Susceptible-Infected-Recovered model on Euclidean network

We consider the Susceptible-Infected-Recovered (SIR) epidemic model on a Euclidean network in one dimension in which nodes at a distance $l$ are connected with probability $P(l) \propto l^{-\delta}$ in addition to nearest neighbors. The topology of the network changes as $\delta$ is varied and its effect on the SIR model is studied. $R(t)$, the recovered fraction of population up to time $t$, and $\tau$, the total duration of the epidemic are calculated for different values of the infection probability $q$ and $\delta$. A threshold behavior is observed for all $\delta$ up to $\delta \approx 2.0$; above the threshold value $q = q_c$, the saturation value $R_{sat}$ attains a finite value. Both $R_{sat}$ and $\tau $ show scaling behavior in a finite system of size $N$; $R_{sat} \sim N^{-\beta/{\tilde{\nu}}} g_1[(q-q_c)N^{1/{\tilde {\nu}}}]$ and $\tau \sim N^{\mu/{\tilde{\nu}}} g_2[(q-q_c)N^{1/\tilde{\nu}}]$. $q_c$ is constant for $0 \leq \delta < 1$ and increases with $\delta$ for $1<\delta\lesssim 2$. Mean field behavior is seen up to $\delta \approx 1.3$; weak dependence on $\delta$ is observed beyond this value of $\delta$.The distribution of the outbreak sizes is also estimated and found to be unimodal for $q < q_c$ and bimodal for $q > q_c$. The results are compared to static percolation phenomenaand also to mean field results for finite systems. Discussions on the properties of the Euclidean network are made in the light of the present results.