta\) values)