Susceptible-Infected-Recovered model on Euclidean network

TL;DR

This paper studies the SIR epidemic model on a one-dimensional Euclidean network with long-range connections, analyzing how network topology affects epidemic thresholds, scaling behavior, and outbreak size distributions.

Contribution

It introduces a detailed analysis of the SIR model on a Euclidean network, revealing threshold behavior, finite-size scaling, and the impact of network topology on epidemic dynamics.

Findings

Threshold behavior observed for $ ext{delta} extless 2.0

Scaling laws for $R_{sat}$ and $ au$ with system size

Distribution of outbreak sizes varies with $q$ and $ ext{delta}$.

Abstract

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.