Figure 1

Simulations of non-Markovian epidemics on networks with $N=1000$ nodes: (a) solid lines show the solution of Eq. (4) and the circles, squares, and diamonds correspond to simulations for homogeneous (random regular) graphs with $⟨k⟩=5,10,15$, respectively; dotted line, $⟨k⟩=5$, and dashed line, $⟨k⟩=15$, lines correspond to purely Markovian epidemics given by Eq. (1); (b) the same as before but for Erdős-Rényi random graphs with $⟨k⟩=5,10,15$; (c) the solid and dashed lines show the solution of pairwise [Eq. (4)] and mean-field [Eq. (5)] models, respectively and, for regular random graphs with $⟨k⟩=5$ and $⟨k⟩=15$. For (a), (b), and (c) the transmission rate is $\tau =0.55$ and the infectious period is fixed, $\sigma =1$. Finally, (d) the diamonds, circles, and squares correspond to simulations using regular random graphs with $⟨k⟩=15$ and using fixed and two different but gamma distributed infectious periods (circle shape $\alpha =2$, scale $\beta =\frac{1}{2}$, square shape $\alpha =\frac{1}{2}$, scale $\beta =2$), respectively. The solid lines correspond to the analytical final size for fixed [Eq. (9)] and general [Eq. (11)] infectious periods, with the dashed line denoting the purely Markovian case. The inset shows the analytical and the simulated final epidemic sizes plotted against the reproduction number.