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Fig. 4 | Applied Network Science

Fig. 4

From: Epidemic spreading and control strategies in spatial modular network

Fig. 4

Epidemic spatial propagation. The average maximum extent of the epidemic, \(\langle r_{max} \rangle\), is measured as a function of time at \(\beta ^{2D}_c\). At early times the epidemic spread locally within the origin city (zeroth circle) for a period of time \(\tau _0\) with a constant \(\langle r_{max} \rangle \sim \zeta\). Afterwards, the epidemic spread to the first circle of cities around the origin city for a period of time \(\tau _1\) and later to the second circle of cities for a period of time \(\tau _2\). The transition time between the circles is denoted by \(\tau _x\). As the epidemic evolves the distinction between circles decreases and identifying the distance of the disease from the origin is less clear. At later times the distinction of circles disappear completely and a clear spatial propagation is observed with \(\langle r_{max} \rangle \sim t^{1/d^{2D}_{min}} = t^{1/1.13}\) (Bunde and Havlin 1991). The reason for the disappearance of the distinction between circles at later times is because the epidemic may spread faster in a given area and slower in another leading to inconclusive distinction between circles. Here we used \(\zeta = 100\), \(K = 4\), \(k_{inter} = 10^{-3}\), \(L = 1000\) and \(\beta ^{2D}_c = 0.407\)

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